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--- abstract: 'People believe that depth plays an important role in success of deep neural networks (DNN). However, this belief lacks solid theoretical justifications as far as we know. We investigate role of depth from perspective of margin bound. In margin bound, expected error is upper bounded by empirical margin error plus Rademacher Average (RA) based capacity term. First, we derive an upper bound for RA of DNN, and show that it increases with increasing depth. This indicates negative impact of depth on test performance. Second, we show that deeper networks tend to have larger representation power (measured by Betti numbers based complexity) than shallower networks in multi-class setting, and thus can lead to smaller empirical margin error. This implies positive impact of depth. The combination of these two results shows that for DNN with restricted number of hidden units, increasing depth is not always good since there is a tradeoff between positive and negative impacts. These results inspire us to seek alternative ways to achieve positive impact of depth, e.g., imposing margin-based penalty terms to cross entropy loss so as to reduce empirical margin error without increasing depth. Our experiments show that in this way, we achieve significantly better test performance.' author: - 'Shizhao Sun$^{1,}$[^1], Wei Chen$^2$, Liwei Wang$^3$, Xiaoguang Liu$^1$' - | Tie-Yan Liu$^2$\ $^1$College of Computer and Control Engineering, Nankai University, Tianjin, 300071, P. R. China\ $^2$Microsoft Research, Beijing, 100080, P. R. China\ $^3$Key Laboratory of Machine Perception (MOE), School of EECS, Peking University, Beijing, 100871, P. R. China\ [email protected], [email protected], [email protected]\ [email protected], [email protected] bibliography: - 'deepnets.bib' title: 'On the Depth of Deep Neural Networks: A Theoretical View' --- Introduction ============ Deep neural networks (DNN) have achieved great practical success in many machine learning tasks, such as speech recognition, image classification, and natural language processing [@hinton2006reducing; @krizhevsky2012imagenet; @hinton2012deep; @ciresan2012multi; @weston2012deep]. Many people believe that the depth plays an important role in the success of DNN  [@srivastava2015training; @simonyan2014very; @lee2014deeply; @romero2014fitnets; @he2015delving; @szegedy2014going]. However, as far as we know, such belief is still lacking solid theoretical justification. On one hand, some researchers have tried to understand the role of depth in DNN by investigating its generalization bound. For example, in [@bartlett1998almost; @karpinski1995polynomial; @goldberg1995bounding], generalization bounds for multi-layer neural networks were derived based on Vapnik-Chervonenkis (VC) dimension. In [@bartlett1998sample; @koltchinskii2002empirical], a margin bound was given to fully connected neural networks in the setting of binary classification. In [@neyshabur2015norm], the capacity of different norm-constrained feed-forward networks was investigated. While these works shed some lights on the theoretical properties of DNN, they have limitations in helping us understand the role of depth, due to the following reasons. First, the number of parameters in many practical DNN models could be very large, sometimes even larger than the size of training data. This makes the VC dimension based generalization bound too loose to use. Second, practical DNN are usually used to perform multi-class classifications and often contains many convolutional layers, such as the model used in the tasks of ImageNet [@deng2009imagenet]. However, most existing bounds are only regarding binary classification and fully connected networks. Therefore, the bounds cannot be used to explain the advantage of using deep neural networks. On the other hand, in recent years, researchers have tried to explain the role of depth from other angles, e.g., deeper neural networks are able to represent more complex functions. In [@hastad1986almost; @delalleau2011shallow], authors showed that there exist families of functions that can be represented much more efficiently with a deep logic circuit or sum-product network than with a shallow one, i.e., with substantially fewer hidden units. In [@bianchini2014complexity; @montufar2014number], it was demonstrated that deeper nets could represent more complex functions than shallower nets in terms of maximal number of linear regions and Betti numbers. However, these works are apart from the generalization of the learning process, and thus they cannot be used to explain the test performance improvement for DNN. To reveal the role of depth in DNN, in this paper, we propose to investigate the margin bound of DNN. According to the margin bound, the expected $0$-$1$ error of a DNN model is upper bounded by the empirical margin error plus a Rademacher Average (RA) based capacity term. Then we first derive an upper bound for the RA-based capacity term, for both fully-connected and convolutional neural networks in the multi-class setting. We find that with the increasing depth, this upper bound of RA will increase, which indicates that depth has its negative impact on the test performance of DNN. Second, for the empirical margin error, we study the representation power of deeper networks, because if a deeper net can produce more complex classifiers, it will be able to fit the training data better w.r.t. any margin coefficient. Specifically, we measure the representation power of a DNN model using the Betti numbers based complexity [@bianchini2014complexity], and show that, in the multi-class setting, the Betti numbers based complexity of deeper nets are indeed much larger than that of shallower nets. This, on the other hand, implies the positive impact of depth on the test performance of DNN. By combining these two results, we can come to the conclusion that for DNN with restricted number of hidden units, arbitrarily increasing the depth is not always good since there is a clear tradeoff between its positive and negative impacts. In other words, with the increasing depth, the test error of DNN may first decrease, and then increase. This pattern of test error has been validated by our empirical observations on different datasets. The above theoretical findings also inspire us to look for alternative ways to achieve the positive impact of depth, and avoid its negative impact. For example, it seems feasible to add a margin-based penalty term to the cross entropy loss of DNN so as to directly reduce the empirical margin error on the training data, without increasing the RA of the DNN model. For ease of reference, we call the algorithm minimizing the penalized cross entropy loss *large margin DNN* (LMDNN)[^2]. We have conducted extensive experiments on benchmark datasets to test the performance of LMDNN. The results show that LMDNN can achieve significantly better test performance than standard DNN. In addition, the models trained by LMDNN have smaller empirical margin error at almost all the margin coefficients, and thus their performance gains can be well explained by our derived theory. The remaining part of this paper is organized as follows. In Section \[sec:def\], we give some preliminaries for DNN. In Section \[sec:generalization\], we investigate the roles of depth in RA and empirical margin error respectively. In Section \[sec:alg\], we propose the large margin DNN algorithms and conduct experiments to test their performances. In Section \[sec:conclusion\], we conclude the paper and discuss some future works. Preliminaries {#sec:def} ============= Given a multi-class classification problem, we denote $\mathcal{X}=\mathbb{R}^d$ as the input space, $\mathcal{Y}=\{1,\cdots,K\}$ as the output space, and $P$ as the joint distribution over $\mathcal{X}\times\mathcal{Y}$. Here $d$ denotes the dimension of the input space, and $K$ denotes the number of categories in the output space. We have a training set $S=\{(x_1,y_1),\cdots,(x_m,y_m)\}$, which is i.i.d. sampled from $\mathcal{X}\times\mathcal{Y}$ according to distribution $P$. The goal is to learn a prediction model $f\in\mathcal{F}:\mathcal{X}\times\mathcal{Y}\to \mathbb{R}$ from the training set, which produces an output vector $(f(x,k);k\in\mathcal{Y})$ for each instance $x\in\mathcal{X}$ indicating its likelihood of belonging to category $k$. Then the final classification is determined by $\arg\max_{k\in\mathcal{Y}} f(x,k)$. This naturally leads to the following definition of the *margin* $\rho(f;x,y)$ of the model $f$ at a labeled sample $(x,y)$: $$\rho(f;x,y)=f(x,y)-\max_{k\neq y} f(x,k).$$ The classification accuracy of the prediction model $f$ is measured by its expected $0$-$1$ error, i.e., $$\begin{aligned} err_P(f)&=\Pr_{(x,y)\sim P}\mathbb{I}_{[\arg\max_{k\in \mathcal{Y}}f(x,k)\neq y]}\\ &=\Pr_{(x,y)\sim P}\mathbb{I}_{[\rho(f;x,y)< 0]}, \end{aligned}$$ where $\mathbb{I}_{[\cdot]}$ is the indicator function. We call the $0$-$1$ error on the training set *training error* and that on the test set *test error*. Since the expected $0$-$1$ error cannot be obtained due to the unknown distribution $P$, one usually uses the test error as its proxy when examining the classification accuracy. Now, we consider using neural networks to fulfill the multi-class classification task. Suppose there are $L$ layers in a neural network, including $L-1$ hidden layers and an output layer. There are $n_l$ units in layer $l$ ($l=1,\dots,L$). The number of units in the output layer is fixed by the classification problem, i.e., $n_L=K$. There are weights associated with the edges between units in adjacent layers of the neural network. To avoid over fitting, people usually constraint the size of the weights, e.g., impose a constraint $A$ on the sum of the weights for each unit. We give a unified formulation for both fully connected and convolutional neural networks. Mathematically, we denote the function space of multi-layer neural networks with depth $L$, and weight constraint $A$ as $\mathcal{F}_A^L$, i.e., $$\begin{aligned} \nonumber \mathcal{F}_A^L=\Big\{(x,k)\to\sum_{i=1}^{n_{L-1}}w_if_i(x); f_i\in\mathcal{F}_A^{L-1},\\ \sum_{i=1}^{n_{L-1}}|w_i|\le A,w_i\in \mathbb{R}\Big\}; \end{aligned}$$ for $l=1,\cdots,L-1$, $$\begin{aligned} \nonumber \mathcal{F}_A^l=\Big\{x\to\varphi\Big(\phi(f_1(x)),\cdots,\phi(f_{p_l}(x))\Big);\\\label{eqn:pooling} f_1,\cdots,f_{p_l}\in\bar{\mathcal{F}}_A^l\Big\}, \end{aligned}$$ $$\label{eqn:weight_sum} \bar{\mathcal{F}}_A^l=\Big\{x \to \sum_{i=1}^{n_{l-1}}w_if_i(x);f_{i}\in\mathcal{F}_A^{l-1},\sum_{i=1}^{n_{l-1}}|w_{i}|\le A, w_i \in \mathbb{R}\Big\};$$ and, $$\mathcal{F}_A^0=\Big\{x\to x_{|i}; i\in\{1,\cdots,d\} \Big\};$$ where $w_i$ denotes the weight in the neural network, $x_{|i}$ is the $i$-th dimension of input $x$. The functions $\varphi$ and $\phi$ are defined as follows: \(1) If the $l$-th layer is a convolutional layer, the outputs of the $(l-1)$-th layer are mapped to the $l$-th layer by means of filter, activation, and then pooling. That is, in Eqn (\[eqn:weight\_sum\]), lots of weights equal $0$, and $n_l$ is determined by $n_{l-1}$ as well as the number and domain size of the filters. In Eqn (\[eqn:pooling\]), $p_{l}$ equals the size of the pooling region in the $l$-th layer, and function $\varphi:\mathbb{R}^{p_l}\to\mathbb{R}$ is called the *pooling function*. Widely-used pooling functions include the max-pooling $\max(t_1,\cdots,t_{p_l})$ and the average-pooling $(t_1+\cdots+t_{p_l})/p_l$. Function $\phi$ is increasing and usually called the *activation function*. Widely-used activation functions include the standard sigmoid function $\phi(t)=\frac{1}{1+e^{-t}}$, the tanh function $\phi(t)=\frac{e^t-e^{-t}}{e^t+e^{-t}}$, and the rectifier function $\phi(t)=\max(0,t)$. Please note that all these activation functions are $1$-Lipschitz. \(2) If the $l$-th layer is a fully connected layer, the outputs of the $(l-1)$-th layer are mapped to the $l$-th layer by linear combination and subsequently activation. That is, in Eqn (\[eqn:pooling\]) $p_{l}=1$ and $\varphi(x)=x$. Because distribution $P$ is unknown and the $0$-$1$ error is non-continuous, a common way of learning the weights in the neural network is to minimize the empirical (surrogate) loss function. A widely used loss function is the cross entropy loss, which is defined as follows, $$C(f;x,y)=-\sum_{k=1}^K z_{k} \ln \sigma(x,k),$$ where $z_{k}=1$ if $k=y$, and $z_{k}=0$ otherwise. Here $\sigma(x,k)=\frac{\exp(f(x,k))}{\sum_{j=1}^{K}\exp(f(x,j))}$ is the softmax operation that normalizes the outputs of the neural network to a distribution. Back-propagation algorithm is usually employed to minimize the loss functions, in which the weights are updated by means of stochastic gradient descent (SGD). The Role of Depth in Deep Neural Networks {#sec:generalization} ========================================= In this section, we analyze the role of depth in DNN, from the perspective of the margin bound. For this purpose, we first give the definitions of empirical margin error and Rademacher Average (RA), and then introduce the margin bound for multi-class classification. Suppose $f\in\mathcal{F}:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}$ is a multi-class prediction model. For $\forall \gamma>0$, the empirical margin error of $f$ at margin coefficient $\gamma$ is defined as follows: $$err_S^\gamma(f)=\frac{1}{m}\sum_{i=1}^m \mathbb{I}_{[\rho(f;x_i,y_i) \le\gamma]}.$$ Suppose $\mathcal{F}:\mathcal{X}\to\mathbb{R}$ is a model space with a single dimensional output. The Rademacher average (RA) of $\mathcal{F}$ is defined as follows: $$R_m(\mathcal{F})=\mathbf{E}_{\mathbf{x},\mathbf{\sigma}}\Big[\sup_{f\in\mathcal{F}}\Big|\frac{2}{m}\sum_{i=1}^{m}\sigma_i f(x_i)\Big|\Big],$$ where $\mathbf{x}=\{x_1,\cdots,x_m\}\sim P_x^m$, and $\{\sigma_1,\cdots,\sigma_m\}$ are i.i.d. sampled with $P(\sigma_i=1)=1/2, P(\sigma_i=-1)=1/2$. [@koltchinskii2002empirical]\[thm:generalization\] Suppose $f\in\mathcal{F}:\mathcal{X}\times\mathcal{Y}\to\mathbb{R}$ is a multi-class prediction model. For $\forall \delta>0$, with probability at least $1-\delta$, we have, $\forall f\in\mathcal{F}$, $$\begin{aligned} \label{eqn:generalization_nn}\nonumber err_P(f)\le & \inf_{\gamma>0}\Big\{err_S^\gamma(f) + \frac{8K(2K-1)}{\gamma} R_m(\tilde{\mathcal{F}}) \\ & +\sqrt{\frac{\log\log_2(2\gamma^{-1})}{m}} +\sqrt{\frac{\log(2\delta^{-1})}{2m}}\Big\}. \end{aligned}$$ where $\tilde{\mathcal{F}}=\{x\to f(\cdot,k);k\in\mathcal{Y},f\in\mathcal{F}\}$ According to the margin bound given in Theorem \[thm:generalization\], the expected $0$-$1$ error of a DNN model can be upper bounded by the sum of two terms, RA and the empirical margin error. In the next two subsections, we will make discussions on the role of depth in these two terms, respectively. Rademacher Average {#subsec:margin_theory} ------------------ In this subsection, we study the role of depth in the RA-based capacity term. In the following theorem, we derive an uniform upper bound of RA for both the fully-connected and convolutional neural networks.[^3] \[thm:generalization\_nn\] Suppose input space $\mathcal{X}=[-M,M]^d$. In the deep neural networks, if activation function $\phi$ is $L_\phi$- Lipschitz and non-negative, pooling function $\varphi$ is max-pooling or average-pooling, and the size of pooling region in each layer is bounded, i.e., $p_l\leq p$, then we have, $$\label{eqn:rademacher_bound} R_m(\mathcal{F}_A^L) \le cM \sqrt{\frac{\ln d}{m}}(pL_\phi A)^{L}.$$ where $c$ is a constant. According to the definition of $\mathcal{F}_A^L$ and RA, we have, $$\begin{aligned} R_m(\mathcal{F}_A^L) = \mathbf{E}_{\mathbf{x},\mathbf{\sigma}}\Big[\sup_{\|\mathbf{w}\|_1\le A,f_j\in\mathcal{F}_A^{L-1}}\Big|\frac{2}{m}\sum_{i=1}^{m}\sigma_i\sum_{j=1}^{n_{L-1}}w_jf_j(x_i)\Big|\Big] \\ = \mathbf{E}_{\mathbf{x},\mathbf{\sigma}}\Big[\sup_{\|\mathbf{w}\|_1\le A,f_j\in\mathcal{F}_A^{L-1}}\Big|\frac{2}{m}\sum_{j=1}^{n_{L-1}}w_j\sum_{i=1}^{m}\sigma_if_j(x_i)\Big|\Big]. \end{aligned}$$ Supposing [$\mathbf{w}=\{w_1,\cdots,w_{n_{L-1}}\}$]{} and [$\mathbf{h}=\{\sum_{i=1}^{m}\sigma_i f_1(x_i),\cdots,\sum_{i=1}^{m}\sigma_if_{n_{L-1}}(x_i)\}$]{}, the inner product [$\langle\mathbf{w},\mathbf{h}\rangle$]{} is maximized when $\mathbf{w}$ is at one of the extreme points of the $l_1$ ball, which implies: $$\begin{aligned} \nonumber R_m(\mathcal{F}_A^L) &\le A \mathbf{E}_{\mathbf{x},\mathbf{\sigma}}\Big[\sup_{f\in\mathcal{F}_A^{L-1}}\Big|\frac{2}{m}\sum_{i=1}^{m}\sigma_i f(x_i)\Big|\Big] \\ &=A R_m(\mathcal{F}_A^{L-1}).\label{eqn:max} \end{aligned}$$ For function class $\mathcal{F}_A^{L-1}$, if the ($L-1$)-th layer is a fully connected layer, it is clear that $R_m(\mathcal{F}_A^{L-1})\le R_m(\phi\circ \bar{\mathcal{F}}_A^{L-1})$ holds. If the ($L-1$)-th layer is a convolutional layer with max-pooling or average-pooling, we have, $$\begin{aligned} \nonumber & R_m(\mathcal{F}_A^{L-1})\\ \nonumber &\le\mathbf{E}_{\mathbf{x},\mathbf{\sigma}}\Big[\sup_{f_1,\cdots,f_{p_{L-1}}\in\bar{\mathcal{F}}_A^{L-1}}\Big|\frac{2}{m}\sum_{i=1}^m\sigma_i \sum_{j=1}^{p_{L-1}}\phi(f_j(x_i))\Big|\Big] \label{eqn:pooling_less} \\ & = p_{L-1}R_m(\phi\circ\bar{\mathcal{F}}_A^{L-1}). \end{aligned}$$ The inequality (\[eqn:pooling\_less\]) holds due to the fact that most widely used activation functions $\phi$ (e.g., standard sigmoid and rectifier) have non-negative outputs. Therefore, for both fully connected layers and convolutional layers, $R_m(\mathcal{F}_A^{L-1})\le p_{L-1} R_m(\phi\circ \bar{\mathcal{F}}_A^{L-1})$ uniformly holds. Further considering the Lipschitz property of $\phi$, we have, $$\label{eqn:lip} R_m(\mathcal{F}_A^{L-1}) \le 2p_{L-1}L_\phi R_m(\bar{\mathcal{F}}_A^{L-1}).$$ Iteratively using maximization principle of inner product in (\[eqn:max\]), property of RA in (\[eqn:pooling\_less\]) and Lipschitz property in (\[eqn:lip\]), considering $p_l\leq p$, we can obtain the following inequality, $$\label{eqn:repeat} R_m(\mathcal{F}_A^{L}) \le (2pL_\phi A)^{L-1}R_m(\bar{\mathcal{F}}_A^1).$$ According to [@bartlett2003rademacher], $R_m(\bar{\mathcal{F}}_A^1)$ can be bounded by: $$\label{eqn:g_bound} R_m(\bar{\mathcal{F}}_A^1) \le cAM\sqrt{\frac{\ln d}{m}},$$ where $c$ is a constant. Combining (\[eqn:repeat\]) and (\[eqn:g\_bound\]), we can obtain the upper bound on the RA of DNN. From the above theorem, we can see that with the increasing depth, the upper bound of RA will increase, and thus the margin bound will become looser. This indicates that depth has its negative impact on the test performance of DNN. Empirical Margin Error {#subsec:margin_exp} ---------------------- In this subsection, we study the role of depth in empirical margin error. To this end, we first discuss representation power of DNN models. In particular, we use the Betti numbers based complexity [@bianchini2014complexity] to measure the representation power. We generalize the definition of Betti numbers based complexity into multi-class setting as follows. The Betti numbers based complexity of functions implemented by multi-class neural networks $\mathcal{F}_A^L$ is defined as $N(\mathcal{F}_A^L)=\sum_{i=1}^{K-1}B(S_i)$, where $B(S_i)$ is the sum of Betti numbers[^4] that measures the complexity of the set $S_i$. Here $S_i=\cap_{j=1,j\neq i}^{K}\{x\in\mathbb{R}^d\mid f(x,i)-f(x,j)\ge 0;f(x,\cdot)\in\mathcal{F}_A^L\}, i=1,\dots,K-1$. As can be seen from the above definition, the Betti numbers based complexity considers classification output and merge those regions corresponding to the same classification output (thus is more accurate than the linear region number complexity [@montufar2014number] in measuring the representation power). As far as we know, only for binary classification and fully connected networks, the bounds of the Betti numbers based complexity was derived [@bianchini2014complexity], and there is no result for the setting of multi-class classification and convolutional networks. In the following, we give our own theorem to fill in this gap. \[thm:multi\_class\_complexity\] For neural networks $\mathcal{F}_A^L$ that has $h$ hidden units. If activation function $\phi$ is a Pfaffian function with complexity $(\alpha,\beta,\eta)$, pooling function $\varphi$ is average-pooling and $d\le h\eta$, then $$\begin{aligned} \nonumber & N(\mathcal{F}_A^L) \le (K-1)^{d+1}2^{h\eta(h\eta-1)/2} \\ &\times O\left(\left(d\left(\left(\alpha+\beta-1+\alpha\beta\right)(L-1)+\beta\left(\alpha+1\right)\right)\right)^{d+h\eta}\right) \end{aligned}$$ We first show that the functions $f(x,\cdot)\in\mathcal{F}_A^L$ are Pfaffian functions with complexity $((\alpha+\beta-1+\alpha\beta)(L-1)+\alpha\beta,\beta,h\eta)$, where $\mathcal{F}_A^L$ can contain both fully-connected layers and convolutional layers. Assume the Pfaffian chain which defines activation function $\phi(t)$ is $(\phi_1(t),\dots,\phi_\eta(t))$, and then $s^l$ is constructed by applying all $\phi_i,1\le i\le\eta$ on all the neurons up to layer $l-1$, i.e., $f^l\in\bar{\mathcal{F}}_A^l,l\in\{1,\dots,L-1\}$. As the first step, we need to get the degree of $f^l$ in the chain $s^l$. Since $f^l=\frac{1}{p_{l-1}}\sum_{k=1}^{n_{l-1}}w_k(\phi (f_{k,1}^{l-1})+\cdots+\phi(f_{k,p_{l-1}}^{l-1}))$ and $\phi$ is a Pfaffian function, $f^l$ is a polynomial of degree $\beta$ in the chain $s^l$. Then, it remains to show that the derivative of each function in $s^l$, i.e., $\frac{\partial \phi_j(f^l)}{\partial x_{|i}}=\frac{d\phi_j(f^l)}{d f^l}\frac{\partial f^l}{\partial x_{|i}}$, can be defined as a polynomial in the functions of the chain and the input. For average pooling, by iteratively using chain rule, we can obtain that the highest degree terms of $\frac{\partial f^l}{\partial x_{|i}}$ are in the form of $\prod_{i=1}^{l-1}\frac{d\phi(f^i)}{df^i}$. Following the lemma 2 in [@bianchini2014complexity], we obtain the complexity of $f(x,\cdot)\in\mathcal{F}_A^L$. Furthermore, the sum of two Pfaffian functions $f_1$ and $f_2$ defined by the same Pffaffian chain of length $\eta$ with complexity $(\alpha_1,\beta_1,\eta)$ and $(\alpha_2,\beta_2,\eta)$ respectively is a Pfaffian function with complexity $\left(\max(\alpha_1,\alpha_2),\max(\beta_1,\beta_2),\eta\right)$ [@gabrielov2004complexity]. Therefore, $f(x,i)-f(x,j),i\neq j$ is a Pfaffian function with complexity $((\alpha+\beta-1+\alpha\beta)(L-1)+\alpha\beta,\beta,h\eta)$. According to [@zell1999betti], since $S_i$ is defined by $K-1$ sign conditions (inequalities or equalities) on Pfaffian functions, and all the functions defining $S_i$ have complexity at most $((\alpha+\beta+\alpha\beta)(L-1)+\alpha\beta,\beta,h\eta)$, $B(S_i)$ can be upper bounded by $(K-1)^{d}2^{h\eta(h\eta-1)/2}\times O(\left(d\left(\left(\alpha+\beta-1+\alpha\beta\right)(L-1)+\beta\left(\alpha+1\right)\right)\right)^{d+h\eta})$. Summing over all $i\in\{1,\dots,K-1\}$, we get the results stated in Theorem \[thm:multi\_class\_complexity\]. Theorem \[thm:multi\_class\_complexity\] upper bounds the Betti numbers based complexity for general activation functions. For specific active functions, we can get the following results: when $\phi=\arctan(\cdot)$ and $d\le 2h$, since $\arctan$ is of complexity $(3,1,2)$, we have $N(\mathcal{F}_A^L) \le (K-1)^{d+1}2^{h(2h-1)}O((d(L-1)+d)^{d+2h})$; when $\phi=\tanh(\cdot)$ and $n\le h$, since $\tanh$ is of complexity $(2,1,1)$, we have $N(\mathcal{F}_A^L) \le (K-1)^{d+1}2^{h(h-1)/2}O((d(L-1)+d)^{d+h})$. Basically, Theorem \[thm:multi\_class\_complexity\] indicates that in the multi-class setting, the Betti numbers based complexity grows with the increasing depth $L$. As a result, deeper nets will have larger representation power than shallower nets, which makes deeper nets fit better to the training data and achieve smaller empirical margin error. This indicates that depth has its positive impact on the test performance of DNN. Actually, above discussions about impact of depth on representation power are consistent with our empirical findings. We conducted experiments on two datasets, MNIST [@lecun1998gradient] and CIFAR-10 [@Krizhevsky09learningmultiple]. To investigate the influence of network depth $L$, we trained fully-connected DNN with different depths and restricted number of hidden units. The experimental results are shown in Figure \[fig:depth\_margin\] and indicate that no matter on which dataset, deeper networks have smaller empirical margin errors than shallower networks for most of the margin coefficients. Discussions ----------- Based on discussions in previous two subsections, we can see that when the depth $L$ of DNN increases, (1) the RA term in margin bound will increase (according to Theorem \[thm:generalization\_nn\]); (2) the empirical margin error in margin bound will decrease since deeper nets have larger representation power (according to Theorem \[thm:multi\_class\_complexity\]). As a consequence, we can come to the conclusion that, for DNN with restricted number of hidden units, arbitrarily increasing depth is not always good since there is a clear tradeoff between its positive and negative impacts on test error. In other words, with the increasing depth, the test error of DNN may first decrease, and then increase. Actually this theoretical pattern is consistent with our empirical observations on different datasets. We used the same experimental setting as that in the subsection \[subsec:margin\_exp\] and repeated the training of DNN (with different random initializations) for 5 times. Figure \[fig:depth\_test\_error\] reports the average and minimum test error of $5$ learned models. We can observe that as the depth increases, the test error first decreases (probably because increased representation power overwhelms increased RA capacity); and then increase (probably because RA capacity increases so quickly that representation power cannot compensate for negative impact of increased capacity). Large Margin Deep Neural Networks {#sec:alg} ================================= From the discussions in Section \[sec:generalization\], we can see that one may have to pay the cost of larger RA capacity when trying to obtain better representation power by increasing the depth of DNN (not to mention that the effective training of very deep neural networks is highly non-trivial [@glorot2010understanding; @srivastava2015training]). Then a nature question is whether we can avoid this tradeoff, and achieve good test performance in an alternative way. To this end, let us revisit the positive impact of depth: it actually lies in that deeper neural networks tend to have larger representation power and thus smaller empirical margin error. Then the question is: can we directly minimize empirical margin error? Our answer to this question is yes, and our proposal is to add a margin-based penalty term to current loss function. In this way, we should be able to effectively tighten margin bound without manipulating the depth. One may argue that widely used loss functions (e.g., cross entropy loss and hinge loss) in DNN are convex surrogates of margin error by themselves, and it might be unnecessary to introduce an additional margin-based penalty term. However, we would like to point out that unlike hinge loss for SVM or exponential loss for Adaboost, which have theoretical guarantee for convergence to margin maximizing separators as the regularization vanishes [@rosset2003margin], there is no optimization consistency guarantee for these losses used in DNN since neural networks are highly non-convex. Therefore, it makes sense to explicitly add a margin-based penalty term to loss function, in order to further reduce empirical margin error during training process. Algorithm Description {#subsec:alg_description} --------------------- We propose adding two kinds of margin-based penalty terms to the original cross entropy loss[^5]. The first penalty term is the gap between the upper bound of margin (i.e., 1)[^6] and the margin of the sample (i.e., $\rho(f; x, y)$). The second one is the average gap between upper bound of margin and the difference between the predicted output for the true category and those for all the wrong categories. It can be easily verified that the second penalty term is an upper bound of the first penalty term. Mathematically, the penalized loss functions can be described as follows (for ease of reference, we call them $C_{1}$ and $C_{2}$ respectively): for model $f$, sample $x,y$, $$\begin{aligned} C_{1}(f;x,y)= & C(f;x,y) + \lambda \Big(1-\rho(f;x,y)\Big)^2,\\ C_{2}(f;x,y)= & C(f;x,y) \\ &+\frac{\lambda}{K-1}\sum_{k\neq y}\Big(1-(f(x,y)-f(x,k))\Big)^2. \end{aligned}$$ We call the algorithms that minimize the above new loss functions *large margin DNN algorithms* (LMDNN). For ease of reference, we denote LMDNN minimizing $C_1$ and $C_2$ as LMDNN-$C_1$ and LMDNN-$C_2$ respectively, and the standard DNN algorithms minimizing $C$ as DNN-$C$. To train LMDNN, we also employ the back propagation method. Experimental Results {#subsec:alg_exp} -------------------- MNIST CIFAR-10 ----------------- -------------------------- --------------------------- -- DNN-$C$ (%) $0.899\pm0.038$ $18.339\pm0.336$ LMDNN-$C_1$ (%) $\mathbf{0.734\pm0.046}$ $\mathbf{17.598\pm0.274}$ LMDNN-$C_2$ (%) $0.736\pm0.041$ $17.728\pm0.283$ : Test error (%) of DNN-$C$ and LMDNNs.[]{data-label="tab:loss_test_error"} Now we compare the performances of LMDNNs with DNN-$C$. We used well-tuned network structures in the Caffe [@jia2014caffe] tutorial (i.e., LeNet[^7] for MNIST and AlexNet[^8] for CIFAR-10) and all the tuned hyper parameters on the validation set. Each model was trained for 10 times with different initializations. Table \[tab:loss\_test\_error\] shows mean and standard deviation of test error over the 10 learned models for DNN-$C$ and LMDNNs after tuning margin penalty coefficient $\lambda$. We can observe that, on both MNIST and CIFAR-10, LMDNNs achieve significant performance gains over DNN-$C$. In particular, LMDNN-$C_1$ reduce test error from $0.899\%$ to $0.734\%$ on MNIST and from $18.399\%$ to $17.598\%$ on CIFAR-10; LMDNN-$C_2$ reduce test error from $0.899\%$ to $0.736\%$ on MNIST and from $18.399\%$ to $17.728\%$ on CIFAR-10. To further understand the effect of adding margin-based penalty terms, we plot empirical margin errors of DNN-$C$ and LMDNNs in Figure \[fig:comp\_margin\]. We can see that by introducing margin-based penalty terms, LMDNNs indeed achieve smaller empirical margin errors than DNN-$C$. Furthermore, the models with smaller empirical margin errors really have better test performances. For example, LMDNN-$C_{1}$ achieved both smaller empirical margin error and better test performance than LMDNN-$C_{2}$. This is consistent with Theorem \[thm:multi\_class\_complexity\], and in return indicates reasonability of our theorem. We also report mean test error of LMDNNs with different margin penalty coefficient $\lambda$ (see Figure \[fig:lambda\_alg\]). In the figure, we use dashed line to represent mean test error of DNN-$C$ (corresponding to $\lambda=0$). From the figure, we can see that on both MNIST and CIFAR-10, (1) there is a range of $\lambda$ where LMDNNs outperform DNN-$C$; (2) although the best test performance of LMDNN-$C_{2}$ is not as good as that of LMDNN-$C_{1}$, the former has a broader range of $\lambda$ that can outperform DNN-$C$ in terms of the test error. This indicates the value of using LMDNN-$C_{2}$: it eases the tuning of hyper parameter $\lambda$; (3) with increasing $\lambda$, test error of LMDNNs will first decrease, and then increase. When $\lambda$ is in a reasonable range, LMDNNs can leverage both good the optimization property of cross entropy loss in training process and the effectiveness of margin-based penalty term, and thus achieve good test performance. When $\lambda$ becomes too large, margin-based penalty term dominates cross entropy loss. Considering that margin-based penalty term may not have good optimization property as cross entropy loss in the training process, the drop of test error is understandable. Conclusion and Future Work {#sec:conclusion} ========================== In this work, we have investigated the role of depth in DNN from the perspective of margin bound. We find that while the RA term in margin bound is increasing w.r.t. depth, the empirical margin error is decreasing instead. Therefore, arbitrarily increasing the depth might not be always good, since there is a tradeoff between the positive and negative impacts of depth on test performance of DNN. Inspired by our theory, we propose two large margin DNN algorithms, which achieve significant performance gains over standard DNN algorithm. In the future, we plan to study how other factors influence the test performance of DNN, such as unit allocations across layers and regularization tricks. We will also work on the design of effective algorithms that can further boost the performance of DNN. Acknowledgments =============== Liwei Wang was partially supported by National Basic Research Program of China (973 Program) (grant no. 2015CB352502), NSFC(61573026), and a grant from MOE-Microsoft Laboratory of Statistics of Peking University. Xiaoguang Liu was partially supported by NSF of China (61373018, 11301288, 11450110409) and Program for New Century Excellent Talents in University (NCET130301). Experiment Settings in Section 3.2 ================================== The MNIST dataset (for handwritten digit classification) consists of $28\times28$ black and white images, each containing a digit $0$ to $9$. There are $60$k training examples and $10$k test examples in this dataset. The CIFAR-10 dataset (for object recognition) consists of $32\times32$ RGB images, each containing an object, e.g., cat, dog, or ship. There are $50$k training examples and $10$k test examples in this dataset. For each dataset, we divide the $10$k test examples into two subsets of equal size, one for validation and the other for testing. In each experiment, we use standard sigmoid activation in hidden layers and train neural networks by mini-batch SGD with momentum and weight decay. All the hyper-parameters are tuned on the validation set. To investigate the influence of the network depth $L$, we train fully-connected DNN models with different depths and restricted number of hidden units. For simplicity and also following many previous works [@simard2003best; @hinton2012improving; @glorot2011deep; @ba2014deep], we assume that each hidden layer has the same number of nodes in the experiment. Specifically, for MNIST and CIFAR-10, the DNN models with depth $2$, $3$, $4$, $5$ and $6$ respectively have $3000$, $1500$, $1000$, $750$ and $600$ units in each hidden layer when the total number of hidden units is $3000$. Experimental Settings in Section 4.2 ==================================== For data pre-processing, we scale the pixel values in MNIST to $[0,1]$, and subtract the per-pixel mean computed over the training set from each image in CIFAR-10. On both datasets, we do not use data augmentation for simplicity. For network structure, we used the well-tuned neural network structures as given in the Caffe tutorial (i.e., LeNet for MNIST and AlexNet) for CIFAR-10). For the training process, the weights are initialized randomly and updated by mini-batch SGD. We use the model in the last iteration as our final model. For DNN-$C$, all the hyper parameters are set by following Caffe tutorial. For LMDNNs, all the hyper parameters are tuned to optimal on the validation set. Finally, we find that by using the following hyper parameters, both DNN-$C$ and LMDNNs can achieve best performance as we reported. For MNIST, we set the batch size as $64$, the momentum as $0.9$, and the weight decay coefficient as $0.0005$. Each neural network is trained for $10$k iterations and the learning rate in each iteration $T$ decreases by multiplying the initial learning rate with a factor of $(1 + 0.0001T) ^ {-0.75}$. For CIFAR-10, we set the batch size as $100$, the momentum as $0.9$, and the weight decay coefficient as $0.004$. Each neural network is trained for $70$k iterations. The learning rate is set to be $10^{-3}$ for the first $60$k iterations, $10^{-4}$ for the next $5$k iterations, and $10^{-5}$ for the other $5$k iterations. [^1]: This work was done when the author was visiting Microsoft Research Asia. [^2]: One related work is [@li2015max], which combines the generative deep learning methods (e.g., RBM) with a margin-max posterior. In contrast, our approach aims to enlarge the margin of discriminative deep learning methods like DNN. [^3]: To the best of our knowledge, an upper bound of RA for fully connected neural networks has been derived before [@bartlett2003rademacher; @neyshabur2015norm], but there is no result available for the convolutional neural networks. [^4]: For any subset $S\subset \mathbb{R}^d$, there exist $d$ Betti numbers, denoted as $b_j(S), 0\le j \le d-1$. Therefore, the sum of Betti numbers is denoted as $B(S)=\sum_{j=0}^{d-1} b_j(S)$. Intuitively, the first Betti number $b_0(S)$ is the number of connected components of the set $S$, while the $j$-th Betti number $b_j(S)$ counts the number of $(j+1)$-dimension holes in $S$ [@bianchini2014complexity]. [^5]: Although we take the most widely-used cross entropy loss as example, these margin-based penalty terms can also be added to other loss functions. [^6]: Please note that, after softmax operation, the outputs are normalized to $[0,1]$ [^7]: <http://caffe.berkeleyvision.org/gathered/examples/mnist.html> [^8]: <http://caffe.berkeleyvision.org/gathered/examples/cifar10.html>

--- abstract: 'For a system with a fixed number of electrons, the total optical sum is a constant, independent of many-body interactions, of impurity scattering and of temperature. For a single band in a metal, such a sum rule is no longer independent of the interactions or temperature, when the dispersion and/or finite bandwidth is accounted for. We adopt such a model, with electrons coupled to a single Einstein oscillator of frequency $\omega _{E}$, and study the optical spectral weight. The optical sum depends on both the strength of the coupling and on the characteristic phonon frequency, $\omega_{E}$. A hardening of $\omega_{E}$, due, for example, to a phase transition, leads to electron undressing and translates into a decrease in the electron kinetic energy and an increase in the total optical sum, as observed in recent experiments in the cuprate superconductors.' address: - | Department of Physics and Astronomy, McMaster University\ Hamilton, Ontario, Canada, L8S 4M1 - | Department of Physics, University of Alberta\ Edmonton, Alberta, Canada, T6J 2J1 author: - 'A. Knigavko and J.P. Carbotte' - 'F. Marsiglio' title: Optical sum increase due to electron undressing --- Introduction ============ Recently there has been considerable interest [@vdmarel1]–[@norman2] in the relationship between the kinetic energy of an electron system and its optical oscillator strength sum rule. The optical sum, $S=\int_{0}^{+\infty}d\nu \mathop{\rm Re}% \left[ \sigma (\nu )\right] ,$ measured in several high-T$_{c}$ cuprates for in–plane conductivity shows noticeable temperature dependence from room temperature all the way down to the zero temperature limit [@vdmarel1]. This dependence is approximately $S \simeq c_0 - c_2 T^{2}$ but the proportionality coefficient $c_2$ seems to change abruptly at the superconducting transition temperature. If one describes the normal state by a tight binding band [@hirsch2; @vdmarel2] with only nearest neighbor hopping, the optical sum is directly related to the negative of the kinetic energy. This also holds approximately when further neighbors are included in the electron dispersion relation [@vdmarel2]. The observed behavior of the optical sum [@vdmarel1; @lobo1] has revived discussion [@hirsch3] of the possibility of kinetic energy-driven superconductivity. In this context a more general issue of importance arises: what is the relationship between the optical sum and essential characteristics of the electronic system ? An understanding of the physical content of the optical sum is achieved through the optical sum rule for a single band: $$\int_{0}^{+\infty }d\nu \mathop{\rm Re}% \left[ \sigma _{\alpha \alpha }(\nu )\right] =-{\pi e^{2} \over \hbar^2} \frac{1}{N}\sum_{k}% \left(\frac{\partial \xi _{k}}{\partial k_{\alpha }} \right)^2 \frac{dn_{k}}{d\xi _{k}}, \label{1}$$ where $e$ is the electronic charge, $N$ is the number of unit cells, $k$ is electron momentum, $\xi _{k}$ is the electron dispersion, $n_{k}$ is the probability of occupation of the state $k$ for a single spin, and $\alpha$ is a Cartesian coordinate. A more familiar form for the right hand side (RHS) of Eq. (\[1\]) is $$\int_{0}^{+\infty }d\nu \mathop{\rm Re}% \left[ \sigma _{\alpha \alpha }(\nu )\right] = {\pi e^{2} \over \hbar^2} \frac{1}{N}\sum_{k}% \left( \partial ^{2}\xi _{k} \over \partial k_{\alpha }^2\right) n_{k}. \label{2}$$ Eq. (\[1\]) is obtained from Eq. (\[2\]) by performing an integration by parts on the momentum $k_\alpha$. The merit in the optical sum rule in either form is that it relates the optical integral on the left to quantities that are easier to analyze. Note, however, that for a parabolic band with infinite bandwidth only Eq. (\[2\]) yields the well known result ${\pi e^2 n \over 2m} \equiv \Omega_P^2/8 $ where $\Omega_P$ is the plasma frequency. This latter expression is valid for a system with interacting electrons, where $n$ is the electron density for [*all*]{} the electrons. In this case the sum rule yields a constant, independent of temperature, and does not give a hint of the underlying interactions. In practice one usually deals with a limited frequency range. Standard optical experiments probe the conduction band, and the sum rule is to be adapted correspondingly. A quadratic dependence of energy on wave vector is often a model of choice for the dispersion of conduction electrons. When combined with the infinite band approximation, it gives the electronic density on the right hand side of Eq. (\[2\]). However, in many cases it is important to account for the finite width of the electronic band (for example the quadratic dispersion definitely cannot be a good description over the whole Brillouin zone). Then the optical sum rule, expressed for a single band, definitely acquires an explicit temperature dependence. Another implicit source of temperature dependence is the quasiparticle occupation number $n_{k}$, which can be strongly affected by many-body interactions amongst the electrons. In this paper we wish to use a simple model to understand the properties of the optical sum for a normal metal (governed by Fermi Liquid Theory) when the different sources which can lead to its deviation from a constant value are included. To this end we assume a constant electron density of states with sharp cut-offs at the band edges [@alexandrov87]–[@dogan1]. We also assume that the electrons are coupled to Einstein oscillators of frequency $\omega_{E}$. The microscopic origin of this oscillator is not specified — it could be a phonon or a spin fluctuation. A system of electrons coupled to an Einstein oscillator provides an important example of an interacting system which is simple enough that it can be analyzed in great detail in order to gain a qualitative understanding of various phenomena [@cappelluti1],[@mars-b],[@knigavko1]. In this model the optical sum is no longer equal to $\Omega _{p}^{2}/8$ but varies with temperature and with interaction strength with the oscillators. We study how the strength of the coupling, denoted by $A$, as well as the value of $\omega _{E}$ affect the value of the optical sum, and the expectation value of the kinetic energy of the electrons, as a function of temperature. In particular we find that, if at some specific temperature $\omega_{E}$ undergoes a sudden hardening so that the mass enlacement parameter $\lambda $ decreases, then the total optical weight increases while the kinetic energy decreases, as observed in experiment. We begin in Sec. II with a brief review of the standard technique for calculating optical conductivity within the Kubo formalism. The model of a constant density of states with bandedge cutoffs allows us to make a clear connection with the conductivity calculations that use the standard “infinite bandwidth” approximation. This model is also well suited for simulating a tight binding dispersion for electrons in two spatial dimensions. We argue that the self consistent treatment of the underlying equations for the electronic self energy is important in this model. Numerical results and a discussion are presented in Sec. III. Formalism ========= To evaluate the left hand side of the optical sum rule, Eq. (\[1\]), we need the frequency dependent conductivity $\sigma (\nu )$. Within linear response theory this is obtained from the appropriate current–current correlation function $\Pi$ [@mars-b] $$\sigma _{\alpha \alpha }(\nu )=\frac{i}{\nu }\Pi_{\alpha \alpha }(i\nu _{n}\rightarrow \nu +i0^{+}), \label{3}$$ where $\alpha$ is a Cartesian coordinate, [*x,y,z*]{}. The response function $\Pi $ is analytically continued from bosonic Matsubara frequencies $i\nu _{n} \equiv 2 i\pi T n$ to the real axis by $i\nu _{n}\rightarrow \nu +i0^{+}$ [@mars1]. On the imaginary axis $\Pi $ is given in the bubble approximation, in terms of the electronic Green’s functions $G\left( k,i\omega _{m}\right) $, by the equation: $$\begin{aligned} \Pi _{\alpha \alpha }(i\nu _{n}) = { 2e^{2} \over \hbar^2} \frac{1}{N}\sum_{k\in BZ} \left( \frac{\partial \xi_{k}}{\partial k_{x}} \right)^{2} T\sum_{m=-\infty }^{+\infty } \phantom{mm} \nonumber\\ \phantom{mmmm} G\left( k,i\omega _{m}+i\nu_{n}\right) G\left( k,i\omega _{m}\right), \label{4}\end{aligned}$$ where $T$ is temperature and $i\omega_{m} \equiv i\pi T (2m-1)$ is the $m$-th fermionic Matsubara frequency. The $k$ sum runs over the first Brillouin zone for the particular band of interest. To evaluate Eq. (\[4\]), the $k$ summation will be replaced by an energy integration (see below). When the Green’s functions in Eq. (\[4\]) are expressed though the electron spectral density $A(k,\omega )=-\frac{1}{\pi } \mathop{\rm Im}% G(k,i\omega _{n}\rightarrow \omega +i0^{+})$, the formula for the real part of the in–plane optical conductivity assumes the form $$\begin{aligned} \mathop{\rm Re}\left[ \sigma _{xx}(\nu )\right] = {2\pi e^{2} \over \hbar^2} {1 \over N} \sum_k \left( \frac{\partial \xi_{k}}{\partial k_{x}}\right)^{2} \int_{-\infty }^{\infty }d \omega \phantom{mmmm} \nonumber\\ \phantom{mmmm} A\left( \xi_k,\omega\right)\,A\left( \xi_k,\omega+\nu\right) \frac{f_{F}\left( \omega\right) -f_{F}(\omega+\nu)}{\nu}, \label{5}\end{aligned}$$ where $f_{F}(\omega )$ is the Fermi–Dirac distribution, and $A\left(\xi_k,\omega\right)$ is the electron spectral function. On the RHS of the optical sum rule, Eq. (\[1\]), the particle occupation number $n_{k}=n(\xi_k)$ is also expressed though the Green’s function: $$n(\xi_k) = \int_{-\infty }^{+\infty }d\omega\,A\left( \xi_k,\omega\right) f_{F}\left(\omega\right) \label{6}$$ and the RHS can be written as $${\rm RHS}=- {\pi e^2 \over \hbar^2} \int_{-\infty }^{+\infty }d\omega f_{F}\left( \omega\right) {1 \over N} \sum_k\left( \frac{\partial \xi_{k}}{\partial k_{x}}\right)^{2} \frac{\partial A\left( \xi_k, \omega\right) }{\partial \xi_k}. \label{7}$$ Eq. (\[7\]) is closer to the starting point of the sum rule derivation, in that the same factor of the Fermi velocity squared, $\left( \partial \xi_{k}/\partial k_{x}\right) ^{2},$ occurs in both equations (\[5\]) and (\[7\]); thus if an approximate band structure is introduced at this step the sum rule will hold exactly. Here we will consider two possible choices for the band structure and hence group velocity. In a model with quadratic dispersion with lower band edge at $\xi = -W/2$ we have $\left( {1 \over \hbar}\frac{\partial \xi_{k}}{\partial k_{x}}\right)^{2}= {2 \over mD} \left( W/2 + \xi \right)$, where $D$ is the dimensionality and $m$ the free electron mass. As is usual, we also adopt a constant density of states, with $g(\xi) = 1/W$ for $-W/2 < \xi < W/2$. Here $W$ is the bandwith, and the density of states obeys the usual sum rule. An integration by parts, assuming this constant density of states, then leads to the result $${\rm RHS}= {2 \over D} {\pi e^2 n\over 2 m} \left[ 1 -{2 \over n}\int_{-\infty}^{\infty}d \omega\,f_F(\omega) A(W/2,\omega) \right], \label{8}$$ where $n$ is the electron density in the band. Note that the result is now temperature dependent, and dependent on interactions. In this expression the electron spectral density $A(\xi,\omega)$ is to be evaluated at the unperturbed band edge $\xi = W/2$. Thus, in the limit of large bandwidth the second term goes to zero, and we are left with a constant result which is within a factor of $2/D$ of the usual sum rule in three dimensions. A precise agreement is in general not expected, since, for a single band the sum rule will depend on the details of the dispersion, etc. For a tight binding band the group velocity depends on wavevector in an essential way. One can introduce a weighted density of states [@mars2], $g_{xx}(\xi) \equiv {1 \over N} \sum_k (\partial \xi_k/\partial k_x)^2 \delta (\xi - \xi_k)$, so that the Brillouin zone sum in Eq.(\[7\]) is reduced to a single energy integration. If only the nearest neighbor hopping is included, in one dimension one finds that $g_{xx}(\xi)/g(\xi)= (2ta)^2 [1 - (\xi/2t)^2]$, where $a$ is the lattice spacing and $t$ is the single particle hopping. Here $g(\xi)$ is the single electron density of states. In higher dimensions one can obtain somewhat more complicated expressions involving complete elliptic integrals of the first and second kind, but the approximation $g_{xx}(\xi)/g(\xi) \approx \left(Wa/(2D)\right)^2 \left[1 - \left({2 \xi / W}\right)^2\right]$ remains excellent, particularly near the band edges. Note that in $D$ dimensions the hopping integral can be expressed through the bandwidth as $t=W/(4D)$. Thus it becomes natural to use the replacement $$\left( {\partial \xi_{k} \over \partial k_{x}}\right) ^{2} = {W\over D} {\hbar^2 \over 2m_b} \left[1 - \left({ \xi \over W/2}\right)^2\right], \label{9}$$ where we introduced the mass $m_b$ of an electron in a tight binding band using the standard definition ${\hbar^2 \over 2m_b} = ta^2$. Substituting this expression into Eq. (\[7\]), we obtain $$\begin{aligned} {\rm RHS} = {2\over D}{\pi e^2 \over 2m_b} \biggl[ - \int_{-\infty}^{+\infty}{d\omega \over W/2} \, \phantom{mmmmmmmm} \nonumber\\[0.03in] \phantom{mmmmmmmm} f_F(\omega)\int_{-W/2}^{+W/2} {d\xi \over W/2} \, \xi A(\xi,\omega) \biggr] \label{10}.\end{aligned}$$ Note that the quantity in the square brackets in Eq. (\[10\]) is just the negative of the kinetic energy, which is a well known result [@hirsch2; @vdmarel2] for a tight-binding model with nearest neighbour hopping only. In everything that follows we will retrict ourselves to half-filling. Then the model has particle-hole symmetry, and Eq. (\[10\]) reduces to $$\begin{aligned} {\rm RHS} = {1\over D}{\pi e^2 \over 2m_b} \biggl[ 1 - 4 \int_{-\infty}^{+\infty}{d\omega \over W/2} \, \phantom{mmmmmmm} \nonumber\\[0.03in] \phantom{mmmmmm} f_F(\omega) \int_0^{W/2}{d\xi \over W/2} \, \xi A(\xi,\omega) \biggr] \label{11}.\end{aligned}$$ Eq. (\[10\]) or Eq. (\[11\]) reduces to the usual result for large bandwidth, within a factor of order unity involving the dimensionality, as in the case with quadratic dispersion. To compute the conductivity given by Eq. (\[5\]) and the optical sum rule given by Eqs. (\[8\]) and (\[11\]) we require the electron self energy $\Sigma (\omega + i \delta)$, which determines the electron Green’s function $G(\xi,\omega + i \delta)$ and spectral function $A(\xi,\omega)$. One possibility is to use a model for the self energy (see the paper by Norman and Pepin [@norman1] where they obtain $\Sigma$ from a fit to APRES data for example). We use a more microscopic approach and assume that the electrons are coupled to bosons which are modeled by Einstein oscillators. While we adopt the formalism for the conventional electron phonon mechanism, we are open to the possibility that the electrons interact with spin fluctuations, and we therefore tacitly assume that this formalism applies in this case as well. The interaction is defined in terms of the electron-boson spectral density, $\alpha ^{2}F(\omega)$, which for an Einstein oscillator is simply a delta function: $\alpha ^{2}F(\omega)=A\,\delta (\omega-\omega_E)$ where $\omega_E$ is the Einstein frequency. The parameter $A$ specifies the strength of the interaction (not to be confused with the spectral function used above); it is given by $A = \lambda \omega_E /2$. This quantity can be conveniently visualized as the area under the $\alpha ^{2}F(\omega)$ curve for arbitrary (i.e. non delta-function-like) electron–boson spectral densities. The parameter $\lambda $ is the usual electron mass enhancement parameter. Two of these three parameters ($\lambda $, $\omega_E$, and $A$) are independent. The self energy equations for $\Sigma(\omega + i \delta) \equiv \Sigma_{1}(\omega + i \delta) + i\Sigma_{2}(\omega + i \delta)$ have the form: $$\begin{aligned} \Sigma_1(\omega + i \delta) =\, A\, P\int_{-\infty }^{\infty }d\omega ^{\prime }\biggl[ \frac{f_{B}(\omega_E) + f_{F}(-\omega^{\prime })}{\omega -\omega_E -\omega^{\prime}} \phantom{mm} \nonumber\\[0.03in] \phantom{mm} +\frac{f_{B}(\omega_E) + f_{F}(\omega ^{\prime })} {\omega +\omega_E -\omega^{\prime }}\biggr] N(\omega^{\prime }), \label{self-eq-1} \\[0.08in] \Sigma_{2}(\omega + i \delta) = -A\, \pi \left[ N(\omega -\omega_E)\left\{ f_{B}(\omega_E) + f_{F}(\omega_E \right.\right. \nonumber \\[0.03in] %\phantom{mmmm} \left. -\omega ) \right\} + \left. N(\omega +\omega_E) \left\{ f_{B}(\omega_E) + f_{F}\left(\omega_E +\omega \right) \right\} \right] , \label{self-eq-2} \\[0.08in] N(\omega ) =\frac{1}{\pi }\biggl[ \tan ^{-1}\frac{\omega -\Sigma _{1}(\omega )+ W/2}{-\Sigma _{2}(\omega )} \phantom{mmmmm} \nonumber\\[0.03in] \phantom{mmm} -\tan ^{-1}\frac{\omega -\Sigma _{1}(\omega )- W/2}{-\Sigma _{2}(\omega )}\biggr] , \label{self-eq-3}\end{aligned}$$ where the symbol $P$ in Eq. (\[self-eq-1\]) denotes the Cauchy principal value of the integral and $f_{B}(\omega_E)$ is the Bose–Einstein distribution function. The form of Eq. (\[self-eq-3\]) is a consequence of the model for the electronic band we have adopted. For an infinite quadratic band the self energy is an explicit function of frequency [@mars1], which is given by Eqs. (\[self-eq-1\])–(\[self-eq-2\]) with $N(\omega)=1$ according to Eqs. (\[self-eq-3\]). However, if one tried to keep using Eqs. (\[self-eq-1\])–(\[self-eq-2\]) with $N(\omega)=const$ to compute the (dimensionless) renormalized density of states $N(\omega) = W \int d \xi g(\xi) \,A\left(\xi,\omega\right) $ for a band of finite width, then no satisfactory result could be produced. In this case it is necessary to solve Eqs. (\[self-eq-1\])–(\[self-eq-3\]) for the self energy $\Sigma (\omega + i \delta)$ self-consistently [@mitrovic1]. The importance of this is illustrated in Fig. 1. Here we show both self-consistent (solid curve) and non self-consistent (dashed curve) results for $N(\omega)$ in the electronic band at reduced temperature $t \equiv T/(W/2) = 0.02$ for $\Omega \equiv \omega_E/(W/2) = 0.1$ and $\lambda =2$. Note that the non self-consistent electron density of states (dashed curve) shows an unphysical saturation for a small range of frequencies above the bare band edge (at a value of $\omega/(W/2) = 1$). On the other hand, the self-consistent density of states gradually decreases with increasing $\omega $ over a range of frequencies equal to a fraction of the bandwidth. Further details of this model will be provided elsewhere [@note]. We also refer the reader to papers [@alexandrov87]–[@dogan1] in which coupling of the electrons to phonons is considered within a Migdal-Eliashberg self-consistent approximation. Before proceeding to the presentation of numerical results for the optical integral, note that there are two ways to calculate it. The easier way is through direct evaluation of Eq. (\[8\]) or (\[11\]), which is the RHS of the optical integral sum rule. The harder way is to evaluate the conductivity (Eq. (\[5\])) and then integrate it explicitly over all frequencies. This latter method gives us an understanding of how the optical spectral weight is distributed in frequency [@note]. We have used both methods, and find agreement with an accuracy of $0.01\%$. {height="5.cm"} {height="10cm"} Discussion of the results ========================= For purposes of presentation, we show results for the optical integral $S$ in dimensionless form, i.e. by omitting the factor that precedes the square brackets in Eq. (\[8\]) for the quadratic dispersion, and in Eq. (\[11\]) for the tight-binding dispersion. Thus, the ‘standard’ value for the sum rule in the ensuing results corresponds to a value 1. All energies are measured in units of $W/2$, half of the bare electronic bandwidth. We also use normalized variables: the normalized frequency of Einstein oscillators, $\Omega \equiv \omega_E/(W/2)$, the normalized temperature, $t \equiv T/(W/2)$, and the normalized area under $\alpha^2F(\omega)$, $a \equiv A/(W/2)$. In Fig. 2 we show zero temperature results for the percent deviation of the total optical spectral weight which results from interactions, i.e. from a finite value of the parameter $a$ which enters the electron boson spectral density. In all cases the deviations are negative but they are plotted as positive as a function of the normalized boson frequency, $\Omega$. The top (bottom) frame applies to the case with quadratic (tight-binding) dispersion. The various curves are labeled by the value of electron mass enhancement parameter $\lambda $ that were used. We see that for a fixed value of $\lambda$ the percent deviation increases as $\omega_E$ increases; similarly, for fixed $\omega_E$ and increasing $\lambda$ the deviation increases. While both band structure models show the same qualitative behavior the effect is larger for the tight binding case. Note that the result in the lower frame also represents the kinetic energy change. The optical sum and the kinetic energy do follow each other but are numerically different for the quadratic band case. In Fig. 3 we consider temperature variations of the optical sum as a function of normalized temperature $t$. Results for $a=0.02$ (top frame) and $a=0.1$ (bottom frame) are shown. The former value corresponds to conventional metals while the latter number is characteristic of strongly coupled systems like the high–T$_{c}$ cuprates. For Fig. 3 we have chosen two representative values of the boson frequency $\omega_E$: in the top frame they correspond to $\lambda =0.8$ and $1$, while in the bottom frame they correspond to $\lambda =2$ and $4$. In both frames we show the optical sum derived from both the quadratic band and the tight binding band, as indicated. In each frame solid and dashed curves are used to distingush between the results for two different values of the electron enhancement parameter. We see that, with $a$ and $\lambda$ parameters fixed, the temperature variations of the optical sum for tight binding are larger than for quadratic bands. {height="10cm"} In all cases the variation of the optical sum with temperature increases as $\lambda$ increases. At the same time the absolute value of the deviation of $S$ from 1 at $t=0$ increases with increasing coupling strength $a$ (compare the top frame with the bottom frame). This is in accordance with the results presented in Fig. 2 where we see that for a fixed $\lambda$ the absolute value of the deviation grows as $\Omega$ increases (remember that $a=\lambda \Omega /2$). From Fig. 3 we conclude that interactions play an essential role in determining the temperature dependence of the optical sum. To understand this point better we return to Eq. (\[2\]) which gives the optical sum as an integral of two factors, the second derivative of the electron dispersion, and the occupation probability, $ n(\xi_{k})$, given by Eq. (\[6\]). The temperature dependence of $ n(\xi_{k})$ derives from two sources: the Fermi function $f_{F}\left(\omega\right)$ and the electron spectral function $A\left(\xi,\omega\right)$. The former factor is always operative, even in the noninteracting case. The latter factor produces an additional temperature dependence only when self energy effects are included (check Eqs. (\[self-eq-1\])–(\[self-eq-2\]) which include temperature through the functions $f_{F}\left(\omega\right)$ and $f_{B}\left(\omega\right)$). To get some idea of the significance of this second source of the temperature dependence we evaluate Eq. (\[6\]) with the thermal factor $f_{F}\left(\omega\right)$ artificially “switched off” and kept in the form valid at $t=0$ (i. e. in the form of the step function), but with the spectral function $A\left(\xi,\omega\right)$ evaluated properly from Eqs. (\[self-eq-1\])–(\[self-eq-2\]) for a range of temperatures. The results are shown in Fig. 4a for $t=0.0, 0.01, 0.03$ and $0.05$. Sharper curves correspond, of course, to lower temperatures. We see that the temperature dependence of $ n(\xi)$ obtained in this way is quite strong. The full temperature dependence of the occupation probability $ n(\xi)$, resulting when both the thermal factor $f_{F}\left(\omega\right)$ and the temperature dependence of $A\left(\xi,\omega\right)$ are accounted for in Eq. (\[6\]), is illustrated in Fig. 4b by dashed curves. We have also included the corresponding results from Fig. 4a so a direct comparison could be made. The conclusion is that the temperature dependence of the self energy is always important for determining the occupation probability $n_{k}(T)$ and therefore plays an essential role in the sum rule. This source of the temperature dependence of $S$ becomes dominant as the temperature increases. This important dependence is omitted in Ref. [@vdmarel2]; in that analysis interactions were not included. {height="10cm"} ![ The percent variation in the optical sum between normalized temperatures $t=0.025$ and $t=0$ vs. the normalized boson frequency $\Omega $ for the values of mass enhancement parameter $\lambda =0.8 ,1, 2$, and $4$. Panel (a) applies to the parabolic band, and panel (b) to the tight binding band. The dashed line shows the 2% loss of $S$ estimated from the data of Molegraaf [*et al*]{} [@vdmarel1]. ](paper-5-fig5-2c.ps){height="10cm"} In Fig. 5 we show the percentage spectral weight lost between $t=0$ and $t=0.025$ as a function of the normalized boson energy $\Omega $ for several values of the mass enhancement parameter $\lambda$. We see that the percentage increases with increasing $\lambda$ but that for a fixed $\lambda $ it decreases with increasing values of $\omega_E $ in the range of parameters considered. In a recent experiment Molegraaf [*et al*]{} [@vdmarel1] have found that the optical sum decreases noticeably from $0$ K to $200$ K in two cuprate superconducting samples. From the data in this reference we estimate the corresponding percentage change to be of the order of 2%. We show this as a horizontal dotted line in Fig. 5. Note that in these experiments the bandwidth $W$ is estimated to be 1.25 eV and $T=200$ K corresponds to $t=.027$, close to the value used in Fig. 5. While our theory is simple with coupling to a single boson only, the experimental observation puts a constraint on allowed values of $\Omega$ and $\lambda$. Only those results that fall close to the dotted line are possible. This still leaves a considerable range of possible parameters. If one has in mind a particular boson model such as phonons or spin fluctuations, $\omega_E $ is further constrained and the optical sum rule can be used to deduce the value of $\lambda$. Alternatively, from a detailed fit to the frequency dependence of the optical conductivity in the cuprates, Schachinger and Carbotte [@carbotte-book] have determined that $\lambda \simeq 2$ in these materials. Reference to the lower frame of Fig. 5 gives an estimate of $\Omega \approx 0.1$, This implies a frequency $\omega_E \approx 62$ meV, which is somewhat high for phonons. Extensive calculations [@carbotte-rmp] for a distributed electron–boson spectra $\alpha ^{2}F(\omega )$ suggest that the appropriate single frequency characterizing the spectrum is $\omega_{\ln }$, which is equal to approximately one half of the maximum phonon energy ($\omega_{D}$). For the cuprates this should be $\lesssim 40\,$ meV. We will not pursue this point here [@note], but instead focus on the primary observation of Molegraaf and coworkers[@vdmarel1]. They find an abrupt jump upwards in the optical spectral weight as the temperature is lowered into the superconducting state in two cuprate materials. They interpret this as indicative of a [*decrease*]{} in the absolute value of the kinetic energy. This is contrary to what is expected in the conventional BCS framework and is suggestive of a novel type of [*kinetic energy driven*]{} superconductivity. Returning to the top frame of Fig. 3 note that if at a critical temperature $T_{c}$ a phase transition occurs in which the boson energy hardens (leaving everything else the same) so that $\lambda $ changes from 1.0 to 0.8 say, the corresponding total optical spectral weight would jump from the solid to the dashed line. This is illustrated in the inset. ‘Undressing’ of the electron’s mass due to hardening of the boson spectrum leads directly to an increase in the optical sum and a decrease in the kinetic energy. We have not considered specifically the superconducting transition. In this case the reduction in kinetic energy due to the undressing process would have to overcompensate for the increase in kinetic energy that occurs when Cooper pairs form. Schachinger, Carbotte and Basov [@carbotte1] (see also reference [@carbotte-book]) have determined the boson spectrum involved for electron interactions in the cuprates from considerations of the frequency dependence of the infrared conductivity. They found that, as the temperature is reduced, the boson spectrum becomes gapped at low frequency with formation of an optical resonance at higher frequency, a process which effectively corresponds to a hardening of the boson spectrum; this change would manifest itself as the ‘undressing’ process described here. A similar conclusion was reached in Ref. [@haslinger03] based on calculations of the condensation energy. Conclusions =========== We have adopted a very simple model for interacting electrons to investigate the dependence of the optical sum on interactions and temperature. The model consists of electrons with bandwidth $W$ with a constant density of states, and with a dispersion given by either a parabolic relation or a tight-binding description. This additional modelling is required for the dispersion in order to correctly describe the group velocity, whose energy dependence is important for satisfying the optical sum rule. These electrons interact with a boson, and we have described this interaction with a (self-consistent) Migdal approximation. The optical conductivity is described by the bubble approximation; one can readily verify that the optical sum rule, which relates an exact two particle response function to an exact single particle property, is satisfied by these two (seemingly) unrelated approximations. A natural interpretation of the sum rule experiments [@vdmarel1; @lobo1; @vdmarel2; @hirsch3] is to infer a novel mechanism for superconductivity that is accompanied by a kinetic energy decrease (absolute value), in contrast to the usual BCS case. Here we have adopted an approach which in this sense is conventional; the deviation from the usual BCS case instead arises because of a change in the boson characteristics at and below the superconducting transition. This possibility was inspired by an analysis of the neutron scattering data which showed a definite change in the spin fluctuation spectrum at $T_c$ [@carbotte-book; @carbotte1]. Here we have modelled these changes as a shift in boson spectral weight from low to high frequency with a concomitant lowering of $\lambda$. As our results show, this leads naturally to an [*increase*]{} in the optical sum, as observed in experiment. Thus, these experiments, along with others [@carbotte-book; @carbotte1; @timusk04] find a consistent explanation in boson mediated superconductivity accompanied by temperature dependent changes in the boson spectral function. Acknowledgments =============== Work supported by the Natural Science and Engineering Research Council of Canada (NSERC) and the Canadian Institute for Advanced Research (CIAR). H.J.A. Molegraaf, C. Presura, D. van der Marel, P.H. Kes, and M. Li, Science 295, 2239 (2002). A.F. Santander-Syro, R.P.M.S. Lobo, W. Bontemps, Z. Konstantinovic, Z.Z. Li, and H. Raffz, Europhys. Lett. 62, 568 (2003). C.C. Homes, S.V. Dordevic, D.A. Bonn, Ruixing Liang, and W.N. Hardy, cond-mat/0303506. D. van der Marel, H.J.A. Molegraaf, C. Presura, and I. Santoso, cond-mat/0302169. A.E. Karakozov, E.G. Maksimov, and O.V. Dolgov, cond-mat/0208170. J.E. Hirsch, Physica C 199, 305 (1992); Physica C 201, 347 (1992). J.E. Hirsch and F. Marsiglio, Phys. Rev. B62, 15131 (2000). J.E. 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--- abstract: | By analyzing the data sets of 17.3 pb$^{-1}$ taken at $\sqrt s= 3.773$ GeV and of 6.5 pb$^{-1}$ taken at $\sqrt s= 3.650$ GeV with the BES-II detector at the BEPC collider, we measure the observed cross sections for the exclusive light hadron final states of $K_S^0K^-\pi^+$, $K_S^0K^-\pi^+\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^-$, $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ and $K_S^0K^-\pi^+\pi^0\pi^0$ produced in $e^+ e^-$ annihilation at the two energy points. We set the upper limits on the observed cross sections and the branching fractions for $\psi(3770)$ decay to these final states at $90\%$ C.L.. author: - | M. Ablikim$^{1}$, J. Z. Bai$^{1}$, Y. Bai$^{1}$, Y. Ban$^{11}$, X. Cai$^{1}$, H. F. Chen$^{15}$, H. S. Chen$^{1}$, H. X. Chen$^{1}$, J. C. Chen$^{1}$, Jin Chen$^{1}$, X. D. Chen$^{5}$, Y. B. Chen$^{1}$, Y. P. Chu$^{1}$, Y. S. Dai$^{17}$, Z. Y. Deng$^{1}$, S. X. Du$^{1}$, J. Fang$^{1}$, C. D. Fu$^{14}$, C. S. Gao$^{1}$, Y. N. Gao$^{14}$, S. D. Gu$^{1}$, Y. T. Gu$^{4}$, Y. N. Guo$^{1}$, K. L. He$^{1}$, M. He$^{12}$, Y. K. Heng$^{1}$, J. Hou$^{10}$, H. M. Hu$^{1}$, T. Hu$^{1}$, G. S. Huang$^{1}$$^{a}$, X. T. Huang$^{12}$, Y. P. Huang$^{1}$, X. B. Ji$^{1}$, X. S. Jiang$^{1}$, J. B. Jiao$^{12}$, D. P. Jin$^{1}$, S. Jin$^{1}$, Y. F. Lai$^{1}$, H. B. Li$^{1}$, J. Li$^{1}$, L. Li$^{1}$, R. Y. Li$^{1}$, W. D. Li$^{1}$, W. G. Li$^{1}$, X. L. Li$^{1}$, X. N. Li$^{1}$, X. Q. Li$^{10}$, Y. F. Liang$^{13}$, H. B. Liao$^{1}$$^{b}$, B. J. Liu$^{1}$, C. X. Liu$^{1}$, Fang Liu$^{1}$, Feng Liu$^{6}$, H. H. Liu$^{1}$$^{c}$, H. M. Liu$^{1}$, J. B. Liu$^{1}$$^{d}$, J. P. Liu$^{16}$, H. B. Liu$^{4}$, J. Liu$^{1}$, R. G. Liu$^{1}$, S. Liu$^{8}$, Z. A. Liu$^{1}$, F. Lu$^{1}$, G. R. Lu$^{5}$, J. G. Lu$^{1}$, C. L. Luo$^{9}$, F. C. Ma$^{8}$, H. L. Ma$^{1}$, L. L. Ma$^{1}$$^{e}$, Q. M. Ma$^{1}$, M. Q. A. Malik$^{1}$, Z. P. Mao$^{1}$, X. H. Mo$^{1}$, J. Nie$^{1}$, R. G. Ping$^{1}$, N. D. Qi$^{1}$, H. Qin$^{1}$, J. F. Qiu$^{1}$, G. Rong$^{1}$, X. D. Ruan$^{4}$, L. Y. Shan$^{1}$, L. Shang$^{1}$, D. L. Shen$^{1}$, X. Y. Shen$^{1}$, H. Y. Sheng$^{1}$, H. S. Sun$^{1}$, S. S. Sun$^{1}$, Y. Z. Sun$^{1}$, Z. J. Sun$^{1}$, X. Tang$^{1}$, J. P. Tian$^{14}$, G. L. Tong$^{1}$, X. Wan$^{1}$, L. Wang$^{1}$, L. L. Wang$^{1}$, L. S. Wang$^{1}$, P. Wang$^{1}$, P. L. Wang$^{1}$, W. F. Wang$^{1}$$^{f}$, Y. F. Wang$^{1}$, Z. Wang$^{1}$, Z. Y. Wang$^{1}$, C. L. Wei$^{1}$, D. H. Wei$^{3}$, Y. Weng$^{1}$, N. Wu$^{1}$, X. M. Xia$^{1}$, X. X. Xie$^{1}$, G. F. Xu$^{1}$, X. P. Xu$^{6}$, Y. Xu$^{10}$, M. L. Yan$^{15}$, H. X. Yang$^{1}$, M. Yang$^{1}$, Y. X. Yang$^{3}$, M. H. Ye$^{2}$, Y. X. Ye$^{15}$, C. X. Yu$^{10}$, G. W. Yu$^{1}$, C. Z. Yuan$^{1}$, Y. Yuan$^{1}$, S. L. Zang$^{1}$$^{g}$, Y. Zeng$^{7}$, B. X. Zhang$^{1}$, B. Y. Zhang$^{1}$, C. C. Zhang$^{1}$, D. H. Zhang$^{1}$, H. Q. Zhang$^{1}$, H. Y. Zhang$^{1}$, J. W. Zhang$^{1}$, J. Y. Zhang$^{1}$, X. Y. Zhang$^{12}$, Y. Y. Zhang$^{13}$, Z. X. Zhang$^{11}$, Z. P. Zhang$^{15}$, D. X. Zhao$^{1}$, J. W. Zhao$^{1}$, M. G. Zhao$^{1}$, P. P. Zhao$^{1}$, B. Zheng$^{1}$, H. Q. Zheng$^{11}$, J. P. Zheng$^{1}$, Z. P. Zheng$^{1}$, B. Zhong$^{9}$ L. Zhou$^{1}$, K. J. Zhu$^{1}$, Q. M. Zhu$^{1}$, X. W. Zhu$^{1}$, Y. C. Zhu$^{1}$, Y. S. Zhu$^{1}$, Z. A. Zhu$^{1}$, Z. L. Zhu$^{3}$, B. A. Zhuang$^{1}$, B. S. Zou$^{1}$\ (BES Collaboration)\ title: | Measurements of the observed cross sections for $e^+e^- \to $ [*exclusive light hadrons*]{} containing $K_S^0$ meson at $\sqrt s=3.773$ and 3.650 GeV --- -0.2cm -0.2cm **INTRODUCTION** ================ The $\psi(3770)$ resonance is expected to decay almost entirely into $D\bar D$ meson pairs since its width is almost two orders of magnitude larger than that of $\psi(3686)$ [@prl39_526]. In recent years, the study of the $\psi(3770)$ non-$D\bar D$ decays becomes an attractive study field in the charmonium energy region due to the existing puzzle that about $38\%$ of $\psi(3770)$ does not decay into $D\bar D$ meson pairs [@hepex_0506051]. To understand the possible excess of the $\psi(3770)$ cross section relative to the $D\bar D$ cross section, BES and CLEO Collaborations made many efforts to study the $\psi(3770)$ non-$D\bar D$ decays. The CLEO Collaboration measured the $e^+e^-\to\psi(3770)\to$ non-$D\bar D$ cross section to be $(-0.01\pm0.08^{+0.41}_{-0.30})$ nb [@prl96_092002]. While the BES Collaboration measured the branching fraction for $\psi(3770)\to$ non$-D\bar D$ decay to be $(15\pm5)\%$ [@plb641_145; @prl97_121801; @plb659_74; @prd76_000000; @pdg07], which indicates that, contrary to what is generally expected, the $\psi(3770)$ might substantially decay into non$-D \bar D$ final states or there are some new structure or physics effects which may partially be responsible for the largely measured non-$D\bar D$ branching fraction of the $\psi(3770)$ decays [@prl101_102004; @plb668_263]. BES Collaboration observed the first non$-D \bar D$ decay mode for $\psi(3770) \to J/\psi\pi^+\pi^-$, and measured its decay branching fraction to be ${\mathcal B}[\psi(3770) \to J/\psi\pi^+\pi^-]= (0.34\pm0.14\pm0.09)\%$ [@hepnp28_325; @plb605_63]. This was confirmed by CLEO Collaboration [@prl96_082004]. Latter, CLEO Collaboration observed more $\psi(3770)$ exclusive non-$D\bar D$ decays, $\psi(3770)\to J/\psi\pi^0\pi^0$, $J/\psi\pi^0$, $J/\psi\eta$ [@prl96_082004], $\gamma\chi_{cJ}(J=0,1,2)$ [@prl96_182002; @prd74_031106] and $\phi\eta$ [@prd74_012005], etc. Summing over these measured branching fractions yields the sum of the branching fractions for the $\psi(3770)$ exclusive non-$D\bar D$ decays not more than 2%. In addition, BES and CLEO Collaborations also attempted to search for other $\psi(3770)$ exclusive charmless decays [@prd70_077101; @prd72_072007; @plb650_111; @plb656_30; @epjc52_805] [@prd74_012005; @prl96_032003; @prd73_012002]. However, the existing results can not clarify the possible excess. For better understanding the origin of the possible excess, search for more $\psi(3770)$ exclusive charmless decays will be helpful. In this Letter, we report measurements of the observed cross sections for the exclusive light hadron final states of $K_S^0K^-\pi^+$ (Throughout the Letter, charge conjugation is implied), $K_S^0K^-\pi^+\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^-$, $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ and $K_S^0K^-\pi^+\pi^0\pi^0$ at the center-of-mass energies of 3.773 and 3.650 GeV with the same method as the one used in our previous works [@plb650_111; @plb656_30; @epjc52_805]. With the measured cross sections at the two energy points, we set the upper limits on the observed cross sections and the branching fractions for $\psi(3770)$ decay to these final states. The measurements are made by analyzing the data set of 17.3 pb$^{-1}$ collected at $\sqrt s= 3.773$ GeV \[called as the $\psi(3770)$ resonance data\] and the data set of 6.5 pb$^{-1}$ collected at $\sqrt s= 3.650$ GeV (called as the continuum data) with the BESII detector at the BEPC collider. BESII detector ============== The BES-II is a conventional cylindrical magnetic detector that is described in detail in Refs. [@nima344_319; @nima458_627]. A 12-layer vertex chamber (VC) surrounding the beryllium beam pipe provides input to the event trigger, as well as coordinate information. A forty-layer main drift chamber (MDC) located just outside the VC yields precise measurements of charged particle trajectories with a solid angle coverage of $85\%$ of 4$\pi$; it also provides ionization energy loss ($dE/dx$) measurements which are used for particle identification. Momentum resolution of $1.7\%\sqrt{1+p^2}$ ($p$ in GeV/$c$) and $dE/dx$ resolution of $8.5\%$ for Bhabha scattering electrons are obtained for the data taken at $\sqrt s= 3.773$ GeV. An array of 48 scintillation counters surrounding the MDC measures the time of flight (TOF) of charged particles with a resolution of about 180 ps for electrons. Outside the TOF, a 12 radiation length, lead-gas barrel shower counter (BSC), operating in limited streamer mode, measures the energies of electrons and photons over $80\%$ of the total solid angle with an energy resolution of $\sigma_E/E=0.22/\sqrt{E}$ ($E$ in GeV) and spatial resolutions of $\sigma_{\phi}=7.9$ mrad and $\sigma_z=2.3$ cm for electrons. A solenoidal magnet outside the BSC provides a 0.4 T magnetic field in the central tracking region of the detector. Three double-layer muon counters instrument the magnet flux return and serve to identify muons with momentum greater than 500 MeV/$c$. They cover $68\%$ of the total solid angle. EVENT SELECTION {#evtsel} ================ In the reconstruction of the $K_S^0K^-\pi^+$, $K_S^0K^-\pi^+\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^-$, $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ and $K_S^0K^-\pi^+\pi^0\pi^0$ final states, the $K^0_S$ and $\pi^0$ mesons are reconstructed through the decays of $K^0_S\to \pi^+\pi^-$ and $\pi^0\to \gamma\gamma$. For each candidate event, we require that at least four charged tracks are well reconstructed in the MDC with good helix fits, and the polar angle of each charged track satisfies $|\rm cos\theta|<0.85$. The charged tracks (except for the $K_S^0$ meson reconstruction) are required to originate from the interaction region $V_{xy}<2.0$ cm ($V_{xy}<8.0$ cm) and $|V_z|<20.0$ cm, where $V_{xy}$ and $|V_z|$ are the closest approaches in the $xy$-plane and the $z$ direction, respectively. The charged particles are identified by using the $dE/dx$ and TOF measurements, with which the combined confidence levels $CL_{\pi}$ and $CL_K $ for pion and kaon hypotheses are calculated. The pion and kaon candidates are required to satisfy $CL_{\pi}>0.001$ and $CL_K >CL_{\pi}$, respectively. To reconstruct $K^0_S$ mesons, we require that the $\pi^+\pi^-$ meson pairs must originate from a secondary vertex which is displaced from the event vertex at least by 4 mm in the $xy$-plane. The photons are selected with the BSC measurements. The good photon candidates are required to satisfy the following criteria: the energy deposited in the BSC is greater than 50 MeV, the electromagnetic shower starts in the first 5 readout layers, the angle between the photon and the nearest charged track is greater than $22^{\circ}$ [@plb_597_39; @plb_608_24], and the opening angle between the cluster development direction and the photon emission direction is less than $37^{\circ}$ [@plb_597_39; @plb_608_24]. For each candidate event, there may be several different charged and/or neutral track combinations satisfying the above selection criteria for exclusive light hadron final states. Each combination is subjected to an energy-momentum conservation kinematic fit. For the processes containing $\pi^0$ meson in the final states, an additional constraint kinematic fit is imposed on $\pi^0\to\gamma\gamma$. Candidates with a fit probability larger than 1$\%$ are accepted. If more than one combination satisfies the selection criteria in an event, only the combination with the longest decay distance of the reconstructed $K_S^0$ mesons is retained. To suppress the background from the $D\bar D$ decays, we use the double tag method [@npb727_395] to remove the $D\bar D$ events. For example, for the $K_S^0 K^-\pi^+\pi^0$ final state, we exclude the all possible events from $D\bar D$ decays by rejecting those in which the $D$ and $\bar D$ mesons can be reconstructed in the decay modes of $D^-\to K_S^0K^-$ and $D^+\to \pi^+\pi^0$, $D^-\to K^-\pi^0$ and $D^+\to K_S^0 \pi^+$, $\bar D^0\to K_S^0 \pi^0$ and $D^0 \to K^-\pi^+$ [@npb727_395]. For the other final states, the events from $D\bar D$ decays are suppressed similarly. The remaining contaminations from $D\bar D$ decays due to particle misidentification or missing photon(s) are accounted by using Monte Carlo simulation, as discussed in Section \[backsub\]. DATA ANALYSIS ============= In the data analysis, these processes containing $K^0_S$ meson in the final state are studied by examining the invariant mass spectra of the $\pi^+\pi^-$ combinations satisfying the above selection criteria for the $K^0_S$ meson reconstruction. The invariant masses of the $\pi^+\pi^-$ combinations are calculated with the momentum vectors from the $K^0_S$ reconstruction. Figure \[fig:data:con\] shows the resulting distribution of the invariant masses of the $\pi^+\pi^-$ combinations from the selected candidates for the $K_S^0K^-\pi^+$, $K_S^0K^-\pi^+\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^-$, $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ and $K_S^0K^-\pi^+\pi^0\pi^0$ final states. In each figure, the peak around the $K^0_S$ nominal mass indicates the production of $e^+e^-\to$ exclusive light hadrons containing $K_S^0$ meson. Fitting the $\pi^+\pi^-$ invariant mass spectra with a Gaussian function for the $K^0_S$ signal and a flat background yields the number of the events for each process observed from the $\psi(3770)$ resonance data and the continuum data. In the fit, the $K^0_S$ mass and its mass resolution are fixed at the values obtained by analyzing Monte Carlo samples. ![ The $\pi^+\pi^-$ invariant mass spectra of the candidates for the (a) $K_S^0K^-\pi^+$, (b) $K_S^0K^-\pi^+\pi^0$, (c) $K_S^0K^-\pi^+\pi^+\pi^-$, (d) $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$, (e) $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ and (f) $K_S^0K^-\pi^+\pi^0\pi^0$ final states selected from the $\psi(3770)$ resonance data (left) and the continuum data (right).[]{data-label="fig:data:con"}](bes_data.eps "fig:"){width="8cm" height="7cm"} (-145,178)[**(a)**]{} (-50,178)[**(a’)**]{} (-145,150)[**(b)**]{} (-50,150)[**(b’)**]{} (-145,122)[**(c)**]{} (-50,122)[**(c’)**]{} (-145,94)[**(d)**]{} (-50,94)[**(d’)**]{} (-145,66)[**(e)**]{} (-50,66)[**(e’)**]{} (-145,38)[**(f)**]{} (-50,38)[**(f’)**]{} (-165,5)[**Invariant mass (GeV/c$^2)$**]{} (-220,60) BACKGROUND SUBTRACTION {#backsub} ====================== Some other events may contribute to the selected candidate events for $e^+e^- \to f$ ($f$ represents exclusive light hadron final state). These include the events from $J/\psi$ and $\psi(3686)$ decays due to ISR returns, the events from the other final states due to misidentifying a pion as a kaon or reverse, and the events from $D\bar D$ decays. The number $N^{\rm b}$ of these contaminations should be subtracted from the number $N^{\rm obs}$ of the candidates for $e^+e^- \to f$. The estimation of them can be done based on Monte Carlo simulation. The details about the background subtraction have been described in Ref. [@plb650_111]. For each background channel except $D\bar D$ decays, 50,000 or 100,000 Monte Carlo events are used in the background estimation. The Monte Carlo sample of each different background channel is from ten to several thousands times larger than the data in size. Monte Carlo study shows that the contaminations from $\psi(3770) \to J/\psi\pi^+\pi^-$, $\psi(3770) \to J/\psi\pi^0\pi^0$, $\psi(3770) \to J/\psi\pi^0$ and $\psi(3770) \to\gamma \chi_{cJ}\hspace{0.1cm} (J=0,1,2)$ can be neglected. Even though we have removed the main contaminations from $D\bar D$ decays in the previous event selection (see section \[evtsel\]), there are still some events from $D\bar D$ decays satisfying the selection criteria for the light hadron final states due to particle misidentification or missing photon(s). The number of these contaminations from $D\bar D$ decays are further removed by analyzing a Monte Carlo sample which is about forty times larger than the $\psi(3770)$ resonance data. The Monte Carlo events are generated as $e^+e^- \to D\bar D$ at $\sqrt s=$ 3.773 GeV, where the $D$ and $\bar D$ mesons are set to decay into all possible final states with the branching fractions quoted from PDG [@pdg07]. Subtracting the number $N^{\rm b}$ of these contaminations from the number $N^{\rm obs}$ of the candidate events, we obtain the net number $N^{\rm net}$ of the signal events for each process. For the $K_S^0K^-\pi^+$ and $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ final states, for which only a few signal events are observed from the continuum data, we set the upper limits $N^{\rm up}$ on the number of the signal events at 90% C.L.. Here, we use the Feldman-Cousins method [@prd57_3873] and assume that the background is absent. The numbers of $N^{\rm obs}$, $N^{\rm b}$ and $N^{\rm net}$ (or $N^{\rm up}$) are summarized in the second, third and fourth columns of Tabs. \[tab:data\] and \[tab:con\]. For each process, the background events in the $\psi(3770)$ resonance data are dominant by $D\bar D$ decays and $\psi(3686)$ decays. While, there is no $D\bar D$ decay in the continuum data, and the $\psi(3686)$ production cross section at $\sqrt s=$ 3.650 GeV is much less than that at $\sqrt s=$ 3.773 GeV. So, the number of the background events in the continnum data can almost be negligible. RESULTS ======== Monte Carlo efficiency ----------------------- To estimate the detection efficiency $\epsilon$ for $e^+e^-\to f$, we use a phase space generator including initial state radiation and vacuum polarization corrections [@yf41_377] with $1/s$ energy dependence in cross section. Final state radiation [@cpc79_291] decreases the detection efficiency not more than 0.5%. Detailed analysis based on Monte Carlo simulation for the BES-II detector [@nima552_344] gives the detection efficiencies for each process at $\sqrt s= 3.773$ and 3.650 GeV, which are summarized in the fifth columns of Tabs. \[tab:data\] and \[tab:con\], where the detection efficiencies do not include the branching fractions for $K_S^0 \to \pi^+\pi^-$ and $\pi^0\to\gamma\gamma$, ${\mathcal B}(K_S^0 \to \pi^+ \pi^-)$ and ${\mathcal B}(\pi^0 \to \gamma\gamma)$. Observed cross sections ----------------------- Let ${\mathcal B}_{\pi^0}$ = ${\mathcal B}(\pi^0 \to \gamma \gamma)$ for the modes of $K_S^0K^-\pi^+\pi^0$ and $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$, ${\mathcal B}_{\pi^0}$ = ${\mathcal B}^2(\pi^0 \to \gamma\gamma)$ for the mode of $K_S^0K^-\pi^+\pi^0\pi^0$ and ${\mathcal B}_{\pi^0}$ = 1 for $K_S^0K^-\pi^+$, $K_S^0K^-\pi^+\pi^+\pi^-$ and $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$, where ${\mathcal B}(\pi^0 \to \gamma\gamma)$ is the branching fraction for the decay of $\pi^0 \to \gamma\gamma$, then the observed cross section for $e^+e^- \to f$ can be determined by $$\sigma_{e^+e^- \to f}=\frac{N^{\rm net}} {{\mathcal L} \times \epsilon \times {\mathcal B}(K_S^0 \to \pi^+ \pi^-) \times {\mathcal B}_{\pi^0}}, \label{eq:sig:net}$$ where ${\mathcal L}$ is the integrated luminosity of the data set, $N^{\rm net}$ is the number of the signal events, $\epsilon$ is the detection efficiency and ${\mathcal B}(K_S^0 \to \pi^+ \pi^-)$ is the branching fraction for the decay of $K_S^0 \to \pi^+ \pi^-$. Inserting these numbers in Eq. (\[eq:sig:net\]), we obtain the observed cross sections for each process at $\sqrt s= 3.773$ and 3.650 GeV. They are summarized in Tabs. \[tab:data\] and \[tab:con\], where the first error is statistical and the second systematic. In the measurements of the observed cross sections, the systematic errors arise from the uncertainties in integrated luminosity of the data set ($2.1\%$ [@plb641_145; @prl97_121801]), photon selection ($2.0\%$ per photon), tracking efficiency ($2.0\%$ per track), particle identification ($0.5\%$ per pion or kaon), kinematic fit ($1.5\%$), $K_S^0$ reconstruction ($1.1\%$ [@plb_608_24]), branching fractions quoted from PDG [@jpg33_1] ($0.03\%$ for ${\mathcal B}(\pi^0 \to \gamma\gamma)$ and $0.07\%$ for ${\mathcal B}(K_S^0 \to \pi^+ \pi^-)$), Monte Carlo modeling ($6.0\%$ [@plb650_111; @plb656_30; @epjc52_805]), Monte Carlo statistics ($1.4\%\sim4.4\%$), background subtraction ($0.0\%\sim3.0\%$) and fit to mass spectrum ($0.4\%\sim 8.5\%$). Adding these uncertainties in quadrature yields the total systematic error $\Delta_{\rm sys}$ for each mode at $\sqrt s= 3.773$ and 3.650 GeV. The upper limit $\sigma_{e^+e^- \to f}^{\rm up}$ on the observed cross sections for the $K_S^0K^-\pi^+$ and $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ final states at $\sqrt s= 3.650$ GeV are set with Eq. (\[eq:sig:net\]) by substituting $N^{\rm net}$ with $N^{\rm up}/(1-\Delta_{\rm sys})$, where $N^{\rm up}$ is the upper limit on the number of the signal event, and $\Delta_{\rm sys}$ is the systematic error in the cross section measurement. Inserting the corresponding numbers in the equation, we obtain the upper limits on the observed cross sections for $e^+e^-\to K_S^0K^-\pi^+$ and $e^+e^-\to K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ at $\sqrt s= 3.650$ GeV, which are also listed in Tab. \[tab:con\]. Upper limits on the observed cross sections and the branching fractions for $\psi(3770) \to f $ ----------------------------------------------------------------------------------------------- If we ignore the possible interference effects between the continuum and resonance amplitudes, and the difference of the vacuum polarization corrections at $\sqrt s= 3.773$ and 3.650 GeV, we can determine the observed cross section $\sigma _{\psi(3770) \to f}$ for $\psi(3770)\to f$ at $\sqrt s=3.773$ GeV by comparing the observed cross sections $\sigma _{e^+e^- \to f}^{3.773 \hspace{0.05cm}\rm GeV}$ and $\sigma _{e^+e^- \to f}^{3.650\hspace{0.05cm}\rm GeV}$ for $e^+e^- \to f$ measured at $\sqrt s= 3.773$ and 3.650 GeV, respectively. It can be given by $$\sigma _{\psi(3770) \to f}= \sigma _{e^+e^- \to f}^{3.773 \hspace{0.05cm}\rm GeV} - f_{\rm co} \times \sigma _{e^+e^- \to f}^{3.650\hspace{0.05cm}\rm GeV}, \label{eq:sig:3770}$$ where $f_{\rm co}=3.650^2/3.773^2$ is the normalization factor to consider the $1/s$ cross section dependence. The results are summarized in the second column of Tab. \[tab:psipp:f\], where the first error is the statistical, the second is the independent systematic arising from the uncertainties in the Monte Carlo statistics, in the fit to the mass spectrum and in the background subtraction, and the third is the common systematic error arising from the other uncertainties as discussed in the subsection B. The upper limit on the observed cross section $\sigma^{\rm up}_{\psi(3770) \to f}$ for $\psi(3770) \to f$ at $\sqrt s=3.773$ GeV is set by shifting the cross section by 1.64$\sigma$, where $\sigma$ is the total error of the measured cross section. The results on $\sigma^{\rm up}_{\psi(3770) \to f}$ are summarized in the third column of Tab. \[tab:psipp:f\]. The upper limit on the branching fraction ${\mathcal B}^{\rm up}_{\psi(3770)\to f}$ for $\psi(3770)\to f$ is set by dividing its upper limit on the observed cross section $\sigma^{\rm up}_{\psi(3770)\to f}$ by the observed cross section $\sigma^{\rm obs}_{\psi(3770)}=(7.15\pm0.27\pm0.27)$ nb [@plb650_111] for the $\psi(3770)$ production at $\sqrt s=3.773$ GeV and a factor $(1-\Delta \sigma^{\rm obs}_{\psi(3770)})$, where $\Delta \sigma^{\rm obs}_{\psi(3770)}$ is the relative error of the $\sigma^{\rm obs}_{\psi(3770)}$. The results on ${\mathcal B}^{\rm up}_{\psi(3770)\to f}$ are summarized in the last column of Tab. \[tab:psipp:f\]. SUMMARY ======= In this Letter, we present the measurements of the observed cross sections for $K_S^0K^-\pi^+$, $K_S^0K^-\pi^+\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^-$, $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$, $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ and $K_S^0K^-\pi^+\pi^0\pi^0$ produced in $e^+ e^-$ annihilation at $\sqrt s= 3.773$ and 3.650 GeV. These cross sections are obtained by analyzing the data sets of 17.3 pb$^{-1}$ taken at $\sqrt s= 3.773$ GeV and of 6.5 pb$^{-1}$ at $\sqrt s= 3.650$ GeV with the BES-II detector at the BEPC collider. By comparing the observed cross sections for each process measured at $\sqrt s= 3.773$ and 3.650 GeV, we set the upper limits on the observed cross sections and the branching fractions for $\psi(3770)$ decay to these final states at $90\%$ C.L.. These measurements provide helpful information to understand the mechanism of the continuum light hadron production and the discrepancy between the observed cross sections for $D\bar D$ and $\psi(3770)$ production. Acknowledgments =============== The BES collaboration thanks the staff of BEPC for their hard efforts. This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10491300, 10225524, 10225525, 10425523, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos. U-11, U-24, U-25, the Knowledge Innovation Project of CAS under Contract Nos. U-602, U-34 (IHEP), the National Natural Science Foundation of China under Contract No. 10225522 (Tsinghua University). [99]{} MARK-I Collaboration, P. A. Rapidis, [*et al*]{}., Phys. Rev. Lett. 39 (1978) 526. G. Rong, D. H. Zhang, J. C. Chen, hep-ex/0506051. CLEO Collaboration, D. Besson, [*et al*]{}., Phys. Rev. Lett. 96 (2006) 092002. BES Collaboration, M. Ablikim [*et al*]{}., Phys. Lett. B 641 (2006) 145. BES Collaboration, M. Ablikim [*et al*]{}., Phys. Rev. Lett. 97 (2006) 121801. BES Collaboration, M. Ablikim [*et al*]{}., Phys. Lett. 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G 33 (2006) 1. $e^+e^- \to$ $N_{\rm}^{\rm obs}$ $N^{\rm b}$ $N^{\rm net}$ $\epsilon$($\%$) $\Delta_{\rm sys}$($\%$) $\sigma^{\rm obs}[{\rm pb}]$ ------------------------------------- --------------------- -------------- --------------- ------------------ -------------------------- ------------------------------ $K^0_S K^-\pi^+$ $18.4 \pm4.6$ $0.1\pm 0.0$ $18.3\pm4.6$ $10.02\pm0.14$ 10.7 $15.2\pm3.8\pm1.6$ $ K^0_S K^-\pi^+\pi^0$ $41.2 \pm6.6$ $1.1\pm0.2$ $40.1\pm6.6$ $3.52\pm0.08$ 11.6 $96.2\pm15.9\pm11.1$ $ K^0_S K^-\pi^+\pi^+\pi^-$ $40.0 \pm 6.5$ $1.0\pm0.2$ $38.9\pm6.5$ $3.56\pm0.06$ 14.2 $91.5\pm15.3\pm13.0$ $K^0_S K^-\pi^+\pi^+\pi^-\pi^0 $ $24.5 \pm 5.2$ $1.5\pm0.3$ $23.0\pm5.2$ $0.77\pm0.03$ 15.2 $253.0\pm57.1\pm38.4$ $K^0_SK^-\pi^+\pi^+\pi^+\pi^-\pi^-$ $4.8 \pm 2.2$ $0.3\pm0.1$ $4.5\pm2.2$ $0.84\pm0.03 $ 18.4 $44.4\pm21.9\pm8.2$ $ K^0_S K^-\pi^+\pi^0\pi^0$ $19.8 \pm4.9$ $2.8\pm0.5$ $17.0\pm4.9$ $0.99\pm0.04$ 14.3 $147.0\pm42.4\pm21.0$ \[tab:data\] ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $e^+e^- \to$ $N_{\rm }^{\rm obs}$ $N^{\rm b}$ $N^{\rm net}({\rm $\epsilon$($\%$) $\Delta_{\rm sys}$($\%$) $\sigma^{\rm obs}(\sigma^{\rm up})[\rm pb]$ or}\hspace{0.5em} N^{\rm up}_{\rm})$ ------------------------------------- ---------------------- ------------- -------------------------------------- ------------------ -------------------------- --------------------------------------------- $K^0_S K^-\pi^+$ $2$ 0.0 $<5.91$ $10.55\pm0.15$ 12.9 $<14.3$ $K^0_S K^-\pi^+\pi^0$ $7.7\pm2.9$ 0.0 $7.7 \pm2.9$ $3.62\pm 0.09$ 11.6 $47.9\pm18.0\pm5.6 $ $K^0_S K^-\pi^+\pi^+\pi^-$ $13.4 \pm3.8$ 0.0 $13.4 \pm 3.8$ $3.66\pm 0.06$ 14.1 $81.4\pm23.1\pm11.5 $ $K^0_S K^-\pi^+\pi^+\pi^-\pi^0 $ $4.6\pm2.5$ 0.0 $4.6\pm2.5$ $0.87\pm 0.03$ 17.2 $119.0\pm64.7\pm20.5$ $K^0_SK^-\pi^+\pi^+\pi^+\pi^-\pi^-$ $0$ 0.0 $<2.44$ $0.95\pm 0.03$ 18.1 $<69.7$ $K^0_S K^-\pi^+\pi^0\pi^0$ $3.3\pm2.0$ 0.0 $3.3\pm2.0$ $1.12\pm 0.05$ 14.1 $67.1\pm40.7\pm9.5$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \[tab:con\] Decay Mode $\sigma_{\psi(3770) \to f}$ (pb) $\sigma^{\rm up}_{\psi(3770) \to f}$(pb) ${\mathcal B}^{\rm up}_{\psi(3770) \to f}$ ------------------------------------- ---------------------------------- ------------------------------------------ -------------------------------------------- $K_S^0K^-\pi^+$ $15.2\pm3.8\pm0.2\pm1.6^*$ $<22.0$ $3.2\times 10^{-3}$ $K_S^0K^-\pi^+\pi^0$ $51.4\pm23.2\pm2.6\pm5.8$ $<90.7$ $13.3\times 10^{-3}$ $K_S^0K^-\pi^+\pi^+\pi^-$ $15.3\pm26.5\pm2.2\pm2.1$ $<59.0 $ $8.7 \times 10^{-3}$ $K_S^0K^-\pi^+\pi^+\pi^-\pi^0$ $141.6\pm83.2\pm14.7\pm20.7$ $<284.3$ $41.8 \times 10^{-3}$ $K_S^0K^-\pi^+\pi^+\pi^+\pi^-\pi^-$ $44.4\pm21.9\pm2.1\pm7.9^*$ $<82.7$ $12.2 \times 10^{-3}$ $K_S^0K^-\pi^+\pi^0\pi^0$ $84.2\pm57.2\pm8.3\pm11.2$ $<180.4$ $26.5 \times 10^{-3}$ \[tab:psipp:f\]

--- abstract: 'We present a flexible statistical modelling framework to deal with multivariate count data along with longitudinal and repeated measures structures. The covariance structure for each response variable is defined in terms of a covariance link function combined with a matrix linear predictor involving known matrices. To specify the joint covariance matrix for the multivariate response vector the generalized Kronecker product is employed. The count nature of the data is taken into account by means of the power dispersion function associated with the Poisson-Tweedie distribution. Furthermore, the score information criterion is extended for selecting the components of the matrix linear predictor. We analyse a dataset consisting of prey animals (the main hunted species, the blue duiker *Philantomba monticola* and other taxa) shot or snared for bushmeat by $52$ commercial hunters over a $33$-month period in Pico Basilé, Bioko Island, Equatorial Guinea. By taking into account the severely unbalanced repeated measures and longitudinal structures induced by the hunters and a set of potential covariates (which in turn affect the mean and covariance structures), our method can be used to indicate whether there was statistical evidence of a decline in blue duikers and other species hunted during the study period. Determining whether observed drops in the number of animals hunted are indeed true is crucial to assess whether species depletion effects are taking place in exploited areas anywhere in the world. We suggest that our method can be used to more accurately understand the trajectories of animals hunted for commercial or subsistence purposes, and establish clear policies to ensure sustainable hunting practices.' author: - | W. H. Bonat[^1]  and J. Olivero[^2]  and M. Grande-Vega[^3]\  and M. A. Farfán[^4]  and J. E. Fa[^5] bibliography: - 'Bonat2016.bib' title: 'Modelling the covariance structure in marginal multivariate count models: Hunting in Bioko Island' --- *Keywords: Multivariate models; Estimating functions; Hunting; Longitudinal data* Introduction ============ Multivariate regression models have been of increased interest in the statistical literature. Recent applications include functional disability data [@Vallier:2014], cognitive functioning [@Anderlucci:2015], evolutionary biology [@Cybis:2015], multi-species distribution [@Hui:2015; @Ovaskainen:2011], social, economic [@Klein:2015; @Klein:2015a] and political sciences [@Lagona:2015] to cite a few. The mentioned methodologies apply latent variables or finite mixture of regression models to describe the covariance structure introduced by the multiple response variables. In contrast to these approaches @Bonat:2016 proposed the multivariate covariance generalized linear models (McGLMs), which explicitly model the marginal covariance matrix combining a covariance link function and a matrix linear predictor composed of known matrices. McGLMs have much in common with the GEE (Generalized Estimating Equations) [@Zeger:1988] approach popular in the analysis of longitudinal data. However, McGLMs were explicitly designed to deal with multiple response variables and allow for a flexible modelling of the covariance structure. On the other hand, current GEE implementations [@Hojsgaard:2006] deal only with one response variable and include a short list of pre-specified covariance structures, such as autoregression and compound symmetry. Generalized linear mixed models (GLMMs) [@Breslow:1993] are flexible models for handling multivariate data [@Verbeke:2014]. GLMMs are computationally demanding, and many algorithms have been proposed in the past three decades, see @McCullogh:1997 and @Fong:2010 for reviews and further references. @Motta:2013 presented a specific example of GLMM for count data. An aspect of GLMMs that gives rise to concern is the general lack of a closed-form expression for the likelihood and the marginal distribution of the data vector. A related question is the special interpretation of parameters inherent from the construction of GLMMs. Thus, the covariate effects are conditional on the latent variables, whereas the correlation structure is marginal for the latent variables rather than for the response variables. The multivariate Poisson [@Tsionas:1999] and negative binomial [@Shi:2014] distributions are suitable approaches to deal with multivariate count data. The multivariate Poisson has the restriction to deal only with equidispersed and positive correlated data. The last restriction is also shared by the multivariate negative binomial model. The assumption of a common error distribution required for these models may not be satisfied in practice, and methods for handling the case of unequal marginal distributions do not seem easily available. Additional methods for specifying models for dependent data include the Gaussian copula marginal regression models [@Masarotto:2012] and the class of hierarchical generalized linear models [@Lee:1996]. In the context of multivariate longitudinal models, besides the modelling of the covariance structure between response variables, we also have to model the longitudinal and repeated measures structures for each response variable, i.e. the within covariance structure. The question of how to model the within covariance structure in the univariate case is often solved by choosing from a short list of options, such as compound symmetry, autoregressive and unstructured [@Diggle:2002]. Such choices are, however, not suitable for the combination of multivariate, repeated measures and longitudinal structures found in the application described in the Section \[dataset\]. It motivates the development of a more general and flexible approach for covariance modelling in multivariate longitudinal count models. In this paper, we adopt the McGLM framework in order to present a multivariate model suitable to deal with count response variables. Our model also relies on the structure of the multivariate discrete dispersion models [@Jorgensen:2014], where the Poisson-Tweedie distribution provides a flexible framework for modelling discrete response variables. In this framework multivariate extensions of the Neyman Type A, P[ó]{}lia-Aepply, negative binomial and Poisson-inverse Gaussian distributions appear as special cases. One advantage of this class of models is that similar to the exponential dispersion models [@Jorgensen1997b] the whole family is described by the power dispersion function, analogous to ordinary Tweedie exponential dispersion models with power variance functions. This fact allows us to specify models based on second-moment assumptions and use the engine of McGLMs for estimation and inference. For further references and regression models based on the Poisson-Tweedie distribution, see @Bonat:2016c. The model is motivated by a data set consisting of the number of blue duikers and other small animals shot or snared by $52$ commercial hunters over a $33$-month period in Pico Basil[é]{}, Bioko Island, Equatorial Guinea [@Vega:2015]. Bushmeat trade is an important resource in the livelihoods of many rural communities in West and central Africa. Overhunting for profit is known to cause immediate reductions in the density of targeted animals [@Fa:2000]. In extreme cases it may precipitate the disappearance of local populations and eventually result in the complete extirpation of a species [@Fa:2009]. It is also known that hunted island animal populations are often at a greater risk of extinction because of their small geographic ranges and usually low population numbers [@Vega:2015]. In Bioko Island, the blue duiker (*Philantomba monticola*) is the most hunted species among $18$ species of mammals and birds consumed as food. The main goal of this data analysis is to investigate whether the number of hunted blue duikers declined during the study period. The data analysis should take into account the severely unbalanced repeated measures and longitudinal structures introduced by the hunters and a set of potential covariates affecting both the mean and covariance structures. Determining whether the decline of hunted animals is instrumental, since it could suggest a reduction in the population of this species, with important applications for establishing policies of sustainable hunting practices. In this scenario, a bivariate count model is useful, since a significant negative correlation could indicate that hunters target another species as a result of the decline in the target species, while a non-significant correlation may push hunter to turn to alternative sources of income. In view of the recent developments in the McGLMs framework the main contributions of this article are: i) introduces a suitable specification of the McGLMs to deal with the combination of longitudinal and repeated measures in the context of multivariate count data. ii) describes how to specify the components of the matrix linear predictor in order to take into account the effects of known covariates in a linear mixed model fashion. iii) extends the score information criterion (SIC) to select the components of the matrix linear predictor. iv) applied the methods to analyse the Hunting data set and v) provides `R` code for constructing the components of the matrix linear predictor as well as fitting the models through the `mcglm` [@Bonat:2016a] package for the `R` statistical software. We present the Hunting data set in Section \[dataset\]. Section \[model\] discusses the model and its properties. We emphasize the specification of the matrix linear predictor. Section \[sic\] extends the score information criterion for selecting the components of the matrix linear predictor. Section \[results\] describes the application of the model to the data. Section \[discussion\] discusses the main results. Finally, Section \[conclusion\] presents the concluding remarks. The data set that is analysed in the paper and the programs that were used to analyse it can be obtained from\ `http://www.leg.ufpr.br/doku.php/publications:papercompanions:hunting bioko2016`. Data set {#dataset} ======== The case study analysed in this paper uses data of animals hunted in the village of Basil[é]{} Fang, Bioko Norte Province, Bioko Island, Equatorial Guinea. The monthly number of blue duikers and other small animals shot or snared were collected from a random sample of $52$ commercial hunters from August $2010$ to September $2013$. For each animal caught, the species, sex, method of capture and altitude were documented. The data set has $1216$ observations. For additional description of the field work, see @Vega:2015. In this analysis, we opted to aggregate the species into two levels blue duikers (`BD`) and other small animals (`OT`), since `BD` is the target species and `OT` are hunted at random. The covariates `sex` (Female, Male) and `method` (Firearm, Snare) are factors with two levels. The covariate `alt` is a factor with $5$ levels ($300\--600$, $601\--900$, $901\--1200$, $1201\--1500$ and $>1500$) indicating the altitude where the animal was caught. Finally, the number of hunter days per month was recorded. It is important, because represents the effort employed by the hunter and should be used as an `offset`(in logarithm scale) for modelling the counts of hunted animals. The study design introduces some sources of dependence in the data. We call `hunter-month` the effect of all observations taken at the same hunter and month. The `hunter` effect is represented by all observations taken at the same hunter. The `longitudinal` effect is introduced by the observations taken at sequentially months. The within covariance for each outcome can also be affected by the covariates in a linear mixed model fashion, see Section \[model\] and @Demidenko:2013 for details. Finally, the correlation between response variables should be taken into account, since it plays an important role in terms of model interpretation. The number of observations per `hunter-month` and `hunters` varied between $1$ and $16$ and $1$ and $104$, respectively. These numbers show the severely unbalanced repeated measures and longitudinal structures present in the data set. ![Histograms (A and F). Taylor plot (hunter mean and variance in double logarithmic scale) (B and G). Boxplots for `sex` (C and H), `method` (D and I) and `alt` (E and J). Individual average (gray) and overall average (black) trajectories (K and L) for `BD` and `OT`, respectively.[]{data-label="fig:descritiva"}](Dataset-001) Histograms in Figure \[fig:descritiva\] suggest that the two error distributions may not be identical, and hint at potential problems with excess of zeroes and overdispersion. Boxplots suggest an effect of all covariates, whereas the approximate linearity of the Taylor plots suggest a variance function of power form. Multivariate longitudinal models for count data {#model} =============================================== Let $\mathbf{Y}_{N \times R} = \{\boldsymbol{Y}_1, \ldots, \boldsymbol{Y}_R\}$ be a response variable matrix and let $\mathbf{M}_{N \times R} = \{\boldsymbol{\mu}_1, \ldots, \boldsymbol{\mu}_R\}$ denote the corresponding matrix of expected values. Let $\boldsymbol{\Sigma}_r$ denote the $N \times N$ covariance matrix within the response variable $r$ for $r = 1, \ldots, R$. Similarly, let $\boldsymbol{\Sigma}_b$ be the $R \times R$ correlation matrix whose components $\rho_{rr^{\prime}}$’s denote the correlation between the response variables $r$ and $r^{\prime}$. The multivariate covariance generalized linear model as proposed by @Bonat:2016 is given by $$\begin{aligned} \label{McGLM} \mathrm{E}(\mathbf{Y}) &=& \mathbf{M} = \{g_1^{-1}(\boldsymbol{X}_1 \boldsymbol{\beta}_1), \ldots, g_R^{-1}(\boldsymbol{X}_R \boldsymbol{\beta}_R)\} \nonumber \\ \mathrm{Var}(\mathbf{Y}) &=& \boldsymbol{C} = \boldsymbol{\Sigma}_R \overset{G} \otimes \boldsymbol{\Sigma}_b \nonumber\end{aligned}$$ where $\boldsymbol{\Sigma}_R \overset{G} \otimes \boldsymbol{\Sigma}_b = \mathrm{Bdiag}(\tilde{\boldsymbol{\Sigma}}_1, \ldots, \tilde{\boldsymbol{\Sigma}}_R)(\boldsymbol{\Sigma}_b \otimes \boldsymbol{I})\mathrm{Bdiag}(\tilde{\boldsymbol{\Sigma}}_1^T, \ldots, \tilde{\boldsymbol{\Sigma}}_R^T)$ is the generalized Kronecker product [@Martinez:2013]. The matrix $\tilde{\boldsymbol{\Sigma}}_r$ denotes the lower triangular matrix of the Cholesky decomposition of $\boldsymbol{\Sigma}_r$. The operator $\mathrm{Bdiag}$ denotes a block diagonal matrix and $\boldsymbol{I}$ denotes an $R \times R$ identity matrix. The functions $g_r$ are link functions, for which we adopt the orthodox $\log$-link function. Let $\boldsymbol{X}_r$ denote an $N \times k_r $ design matrix and $\boldsymbol{\beta}_r$ a $k_r \times 1$ regression parameter vector. Note that, the model has a specific linear predictor for each response variable. In order to specify the covariance within response variables, we adopt the definition of @Jorgensen:2014 for Poisson-Tweedie random vector, i.e. $$\boldsymbol{\Sigma}_r = \mathrm{diag}(\boldsymbol{\mu}_r) + \mathrm{V}(\boldsymbol{\mu}_r;p_r)^{\frac{1}{2}} (\boldsymbol{\Omega}(\boldsymbol{\tau}_r)) \mathrm{V}(\boldsymbol{\mu}_r;p_r)^{\frac{1}{2}}$$ where $\mathrm{V}(\boldsymbol{\mu}_r;p_r) = \mathrm{diag}(\boldsymbol{\mu}_r^{p_r})$, is a diagonal matrix whose main entries are given by the power variance function. This specification is a multivariate representation of the power dispersion function which characterizes the Poisson-Tweedie family, see @Jorgensen:2014 for details. Finally, following the ideas of @Anderson:1973 and @Pourahmadi:2000 we model the dispersion matrix $\boldsymbol{\Omega}(\boldsymbol{\tau}_r)$ as a linear combination of known matrices, i.e. $$\label{linearcovariance} h(\boldsymbol{\Omega}(\boldsymbol{\tau}_r)) = \tau_{r0} Z_{r0} + \cdots + \tau_{rD} Z_{rD}.$$ Here $h$ is the covariance link function, $Z_{rd}$ with $d = 0, \ldots, D$ are known matrices reflecting the covariance structure within the response variable $r$, and $\boldsymbol{\tau}_r = (\tau_{r0}, \ldots, \tau_{rD})$ is a $(D+1) \times 1$ parameter vector. This structure is a natural analogue of the linear predictor of the mean structure, and following @Bonat:2016 we call it a matrix linear predictor. In this paper we focus on the identity covariance link function, since many interesting models appear as special cases. @Demidenko:2013 showed that the covariance structure induced by the orthodox Gaussian linear mixed model is a linear covariance matrix, i.e. has the form of (\[linearcovariance\]). In this sense, the models presented in this paper can been seen as an extension of the Gaussian linear mixed model for handling count data. Furthermore, popular approaches to deal with longitudinal autocorrelated data, as the compound symmetry, moving average and first order autoregressive, are also covariance linear models. In what follows we discuss some of the possibilities for the specification of the matrix linear predictor in the context of longitudinal data. Since the matrix linear predictor is specified for each response variable, suppose without loss of generality that $r=1$. Denote $y_{go}$ an observation $o = 1, \ldots, O_g$ within the group $g = 1, \ldots, G$ and let $\boldsymbol{y}_g$ denote the $O_g$-dimensional vector of measurements from the $g$th group. In particular, for the data set presented in Section \[dataset\] the groups are given by the `Hunters`. Thus, the response variable vector is given by $\boldsymbol{Y} = (\boldsymbol{y}_1, \ldots, \boldsymbol{y}_G)^\top$. Let $A_g$ denote an $O_g \times E$ design matrix composed of the values of $E$ known covariates available to model the covariance structure. Furthermore, let $A_{g,\cdot e}$ denote the $e$th column of the matrix $A_g$. Following @Demidenko:2013 the main effect of the covariate $e$ and the interaction effect between the covariates $e$ and $e^{\prime}$ are included in the covariance model through the symmetric matrices $$A_{g}^{e} = A_{g,\cdot e} A_{g, \cdot e}^{\top} \quad \mbox{and} \quad A_{g}^{e e^{\prime}} = A_{g,\cdot e} A_{g, \cdot e^{\prime}}^T + A_{g,\cdot e^{\prime}} A_{g, \cdot e}^T,$$ respectively. The matrices $A_{g}^{e}$ and $A_{g}^{e e^{\prime}}$ are group specific. To obtain the components of the matrix linear predictor for the entire response variable vector $\boldsymbol{Y}$, we assume independent groups. Thus, the components of the matrix linear predictor that measure the effect of the $e$th covariate and the interaction effect are given by $$\label{matlinear} Z_e = \mathrm{Bdiag}(A_{1}^{e}, \ldots, A_{G}^e) \quad \mbox{and} \quad Z_{ee^{\prime}} = \mathrm{Bdiag}(A_{1}^{ee^{\prime}}, \ldots, A_{G}^{ee^{\prime}}),$$ where as before the operator $\mathrm{Bdiag}$ denotes a block diagonal matrix. The matrices $Z_e$ and $Z_{ee^{\prime}}$ can be included as the $Z_{d}$’s components in the matrix linear predictor, see \[linearcovariance\]. When the main and interaction effects are included in the model, we have $E(E+1)/2$ components. A simplification is obtained by considering only main effects resulting in $E$ components. In general, we reserve the first component of the matrix linear predictor $Z_0$ to an identity matrix, that represents the intercept of the linear covariance model. @Demidenko:2013 showed that some well known covariance structures used to model longitudinal and repeated measures data are linear covariance models. To describe these structures consider a particular group $g$ with three observations. As before to extend the matrices to the entire response variable vector, we assume independent groups and use the $\mathrm{Bdiag}$ operator. The compound symmetry or exchangeable structure is a linear combination of an identity and a matrix of ones, i.e. for this particular group the matrix linear predictor is given by $$\boldsymbol{\Omega}_g(\boldsymbol{\tau}) = \tau_{0} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} + \tau_{1} \begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1 \end{bmatrix} .$$ The Moving Average model of order $p$ MA(p) is also a linear covariance model. The components of the matrix linear predictor associated with the MA(1) and MA(2) structures are given respectively by $$\label{compsyme} A_1 = \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix} \quad \mbox{and} \quad A_2 = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 0 \end{bmatrix}.$$ For longitudinal data analysis, we can use the inverse of Euclidean distance between pairs of observations as a component of the matrix linear predictor, for example $$\label{Eucli} A_1 = \begin{bmatrix} 0 & 1/d_{12} & 1/d_{13} \\ 1/d_{12} & 0 & 1/d_{23} \\ 1/d_{13} & 1/d_{23} & 0 \end{bmatrix},$$ where $d_{ij}$ denotes the Euclidean distances between the observations at time $i$ and $j$. By combining the simple structures described above, we have a flexible set of components to compose the matrix linear predictor for the analysis of longitudinal data. @Demidenko:2013 also showed that the popular first-order autoregression model can be written as a linear covariance model, but using the inverse covariance link function. In this paper, we do not pursue in this covariance link function. The power parameter $p$ plays an important role in the context of multivariate Poisson-Tweedie models, since it is an index which distinguishes between some important discrete distributions. Examples include the Neyman Type A ($p = 1$), P[ó]{}lya-Aeppli ($p=1.5$), negative binomial ($p=2$) and Poisson-inverse Gaussian ($p=3$). The algorithm proposed by @Bonat:2016 allows us to estimate the power parameter, which works as an automatic distribution selection. The score information criterion {#sic} =============================== In this section, we extend the score information criterion (SIC) proposed by @Stoklosa:2014 for the selection of the components of the matrix linear predictor. In order to introduce the SIC, we first present some key components of the estimating function approach used to fit McGLMs. The algorithm and asymptotic theory associated with the estimating function estimators were presented by @Bonat:2016 and implemented in the `mcglm` [@Bonat:2016a] package for the `R` [@R:2015] statistical sofware. The second-moment assumptions of McGLMs motivate us to divide the set of parameters into two subsets $\boldsymbol{\theta} = (\boldsymbol{\beta}^{\top},\boldsymbol{\lambda}^{\top})^{\top}$. In this notation $\boldsymbol{\beta} = (\boldsymbol{\beta}^{\top}_1, \ldots, \boldsymbol{\beta}^{\top}_R)^{\top}$ and $\boldsymbol{\lambda} = (\rho_1, \ldots, \rho_{R(R-1)/2}, p_1, \ldots, p_R, \boldsymbol{\tau}_1^\top, \ldots, \boldsymbol{\tau}_R^\top)^\top$ denote a $K \times 1$ and $Q \times 1$ vector of all regression and dispersion parameters, respectively. Let $\mathcal{Y} = (\boldsymbol{Y}_1^\top, \ldots, \boldsymbol{Y}_R^\top)^\top$ and $\mathcal{M} = (\boldsymbol{\mu}_1^\top, \ldots, \boldsymbol{\mu}_R^\top)^\top$ denote the $NR \times 1$ stacked vector of the response variable matrix $\mathbf{Y}_{N \times R}$ and expected values matrix $\mathbf{M}_{N \times R}$ by columns, respectively. The regression coefficients are estimated by using the orthodox quasi-score function [@Bonat:2016; @Zeger:1988]. The dispersion parameters are estimated based on the Pearson estimating function, defined by the components $$\label{Pearson} \psi_{\boldsymbol{\lambda}_i}( \boldsymbol{\beta}, \boldsymbol{\lambda}) = \mathrm{tr}(W_{\boldsymbol{\lambda}_i}(\boldsymbol{r}^\top\boldsymbol{r} - \boldsymbol{C})) \quad \text{for} \quad i = 1,\ldots,Q,$$ where $W_{\boldsymbol{\lambda}_i} = -\partial \boldsymbol{C}^{-1} / \partial \boldsymbol{\lambda}_i$ and $\boldsymbol{r} = \mathcal{Y} - \mathcal{M}$. Two key components of an estimating function approach are the sensitivity and variability matrices. The entry $(i,j)$ of the $Q \times Q$ sensitivity matrix of $\psi_{\boldsymbol{\lambda}}$ is given by, $$\label{Slambda} \mathrm{S}_{\boldsymbol{\lambda}_{ij}} = \mathrm{E} \left ( \frac{\partial}{\partial \boldsymbol{\lambda}_i} \psi_{\boldsymbol{\lambda}_j} \right ) = -\mathrm{tr} \left (W_{\boldsymbol{\lambda}_i} \boldsymbol{C} W_{\boldsymbol{\lambda}_j} \boldsymbol{C} \right).$$ Similarly, the entry $(i,j)$ of the $Q \times Q$ variability matrix of $\psi_{\boldsymbol{\lambda}}$ is given by $$\label{Vl} \mathrm{V}_{\boldsymbol{\lambda}_{ij}} = \mathrm{Cov}(\psi_{\boldsymbol{\lambda}_i},\psi_{\boldsymbol{\lambda}_j}) = 2\mathrm{tr}(W_{\boldsymbol{\lambda}_i} \boldsymbol{C} W_{\boldsymbol{\lambda}_j} \boldsymbol{C}) + \sum_{l=1}^{NR} k^{(4)}_l (W_{\boldsymbol{\lambda}_i})_{ll} (W_{\boldsymbol{\lambda}_j})_{ll},$$ where $k^{(4)}_l$ denotes the fourth cumulant of $\mathcal{Y}_l$. In order to keep the model based on second-moment assumptions only, we following @Bonat:2016 use the empirical fourth cumulant. @Stoklosa:2014 in the context of generalized estimating equations (GEE) proposed the score information criterion (SIC) to be used with forward selection algorithms in the cases where we have a large number of covariates to compose the linear predictor. The SIC is based on the score statistics, what becoming such criterion convenient, since it can be computed for all candidate models without actually fitting them. Suppose without loss of generality that $r = 1$ and fixed power parameter. In that case, the vector of dispersion parameters simplify to $\boldsymbol{\lambda} = \boldsymbol{\tau}$, since we have no correlation neither power parameters. For a given mean structure, suppose that the parameter vector $\boldsymbol{\tau}$ can be partitioned as $\boldsymbol{\tau} = (\boldsymbol{\tau}_{1}^\top, \boldsymbol{\tau}_{2}^\top)^\top$, whose dimension are $(Q - s) \times 1$ and $s \times 1$, respectively. The Pearson estimating function $\psi_{\boldsymbol{\lambda}}$ and its sensitivity and variability matrices, can also be partitioned to $\psi_{\boldsymbol{\lambda}}(\boldsymbol{\beta},\boldsymbol{\tau}) = (\psi_{\boldsymbol{\lambda}_1}(\boldsymbol{\beta},\boldsymbol{\tau}_1)^\top, \psi_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta},\boldsymbol{\tau}_2)^\top)^\top$, $$\mathrm{S}_{\boldsymbol{\lambda}} = \begin{pmatrix} \mathrm{S}_{\boldsymbol{\lambda}_{11}} & \mathrm{S}_{ \boldsymbol{\lambda}_{12}} \\ \mathrm{S}_{\boldsymbol{\lambda}_{21}} & \mathrm{S}_{\boldsymbol{\lambda}_{22}} \end{pmatrix},$$ and $$\mathrm{V}_{\boldsymbol{\lambda}} = \begin{pmatrix} \mathrm{V}_{\boldsymbol{\lambda}_{11}} & \mathrm{V}_{ \boldsymbol{\lambda}_{12}} \\ \mathrm{V}_{\boldsymbol{\lambda}_{21}} & \mathrm{V}_{\boldsymbol{\lambda}_{22}} \end{pmatrix},$$ respectively. The null hypothesis $H_0$ is $\boldsymbol{\tau}_{2} = \boldsymbol{0}$. Let $\tilde{\boldsymbol{\tau}} = (\hat{\boldsymbol{\tau}}_{1}^\top, \boldsymbol{0}^\top)^\top$ be the vector of Pearson estimates under $H_0$. Note that, only the base model containing $\hat{\boldsymbol{\tau}}_{1}$ parameters has to be fitted. In practical situations, this model can contain only a simple intercept. The Pearson estimating function takes the form $$\psi_{\boldsymbol{\lambda}}(\beta,\tilde{\boldsymbol{\tau}}) = (\psi_{\boldsymbol{\lambda}_1}^\top(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}}), \psi_{\boldsymbol{\lambda}_2}^\top(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}}))^\top = (\boldsymbol{0}^\top, \psi_{\boldsymbol{\lambda}_2}^\top(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}}))^\top.$$ The generalized score statistic is given by $$\label{TU} T_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta}, \tilde{\boldsymbol{\tau}}) = \psi_{\boldsymbol{\lambda}_2}^\top(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}}) \mathrm{Var}(\psi_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}}))^{-1} \psi_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}})$$ where $$\begin{aligned} \label{V2} \mathrm{Var}(\psi_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}})) &= \mathrm{V}_{\boldsymbol{\lambda}_{22}} - \mathrm{S}_{\boldsymbol{\lambda}_{21}}\mathrm{S}_{\boldsymbol{\lambda}_{11}}^{-1}\mathrm{V}_{ \boldsymbol{\lambda}_{12}} - \mathrm{V}_{ \boldsymbol{\lambda}_{12}}\mathrm{S}_{\boldsymbol{\lambda}_{11}}^{-1}\mathrm{S}_{ \boldsymbol{\lambda}_{12}} \\ & + \mathrm{S}_{\boldsymbol{\lambda}_{21}}\mathrm{S}_{\boldsymbol{\lambda}_{11}}^{-1}\mathrm{V}_{\boldsymbol{\lambda}_{11}}\mathrm{S}_{\boldsymbol{\lambda}_{11}}^{-1}\mathrm{S}_{ \boldsymbol{\lambda}_{12}} \end{aligned}$$ is the variance of the subvector $\psi_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}})$. Under the null hypothesis, $T_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta}, \tilde{\boldsymbol{\tau}})$ has a chi-square distribution with $s$ degrees of freedom. In practice, all quantities in (\[TU\]) are evaluated at the Pearson estimates under the null hypotheses. If $H_0$ were true, then $\psi_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta},\tilde{\boldsymbol{\tau}})$ that is the Pearson estimating function for $\boldsymbol{\tau}_2$ would be close to zero when evaluated under the null. Large values of $T_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta}, \tilde{\boldsymbol{\tau}})$ would argue against $H_0$. The main idea behind SIC is to use (\[TU\]) as a quadratic approximation to the log-likelihood ratio statistic. The so-call one-step SIC is defined by $$SIC^{(1)}(\boldsymbol{\beta}, \boldsymbol{\tau}) = - T_{\boldsymbol{\lambda}_2}(\boldsymbol{\beta}, \tilde{\boldsymbol{\tau}}) + \delta |\boldsymbol{\tau}|.$$ Note that this criterion is a function of $\tilde{\boldsymbol{\tau}}$ only, thus only the base model needs to be fitted. As point out by @Stoklosa:2014 the approximation of score statistics to likelihood ratio statistics can be poor when there is a significant departure from the null model. Hence an improved approximation might calculate the score statistic in one-parameter increments, i.e. $$SIC(\boldsymbol{\beta}, \boldsymbol{\tau}) = - \sum_{s = 1}^{|\boldsymbol{\tau}_2|} \underset{\tau_{(s)} \in \boldsymbol{\tau}_2^{\backslash s-1}}{\mathrm{max}} \{ T_{\boldsymbol{\lambda}_2(s)}(\boldsymbol{\beta}, \tilde{\boldsymbol{\tau}}_{s-1}) \} + \delta |\boldsymbol{\tau}|$$ where $\boldsymbol{\tau}^\top_s = (\boldsymbol{\tau}^\top_{s-1}, \tau_s)$ and $\boldsymbol{\tau}_2^{\backslash s-1} = {\boldsymbol{\tau}_2 \cap \boldsymbol{\tau}^c_{s-1} }$ where $\boldsymbol{\tau}^c_{s-1}$ is the complement set of $\boldsymbol{\tau}_{s-1}$. In summary, we sequentially add new parameters selected from $\boldsymbol{\tau}_2$, these are $\tau_{(s)}$ for $s = 1, \ldots, |\boldsymbol{\tau}_2|$, in the order that maximizes the score statistic (\[TU\]) in each step. In that case no more than $|\boldsymbol{\tau}_2|$ models will be fitted to reach the final model. In this paper we consider the penalties $\delta = 2$, as it is analogous to the *Akaike* information criterion. It is also possible to use $\delta = \log N$ to have an analogous to the *Bayesian* information criterion. Results ======= In this section, we apply the McGLM for multivariate count data to analyse the data set presented in Section \[dataset\]. The second-moment assumptions of the McGLM require the specification of a linear predictor and a matrix linear predictor for each response variable. In this application, for composing the linear predictor we have three covariates `sex`, `method` and `alt` along with the time trend `month`. We considered interaction terms up to second order between the four main effects. The time trend was modelled as a polynomial of third and fourth degrees for `BD` and `OT`, respectively. Such choices were based on exploratory analysis and preliminary fits as we shall explain better in the Section \[discussion\]. In all fitted models the number of hunter days (in logarithm scale) was used as an `offset`. To specify the matrix linear predictor, we have the repeated measures structures represented by the `Hunter` and `Hunter-Month` effects. The `Longitudinal` effect introduced by the observations taken at sequentially months and the three covariates, `sex`, `method` and `alt`. For the repeated measures effects we assumed a compound symmetry (of ones) structure,see (\[compsyme\]). The longitudinal effect was modelled using the inverse of Euclidean distances,see (\[Eucli\]). Finally, the covariates are included in the covariance model in a linear mixed model fashion, see \[matlinear\]. In this application for model parsimony and since we have only categorical covariates to compose the matrix linear predictor, we considered only main effects. For clarity, consider a particular `Hunter` that represents the group structure described in the Section \[model\]. Furthermore, consider that we have four observations (two for the first month and two for the second month). Consider also for simplicity that we have the values of a covariate $\boldsymbol{e} = (e_1, e_2, e_3, e_4)$. In that case, the matrix linear predictor has the following form $$\begin{aligned} \boldsymbol{\Omega}(\boldsymbol{\tau}) = \tau_0 \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} + \tau_1 \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix} + \tau_2 \begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix} + \\ \tau_3 \begin{bmatrix} 0 & 0 & 1/d_{12} & 1/d_{12} \\ 0 & 0 & 1/d_{12} & 1/d_{12} \\ 1/d_{12} & 1/d_{12} & 0 & 0 \\ 1/d_{12} & 1/d_{12} & 0 & 0 \end{bmatrix} + \tau_4 \begin{bmatrix} e_1^2 & e_1 e_2 & e_1 e_3 & e_1 e_4 \\ e_1 e_2 & e_2^2 & e_2 e_3 & e_2 e_4 \\ e_1 e_3 & e_2 e_3 & e_3^2 & e_3 e_4 \\ e_1 e_34& e_2 e_4 & e_3 e_4 & e_4^2 \end{bmatrix},\end{aligned}$$ where $\tau_0$ is the `intercept` of the covariance linear model. The parameters $\tau_1$, $\tau_2$, $\tau_3$ and $\tau_4$ measure the `Hunter`, `Hunter-Month`, `Longitudinal` and covariate effects, respectively. We employed a stepwise procedure for selecting the components of the linear and matrix linear predictors. The SIC using penalty $\delta = 2$ and the Wald test were used in the forward and backward steps, respectively. We defined as stop criterion SIC $> 0$, since in that case the penalty is larger than the score statistics. Our strategy to select the final model consists of: i) select the components of the linear predictor for each response variable fixing the covariance structure assuming independent observations, i.e. $Z_0 = \mathrm{I}$. ii) select the components of the matrix linear predictor for each response variable fixing the mean structure obtained in step (i). iii) fit the multivariate model and iv) remove non-significant effects in both linear and matrix linear predictors if any. In this application after fit the multivariate model all covariates selected to compose the linear and matrix linear predictors were significant. Supplemenaty Tables S$1$ and S$2$ present the step-by-step procedure. Table \[Wald\] presents the Wald statistics for the components of the selected linear predictor for each response variable obtained by fitting the final multivariate model. The selected matrix linear predictors were composed of a diagonal matrix (`Intercept`) combined with the `Hunter-Month`, `Method` and `Longitudinal` effects for `BD` and only the `Hunter-Month` effect for `OT`. -------------- ------ ----------- --------- -------------- ------ ----------- --------- Effects Df $\chi^2$ p-value Effects Df $\chi^2$ p-value `method` $1$ $6.986$ $0.008$ `method` $1$ $1.766$ $0.183$ `alt` $4$ $138.262$ $0.000$ `alt` $4$ $128.042$ $0.000$ `sex` $1$ $247.843$ $0.000$ `sex` $1$ $15.927$ $0.000$ `month` $3$ $25.791$ $0.000$ `month` $4$ $10.150$ $0.038$ `method:alt` $4$ $58.688$ $0.000$ `method:alt` $4$ $26.455$ $0.000$ `alt:month` $12$ $43.898$ $0.000$ `alt:sex` $4$ $13.238$ $0.012$ $-$ $-$ $-$ $-$ `alt:month` $16$ $90.365$ $0.000$ -------------- ------ ----------- --------- -------------- ------ ----------- --------- : Wald statistics ($\chi^2$), degrees of freedom (Df) and p-values for the components of the selected linear predictor for each response variable.[]{data-label="Wald"} The results in Table \[Wald\] show that the `method` effect for the response variable `OT` was non-significant, but given its highly significant interaction with `alt` we opted to keep this effect in the model. Table \[covariance\] shows the estimates, standard errors (SE) and Z-statistics for the power and dispersion parameters for the final model. ---------------- ---------- ---------- -------------- ---------- --------- -------------- Effects Estimate SE Z-statistics Estimate SE Z-statistics `power` $1.165$ $0.115$ $10.108$ $1.453$ $0.251$ $5.777$ `Intercept` $0.474$ $0.142$ $3.345$ $0.686$ $0.184$ $3.737$ `Hunter-Month` $0.722$ $0.151$ $4.792$ $0.294$ $0.093$ $3.163$ `Method` $0.928$ $0.258$ $3.603$ $-$ $-$ $-$ `Longitudinal` $-0.155$ $0.0424$ $-3.660$ $-$ $-$ $-$ ---------------- ---------- ---------- -------------- ---------- --------- -------------- : Power and dispersion parameter estimates, standard errors (SE) and Z-statistics for the components of the selected matrix linear predictor for each response variable.[]{data-label="covariance"} The estimates of the power parameters suggest that the Neyman Type A ($p=1$), which indicates a zero inflation relative to the Poisson distribution is a suitable choice for both response variables. For the response variable `OT` the P[ó]{}lya-Aeppli ($p=1.5$) can also be suggested. The correlation between response variables was weak $-0.0532$ ($0.0287$) and not significantly different from $0$. It is interesting to highlight that the $\boldsymbol{\Omega}$ matrix describes the part of the covariance structure that does not depend on the mean structure. Thus, it is interesting to interpret the parameters that compose this matrix in terms of the correlation introduced by its components. For example, the correlation introduced by the `Hunter-Month` effect is $0.604 (0.0594)$ and $0.299 (0.102)$ for `BD` and `OT`, respectively. These numbers are easily obtained by $\hat{\tau}_1/(\hat{\tau}_0 + \hat{\tau}_1)$. Similarly, the correlation between observations taken at the same hunter by the method snare is $0.652 (0.074)$. Note that, since the `Hunter` effect was not significant the reference level is the `Intercept` i.e. independence. Thus, we have no evidence of dependence between observations taken at the same hunter by the method firearm. Finally, the correlation introduced by the `Longitudinal` effect is $-0.487 (0.203)$ for lag equals $1$. The numbers in the brackets denote the standard error computed using the delta method. Figures \[fig:bd\] and \[fig:ot\] present the fitted values and $95\%$ confidence intervals for the response variables `BD` and `OT`, respectively. We plot the observed values divided by the `offset` and the fitted values were computed fixing the `offset` equals $1$. Supplementary Tables S$3$ and S$4$ present the estimates and standard errors for the regression coefficients associated with the response variables `BD` and `OT`, respectively. ![Fitted values and $95\%$ confidence intervals by altitude, method of capture and sex for the response variable `BD`.[]{data-label="fig:bd"}](FittedBD) ![Fitted values and $95\%$ confidence intervals by altitude, method of capture and sex for the response variable `OT`.[]{data-label="fig:ot"}](FittedOT) Figure \[fig:bd\] shows that for all altitudes the number of hunted blue duikers increases from the beginning to the middle of the data collection, when a clear decreases start with sensible differences in the threshold point among the levels of the covariate `alt`. Altitudes $4$ and $5$ present the largest numbers of caught animals while altitudes $1$ and $2$ the smallest ones. Similar we have seen for `BD` Figure \[fig:ot\] shows a clear time trend for the response variable `OT` in the altitudes $1$ and $2$. Altitudes $3$ and $4$ show a different pattern with a slightly increase at the end of the experiment. Altitudes $1$ and $2$ present the largest numbers of other animals hunted by both methods and sexes. The smallest numbers appear in altitudes $3$ and $4$ using firearms. In general the number of females hunted is bigger than males and the most effective method of capture depends on the altitude. It is important to highlight that despite of the differences in terms of altitudes, sexes and methods seem small in its magnitude judging by the results presented in Figures \[fig:bd\] and \[fig:ot\]. Such impression is due to the fact that, such results were obtained by fixing the number of hunter days (`offset`) equals $1$. Thus, the differences tend to be amplified while the number of hunter days increases. Furthermore, the regression coefficients associated with these effects are in general significantly different from $0$ (see Tables \[Wald\], S$3$ and S$4$). Discussion ========== This section discusses the results presented in Section \[results\]. The main data analysis goal was to determine if there was evidence of depletion in the population of blue duikers and other small animals based on data of hunted animals. To detect such a depletion effect, we included in the model a special term that represented the time trend for which we allowed a flexible functional form through a polynomial of degree three and four for the response variables `BD` and `OT`, respectively. To control other effects that were not of main interest, we included in the model the effects of covariates such as sex, method of capture and altitude. The irregular activity of the hunters introduces severely unbalanced repeated measures and longitudinal structures that were modelled through a matrix linear predictor composed of known matrices. Although these effects are not of main interest, they help us to understand the complex dynamics of hunting activity and provide us with insights of the general aspects of the population of the targeted taxa. In what follows we discuss the effect of all covariates. The results presented in Section \[results\] showed that for both response variables (`BD` and `OT`), methods (snare and firearm) and all altitudes, the number of females hunted was larger than males. Since hunters do not target any particular animal, this bias in sexes hunted could be a function of a greater hunting susceptibility of females or that there are more females in the population than males. With regards to the method of capture, our results showed that this covariate presents a highly significant interaction with the covariate altitude. For the response variable `BD` the regression coefficients presented in the Supplementary Table S$3$, show that the method firearm is the most effective in altitude $1$, while the method snare is the most effective in altitude $5$. For altitudes $2$ to $4$ the differences between the methods of capture are not significant. Regarding the response variable `OT` the method snare is the most effective in altitudes $2$ and $3$, while the method firearm is the most effective in altitude $5$. In the altitudes $1$ and $4$ there is no difference between the methods. The covariate altitude reflected different hunting pressure at variable elevations in the study areas. Blue duikers may be overhunted in lower altitudes ($1$ and $2$) because of the proximity to human settlements, which increases hunting pressure. It may explain why the number of blue duikers is lower in altitudes $1$ and $2$. On the other hand, in altitudes $3$ to $5$ we presume that more animals are hunted because these areas are less exploited areas. The opposite situation appears for other small animals, this result may indicate a depletion effect. Often, when the bigger animals (such as blue duikers in Bioko) are hunted out, which may be happening in altitudes $1$ and $2$, smaller ones tend to increase in numbers. This phenomenon is known as density compensation [@Fa:2009]. While modelling the covariance structure we detected a significant effect of the covariate `Hunter-Month` for `BD` and `OT`. This effect is clearly due to the way that the data were collected and the arbitrary monthly aggregation. For the response variable `BD` in addition to the `Hunter-Month` effect, the longitudinal structure showed a significant negative effect. This result indicates that hunters may be affecting the prey population. Hence, some time is required for the population to recover and may indicate overexploitation of the hunted blue duikers population. A strong correlation between observations taken by the method snare was detected, but none appeared between observations taken by the method firearm. Such result is expected since the use of firearms to hunt is more effective when killing larger animals, so we would expect that the number of prey to decline with hunting effort with guns. This effect was detected by the longitudinal effect. On the other hand, because the method snare requires a much more continuous effort, the observations are more similar and consequently correlated along the study period. This mix of methods of capture could explain that months with a large number of animals hunted were followed by months with a smaller number of animals taken, explaining the negative longitudinal effect detected. Finally, the time trend showed that for the response variable `BD` the number of hunted animals increases from the beginning to the middle of the data collection, followed by an intense decline after that. The maximum number of animals hunted appeared around the months $20$ and $14$ for altitudes $1$ to $2$ and $3$ to $5$, respectively. A possible explanation for this result could be that at the start of the study period the blue duiker population in the region were more numerous, but following intensive hunting the population starts to decrease and consequently the number of hunted animals also falls. Another explanation could be that there is interannual variation in numbers which may be related to changes in climate and by consequence productivity of the forest, but we have no additional data to confirm this hypothesis. The significant decline after the middle of the study period provides support for an overhunting effect. The temporal pattern detected for the response variable `OT` is more volatile mainly in altitudes $1$ and $2$, indicating that the number of `OT` animals hunted could have been affected by many factors, including the availability of other species as well as economic and climate conditions. This volatile pattern may also explain the weak and non-significant correlation between `OT` and `BD`. Modelling the time trend through a polynomial function was a data-driven decision based on exploratory analysis and preliminary fits. The preliminary fits consisted of fitting models using B-splines basis as implemented in the package `splines` for the `R` statistical software. To select the number of degrees of freedom required for the B-splines basis, we fitted models using different degrees of freedom and check the significance of their regression coefficients using Wald test. Based on this procedure, we obtained that for the response variables `BT` and `OT` three and four degrees of freedom were enough to provide a suitable fit. Furthermore, based on the behaviour of the fitted values and given the low number of degrees of freedom required by the B-spline basis, we detected that a simple polynomial could provide a suitable fit. Thus, we fitted the model changing the B-spline basis by polynomial of three and four degrees of freedom for the response variables `BT` and `OT`, respectively. We compared the fitted model with the one obtained by using the B-spline basis in terms of Gaussian pseudo-likelihood (GPL) [@Carey:2011]. GPL is a measure similar to the log-likelihood value in the context of maximum likelihood estimation. Thus, larger values indicate better fit. The value of the GPL for the model presented in the Section \[results\] was $-4463.330$. Similarly, the value of the GPL for the model fitted using the B-splines basis was $-4462.270$. The GPL indicated that the fits are quite similar. Furthermore, we also compared the fitted values obtained from both models that were virtually the same. Thus, we opted to present the model fitted using the polynomial. The advantage of the polynomial is that it is more familiar to applied researchers than the B-spline basis. To provide more sources of evidence that the data support the model presented in the Section \[results\] we fitted models using linear and quadratic time trends. The value of the GPL for the model fitted by using the linear trend was $-4572.300$. Similarly, the value of the GPL for the model fitted by using the quadratic trend was $-4477.670$. Thus, we have clear evidences that the model presented in the Section \[results\] provides the best fit among the polynomial alternatives considered to describe the time trend. Furthermore, the same conclusion is obtained when penalizing the Gaussian pseudo log-likelihood with penalties compatible with the Akaike and Kullback-Leibler information criterion [@Bonat:2016a]. Concluding remarks {#conclusion} ================== We presented a flexible class of multivariate models for handling count data. The models were motivated by a data set consisting of the number of blue duikers and other small animals shot or snared by $52$ commercial hunters in Bioko Island, Equatorial Guinea. The analysis of the data showed interesting features as overdispersion, excess of zeroes and negatively correlated response variables, which in turn allowed to show the flexibility of our models. In our framework overdispersion and excess of zeroes are taken into account by means of a dispersion function. It is similar to a variance function in the context of generalized linear models. The dispersion function allows to specify models based only on second-moment assumptions and adopts an estimating function approach for parameter estimation and inference. The advantage of the estimating function approach is that the estimation procedure relies on a simple and efficient Newton scoring algorithm. In this paper, we adopted the dispersion function associated with the Poisson-Tweedie distribution, since important discrete distributions as the Neyman Type A, negative binomial and Poisson-inverse Gaussian appear as special cases. The marginal covariance structure within response variables is specified by means of a matrix linear predictor composed of known matrices. This specification easily deals with the combination of unbalanced repeated measures and longitudinal structures as well as the effects of the covariates in a linear mixed model fashion. The flexibility of this structure comes with the issue to select its components. In this paper, we extended the SIC to guide the selection of the matrix linear predictor components. The great advantage of the SIC is its simplicity. Since the SIC is based on the score statistics it can be computed without actually fitting all the candidate models. The strategy employed in this paper for selecting the components of the linear and matrix linear predictors consisted of combining the SIC and Wald statistics in a stepwise procedure applied independently for the mean and covariance structures. In the first step, we selected the components of the linear predictor for each response variable assuming independent observations. In fact, in this step we are purposely ignoring the correlation between and within response variables. It is well known that in the presence of correlation the standard errors associated with the regression parameters are underestimated. In this way, we avoid to remove important covariates of the analysis. In the second step, we fixed the linear predictor as obtained in the first step and selected the components of the matrix linear predictor. As the linear predictor potentially contains all significant covariates, we avoid that missing covariates affect the selection of the matrix linear predictor components. In the last step, we fit the multivariate model and remove any non-significant effect. Finally, the joint covariance matrix is specified by using the generalized Kronecker product. This specification combined with the possibility to estimate the power parameter for each marginal response variable allow our models easily deal with negatively correlated and unequal marginal response variables, overcoming the main limitations of the multivariate Poisson and negative binomial models. The main limitation of the models presented in this paper is the general lack of algorithms for simulation. Recent work of @Baccini:2015 discussed the problems involving the simulation of univariate Poisson-Tweedie distributions. The related topic of simulation of the multivariate Tweedie distributions was addressed recently by @Cuenin:2015, but the extension to multivariate Poisson-Tweedie distributions specified by general covariance structures in high dimension, as used in this paper, still requires further theoretical and computational developments. Supplement material {#supplement-material .unnumbered} =================== Dataset and R code for the analysis are available at the paper companion page at `http://www.leg.ufpr.br/doku.php/publications:papercompanions:hunting bioko2016`. The authors thank Professors Elias Teixeira Krainski, Walmes Marques Zeviani, Fernando Poul Mayer and Paulo Justianiano Ribeiro Jr for their comments and suggestions that substantially improve the article. The first author is supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior)-Brazil. [^1]: Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark. Department of Statistics, Paraná Federal University, Centro Politécnico, Curitiba 81531980, CP 19081, Paraná, Brazil. E-mail: [email protected] [^2]: Universidad de Málaga, Grupo de Biogeografía, Diversidad y Conservación, Departamento de Biologia Animal, Facultad de Ciencias, Campus de Teatinos s/n, 29071 Málaga, Spain. [^3]: Research Group for Sustainable Management Silvanet, Faculty of Forestry, Technical University of Madrid, Ciudad Universitaria, 28040 Madrid, Spain and Asociación Ecotono, Paseo de la Habana 109. 2 A 28036 Madrid, Spain. [^4]: Universidad de Málaga, Grupo de Biogeografía, Diversidad y Conservación, Departamento de Biología Animal, Facultad de Ciencias, Campus de Teatinos s/n, 29071 Málaga, Spain and BioGea Consultores, C/Navarro Ledesma 243, Portal 4-3 C, 29010 Málaga, Spain. [^5]: Division of Biology and Conservation Ecology, School of Science and the Environment, Manchester Metropolitan University, Manchester M1 5GD, UK and Center for International Forestry Research (CIFOR), CIFOR Headquarters, Bogor 16115, Indonesia.

--- abstract: 'Using the classical Lazard’s elimination theorem, we obtain a decomposition theorem for Lie algebras defined by generators and relations of a certain type. This is a preprint version of the paper appearing in Communications in Algebra Volume 32, Issue 10, 2004.' address: - 'Department of Mathematics, College of Charleston, Charleston SC 29424, [email protected]' - 'Department of Mathematics Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, [email protected] ' author: - Elizabeth Jurisich and Robert Wilson title: 'A generalization of Lazard’s elimination theorem' --- Introduction ============ This paper grew out of, and has its main application in, the theory of generalized Kac-Moody Lie algebras over a field $\Phi$. Borcherds initiated the study of these algebras in [@2], and applied the theory in his proof of the Conway-Norton conjectures [@3]. Generalized Kac-Moody Lie algebras may be defined by generators and relations (see [@7]). While studying these algebras it is natural to consider an algebra of the form $$L(V \oplus W)/I$$ where $L(X)$ denotes the free Lie algebra on the vector space $X$, $V$ and $W$ are vector spaces with $V \cap W = (0)$, $L(V \oplus W)$ is graded by giving nonzero elements of $V$ degree zero and nonzero elements of $W$ degree one, and where $I$ is an ideal generated by homogeneous elements of degree zero or one. Theorem 1 gives the structure of such an algebra. We write $U({\mathfrak}r)$ for the universal enveloping algebra of the Lie algebra ${\mathfrak}r$. The algebra $U({\mathfrak}r)$ acts on ${\mathfrak}r$ via the adjoint action; $ a \cdot b$ denotes the image of $b$ in ${\mathfrak}r$ under the action of $a \in U({\mathfrak}r)$. Our main result, Theorem 1, generalizes and is proved using the following theorem of Lazard [@10], [@4 Proposition 10]. Let $M = U(L(V))\cdot W \subset L(V \oplus W)$. Note that $M$ is a $U(L(V))$-module and so is an $L(V)$-module. The ideal of $L(V \oplus W) $ generated by $W$ is isomorphic to $L(M)$ and therefore $L(V \oplus W) \cong L(V) \ltimes L(M)$. R. Block has pointed out to us that this theorem is proven in [@4] only for finite dimensional $V$ and $W$. Work of Block and Leroux [@1] shows that the theorem holds in general. Several generalizations of Lazard’s elimination theorem are known: [@1] (giving a general category theoretic result which in the special case of a free Lie algebra gives Lazard’s theorem), [@6] (treating certain generalized Kac-Moody Lie algebras), [@7] (treating all generalized Kac-Moody Lie algebras), [@11] (treating the case in which $I$ is generated by elements of degree zero). All of these results on Lie algebras are contained in Theorem 1. In the special case pertinent to the Conway-Norton conjectures this theorem yields the decomposition appearing in [@6]. This decomposition simplifies part of the proof of the Conway-Norton conjectures [@6; @9]. [@5], which treats the case in which $I$ is generated by a collection, $\theta$, of elements of the form $[r,s]$ where $r,s \in$ a basis for $V \oplus W$, follows from Theorem 1 only for certain $\theta$ (those in which, for every pair $r,s$, we have $r,s \in V \bigcup W$ and at least one of $r,s$ is in $V$). Section 2 contains the statement and proof of Theorem 1. Section 3 discusses applications to generalized Kac-Moody Lie algebras. Main Result =========== If ${\mathfrak}r$ is a Lie algebra and $ S \subset {\mathfrak}r$ we let ${\langle}S {\rangle}_{\mathfrak}r$ denote the ideal of ${\mathfrak}r$ generated by $S$. Of course ${\langle}S {\rangle}_{\mathfrak}r = U({\mathfrak}r) \cdot S$. If $U \supseteq W$ then $L(U)/{{\langle}W {\rangle}}_{L(U)} \cong L(U /W)$. Write $U= V \oplus W$, so $V \cong U /W$. Then $$\begin{aligned} L(U)/ {\langle}W {\rangle}_{L(U)} & = L(V \oplus W)/{\langle}W {\rangle}_{L(U)} = (L(V) \ltimes {\langle}W {\rangle}_{L(U)})/{\langle}W {\rangle}_{L(U)} \\ &\cong L(V) \cong L(U/W).\end{aligned}$$ Now, as in Lazard’s Elimination Theorem, let $M = U(L(V))\cdot W \subseteq L(V \oplus W).$ Then, as a corollary of that theorem we obtain: 1. $U(L(V \oplus W)) = U(L(M))U(L(V))$ 2. $U(L(V\oplus W)) = U(L(V)) + U(L(M)) M U(L(V))$. By the elimination theorem $L(V \oplus W) = L(V) \ltimes L(M)$ (where we identify ${\langle}W {\rangle}_{L(V \oplus W)}$ with $L(M)$), so (1) follows by the Poincaré-Birkhoff-Witt Theorem. Also, as $\Phi $ is the base field $$\begin{aligned} U(L(M))&= \Phi + U(L(M))L(M)\\ &=\Phi + U(L(M))M\end{aligned}$$ so (2) follows from (1). Now let $A \subset L(V) \subset L(V\oplus W)$, $B \subset M \subset L(M) \subset L(V \oplus W)$. Thus, if $L(V \oplus W)$ is graded by giving nonzero elements of $V$ degree zero and nonzero elements of $W$ degree one, then $A$ is an arbitrary subspace of elements of degree zero and $B$ is an arbitrary subspace of elements of degree one. Write ${\mathfrak}g = L(V)/ {\langle}A {\rangle}_{L(V)}$, $M_1 = \{ [M, {\langle}A{\rangle}_{L(V)}] + U(L(V))\cdot B \}$ and $N = M/M_1$. The following theorem, our main result, gives the structure of the quotient algebra $L(V \oplus W)/ {\langle}A, B{\rangle}_{L(V \oplus W)}$. 1. The space $N$ is a ${\mathfrak}g$-module. 2. $L(N)$ is isomorphic as a ${\mathfrak}g$-module to the ideal of $L(V \oplus W)/ {\langle}A, B{\rangle}_{L(V \oplus W)}$ generated by the image of $W$. 3. $L(V \oplus W)/ {\langle}A, B{\rangle}_{L(V \oplus W)} \cong {\mathfrak}g \ltimes L(N)$. To prove (1), note $M= U(L(V))\cdot W$ is an $L(V)$-module as is $U(L(V))\cdot B$. Since $M$ and ${\langle}A{\rangle}_{L(V)}$ are $L(V)$-modules, so is $[M, {\langle}A {\rangle}_{L(V)}]$. Thus $N$ is an $L(V)$-module. Since $[{\langle}A{\rangle}_{L(V)}, M] \subset M_1$, $N$ is a ${\mathfrak}g$-module. We now prove (2): $$\begin{aligned} {\langle}A, B{\rangle}_{L(V \oplus W)} &= {\langle}A {\rangle}_{L(V \oplus W)} + {\langle}B {\rangle}_{L(V\oplus W)} \\ &= U(L(V \oplus W))\cdot A + U(L(V \oplus W))\cdot B.\end{aligned}$$ By Lemma 2 this is equal to $$\begin{aligned} U(L(V))\cdot A &+ U(L(M))MU(L(V))\cdot A + U(L(M))U(L(V))\cdot B \\ &= {\langle}A {\rangle}_{L(V)} + {\langle}[ M, {\langle}A {\rangle}_{L(V)}] {\rangle}_{L(M)} + {\langle}U(L(V))\cdot B{\rangle}_{L(M)}\\ &= {\langle}A {\rangle}_{L(V)} + {\langle}M_1 {\rangle}_{L(M)}. \label{E:*}\tag{*}\end{aligned}$$ Now the ideal of $L(V \oplus W)/ {\langle}A, B {\rangle}_{L(V \oplus W)}$ generated by $W$ is $$\begin{aligned} (L(M) &+ {\langle}A, B {\rangle}_{L(V \oplus W)})/ {\langle}A, B {\rangle}_{L(V \oplus W)}\\ &\cong L(M)/(L(M) \cap {\langle}A, B {\rangle}_{L(V \oplus W)}).\end{aligned}$$ By equation (\[E:\*\]) this is equal to $L(M)/ {\langle}M_1 {\rangle}_{L(M)}$. By Lemma 1 this is $L(M/M_1) = L(N)$. Furthermore, $L(V \oplus W) / {\langle}A, B {\rangle}_{L(V \oplus W)} \cong (L(V) \ltimes L(M))/{\langle}A, B {\rangle}_{L(V \oplus W)}$. By equation (\*) this is isomorphic to $$L(V)/ {\langle}A {\rangle}_{L(V)} \ltimes L(M)/ {\langle}M_1 {\rangle}_{L(M)} = {\mathfrak}g \ltimes L(N),$$ so we have proven (3). Applications ============ Let $I$ be an index set which is finite or countably infinite and let $R \subset I \times I$. Let ${\mathfrak}n$ be the Lie algebra with generators $X= \{x_i | i\in I\}$ and relations $({\text{ad}\thinspace}x_i)^{n_{ij} }x_j$ for $(i,j)\in R$. We may assume that $R$ does not contain diagonal elements $(i,i)\in I$ because $[x_i,x_i]=0 $ in $L(X)$. If $J\subset I$, let ${\mathfrak}n_J$ denote the subalgebra of ${\mathfrak}n$ generated by the $x_i$ for $i \in J$. Theorem 1 gives: Let ${\mathfrak}n, I, R$ be as above. Suppose that for some choice of $S,T \subset I$, $I = S\cup T$ (disjoint union), and $i,j \in T$ with $i \neq j$ implies $(i,j) \notin R$. Then ${\mathfrak}n \cong {\mathfrak}n_S \ltimes L(U({\mathfrak}n_S)\cdot W)$ where $W$ denotes the vector space spanned by the $x_i$ for $i \in T$. Let $V$ be the vector space with basis $x_i , i \in S$. Take $A = \{ ({\text{ad}\thinspace}x_i)^{n_{ij}}~x_j | i, j \in S\}$ and $B = \{ ({\text{ad}\thinspace}x_i)^{n_{ij}}x_j | i \in S, j \in T\}$. Then Theorem 1 gives the above decomposition, where we write $N$ as the ${\mathfrak}n_S$-module in $L(V \oplus W)/\langle A \rangle_{L(V \oplus W)}$ generated by $W$. Let ${\mathfrak}l$ be a generalized Kac-Moody algebra associated to a symmetrizable matrix $(a_{ij})_{i,j\in I}$. By Proposition 1.5 [@6] one has ${\mathfrak}l = {\mathfrak}n^+ \oplus {\mathfrak}h \oplus {\mathfrak}n^-$. Because the radical (the maximal graded ideal not intersecting ${\mathfrak}h$) is zero (see [@6],[@7]), the subalgebras ${\mathfrak}n^\pm$ can be written as ${\mathfrak}n$ above, choosing $X= \{x_i = e_i | i \in I\}$ for ${\mathfrak}n^+$ and $X= \{x_i = f_i | i \in I\}$ for ${\mathfrak}n^-$ where $e_i$ and $f_i$, $i \in I$ are the Chevalley generators of ${\mathfrak}l$. The Serre relations $({\text{ad}\thinspace}x_i)^{n_{ij}}x_j=0$ occur whenever $a_{ii}>0$, or when $a_{ii}\leq 0$ and $a_{ij} =0$. If we take $R$ to be the set corresponding to the occurrence of Serre relations and $S, T$ as in Theorem 2, then applying Theorem 2 to both ${\mathfrak}n^+$ and ${\mathfrak}n^-$ gives Theorem 3.19 of [@7]: [Corollary 3]{} Let ${\mathfrak}l$ be a generalized Kac-Moody algebra associated to a symmetrizable matrix $(a_{ij})_{i,j\in I}$. Let $R$ denote the set $\{(i,j)| a_{ii}>0 \text{ or } a_{ii}\leq 0 \text{ and } a_{ij} =0\}\subset I \times I$. Choose $S,T$ so that $I = S \cup T$ (disjoint union), and $i,j \in T$ implies $(i,j) \notin R$. Let ${\mathfrak}l_{1}$ be the subalgebra of ${\mathfrak}l$ generated by the $e_i$ and $f_i$ with $i \in S$. Then ${\mathfrak}l = {\mathfrak}u^+ \oplus ({\mathfrak}l_{1}+ {\mathfrak}h) \oplus {\mathfrak}u^-$, where ${\mathfrak}u^-$ is the free Lie algebra on the direct sum of the standard highest weight ${\mathfrak}l_{1}$-modules ${\mathcal U}({\mathfrak}n^-_{S})\cdot f_j$ for $j \in T$ and ${\mathfrak}u^+$ is the free Lie algebra on the direct sum of the standard lowest weight ${\mathfrak}l_{1}$-modules ${\mathcal U}({\mathfrak}n^+_{S})\cdot e_j$ for $j \in T$. Theorem 5.1 of [@6] is a special case of this Corollary. As noted in [@7] one can iterate this decomposition until ${\mathfrak}l_1$ is a semi-simple or Kac-Moody subalgebra. The results of [@5] on free partially commutative Lie algebras may be obtained as the case where all $a_{ii}<0$. This decomposition is used in [@6] to obtain the denominator identity for the algebra ${\mathfrak}l$ from the identity for the subalgebra ${\mathfrak}l_1$. Conversely, one can prove Corollary 3 using the denominator and character formulas for generalized Kac-Moody algebras. (This is the proof in [@7].) Other applications include computing the homology of the Lie algebra over a standard module, and determining a class of completely reducible modules [@8]. [99]{} R. Block and P. Leroux, *Free Hopf module algebras and related functors*, unpublished preprint 1972, revised version. R. Borcherds, *Generalized Kac-Moody Lie algebras*, J. Algebra, 1988, 115, 501–512. R. Borcherds, *Monstrous moonshine and monstrous Lie superalgebras*, Invent. Math. 1992, 109, 405–444. N. Bourbaki, Lie Groups and Lie Algebras, Hermann Paris, France 1975. G. Duchamp and D. Krob, *Free partially commutative structures*, Journal of Algebra, 1993, 156, 318-361. E. Jurisich, *Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra*, Journal of Pure and Applied Algebra, 1998, 126, 233–266. E. Jurisich, *An exposition of generalized Kac-Moody algebras*, in Lie algebras and their representations, Contemporary Math., Amer. Math. Soc. 1996, 194, 121–160. E. Jurisich, *An Equivalence of categories of modules for generalized Kac-Moody algebras* to appear. E. Jurisich, J. Lepowsky, R. Wilson, *Realizations of the Monster Lie algebra*, Selecta Mathematica, New Series, 1995, 1, 129–161. M. Lazard, *Groupes, anneaux de Lie et problème de Burnside*, Inst. Mat. dell’Università Roma 1960. J. Lepowsky and R. L. Wilson, *On Hopf algebras and the elimination theorem for free Lie algebras*, in Moonshine the Monster, and related topics, Contemporary Math., Amer. Math. Soc., 1996, 193, 191-202.

--- abstract: 'We report measurements of long ranged supercurrents through ferromagnetic and fully spin-polarized CrO$_2$ deposited on TiO$_2$ substrates. In earlier work, we found supercurrents in films grown on sapphire but not on TiO$_2$. Here we employed a special contact arrangement, consisting of a Ni/Cu sandwich between the film and the superconducting amorphous Mo$_{70}$Ge$_{30}$ electrodes. The distance between the contacts was almost a micrometer, and we find the critical current density to be significantly higher than found in the films deposited on sapphire. We argue this is due to spin mixing in the Ni/Cu/CrO$_2$ layer structure, which is helpful in the generation of the odd-frequency spin triplet correlations needed to carry the supercurrent.' author: - 'M. S. Anwar' - 'M. Veldhorst' - 'A. Brinkman' - 'J. Aarts,$^1$' date: Date of submission title: 'Long range spin supercurrents in ferromagnetic CrO$_2$ using a multilayer contact structure' --- Conventional spin-singlet Cooper pairs from a superconductor (S) dephase over a coherence length $\xi_F=\sqrt{\hbar D_F/h_{ex}}$ (dirty limit) in a ferromagnet (F) under the influence of its exchange field $h_{ex}$ (and $D_F$ the diffusion constant in the F-metal). Even for weak ferromagnets, $\xi_F$ is only a few nm. Such dephasing would not occur with equal-spin triplet Cooper pairs, leading to a long range proximity (LRP) effect in the ferromagnet. It was predicted that triplet correlations can be induced at an S/F interface when $h_{ex}$ is inhomogeneous [@Bergeret01; @Kadigrobov01; @Eschrig08], for instance from domain walls or unaligned magnetic moments. This should also allow a Josephson current in an S/F/S geometry. To observe this, both interfaces are required to show similar inhomogeneities [@Houzet07] as for instance in an S/F$_1$/F/F$_2$/S trilayer in which the magnetizations of the F$_1$, F$_2$ layers are non-collinear with the central F layer.\ Early work on CrO$_2$ [@Keizer06] and Holmium [@Sosnin06] gave the first indications for such LRP effects in ferromagnets. In the first case, a supercurrent was measured in devices where superconducting electrodes of NbTiN with separations up to 1 $\mu$m were placed on unstructured 100 nm thick films of CrO$_2$ (a half metallic ferromagnet or HMF) which were grown on TiO$_2$ substrates. In the second case, the LRP effect was observed in ferromagnetic Ho wires of lengths up to 150 nm using an Andreev interferometer geometry. More recently, LRP effect were reported using Josephson junctions where a Co central layer was used in combination with PdNi, CuNi or Ni layers [@Khaire10; @Khasawneh11]; and where a Co layer was used together with Ho layers to provide magnetic inhomogeneity [@Robinson10]. Signatures of LRP effect were also observed with the Heusler Compound Cu$_{2}$MnAl [@Sprungmann10] and in Co nanowires [@Wang10]. At the same time, the observation of supercurrents over a length of 700 nm through CrO$_2$ deposited on sapphire substrates was reported [@Anwar10; @Anwar11].\ The experiments with Co junctions were up to Co thicknesses of 50 nm. Since Co is not fully spin polarized the triplet decay is mainly set by the spin diffusion length, and can be expected to be of the order of 100 nm. That makes the CrO$_{2}$ case with its significantly larger decay length of special interest, but in the previous experiments the reproducibility was an issue. In particular, it was not clear where the inhomogeneous magnetization resides which is needed for the triplet generation. Also, in our previous work we did not succeed in finding supercurrents in films deposited on TiO$_2$. Here we report on observing long ranged supercurrents in CrO$_2$ grown on TiO$_2$, using 2 nm Ni as an extra layer in the contact geometry to induce an artificial magnetic inhomogeneity, and 5 nm Cu to magnetically decouple the Ni and the CrO$_2$. We find much stronger supercurrents than in the case of sapphire, indicating that with the Ni/Cu sandwich we have a good generator for triplet Cooper pairs.\ The devices were fabricated in a lateral geometry using 60 nm thick a-Mo$_{70}$Ge$_{30}$ superconducting contacts (transition temperature $T_{c}$ = 6 K) deposited on unstructured 100 nm thick CrO$_{2}$ films grown on TiO$_2$ substrates. We made the devices through a lift-off mask using a bilayer resist. Ar-ion etching was applied immediately prior to deposition, in order to remove the Cr$_{2}$O$_{3}$ on the film surface, and the Cu/Ni/Mo$_{70}$Ge$_{30}$ sandwiches were sputtered [*in situ*]{}. Two junctions were made on each sample, perpendicular to each other, and both junctions were measured independently. More details can be found in Refs.[@Anwar10; @Anwar11]\ A supercurrent was measured successfully in three devices out of five, named A$_T$, B$_T$ and C$_T$. On A$_T$ (30 $\mu$m wide leads) both junctions showed a supercurrent. We call them A$_T$-a (600 nm gap) and A$_T$-b (800 nm gap). Samples B$_T$ and C$_T$ were prepared with 5 $\mu$m wide leads, in order to lower the absolute value of the currents, and a gap of 700 nm. Here only one junction was showing a measurable critical current on each sample. Sample C$_T$ was measured in two different cryostats, one with extra filtering to minimize to amplifier contribution to the data in the zero-voltage branch. A drawback still is the limited lifetime of the samples. The supercurrent disappears after a few cool-downs, possibly due to the effect of thermal cycling on the films.\ For sample A$_T$, the resistance $R$ as function of temperature $T$ is given in Fig. \[rtmultilayer\]a and shows a sharp down-jump at $T_c$. For junction B$_T$ (Fig. \[rtmultilayer\]b), $R(T)$ shows a small dip at 6 K, followed by an up-jump, a flat part, and then a slow decrease. For junction C$_T$ the behavior is similar but with a larger up-jump to 0.7 $\Omega$, similar to our sapphire-based devices [@Anwar11].\ ![Resistance $R$ versus Temperature $T$, (a) for junctions A$_T$-a (gap 600 nm; electrode width 30 $\mu$m) and A$_T$-b (gap 800 nm); (b) for junction B$_T$ (gap 700 nm, electrode width 5 $\mu$m).[]{data-label="rtmultilayer"}](CrO2-Ni_Fig1){width="8cm"} Figure \[IVT\]a shows an $I$-$V$ characteristic for sample A$_T$-b, measured at 4.2 K. There is a zero-resistance branch up to a well-defined current of about 3 mA at which a finite voltage develops. On larger scales a bend in the curve is seen, followed by another transition at 15 mA to Ohmic behavior with R$_{N}$=100 m$\Omega$. Figure \[IVT\]b shows $I$-$V$ data measured on sample B$_T$ at 3 K. The value for $I_c$ is 1.2 mA, and the resistive branch has a value of 80 m$\Omega$, in very reasonable agreement with the normal state resistance. The residual resistance below $I_c$ is a few m$\Omega$. Sample C$_T$ was first measured at 4.2 K in a cryostat with well-filtered leads. Here, the $I$-$V$ characteristic showed sharp switching and some hysteretic behavior, with $I_c$ of the order of 0.5 mA. The residual resistance below $I_c$ is 3 m$\Omega$. $I_c(T)$ was defined by a 1 $\mu$V criterion and measured for junction B$_T$ and C$_T$ in the temperature range of 2.5 K to 6 K. As shown in Fig. \[icttio2\] for sample B$_T$ the behavior is almost linear. For sample A$_T$ we first measured the field dependence of $I_c$ at 4.2 K, but we did not measure $I_c(T)$ because the supercurrent disappeared after the third cool-down. The measurement on sample C$_T$ is also shown in Fig. \[icttio2\]. In a subsequent measurement, $I_c$ had gone down to 70 $\mu A$, illustrating the fragility of the sample, but $I_c(T)$ also showed a linear increase. Figure \[ichtio2\]a illustrates the effect of a magnetic field $H_a$ on $I_c$ at 4.2 K for both junctions A$_T$-a and A$_T$-b, with $H_a$ in the plane of the junction and $\perp I$. It shows that $I_c$ in the case of A$_T$-a is quite sensitive to $H_a$, with an initial fast decrease below 60 mT, but less so in the case of A$_T$-b. Figure \[ichtio2\]b presents $I_c(H_a)$ at 3 K for junction B$_T$ in three different configurations, $H_a$ in-plane and $\parallel, \perp I$, and $H_a$ out-of-plane. Here the field-in-plane data show a relatively slow decrease, while the field-out-of-plane data show a small sharp peak, followed by a shoulder around 100 mT. Neither for A$_T$-a,b nor for B$_T$ there is evidence for a Fraunhofer pattern. ![Current $I$ versus Voltage $V$ measured (a) for junction A$_T$-b at 4.2 K, and (b) for junction B$_T$ at 3 K.[]{data-label="IVT"}](CrO2-Ni_Fig2){width="9cm"} ![$I_c(T)$ for junction B$_T$. The open symbol is $I_c$ at 4.2 K for junction C$_T$, as follows from Fig.\[IVT\] Inset: plot of (ln(I$_c$)-(3/2 ln(T)) versus $\sqrt{T}$ to determine the Thouless energy E$_{th}$ = 54 $\mu$eV.[]{data-label="icttio2"}](CrO2-Ni_Fig3){width="6cm"} The claim from the measurements is that large supercurrents are now flowing through the CrO$_2$ bridge. In discussing these results we address the following issues. We compare the residual resistance in the supercurrent measurements with the normal state resistance of the bridge; we discuss the possibility of depairing currents in the superconducting leads; a Thouless analysis is performed; and we discuss the effects of applying a magnetic field.\ The $I_c$’s measured here can be compared with our previous measurements [@Anwar10; @Anwar11] on sapphire-based junctions. The current density at 4.2 K, (d$_{CrO_2}$ $\approx$ 100 nm, junction width 30 $\mu$m and 5 $\mu$m, current $\approx$ 3 mA and 0.5 mA respectively) is of the order of 1 $\times$ 10$^{9}$ A/m$^{2}$ for A$_T$, B$_T$, as well as C$_T$. In all cases, it is 100 times larger than that of sapphire-based junctions, and of similar magnitude as in the earlier observations of Keizer [*et al.*]{} [@Keizer06]. This suggests that a uniform spin active interface is present at the interface, due to the additional 2 nm Ni layer.\ ![Critical current $I_c$ versus applied field $(H_a)$ (a) at 4.2 K for junctions A$_T$-a ($\bigcirc$) and A$_T$-b ($\Box$) with $H_a$ in-plane and $\perp$ current $I$; (b) for junction B$_T$ at 3 K in three different configurations, in-plane $H_a \parallel I$ ($\Box$), $H_a \perp I$ ($\bigcirc$), and out-of-plane $H_a \perp I$ ($\triangle$). The vertical dotted lines indicate the field at 200 mT for reference purposes.[]{data-label="ichtio2"}](CrO2-Ni_Fig4){width="8cm"} An important question is whether the $I-V$ characteristics such as shown in Fig. \[IVT\] are truly from the CrO$_2$ bridge, and not just the superconducting contacts. For this we take another look at the normal resistance of the bridge. Taking $\rho_{CrO_2}$=10 $\mu\Omega$cm, a film thickness of 100 nm, a bridge width of 5 $\mu$m, and a junction length of 700 nm, $R_N$ comes out to be 140 m$\Omega$ (25 m$\Omega$ for the 30 $\mu$m wide contacts). This is significantly higher than what is measured in the zero-voltage branch of the $I$-$V$ characteristics, where it is not more than a few m$\Omega$. Note that the measured resistance above $T_c$ is higher than the above estimate. This is because, when the superconducting leads become normal, the geometry of the sample is a very different one, with both high resistance MoGe and low-resistance CrO$_2$ contributing.\ Another issue is how close $I_c$ comes to the depairing current $I_{dp}$ of the superconducting leads. For the sapphire-based junctions with their low $I_c$ values this was not relevant. The value for $J_{dp}$ of a-MoGe is about 4 $\times$ 10$^{10}$ A/m$^2$ at 4.2 K [@Rusanov04]. Taking into account that the thickness of the lead (40 nm) is smaller than that of the bridge, the current density in the lead at the measured $I_c$ for all junctions is about $2.5 \times 10^9$ A/m$^2$, still an order of magnitude smaller than $J_{dp}$. This probably explains, however, the second transition seen in Fig.\[IVT\]a, which takes place at a 5 times higher current density.\ Although $I_c(T)$ is quite linear, the Thouless energy of the junction can be estimated from a plot of (ln($I_c$)-(3/2 ln(T)) versus $\sqrt{T}$ (see inset of Fig.\[icttio2\]). For junction B$_T$ we find E$_{th}$ = 54 $\mu$eV, not much different from that of sapphire based junctions [@Anwar10; @Anwar11]. From E$_{Th}$ we can estimate $I_c$ at 4.2 K using theoretical results for a long junction [@dubos01]. For (k$_B T$/E$_{Th}$) $\approx$ 7.6, we find from Ref.[@dubos01] that $I_c R_N$ $\approx$ $E_{Th}$ $\approx$ 54 $\mu$V, which with $R_N$ = 60 m$\Omega$ leads to $I_c$ = 0.9 mA , quite close to the measured value.\ The magnetic field effects are complicated. For $H_a \parallel I$ the junctions A$_T$-a,b are more sensitive to the field than B$_T$. For junctions A$_T$-a,b the first sharp decrease at 60 mT might correspond to the first flux quantum, which is a reasonable value according to the dimensions of the junctions, but no such behavior is seen for B$_T$. The suppression of $I_c$ is stronger than in the earlier work. Taking a 200 mT field as a reference point, the suppression is over 70% for A$_T$, and still almost 30% for B$_T$, compared to 10% in the sapphire-based junctions. This points to a diminishing effectiveness of the Ni/Cu layer, although it might be argued that the effect should be even stronger: in 200 mT both the CrO$_2$ and Ni magnetization should be saturated and aligned, removing a possible source of magnetic inhomogeneity. Instead, the supercurrents were not even quenched in 500 mT. It suggests that there is a residual magnetic inhomogeneity residing in the Ni/Cu/CrO$_2$ sandwich, which is not removed by the magnetic field. This needs further study.\ In conclusion, a Ni/Cu sandwich on top of ferromagnetic CrO$_2$ deposited on TiO$_2$ substrates leads to strong supercurrents over a distance of almost 1 $\mu$m. The Ni/Cu sandwich appears to furnish spin mixing and triplet generation similar to what was found in Co-based junctions.\ This work is part of the research program of the Stichting F.O.M., which is financially supported by NWO. M.S.A. acknowledges the financial support of the Higher Education Commission (HEC) Pakistan. [99]{} F. S. 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--- abstract: 'The Kumjian-Pask algebras are path algebras associated to higher-rank graphs, and generalize the Leavitt path algebras. We study the center of simple Kumjian-Pask algebras and characterize commutative Kumjian-Pask algebras.' address: - | Department of Mathematics\ Kansas State University\ 138 Cardwell Hall\ Manhattan, KS 66506-2602\ USA. - | Department of Mathematics and Statistics\ University of Otago\ Dunedin 9054\ New Zealand. author: - 'Jonathan H. Brown' - Astrid an Huef date: '12 September, 2012' --- Introduction ============ Let $E$ be a directed graph $E$ and let ${\mathbb{F}}$ be a field. The Leavitt path algebras $L_{\mathbb{F}}(E)$ of $E$ over ${\mathbb{F}}$ were first introduced in [@AA] and [@AMP], and have been widely studied since then. Many of the properties of Leavitt path algebras can be inferred from properties of the graph, and for this reason provide a convenient way to construct examples of algebras with a particular set of attributes. The Leavitt path algebras are the algebraic analogues of the graph $C^*$-algebras associated to $E$. In [@Tom11], Tomforde constructed an analogous Leavitt path algebra $L_R(E)$ over a commutative ring $R$ with $1$, and introduced more techniques from the graph $C^*$-algebra setting to study it. In [@A-PCaHR11], Aranda Pino, Clark, an Huef and Raeburn generalized Tomforde’s construction and associated to a higher-rank graph $\Lambda$ a graded algebra $\operatorname{KP}_R(\Lambda)$ called the *Kumjian-Pask algebra*. Example 7.1 of [@A-PCaHR11] shows that even the class of commutative Kumjian-Pask algebras over a field is strictly larger than the class of Leavitt path algebras over that field. In this paper we study the center of Kumjian-Pask algebras. In §\[motivation\] we work over ${\mathbb{C}}$ and show how the embedding of $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ in the $C^*$-algebra $C^*(\Lambda)$ can be used together with the Dauns-Hofmann theorem to deduce that the center of a simple Kumjian-Pask algebra is either $\{0\}$ or isomorphic to ${\mathbb{C}}$. More generally, it follows from Theorem \[thm: ZKP fin\], that the center of a “basically simple” (see page ) Kumjian-Pask algebra $\operatorname{KP}_R(\Lambda)$ is either zero or is isomorphic to the underlying ring $R$. Thus our Theorem \[thm: ZKP fin\] generalizes the analogous theorem for Leavitt path algebras over a field [@A-PC11 Theorem 4.2], but our proof techniques are very different and more informative. Indeed, the Kumjian-Pask algebra is basically simple if and only if the graph $\Lambda$ is cofinal and aperiodic, and our proofs show explicitly which of these graph properties are needed to infer various properties of elements in the center. In Proposition \[prop com\] we show that a Kumjian-Pask algebra of a $k$-graph $\Lambda$ is commutative if and only it is a direct sum of rings of Laurant polynomials in $k$-indeterminates, if and only if $\Lambda$ is a disjoint union of copies of the category ${\mathbb{N}}^k$. This generalizes Proposition 2.7 of [@A-PC11]. Preliminaries ============= We view ${\mathbb{N}}^k$ as a category with one object and composition given by addition. We call a countable category $\Lambda=(\Lambda^0, \Lambda, r, s)$ a $k$-graph if there exists a functor $d:\Lambda\to {\mathbb{N}}^k$ with the *unique factorization property*: for all $\lambda\in \Lambda$, $d(\lambda)=m+n$ implies there exist unique $\mu,\nu\in \Lambda$ such that $d(\mu)=m, d(\nu)=n$ and $\lambda=\mu\nu$. Using the unique factorization property, we identify the set of objects $\Lambda^0$ with the set of morphisms of degree $0$. Then, for $n\in {\mathbb{N}}^k$, we set $\Lambda^n:=d{^{-1}}(n)$, and call $\Lambda^n$ the paths of shape $n$ in $\Lambda$ and $\Lambda^0$ the vertices of $\Lambda$. A path $\lambda\in \Lambda$ is *closed* if $r(\lambda)=s(\lambda)$. For $V,W\subset \Lambda^0$, we set $V\Lambda:=\{\lambda\in \Lambda: r(\lambda)\in V\}$, $\Lambda W:=\{\lambda\in \Lambda: s(\lambda)\in W\}$ and $V\Lambda W:= V\Lambda\cap \Lambda W$; the sets $V\Lambda^n, \Lambda^n W$ and $V\Lambda^n W$ are defined similarly. For simplicity we write $v\Lambda$ for $\{v\} \Lambda$. A $k$-graph $\Lambda$ is *row-finite* if $|v\Lambda^n|<\infty$ for all $v\in \Lambda^0$ and $n\in {\mathbb{N}}^k$ and has *no sources* if $v\Lambda^n\neq \emptyset$ for all $v\in\Lambda^0$ and $n\in {\mathbb{N}}^k$. We assume throughout that $\Lambda$ is a row-finite $k$-graph with no sources. Following [@RS07 Lemma 3.2(iv)], we say that a $k$-graph $\Lambda$ is *aperiodic* if for every $v\in \Lambda^0$ and $m\neq n\in {\mathbb N}^k$ there exists $\lambda\in v\Lambda$ such that $d(\lambda)\geq m\vee n$ and $$\lambda(m,m+d(\lambda)-(m\vee n))\neq \lambda(n,n+d(\lambda)-(m\vee n)).$$ This formulation of aperiodicity is equivalent to the original one from [@KumPas00 Definition 4.3] when $\Lambda$ is a row-finite graph with no sources, but is often more convenient since it only involves finite paths. Let $\Omega_k:=\{(m,n)\in {\mathbb{N}}^k: m\leq n\}$. As in [@KumPas00 Definition 2.1] we define an *infinite path* in $\Lambda$ to be a degree-preserving functor $x:\Omega_k\to \Lambda$, and denote the set of infinite paths by $\Lambda^\infty$. As in [@KumPas00 Definition 4.1] we say $\Lambda$ is *cofinal* if for every infinite path $x$ and every vertex $v$ there exists $m\in {\mathbb{N}}^k$ such that $v\Lambda x(m)\neq\emptyset$. For each $\lambda\in \Lambda$ we introduce a *ghost path* $\lambda^*$; for $v\in \Lambda^0$ we take $v^*=v$. We write $G(\Lambda)$ for the set of ghost paths and $G(\Lambda^{\neq 0})$ if we exclude the vertices. Let $R$ be a commutative ring with $1$. Following [@A-PCaHR11 Definition 3.1], a *Kumjian-Pask $\Lambda$-family* $(P,S)$ in an $R$-algebra $A$ consists of functions $P:\Lambda^0\to A$ and $S:\Lambda^{\neq 0}\cup G(\Lambda^{\neq 0})\to A$ such that 1. $\{P_v:v\in \Lambda^0\}$ is a set of mutually orthogonal idempotents; 2. for $\lambda, \mu\in \Lambda^{\neq 0}$ with $r(\mu)=s(\lambda)$, $$S_{\lambda}S_\mu=S_{\lambda\mu},\ S_{\mu^*}S_{\lambda^*}=S_{(\lambda\mu)^*},\ P_{r(\lambda)}S_\lambda=S_\lambda=S_\lambda P_{s(\lambda)},\ P_{s(\lambda)}S_{\lambda^*}=S_{\lambda^*}=S_{\lambda^*}P_{r(\lambda)};$$ 3. for all $\lambda,\mu\in \Lambda^{\neq 0}$ with $d(\lambda)=d(\mu)$, we have $S_{\lambda^*}S_\mu=\delta_{\lambda,\mu}P_{s(\lambda)}$; 4. for all $v\in \Lambda^0$ and $n\in {\mathbb{N}}^k\setminus\{0\}$, we have $P_v=\sum_{\lambda\in v\Lambda^n} S_\lambda S_{\lambda^*}$. By [@A-PCaHR11 Theorem 3.4] there is an $R$-algebra $\operatorname{KP}_R(\Lambda)$, generated by a nonzero Kumjian-Pask $\Lambda$-family $(p, s)$, with the following universal property: whenever $(Q,T)$ is a Kumjian-Pask $\Lambda$-family in an $R$-algebra $A$, then there is a unique $R$-algebra homomorphism $\pi_{Q,T}:\operatorname{KP}_R(\Lambda)\to A$ such that $$\label{defpiqt} \pi_{Q,T}(p_v) = Q_v,\ \pi_{Q,T}(s_{\lambda}) = T_{\lambda} \text{\ and \ } \pi_{Q,T}(s_{\mu^*}) =T_{\mu^*}\text{\ for $v\in \Lambda^0$ and $\lambda,\mu\in\Lambda^{\neq 0}$.}$$ Also by Theorem 3.4 of [@A-PCaHR11], the subgroups $$\operatorname{KP}_R(\Lambda)_n:=\operatorname{span}_R\{s_\lambda s_{\mu^*}:\lambda,\mu\in \Lambda\text{\ and\ }d(\lambda)-d(\mu)=n\}\quad\quad (n\in{\mathbb{Z}}^k)$$ give a ${\mathbb{Z}}^k$-grading of $\operatorname{KP}_R(\Lambda)$. Let $S$ be a ${\mathbb{Z}}^k$-graded ring; then by the graded-uniqueness theorem [@A-PCaHR11 Theorem 4.1], a graded homomorphism $\pi:\operatorname{KP}_R(\Lambda)\to S$ such that $\pi(rp_v)\neq 0$ for nonzero $r\in R$ is injective. We will often write elements $a\in \operatorname{KP}_R(\Lambda)\setminus\{0\}$ in the *normal form* of [@A-PCaHR11 Lemma 4.2]: there exists $m\in {\mathbb{N}}^k$ and a finite $F\subset \Lambda\times \Lambda^m$ such that $a=\sum_{(\alpha,\beta)\in F} r_{\alpha, \beta} s_\alpha s_{\beta^*}$ where $r_{\alpha,\beta}\in R\setminus\{0\}$ and $s(\alpha)=s(\beta)$. Motivation ========== When $A$ is a simple $C^*$-algebra (over ${\mathbb{C}}$, of course), it follows from the Dauns-Hofmann Theorem (see, for example, [@tfb Theorem A.34]) that the center $Z(A)$ of $A$ is isomorphic to ${\mathbb{C}}$ if $A$ has an identity and is $\{0\}$ otherwise. Let $\Lambda$ be a row-finite $k$-graph without sources. In this short section we deduce that the center of a simple Kumjian-Pask algebra $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ is either isomorphic to ${\mathbb{C}}$ or is $\{0\}$. \[lem c\*\] Suppose $A$ is a simple $C^*$-algebra. If $A$ has an identity, then $z\mapsto z1_{A}$ is an isomorphism of ${\mathbb{C}}$ onto the center $Z(A)$ of $A$. If $A$ has no identity, then $Z(A)=\{0\}$. Since $A$ is simple, $\operatorname{Prim}A =\{\star\}$, and $f\mapsto f(\star)$ is an isomorphism of $C_b(\operatorname{Prim}A)$ onto ${\mathbb{C}}$. By the Dauns-Hofmann Theorem, $C_b(\operatorname{Prim}A)$ is isomorphic to the center $Z(M(A))$ of the multiplier algebra $M(A)$ of $A$. Putting the two isomorphisms together gives an isomorphism $z\mapsto z1_{M(A)}$ of ${\mathbb{C}}$ onto $Z(M(A))$. Now suppose that $A$ has an identity. Then $M(A)=A$, and it follows from the first paragraph that $Z(A)$ is isomorphic to ${\mathbb{C}}$. Next suppose that $A$ does not have an identity. Let $a\in Z(A)$, and let $u_\lambda$ be an approximate identity in $A$ and $m\in M(A)$. Then $ma=\lim (mu_\lambda) a=a\lim (mu_\lambda)=am$. Thus $Z(A)\subset Z(M(A))$. Now $Z(A)\subset Z(M(A))\cap A=\{z 1_{M(A)}:z\in {\mathbb{C}}\}\cap A=\{0\}$. \[lem-shor\] Let $D$ be a dense subalgebra of a $C^*$-algebra $A$. Then $Z(A)\cap D= Z(D)$. Trivially, $Z(A)\cap D\subset Z(D)$. To see the reverse inclusion, let $a\in Z(D)$. Let $b\in A$ and choose $\{d_\lambda\}\subset D$ such that $d_\lambda\to b$. Then $ba=\lim_\lambda d_\lambda a=\lim_\lambda a d_\lambda=ab$. Now $a\in Z(A)\cap Z(D)\subset Z(A)\cap D$, and hence $Z(A)\cap D= Z(D)$. By [@A-PCaHR11 Theorem 6.1], $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ is simple if and only if $\Lambda$ is cofinal and aperiodic, so in the next corollary $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ is simple. Also, $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ has an identity if and only if $\Lambda^0$ is finite (see Lemma \[lem: KP unital\] below). \[cor complex\] Suppose that $\Lambda$ is a row-finite, cofinal, aperiodic $k$-graph with no sources. If $\Lambda^{0}$ is finite, then $z\mapsto z1_{\operatorname{KP}_{\mathbb{C}}(\Lambda)}$ is an isomorphism of ${\mathbb{C}}$ onto the center $Z(\operatorname{KP}_{\mathbb{C}}(\Lambda))$ of $\operatorname{KP}_{\mathbb{C}}(\Lambda)$. If $\Lambda^{0}$ is infinite, then $\operatorname{KP}_{\mathbb{C}}(\Lambda)=\{0\}$. Let $(p,s)$ be a generating Kumjian-Pask $\Lambda$-family for $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ and $(q,t)$ a generating Cuntz-Krieger $\Lambda$-family for $C^*(\Lambda)$. Then $(q,t)$ is a Kumjian-Pask $\Lambda$-family in $C^*(\Lambda)$, and the universal property of $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ gives a $*$-homomorphism $\pi_{q,t}:\operatorname{KP}_{\mathbb{C}}(\Lambda)\to C^*(\Lambda)$ which takes $s_\mu s_{\nu^*}$ to $t_\mu t_\nu^*$. It follows from the graded-uniqueness theorem that $\pi_{q,t}$ is a $*$-isomorphism onto a dense $*$-subalgebra of $C^*(\Lambda)$ (see Proposition 7.3 of [@A-PCaHR11]). Since $\Lambda$ is aperiodic and cofinal, $C^*(\Lambda)$ is simple by [@RS07 Theorem 3.1]. Now suppose that $\Lambda^0$ is finite. Then $\operatorname{KP}_{\mathbb{C}}(\Lambda)$ has identity $1_{\operatorname{KP}_{\mathbb{C}}(\Lambda)}=\sum_{v\in\Lambda^0} p_v$ and $C^*(\Lambda)$ has identity $1_{C^*(\Lambda)}=\sum_{v\in\Lambda^0} q_v$, and $\pi_{q,t}$ is unital. By Lemma \[lem c\*\], $Z(C^*(\Lambda))=\{z1_{C^*(\Lambda)}: z\in{\mathbb{C}}\}$. By Lemma \[lem-shor\], $Z(\pi_{q,t}(\operatorname{KP}_{\mathbb{C}}(\Lambda)))=Z(C^*(\Lambda))\cap \pi_{q,t}(\operatorname{KP}_{\mathbb{C}}(\Lambda))=\{z1_{C^*(\Lambda)}: z\in{\mathbb{C}}\}$. Since $\pi_{q,t}$ is unital, $Z(\operatorname{KP}_{\mathbb{C}}(\Lambda))$ is isomorphic to ${\mathbb{C}}$ as claimed. Next suppose that $\Lambda^0$ is infinite. Then $Z(C^*(\Lambda))=\{0\}$ and $Z(\pi_{q,t}(\operatorname{KP}_{\mathbb{C}}(\Lambda)))=Z(C^*(\Lambda))\cap \pi_{q,t}(\operatorname{KP}_{\mathbb{C}}(\Lambda))=\{0\}$, giving $\operatorname{KP}_{\mathbb{C}}(\Lambda)=\{0\}$. The center of a Kumjian-Pask algebra ==================================== Our goal is to extend Corollary \[cor complex\] to Kumjian-Pask algebras over arbitrary rings. Throughout $R$ is a commutative ring with $1$ and $\Lambda$ is a row-finite $k$-graph with no sources. We will need Lemma \[lem: lin inde\] several times. For notational convenience, for $v\in \Lambda^0$, $s_v$ or $s_{v^*}$ means $p_v$. So when $m=0$ Lemma \[lem: lin inde\], says that the set $\{s_{\alpha}: \alpha\in\Lambda\}$ is linearly independent. \[lem: lin inde\] Let $m\in {\mathbb{N}}^k$. Then $\{s_{\alpha}s_{\beta^*}: \text{$s(\alpha)=s(\beta)$ and $d(\beta)=m$}\}$ is a linearly independent subset of $\operatorname{KP}_R(\Lambda)$. Let $F$ be a finite subset of $\{(\alpha,\beta)\in \Lambda\times \Lambda^m : s(\alpha)=s(\beta) \}$, and suppose that $\sum_{(\alpha,\beta)\in F}r_{\alpha,\beta}s_{\alpha}s_{\beta^*}=0$. Fix $(\sigma,\tau)\in F$. Since all the $\beta$ have degree $m$, using (KP3) twice we obtain $$\begin{aligned} 0&=s_{\sigma^*}\Big( \sum_{(\alpha,\beta)\in F}r_{\alpha,\beta}s_{\alpha}s_{\beta^*}\Big) s_\tau\notag \\ &=r_{\sigma,\tau}p_{s(\sigma)}+\sum_{(\alpha,\beta)\in F\setminus\{(\sigma,\tau)\}}r_{\alpha,\tau} s_{\sigma^*}s_\alpha s_{\beta^*}s_\tau\notag\\ &=r_{\sigma,\tau}p_{s(\sigma)}+\sum_{\substack{(\alpha,\tau)\in F\\ \alpha\neq \sigma}}r_{\alpha,\tau} s_{\sigma^*}s_\alpha.\label{eq-independence}\end{aligned}$$ If $d(\sigma)=d(\alpha)$ and $\sigma\neq \alpha$, then $s_{\sigma^*}s_\alpha=0$ by (KP3). If $d(\sigma)\neq d(\alpha)$ then, by [@A-PCaHR11 Lemma 3.1], $s_{\sigma^*}s_\alpha$ is a linear combination of $s_\mu s_{\nu^*}$ where $d(\mu)-d(\nu)=d(\alpha)-d(\sigma)$. It follows that the $0$-graded component of is $r_{\sigma,\tau}p_{s(\sigma)}$. Thus $0=r_{\sigma,\tau}p_{s(\sigma)}$. But $p_{s(\sigma)}\neq 0$ by Theorem 3.4 of [@A-PCaHR11]. Hence $r_{\sigma,\tau}=0$. Since $(\sigma, \tau)\in F$ was arbitrary, it follows that $\{s_{\alpha}s_{\beta^*}: \text{$s(\alpha)=s(\beta)$ and $d(\beta)=m$}\}$ is linearly independent. The next lemma describes properties of elements in the center of $\operatorname{KP}_R(\Lambda)$. \[lem: ZKP arb\] Let $a\in Z(\operatorname{KP}_R(\Lambda))\setminus\{0\}$ be in normal form $\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta} s_{\alpha}s_{\beta^*}$. 1. \[cond: ZKP arb r=r\] If $(\sigma,\tau)\in F$, then $r(\sigma)=r(\tau)$. 2. \[cond: ZKP arb supsat\] Let $W=\{v\in\Lambda^0: ~\exists (\alpha,\beta)\in F\text{~~with~~}v=r(\beta) \}.$ If $\mu\in \Lambda W$, then $r(\mu)\in W$. 3. \[cond: ZKP arb r=s\] If $(\sigma,\tau)\in F$, then there exists $(\alpha,\beta)\in F$ such that $r(\alpha)=r(\beta)=s(\sigma)=s(\tau).$ 4. \[cond: ZKP arb closed path\] There exists $l\in {\mathbb{N}}\setminus\{0\}$ and $\{(\alpha_i,\beta_i)\}_{i=1}^l\subset F$ such that $\beta_1\cdots \beta_l$ is a closed path in $\Lambda.$ Let $(\sigma,\tau)\in F$. By [@BH Lemma 2.3] we have $0\neq s_{\sigma^*}as_\tau$. Since $a\in Z(\operatorname{KP}_R(\Lambda))$ $$0\neq s_{\sigma^*}p_{r(\sigma)}ap_{r(\tau)}s_\tau=s_{\sigma^*}ap_{r(\sigma)}p_{r(\tau)}s_\tau=\delta_{r(\sigma),r(\tau)}s_{\sigma^*}as_\tau.$$ Hence $r(\sigma)=r(\tau)$. By way of contradiction, assume there exists $\mu\in \Lambda W$ such that $r(\mu)\notin W$. Then $p_v p_{r(\mu)}=0$ for all $v\in W$. Thus $$ap_{r(\mu)}= \sum_{(\alpha,\beta)\in F} r_{\alpha,\beta} s_{\alpha}s_{\beta^*}p_{r(\beta)}p_{r(\mu)}=0.$$ Since $a\in Z(\operatorname{KP}_R(\Lambda))$ we get $s_\mu a=as_\mu=ap_{r(\mu)}s_\mu=0$. Since $s(\mu)\in W$, there exist $(\alpha',\beta')\in F$ with $r(\beta')=s(\mu)$. Then $r(\alpha')=s(\mu)$ also by . Thus $S:=\{(\alpha,\beta)\in F: s(\mu)=r(\alpha)\}$ is non-empty, and $$\begin{aligned} 0=s_\mu a=\sum_{(\alpha,\beta)\in S} r_{\alpha,\beta} s_{\mu\alpha}s_{\beta^*}. \end{aligned}$$ But $\{s_{\mu\alpha}s_{\beta^*}: (\alpha,\beta)\in S\}$ is linearly independent by Lemma \[lem: lin inde\], and hence $r_{\alpha,\beta}=0$ for all $(\alpha,\beta)\in S$. This contradicts the given normal form for $a$. Let $(\sigma,\tau)\in F$. Then $s(\sigma)=s(\tau)$ by definition of normal form. By Lemma 2.3 in [@BH] we have $s_{\sigma^*} a s_\tau\neq 0$. Since $a\in Z(\operatorname{KP}_R(\Lambda))$, $$\label{eq com}0\neq s_{\sigma^*} a s_\tau=as_{\sigma^*} s_\tau=\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta} s_{\alpha}s_{\beta^*}s_{\sigma^*}s_\tau=\sum_{\substack{(\alpha,\beta)\in F\\r(\beta)=s(\sigma)}} r_{\alpha,\beta} s_{\alpha}s_{(\sigma\beta)^*}s_\tau.$$ In particular, the set $\{(\alpha,\beta)\in F: r(\beta)=s(\sigma)\}$ is nonempty. So there exists $(\alpha',\beta')\in F$ such that $r(\beta')=s(\sigma)$. Since $r(\alpha')=r(\beta')$ from , we are done. Let $M=|F|+1$. Using there exists a path $\beta_1\dots\beta_M$ such that, for $1\leq i\leq M$, there exists $\alpha_i\in\Lambda$ with $(\alpha_i,\beta_i)\in F$. Since $M>|F|$, there exists $i<j\in\{1,\dots,M\}$ such that $\beta_i=\beta_j$. Then $\beta_i\dots\beta_{j-1}$ is a closed path. The next corollary follows from Lemma \[lem: ZKP arb\]. Let $\Lambda$ be a row-finite $k$-graph with no sources and $R$ a commutative ring with $1$. If $\Lambda$ has no closed paths then the center $Z(\operatorname{KP}_R(\Lambda))=\{0\}$. The next lemma provides a description of elements of the center of $\operatorname{KP}_R(\Lambda)$ when $\Lambda$ is cofinal. \[lem: ZKP cof\] Suppose that $\Lambda$ is cofinal. Let $a\in Z(\operatorname{KP}_R(\Lambda))\setminus\{0\}$ be in normal form $\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta} s_{\alpha}s_{\beta^*}$. Then $\{v\in\Lambda^0: \exists (\alpha,\beta)\in F\text{~~with~~} v=r(\beta)\}=\Lambda^0$. Write $W:=\{v\in\Lambda^0: \exists (\alpha,\beta)\in F\text{~~with~~} v=r(\beta)\}$. Since $\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta} s_{\alpha}s_{\beta^*}$ is in normal form, $F\subset \Lambda\times \Lambda^m$ for some $m\in{\mathbb{N}}^k$. Let $n=m\vee (1,1,\ldots, 1)$. By (KP4), for each $(\alpha,\beta)\in F$, we have $s_{\alpha}s_{\beta^*}=\sum_{\mu\in s(\alpha)\Lambda^{n-m}} s_{(\alpha\mu)}s_{(\beta\mu)^*}$. By “reshaping” each pair of paths in $F$ in this way, collecting like terms and dropping those with zero coefficients, we see that there exists $G\subset \Lambda\times \Lambda^n$ and $r'_{\gamma,\eta}\in R\setminus\{0\}$ such that $a=\sum_{(\gamma,\eta)\in G} r'_{\gamma,\eta} s_\gamma s_{\eta^*}$ is also in normal form. By construction, $W'=\{v\in\Lambda^0: \exists (\gamma,\eta)\in G\text{~~with~~} v=r(\eta)\}\subset W$. Let $v\in \Lambda^0$. Using Lemma \[lem: ZKP arb\], there exists $\{(\gamma_i,\eta_i)\}_{i=1}^l\subset G$ such that $\eta_1\cdots \eta_l$ is a closed path. Since $d(\eta_i)\geq (1,1,\ldots ,1)$ for all $i$, $x:=\eta_1\cdots\eta_l\eta_1\cdots\eta_l\eta_1\cdots$ is an infinite path. By cofinality, there exist $q\in{\mathbb{N}}^k$ and $\nu\in v\Lambda x(q)$. By the definition of $x$, there exist $q'\geq q$ and $j$ such that $x(q')=r(\eta_j)$. Let $\lambda=x(q,q')$. Then $\nu\lambda\in v\Lambda W'$. By Lemma \[lem: ZKP arb\], $v=r(\nu\lambda)\in W'$ as well. Thus $W'=\Lambda^0$, and since $W'\subset W$ we have $W=\Lambda^0$. The next lemma provides a description of elements of the center of $\operatorname{KP}_R(\Lambda)$ when $\Lambda$ is aperiodic. \[lem: ZKP aper\] Suppose that $\Lambda$ is aperiodic and $a\in Z(\operatorname{KP}_R(\Lambda))\setminus\{0\}$. Then there exist $n\in {\mathbb{N}}^k$ and $G\subset \Lambda^n$ such that $a=\sum_{\alpha\in G} r_{\alpha} s_{\alpha}s_{\alpha^*}$ is in normal form. Suppose $a\in Z(\operatorname{KP}_R(\Lambda))\setminus\{0\}$ with $a=\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta} s_{\alpha} s_{\beta^*}$ in normal form. Let $(\sigma,\tau)\in F.$ From Lemma 2.3 in [@BH] we know that $s_{\sigma^*} a s_\tau\neq 0$. Let $m=\vee_{(\alpha,\beta)\in F}(d(\alpha)\vee d(\beta)).$ Since $\Lambda$ is aperiodic, by [@HRSW Lemma 6.2], there exist $\lambda\in s(\sigma)\Lambda$ with $d(\lambda)\geq m$ such that $$\label{eq aper} \left.\begin{array}{l}\alpha,\beta\in \Lambda s(\sigma),~d(\alpha),d(\beta)\leq m\\ \text{and}~~\alpha\lambda(0,d(\lambda))=\beta\lambda(0,d(\lambda))\end{array}\right\} \implies \alpha=\beta.$$ The same argument as in [@A-PCaHR11 Proposition 4.9] now shows that $s_{\lambda^*}s_{\sigma^*}as_{\tau}s_{\lambda}\neq 0$. Since $a\in Z(\operatorname{KP}_R(\Lambda))$, $0\neq s_{\lambda^*}s_{\sigma^*}as_{\tau}s_{\lambda}=as_{\lambda^*}s_{\sigma^*}s_{\tau}s_{\lambda}=as_{(\sigma\lambda)^*}s_{\tau\lambda}$. Thus $$\begin{aligned} 0\neq s_{(\sigma\lambda)^*}s_{\tau\lambda} &=s_{\sigma\lambda(d(\lambda), d(\lambda)+d(\sigma))^*}s_{\sigma\lambda(0,d(\lambda))^*}s_{\tau\lambda(0,d(\lambda))}s_{\tau\lambda(d(\lambda),d(\lambda)+d(\tau))}\\ &=\delta_{\sigma,\tau} s_{\sigma\lambda(d(\lambda), d(\lambda)+d(\sigma))^*}s_{\tau\lambda(d(\lambda),d(\lambda)+d(\tau))}\quad\text{using \eqref{eq aper}}.\end{aligned}$$ Thus $\sigma=\tau$. Since $(\sigma,\tau)\in F$ was arbitrary we have $\alpha=\beta$ for all $(\alpha,\beta)\in F$. Let $G=\{\alpha\in \Lambda: (\alpha,\alpha)\in F\}$ and write $r_{\alpha}$ for $r_{\alpha,\alpha}$. Note $G\subset\Lambda^n$ because $F\subset \Lambda\times\Lambda^n$ for some. Thus $a=\sum_{\alpha\in G} r_{\alpha}s_{\alpha}s_{\alpha^*}$ in normal form as desired. Our main theorem (Theorem \[thm: ZKP fin\]) has two cases: $\Lambda^0$ finite and infinite. \[lem: KP unital\] $\operatorname{KP}_R(\Lambda)$ has an identity if and only if $\Lambda^0$ is finite. If $\Lambda^0$ is finite, then $\sum_{v\in \Lambda^0} p_v$ is an identity for $\operatorname{KP}_R(\Lambda)$. Conversely, assume that $\operatorname{KP}_R(\Lambda)$ has an identity $1_{\operatorname{KP}_R(\Lambda)}$. By way of contradiction, suppose that $\Lambda^0$ is infinite. Write $1_{\operatorname{KP}_R(\Lambda)}$ in normal form $\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta}s_{\alpha}s_{\beta^*}$. Since $F$ is finite, so is $W:=\{v\in\Lambda^0:\exists (\alpha,\beta)\in F\text{~~with~~}v=r(\beta)\}$. Thus there exists $w\in \Lambda^0\setminus W$. Now $p_w=1_{\operatorname{KP}_R(\Lambda)}p_w=\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta}s_{\alpha}s_{\beta^*}p_w=0$ because $w\neq r(\beta)$ for any of the $\beta$. This contradiction shows that $\Lambda^0$ must be finite. \[thm: ZKP fin\] Let $\Lambda$ be a row-finite $k$-graph with no sources and $R$ a commutative ring with $1$. 1. \[lem: ZKP fin 1\] Suppose $\Lambda$ is aperiodic and cofinal, and that $\Lambda^0$ is finite. Then $Z(\operatorname{KP}_R(\Lambda))=R1_{\operatorname{KP}_R(\Lambda)}$. 2. \[lem: ZKP fin 2\] Suppose that $\Lambda$ is cofinal and $\Lambda^0$ is infinite. Then $Z(\operatorname{KP}_R(\Lambda))=\{0\}$. Suppose $\Lambda$ is aperiodic and cofinal, and that $\Lambda^0$ is finite. Let $a\in Z(\operatorname{KP}_R(\Lambda))\setminus\{0\}$. Since $\Lambda$ is aperiodic, by Lemma \[lem: ZKP aper\], there exist $G\subset \Lambda^n$ such that $a=\sum_{\alpha\in G} r_{\alpha} s_{\alpha}s_{\alpha^*}$ is in normal form. Since $\Lambda$ is row-finite and $\Lambda^0$ is finite, $\Lambda^n$ is finite. We claim that $G=\Lambda^n$. By way of contradiction, suppose that $G\neq \Lambda^n$, and let $\lambda\in \Lambda^n\setminus G.$ Then $as_\lambda=0$ by (KP3). But since $a\in Z(\operatorname{KP}_R(\Lambda))$, $$0=as_\lambda=s_\lambda a=\sum_{\substack{\alpha\in G\\r(\alpha)=s(\lambda)}} r_\alpha s_{\lambda\alpha}s_{\alpha^*}.$$ Since $\Lambda$ is confinal, $\{r(\alpha):\alpha\in G\}=\Lambda^0$ by Lemma \[lem: ZKP cof\]. Thus $S=\{\alpha\in G: r(\alpha)=s(\lambda)\}\neq \emptyset$. But $\{s_{\lambda\alpha}s_{\alpha^*}:\alpha\in S\}$ is linearly independent by Lemma \[lem: lin inde\]. Thus $r_{\alpha}=0$ for $\alpha\in S$, contradicting our choice of $\{r_{\alpha}\}$. It follows that $G=\Lambda^n$ as claimed, and that $$a=\sum_{\alpha\in \Lambda^n} r_{\alpha} s_{\alpha}s_{\alpha^*}.$$ Next we claim that $r_\mu=r_\nu$ for all $\mu, \nu\in \Lambda^n$. Let $\mu,\nu\in \Lambda^n$. Let $x\in s(\mu)\Lambda^\infty$. Since $\Lambda$ is cofinal, there exists $m\in {\mathbb{N}}^k$ and $\gamma\in s(\nu)\Lambda s(x(m))$. Set $\eta=x(0,m)$. Now $$\begin{aligned} r_\mu s_{\nu\gamma} s_{(\mu\eta)^*} &=r_\mu s_{\nu\gamma} s_{\eta^*} s_{\mu^*}=s_{\nu\gamma} s_{\eta^*}\sum_{\alpha\in\Lambda^n} r_\mu s_{\mu^*}s_\alpha s_{\alpha^*}\\ &=s_{\nu\gamma} s_{\eta^*}s_{\mu^*}\sum_{\alpha\in\Lambda^n} r_\alpha s_\alpha s_{\alpha^*} =s_{\nu\gamma} s_{(\mu\eta)^*}a\\ &=as_{\nu\gamma} s_{(\mu\eta)^*} =\sum_{\alpha\in \Lambda^n} r_\alpha s_\alpha s_{\alpha^*}s_{\nu}s_{\gamma} s_{(\mu\eta)^*} =r_\nu s_{\nu\gamma} s_{(\mu\eta)^*}.\end{aligned}$$ Since $s_{\nu\gamma} s_{(\mu\eta)^*}\neq 0$ this implies $r_\mu=r_\nu$. Let $r=r_\mu$. Now $$a=\sum_{\alpha\in \Lambda^n} rs_\alpha s_{\alpha^*}=\sum_{v\in \Lambda^0} \sum_{\alpha\in v\Lambda^n} r s_\alpha s_{\alpha^*}=r\sum_{v\in \Lambda^0} p_v=r1_{\operatorname{KP}_R(\Lambda)}$$ as desired. Suppose there exists $a\in Z(\operatorname{KP}_R(\Lambda))\setminus\{0\}$. Write $a=\sum_{(\alpha,\beta)\in F} r_{\alpha,\beta} s_{\alpha}s_{\beta^*}$ in normal form. Then Lemma \[lem: ZKP cof\] gives that $\{v\in\Lambda^0: \exists (\alpha,\beta)\in F\text{~~with~~} v=r(\beta)\}=\Lambda^0$, contradicting that $F$ is finite. Simplicity of $C^*(\Lambda)$ played an important role in §\[motivation\]. To reconcile this with Theorem \[thm: ZKP fin\], recall from [@Tom11] that an ideal $I\in \operatorname{KP}_R(\Lambda)$ is *basic* if $rp_v\in I$ for $r\in R\setminus \{0\}$ then $p_v\in I$, and that $\operatorname{KP}_R(\Lambda)$ is *basically simple*\[page-basic\] if its only basic ideals are $\{0\}$ and $\operatorname{KP}_R(\Lambda)$. By [@A-PCaHR11 Theorem 5.14], $\operatorname{KP}_R(\Lambda)$ is basically simple if and only if $\Lambda$ is cofinal and aperiodic (and by [@A-PCaHR11 Theorem 6.1], $\operatorname{KP}_R(\Lambda)$ is simple if and only if $R$ is a field and $\Lambda$ is cofinal and aperiodic). Thus Theorem \[thm: ZKP fin\] is in the spirit of Corollary \[cor complex\]. Commutative Kumjian-Pask algebras ================================= We view ${\mathbb{N}}^k$ as a category with one object $\star$ and composition given by addition, and use $\{e_i\}_{i=1}^k$ to denote the standard basis of ${\mathbb{N}}^k$. \[ex: Bk\] Let $d:{\mathbb{N}}^k\to {\mathbb{N}}^k$ be the identity map. Then $({\mathbb{N}}^k, d)$ is a $k$-graph. By [@A-PCaHR11 Example 7.1], $\operatorname{KP}_R({\mathbb{N}}^k)$ is commutative with identity $p_\star$, and $\operatorname{KP}_R({\mathbb{N}}^k)$ is isomorphic to the ring of Laurent polynomials $R[x_1, x_1{^{-1}}, \ldots, x_k, x_k{^{-1}}]$ in $k$ commuting indeterminates. \[lem: dir sum\] Suppose $\Lambda=\Lambda_1\bigsqcup \Lambda_2$ is a disjoint union of two $k$-graphs. Then $\operatorname{KP}_R(\Lambda)=\operatorname{KP}_R(\Lambda_1)\oplus \operatorname{KP}_R(\Lambda_2)$. For each $i=1,2$, let $(q^i, t^i)$ be the generating Kumjian-Pask $\Lambda_i$-family of $\operatorname{KP}_R(\Lambda_i)$, and let $(p, s)$ be the generating Kumjian-Pask $\Lambda$-family of $\operatorname{KP}_R(\Lambda)$. Restricting $(p,s)$ to $\Lambda_i$ gives a $\Lambda_i$-family in $\operatorname{KP}_R(\Lambda)$, and hence the universal property for $\operatorname{KP}_R(\Lambda_i)$ gives a homomorphism $\pi_{p,s}^i:\operatorname{KP}_R(\Lambda_i)\to \operatorname{KP}_R(\Lambda)$ such that $\pi_{p,s}^i\circ (q^i, t^i)=(p,s)$. Each $\pi_{p,s}^i$ is graded, and the graded uniqueness theorem ([@A-PCaHR11 Theorem 4.1]) implies that $\pi_{p,s}^i$ is injective. We now identify $\operatorname{KP}_R(\Lambda_i)$ with its image in $\operatorname{KP}_R(\Lambda)$. If $\mu\in \Lambda_1$ and $\lambda\in \Lambda_2$, then $s_\mu s_\lambda=s_\mu p_{s(\mu)} p_{r(\lambda)} s_\lambda =0$. Similarly $s_\lambda s_\mu, s_{\mu^*} s_{\lambda^*}, s_{\lambda^*} s_{\mu^*}, s_\lambda s_{\mu^*}, s_{\mu^*} s_\lambda, s_{\lambda^*} s_\mu, s_\mu s_{\lambda^*}$ are all zero. Thus $\operatorname{KP}_R(\Lambda_1)\operatorname{KP}_R(\Lambda_2)=0=\operatorname{KP}_R(\Lambda_2)\operatorname{KP}_R(\Lambda_1)$, and the internal direct sum $\operatorname{KP}_R(\Lambda_1)\oplus \operatorname{KP}_R(\Lambda_2)$ is a subalgebra of $\operatorname{KP}_R(\Lambda)$. Finally, $\operatorname{KP}_R(\Lambda_1)\oplus \operatorname{KP}_R(\Lambda_2)$ is all of $\operatorname{KP}_R(\Lambda)$ since the former contains all the generators of later. This gives the result. \[prop com\] Let $\Lambda$ be a row-finite $k$-graph with no sources and $R$ a commutative ring with $1$. Then the following conditions are equivalent: 1. \[it 1\]$\operatorname{KP}_R(\Lambda)$ is commutative; 2. \[it 2\] $r=s$ on $\Lambda$ and $r|_{\Lambda^n}$ is injective; 3. \[it 3\] $\Lambda\cong\bigsqcup_{v\in \Lambda^0} {\mathbb{N}}^k$; 4. \[it 4\] $\operatorname{KP}_R(\Lambda)\cong \bigoplus_{v\in \Lambda^0} R[x_1, x_1{^{-1}},\ldots, x_k, x_k{^{-1}}]$. $\eqref{it 1}\Rightarrow \eqref{it 2}$. Suppose that $\operatorname{KP}_R(\Lambda)$ is commutative. By way of contradiction, suppose there exists $\lambda\in \Lambda$ such that $s(\lambda)\neq r(\lambda)$. Then $s_{\lambda^*}s_\lambda=s_\lambda s_{\lambda^*}$, and $$p_{s(\lambda)}=p_{s(\lambda)}^2=p_{s(\lambda)}s_{\lambda^*} s_{\lambda}=p_{s(\lambda)}s_\lambda s_{\lambda^*}=p_{s(\lambda)}p_{r(\lambda)}s_\lambda s_{\lambda^*}=0.$$ But $p_v\neq 0$ for all $v\in\Lambda^0$ by [@A-PCaHR11 Theorem 3.4]. This contradiction gives $r=s$. Next, supppose $\lambda,\mu\in \Lambda^n$ with $\lambda\neq \mu$. By way of contradiction, suppose that $r(\lambda)=r(\mu)$. Since $r=s$, $r(\lambda)=s(\lambda)=s(\mu)=r(\mu)$. Then $$s_\lambda=p_{r(\lambda)}s_\lambda=p_{s(\lambda)} s_\lambda =p_{s(\mu)}s_\lambda=s_{\mu^*}s_\mu s_\lambda=s_{\mu^*} s_\lambda s_\mu =0$$ by (KP3). Now $p_{s(\lambda)}=0$, contradicting that $p_v\neq 0$ for all $v\in\Lambda^0$ by [@A-PCaHR11 Theorem 3.4]. Thus $r$ is injective on $\Lambda^n$. $\eqref{it 2}\Rightarrow \eqref{it 3}$ Assume that $r=s$ on $\Lambda$ and that $r|_{\Lambda^n}$ is injective. Since $r=s$, $\{v\Lambda v\}_{v\in \Lambda^0}$ is a partition of $\Lambda$. Since $r$ is injective on $\Lambda^{e_i}$, the subgraph $v\Lambda^{e_i} v$ has a single vertex $v$ and single edge $f^v_i$. Thus $f_i^v\mapsto e_i$ defines a graph isomorphism $v\Lambda v\to {\mathbb{N}}^k$. Hence $\Lambda=\bigsqcup_{v\in \Lambda^0} v\Lambda v\cong \bigsqcup_{v\in \Lambda^0} {\mathbb{N}}^k$. $\eqref{it 3}\Rightarrow \eqref{it 4}$ Assume that $\Lambda\cong\bigsqcup_{v\in \Lambda^0} {\mathbb{N}}^k$. By Lemma \[lem: dir sum\], $\operatorname{KP}_R(\Lambda)$ is isomorphic to $\bigoplus \operatorname{KP}_R({\mathbb{N}}^k)$, and by Example \[ex: Bk\] each $\operatorname{KP}_R({\mathbb{N}}^k)$ is isomorphic to $R[x_1,x_1{^{-1}},\ldots, x_k, x_k{^{-1}}]$. $\eqref{it 4}\Rightarrow \eqref{it 1}$ Follows since $\bigoplus R[x_1,x_1{^{-1}},\ldots, x_k, x_k{^{-1}}]$ is commutative. [1]{} G. Abrams and G. Aranda Pino, *The Leavitt path algebra of a graph*, J. Algebra **293** (2005), 319–334. P. Ara, M.A. Moreno and E. Pardo, *Nonstable $K$-theory for graph algebras*, Algebr. Represent. Theory **10** (2007), 157–178. G. Aranda Pino, John Clark, A. an Huef and I. Raeburn, *Kumjian-[P]{}ask algebras of higher-rank graphs*, to appear in Trans. Amer. Math. Soc. G. Aranda Pino and K. Crow, *The center of a [L]{}eavitt path algebra*, Rev. Mat. Iberoam. **27** (2011), 621–644. J.H. Brown and A. an Huef, *The socle and semisimplicity of a Kumjian-Pask algebra*, arXiv:1201.1888. R. Hazlewood, I. Raeburn, A. Sims and S.B.G. Webster, *On some fundamental results about higher-rank graphs and their ${C}^*$-algebras*, to appear in Proc. Edinburgh. Math. Soc. A. Kumjian and D. Pask, *Higher rank graph [$C^\ast$]{}-algebras*, New York J. Math. **6** (2000), 1–20. I. Raeburn and D.P. Williams, Morita equivalence and continuous-trace [$C\sp *$]{}-algebras, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. D.I. Robertson and A. Sims, *Simplicity of [$C^\ast$]{}-algebras associated to higher-rank graphs*, Bull. Lond. Math. Soc. **39** (2007), 337–344. M. Tomforde, *Leavitt path algebras with coefficients in a commutative ring*, J. Pure Appl. Algebra **215** (2011), 471–484.

--- abstract: 'Design of Space-Time Block Codes (STBCs) for Maximum Likelihood (ML) reception has been predominantly the main focus of researchers. However, the ML decoding complexity of STBCs becomes prohibitive large as the number of transmit and receive antennas increase. Hence it is natural to resort to a suboptimal reception technique like linear Minimum Mean Squared Error (MMSE) receiver. Barbarossa *et al* and Liu *et al* have independently derived necessary and sufficient conditions for a full rate linear STBC to be MMSE optimal, i.e achieve least Symbol Error Rate (SER). Motivated by this problem, certain existing high rate STBC constructions from crossed product algebras are identified to be MMSE optimal. Also, it is shown that a certain class of codes from cyclic division algebras which are special cases of crossed product algebras are MMSE optimal. Hence, these STBCs achieve least SER when MMSE reception is employed and are fully diverse when ML reception is employed.' author: - 'G. Susinder Rajan and B. Sundar Rajan [^1]' title: 'MMSE Optimal Algebraic Space-Time Codes' --- Crossed product algebra, division algebra, space-time codes, MMSE receiver Introduction ============ Space-Time coding is known to be an efficient coding technique to combat fading and/or exploit the increased capacity gains offered by Multiple Input Multiple Output (MIMO) systems. But the ML decoding complexity of STBCs becomes prohibitively large as the number of transmit and receive antennas increase. The sphere decoder helps to some extent in reducing the complexity but is still far away from practicality for large number of transmit antennas. In [@TJC; @Jaf; @KhR], orthogonal designs, single and double symbol ML decodable STBCs have been proposed to solve this problem. But unfortunately, the rate of such codes decay with increase in the number of transmit antennas and they are information lossy for more than one receive antenna. This led to the study of suboptimal reception strategies such as linear MMSE (Minimum Mean Square Error) and linear ZF (Zero Forcing) receivers [@LZW1]-[@ZLW2]. It is then natural to address the question of how to design STBCs which are optimal for a linear MMSE receiver. This problem was addressed in [@LZW1]-[@BaF]. \[sec1\] \[defn\_mmse\_optimal\] A $n \times n$ linear STBC $S$ in $k$ complex variables $x_1,\dots,x_k$ given by $S = \sum_{i=1}^{k} x_{i}A_{i}$ is called a unitary trace-orthogonal STBC if the set of $n\times n$ matrices $A_i,i=1,\dots,k$ satisfy the following conditions $$\begin{aligned} \label{unitary} A_iA_i^H&=&\frac{n}{k}I_n\\ \label{torthogonal} Tr(A_i^HA_j)&=&0, \forall\ i\neq j\end{aligned}$$ If $k=n^2$, it will be referred to as full rate transmission. It was shown in [@LZW1]-[@BaF] that if full rate transmission is considered, unitary trace-orthogonality is a necessary and sufficient condition for a linear STBC to achieve minimum bit error rate when the variables $x_1,\dots,x_k$ take values from a QPSK (Quadrature Phase Shift Keying) constellation. Further, it was shown that full rate unitary trace orthogonal STBCs achieve MMSE when other two-dimensional constellations are used. Also, it was shown that at high SNR, the predominant metric that decides probability of symbol error is optimized only by unitary trace orthogonal STBCs. Henceforth, we thus refer to full rate unitary trace orthogonal STBCs as MMSE optimal STBCs. Few constructions of such codes are given in [@LZW2]-[@Bar]. However, these constructions were based on matrix manipulations and lacked an algebraic theory behind them. The contributions of this paper are as follows. - Provide sufficient conditions as to when STBCs obtained from left regular representation of crossed product algebras are MMSE optimal. Using these sufficient conditions, a new class of MMSE optimal STBCs is constructed for arbitrary number of transmit antennas. Since the code constructions are algebraic, the description of the code becomes elegant and it also simplifies the study of their properties. - By restricting to a certain class of cyclic division algebras [@SRS1], STBCs which are simultaneously MMSE optimal as well as fully diverse for ML reception are identified. Not all division algebra based codes [@SRS1]-[@EKPKL] are MMSE optimal. In particular, it is shown that the famous Golden code [@ORBV] is not MMSE optimal. Few of the existing code constructions [@LZW2; @Jing_thesis; @Bar] are also shown to be special cases of certain codes from cyclic algebras [@SRS2; @SRS1]. Organization of the paper ------------------------- In Section \[sec2\], a description of our main algebraic tool, i.e., crossed product algebras is provided and an explicit construction of STBCs from crossed product algebras is given. In Section \[sec3\], we identify sufficient conditions as to when STBCs from crossed product algebras are MMSE optimal. Then, we focus on a proper subclass of crossed product algebras called cyclic algebras and it is shown that a certain class among them are MMSE optimal as well. Few illustrative examples of code constructions are provided and the decoding procedure for these codes is briefly discussed. Simulation results comprise Section \[sec4\] and discussions on future work constitute Section \[sec5\]. STBCs from Crossed Product Algebras {#sec2} =================================== In this section, we briefly review the construction of STBCs from crossed product algebras as given in [@SRS2]. We refer the readers to [@SRS2] for a detailed explanation of crossed product algebras. Let $F$ be a field. Then, an associative $F$-algebra $A$ is called a central simple algebra if the center of $A$ is $F$ and $A$ is a simple algebra, i.e., $A$ does not have nontrivial two-sided ideals. Simple examples of central simple algebras are division algebras and matrix algebras over fields. It is well known that the dimension $[A:F]$ of $A$ over its center is always a perfect square, say $n^2$ [@SRS2; @Herstein]. The square root of $[A:F]$ is called the degree of $A$. Let $K$ be a strictly maximal subfield of $A$, i.e., $K\subset A$ and $K$ is not contained in any other subfield of $A$ and the centralizer of $K$ in $A$ is $K$ itself. It is well known that $[K:F]=n$, the degree of the algebra. In addition, let the extension $K/F$ be a Galois extension and let $G=\left\{\sigma_0=1,\sigma_1,\sigma_2,\ldots \sigma_{n-1}\right\}$ be the Galois group of $K/F$. Let $\phi$ be a map from $G \times G$ to $K$\\$\left\{0\right\}$ called the cocycle which satisfies the cocycle condition as shown below: $$\phi(\sigma,\tau\gamma)\phi(\tau,\gamma)=\phi(\sigma\tau,\gamma)\gamma(\phi(\sigma,\tau)),~\forall \sigma,\tau,\gamma \in G.$$ Then, the algebra $A$ is called a Crossed Product Algebra if $$A=\bigoplus_{\sigma_i \in G}u_{\sigma_i}K$$ where, equality and addition are component-wise and where $u_{\sigma}$ are symbols such that i) $\sigma(k)=u_{\sigma}^{-1}ku_{\sigma}$ and ii) $u_{\sigma}u_{\tau}=u_{\sigma\tau}\phi(\sigma,\tau)$ for all $k \in K,\sigma,\tau \in G$. It is clear that $A$ can be seen as a right $K$-space of dimension $n$ over $K$. Also multiplication between two elements of $A$, say $a=\sum_{i=0}^{n-1}u_{\sigma_i}k_{\sigma_i}$ and $a'=\sum_{j=0}^{n-1}u_{\sigma_j}k_{\sigma_j}^{'}$ is given by $$\left(\sum_{i=0}^{n-1}u_{\sigma_i}k_{\sigma_i}\right)\left(\sum_{j=0}^{n-1}u_{\sigma_j}k_{\sigma_j}^{'}\right)= \sum_{l=0}^{n-1}u_{\sigma_l}k_{\sigma_l}^{''}$$ where, $k_{\sigma_l}^{''}=\sum_{\sigma_i\sigma_j=\sigma_l}\phi(\sigma_i,\sigma_j)\sigma_j(k_{\sigma_i})k_{\sigma_j}^{'}$ We will denote this crossed product algebra $A$ by $\left(K,G,\phi\right)$. The field $K$ can be seen as an $n$-dimensional $F$-vector space. Let $B= \left\{ t_0,t_1,\dots t_{n-1} \right\}$ be a basis of $K$ over $F$. Then, the left regular representation [@SRS2] of $A$ in $End_K(A)$[^2] is given by the map $L:A \mapsto End_K(A)$ which is defined as follows $$L\left(a\right)=\lambda_a\ \mathrm{where,}\ \lambda_a\left(u\right)=au,\forall u \in A.$$ The matrix representation $M_a$ of the linear transformation $\lambda_a$ with respect to the basis $\left\{u_{\sigma_i}:\sigma_i\in G\right\}$ is given by $\eqref{Ma}$ where, $f_{\sigma_j}^{(i)}\in F, \forall\ 0\leq i,j\leq n-1$, $\mu_{i,j}=\sigma_i\sigma_j^{-1}$, $\beta_i^{(j)}=\phi(\sigma_i\sigma_j^{-1},\sigma_j)$ and $\alpha$ is a scaling factor to normalize the average total power of a codeword to $n^2$. [$$\label{Ma} M_a=\frac{1}{\sqrt{\alpha}} \left[ \begin{array}{ccccc} \sum_{i=0}^{n-1}f_{\sigma_{0}}^{(i)}t_i & \beta_0^{(1)}\sum_{i=0}^{n-1}f_{\mu_{0,1}}^{(i)}\sigma_1(t_i) & \beta_0^{(2)}\sum_{i=0}^{n-1}f_{\mu_{0,2}}^{(i)}\sigma_2(t_i) & \cdots & \beta_0^{(n-1)}\sum_{i=0}^{n-1}f_{\mu_{0,n-1}}^{(i)}\sigma_{n-1}(t_i)\\ \sum_{i=0}^{n-1}f_{\sigma_{1}}^{(i)}t_i & \beta_1^{(1)}\sum_{i=0}^{n-1}f_{\mu_{1,1}}^{(i)}\sigma_1(t_i) & \beta_1^{(2)}\sum_{i=0}^{n-1}f_{\mu_{1,2}}^{(i)}\sigma_2(t_i) & \cdots & \beta_1^{(n-1)}\sum_{i=0}^{n-1}f_{\mu_{1,n-1}}^{(i)}\sigma_{n-1}(t_i)\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \sum_{i=0}^{n-1}f_{\sigma_{n-1}}^{(i)}t_i & \beta_{n-1}^{(1)}\sum_{i=0}^{n-1}f_{\mu_{0,1}}^{(i)}\sigma_1(t_i) & \beta_{n-1}^{(2)}\sum_{i=0}^{n-1}f_{\mu_{0,2}}^{(i)}\sigma_2(t_i) & \cdots & \beta_{n-1}^{(n-1)}\sum_{i=0}^{n-1}f_{\mu_{0,n-1}}^{(i)}\sigma_{n-1}(t_i)\\ \end{array} \right]$$ ]{} Thus we have obtained a full rate linear STBC $M_a$ in variables $f_{\sigma_j}^{(i)}, 0\leq i,j\leq n-1$ from the crossed product algebra $A$. $M_a$ can expressed in a linear dispersion form where, the matrices $W_{i,j}$ are called the ’weight matrices’ of $M_a$. Then, we have $$\label{Wij} W_{i,j}=\frac{1}{\sqrt{\alpha}}P_jQ_i,\ \mathrm{where}\ Q_i=\left[\begin{array}{cccc} t_i & 0 & \cdots & 0\\ 0 & \sigma_1(t_i) & \ddots & \vdots\\ \vdots & \ddots & \ddots & 0\\ 0 & \cdots & 0 & \sigma_{n-1}(t_i) \end{array} \right]$$ and the matrix $P_j$ can be described as follows. Let us index the rows and columns of $P_j$ with the elements of $G$. Then the $(\sigma_k,\sigma_l)$-th entry of $P_j$ is equal to $\phi(\sigma_j,\sigma_l)$ if $\sigma_j\sigma_l=\sigma_k$ and $0$ otherwise. The matrices $P_j$ and $Q_i$ are nothing but the images of $u_{\sigma_j}$ and $t_i$ respectively under the map $L$. Note that the $P_j$ matrices are known as permutation matrices and are commonly used for group representation. MMSE Optimal STBCs {#sec3} ================== In this Section, we identify sufficient conditions as to when STBCs from crossed product algebras are MMSE optimal. Then, we focus on a proper subclass of crossed product algebras called cyclic algebras and obtain a class of STBCs meeting the required conditions for MMSE optimality. Finally, the decoding procedure for the codes in this paper is discussed and its simplicity as compared to ML decoding is highlighted. \[thm\_mmse\] The STBC $M_a$ constructed as shown in using the crossed product algebra $A=\left(K,G,\phi\right)$ is MMSE optimal if $$\begin{aligned} \label{cond1} \vert\sigma_j(t_i)\vert=\vert t_i\vert&=&\vert\phi(\sigma_i,\sigma_j)\vert=1,\forall\ 0\leq i,j \leq n-1\\ \label{cond2} \mathrm{and}\ \sum_{i=0}^{n-1}\sigma_j(t_i)(\sigma_{j'}(t_i))^{*}&=&0,\ \mathrm{if}\ j\neq j'. \end{aligned}$$ We need to show that the weight matrices of $M_a$ satisfy and . Equation implies that the matrices $P_j$ and $Q_i$ are scaled unitary matrices. The scaling factor $\alpha$ here equals $n$. Therefore $W_{i,j}W_{i,j}^H=\frac{I_n}{n}$ which implies is satisfied. It can be shown [@ZLW1] that the condition in is equivalent to the condition that the matrix $\Phi$ as shown in satisfies $\Phi\Phi^H=nI_n^2$. $$\label{phi} \Phi=\left[\begin{array}{ccccccc}vec(W_{0,0}) & vec(W_{1,0}) & \dots & vec(W_{n-1,0}) & vec(W_{0,n-1}) & \dots & vec(W_{n-1,n-1})\end{array}\right]$$ The $(k,l)$th element of $\Phi\Phi^H$ is given by $\sum_{a=0}^{n-1}\phi(\sigma_i\sigma_j^{-1},\sigma_j)\sigma_j(t_a)\left(\phi\left(\sigma_{i'}\sigma_{j'}^{-1},\sigma_{j'}\right)\sigma_{j'}(t_a)\right)^*$, which simplifies to $\phi(\sigma_i\sigma_j^{-1},\sigma_j)\phi\left(\sigma_{i'}\sigma_{j'}^{-1},\sigma_{j'}\right)\sum_{a=0}^{n-1}\sigma_{j}(t_a)(\sigma_{j'}(t_a))^*$ which is equal to zero from the statement of the theorem. If $k=l$, then we have $(\Phi\Phi^H)_{k,k}=\sum_{a=0}^{n-1}\vert\sigma_j(t_a)\vert^2=n$. Thus, $\Phi\Phi^H=nI_n^2$ which in turn implies is satisfied. Theorem \[thm\_mmse\] gives conditions on the basis of a Galois extension and on the cocycle which result in MMSE optimal STBCs. STBCs from Cyclic Algebras -------------------------- In this subsection, using Theorem \[thm\_mmse\], we identify an existing STBC construction [@SRS2; @SRS1] based on cyclic algebras to be MMSE optimal. An $F$-central simple algebra is called a cyclic algebra, if $A$ has a strictly maximal subfield $K$ which is a cyclic extension of the center $F$. Clearly, a cyclic algebra is a crossed product algebra. Let $\sigma$ be a generator of the Galois group $G$. If $u_{\sigma^i},i=0,1,\dots,n-1$ is a basis for the algebra $A$ over $K$, then we have $$\begin{array}{rcl} u_{\sigma^i}&=&u_{\sigma}^i\\ \mathrm{and}\ \phi(\sigma^i,\sigma^j)&=&\left\{\begin{array}{l}1,\quad \mathrm{if}\ i+j<n\\ \delta,\quad \mathrm{if}\ i+j\geq n\end{array}\right. \end{array}$$ where, $u_{\sigma}^n=\delta$. Since the cocycle can now be described by just one element $\delta$ and similarly $G$ can be described by $\sigma$, we denote the crossed product algebra $(K,G,\phi)$ with $(K,\sigma,\delta)$. Thus, with $z=u_{\sigma}$, we have $A=(K,\sigma,\delta)=\bigoplus_{i=0}^{n-1}z^iK$ where, $z^n=\delta$ and $kz=z\sigma(k),\forall k\in K$. Note that if the smallest positive integer $t$ such that $\delta^t$ is the norm of some element in $K\backslash\left\{0\right\}$ is $n$, then the cyclic algebra $A=(K,\sigma,\delta)$ is a cyclic division algebra [@SRS1]. \[cons\_CDA\_trans\] Let $K/F$ be a cyclic extension of degree $n$ with $K=F(t_{n}=t^{1/n})$, $t,\omega_n\in F$, $\vert t\vert=1$. Here $\omega_n$ denotes the $n$th root of unity and $\sigma:t_n\mapsto\omega_nt_n$ is the generator of the Galois group. Let $\delta$ be a transcendental element over $K$. From Theorem \[thm\_mmse\], the STBC arising from the cyclic division algebra $(K(\delta)/F(\delta),\sigma,\delta)$ is MMSE optimal since it satisfies the following identities $$\begin{array}{c} \vert t\vert=\vert \delta\vert=\vert\sigma^i(t_n)\vert=1,\ i=0,1,\dots,n-1\\ \mathrm{and}\ \sum_{i=0}^{n-1}(t_n)^i(\sigma^k(t_n^i))^*=0,\ \mathrm{if}\ k\neq 0. \end{array}$$ The MMSE optimal STBC $M_a$ is given by $M_a=\sum_{j=0}^{n-1}\sum_{i=0}^{n-1}f_{j}^{(i)}W_{i,j},~f_{j}^{(i)}\in F$ where, the weight matrix $W_{i,j}=t_n^iP^jQ^i$. The matrices $P$ and $Q$ are as shown below: $$P=\left( \begin{array}{ccccc} 0 & \dots & \dots & 0 & \delta\\ 1 & 0 & \dots & 0 & 0\\ 0& 1 & \ddots & \vdots & \vdots\\ \vdots & \ddots & \ddots & 0 & \vdots\\ 0 & \dots & 0 & 1 & 0 \end{array} \right),\ Q=\left( \begin{array}{ccccc} 1 & 0 & \dots & 0 & 0\\ 0 & \omega_n & \ddots & 0 & \vdots\\ \vdots & \ddots & \omega_n^2 & \ddots & \vdots\\ \vdots & 0 & \ddots & \ddots & 0\\ 0 & \dots & \ldots & 0 & \omega_n^{n-1} \end{array} \right).$$ We would like to emphasize here that the codes in [@LZW2; @Jing_thesis; @Bar] can be obtained as a special case of the above construction by simply choosing $\delta=1$. If $\delta=1$ then the algebra $A$ will be a cyclic algebra but is not guaranteed to be a division algebra. Also, we would like to point out that there are cyclic division algebra based STBC constructions in the literature [@ORBV; @KiR; @EKPKL] which opt to carefully choose the element $\delta$ to be from $F^*$ (rather than transcendental as in Construction \[cons\_CDA\_trans\]) for other benefits such as achieving the diversity-multiplexing gain tradeoff. Some of those codes are now known as perfect STBCs [@ORBV]. It is important to note that not all cyclic division algebra based codes satisfy and . In fact there exist perfect STBCs which are not MMSE optimal. A concrete example of such a code is the best known $2$ transmit antenna STBC for ML reception, i.e., the famous Golden code. This is illustrated in the following example. The codewords of the Golden code are given by $\frac{1}{\sqrt{5}}\left[\begin{array}{cc} \alpha(a+b\theta) & \alpha(c+d\theta)\\ i\bar{\alpha}(c+d\bar{\theta}) & \bar{\alpha}(a+b\bar{\theta}) \end{array}\right]$ where, $a,b,c,d\in\mathbb{Z}[j]$, $\theta=\frac{1+\sqrt{5}}{2}$, $\bar{\theta}=\frac{1-\sqrt{5}}{2}$, $\alpha=1+j(1-\theta)$, $\bar{\alpha}=1+j(1-\bar{\theta})$. The weight matrices of the Golden code are given as follows: $$\frac{1}{\sqrt{5}}\left[\begin{array}{cc}\alpha & 0\\0 & \bar{\alpha}\end{array}\right], \frac{1}{\sqrt{5}}\left[\begin{array}{cc}\alpha\theta & 0\\0 & \bar{\alpha}\bar{\theta} \end{array}\right], \frac{1}{\sqrt{5}}\left[\begin{array}{cc}0 & \alpha\\ j\bar{\alpha} & 0\end{array}\right], \frac{1}{\sqrt{5}}\left[\begin{array}{cc}0 & \alpha\theta\\ j\bar{\alpha}\bar{\theta} & 0\end{array}\right].$$ Clearly, the weight matrices of the Golden code are not scaled unitary which is a necessary condition for MMSE optimality (see of Definition \[defn\_mmse\_optimal\]). This is because the crossed product algebra associated with the Golden code fails to satisfy . Hence the Golden code is not MMSE optimal. \[eg\_sim\] This example illustrates our construction procedure for $n=2$. Let $F=\mathbb{Q}(j,t)$, where $t$ is transcendental over $\mathbb{Q}(j)$. Then $K=F(t_2=\sqrt{t})$ is a cyclic extension of $F$ of degree $2$. The generator of the Galois group is given by $\sigma:t_2\mapsto-t_2$. Let $\delta$ be any transcendental element over $K$. Then $(K(\delta)/F(\delta),\sigma,\delta)$ is a cyclic division algebra. For example, we can choose $t=e^{j}$ and $\delta=e^{j\sqrt{5}}$. Then, we have $M_a=\frac{1}{\sqrt{2}}\left[\begin{array}{lr}f_0^{(0)}+f_0^{(1)}t_2 & \delta(f_1^{(0)}-f_1^{(1)}t_2)\\f_1^{(0)}+f_1^{(1)}t_2 & f_0^{(0)}-f_0^{(1)}t_2 \end{array}\right]$. This is an example of a MMSE optimal code which is not obtainable from a cyclic division algebra. Let $n=4$ and $F=\mathbb{Q}(j,x,y)$ where $x$ and $y$ are two transcendental numbers independent over $\mathbb{Q}(j)$. We choose these transcendental numbers to lie on the unit circle. Then $K=F(\sqrt{x},\sqrt{y})$ is a Galois extension of $F$ with the Galois group $G=\langle\sigma_x,\sigma_y\rangle$, where $\sigma_x:\sqrt{x}\mapsto-\sqrt{x}$ and $\sigma_y:\sqrt{y}\mapsto-\sqrt{y}$. The cocyle $\phi$ is defined as follows: $$\begin{array}{c} \phi(\sigma_x,\sigma_x)=\phi(\sigma_x\sigma_y,\sigma_x)=\delta_1,~\phi(\sigma_y,\sigma_y)=\phi(\sigma_x\sigma_y,\sigma_y)=\delta_2,\\ \phi(\sigma_x,\sigma_y)=1~\mathrm{and}~\phi(\sigma_x\sigma_y,\sigma_x\sigma_y)=\delta_1\delta_2. \end{array}$$ Then, the algebra $(K(\delta_1,\delta_2),G,\phi)=K(\delta_1,\delta_2)\oplus u_{\sigma_x}K(\delta_2,\delta_2)\oplus\ u_{\sigma_y}K(\delta_1,\delta_2)\oplus u_{\sigma_x}u_{\sigma_y}K(\delta_1,\delta_2)$ is a crossed product algebra where, $\delta_1,\delta_2$ are independent transcendental numbers over $K$. Also, we choose to pick $\delta_1$ and $\delta_2$ to lie on the unit circle. The matrix representation of this crossed product algebra will give rise to an MMSE optimal STBC and has codewords of the form $\frac{1}{\sqrt{\alpha}}\left[\begin{array}{lccr}k_{0,0} & \delta_2\sigma_y(k_{0,1}) & \delta_1\sigma_x(k_{1,0}) & \delta_1\delta_2\sigma_x\sigma_y(k_{1,1})\\ k_{0,1} & \sigma_y(k_{0,0}) & \delta_1\sigma_x(k_{1,1}) & \delta_1\sigma_x\sigma_y(k_{1,0})\\ k_{1,0} & \delta_2\sigma_y(k_{1,1}) & \sigma_x(k_{0,0}) & \delta_2\sigma_x\sigma_y(k_{0,1})\\ k_{1,1} & \sigma_y(k_{1,0}) & \sigma_x(k_{0,1}) & \sigma_x\sigma_y(k_{0,0}) \end{array}\right]$ where each $k_{i,j},0\leq i,j\leq 1$ is given by $k_{i,j}=f_{i,j}^{(0)}+f_{i,j}^{(1)}\sqrt{x}+f_{i,j}^{(2)}\sqrt{y}+f_{i,j}^{(3)}\sqrt{xy}$ and $f_{i,j}^{(l)}\in\mathbb{Q}(j)\subset F$. Decoding procedure ------------------ In this subsection, the decoding procedure for the codes in this paper is briefly explained and its receiver simplicity compared to ML reception is highlighted. Let the encoded matrix $X=\sum_{j=0}^{n-1}\sum_{i=0}^{n-1}f_{j}^{(i)}W_{i,j}$. Let the number of receive antennas be $m$. We assume that $m\geq n$ in the sequel otherwise there will be an error floor [@FaB] when linear MMSE reception is employed. The received matrix $Y$ can be expressed as $Y=HX+N$, where $H$ is the channel matrix of size $m\times n$ and $N$ is the $m\times n$ matrix representing the additive noise at the receiver whose entries are i.i.d. $\mathcal{CN}(0,1)$. Then, the linear MMSE receiver can be implemented in its simplest form as a *symbol-by-symbol decoder* [@FaB], as described below: $$\label{eqn_mmse} \hat{f}_{j}^{(i)}=tr(W_{i,j}^HJY)$$ with $J=(H^HH+\frac{1}{\rho}I_n)^{-1}H^H$ where, $\rho$ is the Signal to Noise ratio (SNR) or equivalently in this case it is the average energy of the complex constellation used. Computation of $\hat{f}_{j}^{(i)}$ is then followed by hard decision, i.e., it is decoded to the nearest point (in the sense of Euclidean distance) in the constellation. Note that the decoding complexity is *linear* in the size of the signal set which is far less compared to the complexity of sphere decoding. Simulation Results {#sec4} ================== In this section, we compare the error performance of the proposed codes with that of the previously known MMSE optimal STBC in [@LZW2] under linear MMSE reception. In [@FaB], it has been shown that a diversity order of $m-n+1$ is achieved by MMSE optimal STBCs when a linear MMSE or linear Zero forcing receiver is employed. On the other hand, it is well known that under ML decoding a diversity order of $mn$ is possible if the STBC is fully diverse. The codes constructed in this letter have this property as well. Fig. \[fig\_mmse\] shows the bit error rate performance of the MMSE optimal STBC given in Example \[eg\_sim\] with QPSK constellation and the previously known MMSE optimal STBC in [@LZW2] under linear MMSE decoding with the number of receive antennas being equal to $4$. For linear MMSE decoding, the symbol-by-symbol decoder in was utilized. Observe from Fig. \[fig\_mmse\] that the performance of the proposed code is almost same as that of the previously known MMSE optimal STBC in [@LZW2]. It is important to note that the error probability under linear MMSE reception as shown in Fig. \[fig\_mmse\] is the optimal [@LZW1]-[@BaF] among all STBCs with full rate transmission. Discussion {#sec5} ========== The algebraic framework of crossed product algebras is quite general in nature. For instance, MMSE optimal STBCs can also be constructed from tensor products of division algebras and Brauer division algebras. We refer the readers to [@SRS2] for more details on these constructions. It will also be interesting to study the design of optimal STBCs for linear ZF receivers. Some initial work in this direction has been reported in [@ZLW2]. Acknowledgements {#acknowledgements .unnumbered} ================ The authors thank Kiran and Shashidhar of Beceem Communications for useful discussions on the subject of this matter. The authors thank Jing Liu and Prof. Sergio Barbarossa for providing us with preprints of their recent works [@LZW2; @Jing_thesis; @ZLW2; @FaB]. The authors are grateful to the anonymous reviewers for providing constructive comments and useful remarks on perfect space-time codes which helped in improving the presentation of this letter. [10]{} V. Tarokh, H. Jafarkhani and A. R. Calderbank, “Space-Time block codes from Orthogonal designs,” *IEEE Trans. Inform. Theory*, vol. 45, pp. 1456-1467, July 1999. H. Jafarkhani, “ Quasi-Orthogonal Space-Time Block Code,” *IEEE Transactions on Communications*, vol. 49, no. 1, January 2001. Md. Zafar Ali Khan and B. Sundar Rajan, “Single-Symbol Maximum-Likelihood Decodable Linear STBCs,” *IEEE Trans. Inform. Theory*, vol. 52, no. 5, pp.2062-2091, May 2006. Jing Liu, Jian-Kang Zhang and Kon Max Wong, “Design of Optimal Orthogonal Linear Codes in MIMO Systems for MMSE Receiver,” Proceedings of *IEEE International Conference on Acoustics, Speech and Signal Processing*, vol. 4, pp. 725-728, Montreal, Canada, May 17-21, 2004. Jing Liu, Jian-Kang Zhang and Kon Max Wong, “On the Design of Minimum BER Linear Space-Time Block Codes for MIMO Systems Equipped with MMSE Receivers,” *IEEE Trans. Signal Processing*, vol. 54, no. 8, pp. 3147-3158, August 2006. J.K.Zhang, J.Liu, K.M.Wong, “Trace-orthogonal full diversity cyclotomic space-time codes,” Chapter 5 in Space-Time Processing for MIMO Communications, John Wiley Sons, 2005. Jing Liu, “Linear STBC Design for MIMO Systems with MMSE Receivers,” Master’s Thesis, McMaster University. Sergio Barbarossa, Antonio Fasano, “Trace-orthogonal Space-Time Coding for Multiuser Systems,” Proceedings of *IEEE International Conference on Acoustics, Speech and Signal Processing*, vol. 3, pp. 1093-1096, Philadelphia, USA, March 18-23, 2005. J.-K. Zhang, J. Liu, and K. M. Wong, “Linear Toeplitz space-time block codes,” Proceedings of *ISIT 2005*, pp. 1942-1946, Adelaide, September 4-9, 2005. Antonio Fasano, Sergio Barbarossa, “Iterative MMSE decoder for Trace-Orthogonal Space-Time Coding,” Proceedings of *IEEE International Conference on Acoustics, Speech and Signal Processing*, vol. 4, pp. 181-184, Toulouse, France, May 14-19, 2006. Sergio Barbarossa, “Trace-orthogonal design of MIMO systems with simple scalar detectors, full diversity and (almost) full rate," Proceedings of *2004 IEEE 5th Workshop on Signal Processing Advances in Wireless Communications*, Lisbon, Portugal, July 11-14, 2004. V. Shashidhar, B.Sundar Rajan, B.A. Sethuraman, “Information-Lossless Space-Time Block Codes from Crossed-Product Algebras,” *IEEE Trans. Inform. Theory*, vol. 52, no. 9, pp. 3913-3935, September 2006. B. A. Sethuraman, B. S. Rajan, and V. Shashidhar, “Full-diversity, High-rate Space-Time Block Codes from Division Algebras,” *IEEE Trans. Inform. Theory*, vol. 49, no. 10, pp. 2596-2616, October 2003. F. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo, “Perfect Space Time Block Codes,” *IEEE Transactions on Information Theory*, vol. 52, no. 9, September 2006. T. Kiran and B. Sundar Rajan, “STBC-schemes with non-vanishing determinant for certain number of transmit antennas,” *IEEE Transactions on Information Theory*, Vol.51, No.8, pp.2984-2992, August 2005. Petros Elia, K.Raj Kumar, Sameer.A Pawar, P.Vijay Kumar, Hsiao-feng Lu, “Explicit, Minimum-Delay Space-Time Codes Achieving the Diversity Multiplexing Gain Tradeoff,” *IEEE Trans. Inform Theory*, vol. 52, no. 9, pp. 3869-3884, September 2006. I. N. Herstein, *Non-Commutative Rings*, Washington, DC: Math. assoc. Amer., 1968, Carus Mathematical Monographs. ![Error performance comparison of the proposed MMSE optimal STBC with that of [@LZW2] in a $2\times 4$ MIMO system with a linear MMSE receiver[]{data-label="fig_mmse"}](simulation) [^1]: This work was supported through grants to B.S. Rajan; partly by the DRDO-IISc program on Advanced Research in Mathematical Engineering, and partly by the Council of Scientific & Industrial Research (CSIR, India) Research Grant (22(0365)/04/EMR-II). Part of the material in this letter has been published in the Proceedings of Thirteenth National Conference on Communications (NCC 2007) held at IIT Kanpur, January 27-29, 2007. G. Susinder Rajan and B. Sundar Rajan are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560012, India. Email:{susinder,bsrajan}@ece.iisc.ernet.in. [^2]: $End_K(A)$ denotes the set of all $K$ linear maps from $A$ to $A$.

--- bibliography: - 'MTRS.bib' title: | Mixed topology ring states for Hall effect and orbital magnetism\ in skyrmions of Weyl semimetals --- Acknowledgements ================ We gratefully acknowledge computing time on the supercomputers JUQUEEN and JURECA at Jülich Supercomputing Center, and at the JARA-HPC cluster of RWTH Aachen. We acknowledge funding under SPP 2137 “Skyrmionics" of Deutsche Forschungsgemeinschaft (DFG), DARPA TEE program through grant MIPR\# HR0011831554 from DOI. This work has been also supported by the DFG through the Collaborative Research Center SFB 1238.

--- abstract: 'In this paper we define a cohomology for an arbitrary $K$-linear semistrict semigroupal 2-category $(\text\Cg,\otimes)$ (called for short a Gray semigroup) and show that its first order (unitary) deformations, up to the suitable notion of equivalence, are in bijection with the elements of the second cohomology group. Fundamental to the construction is a double complex, similar to the Gerstenhaber-Schack double complex for bialgebras, the role of the multiplication and the comultiplication being now played by the composition and the tensor product of 1-morphisms. We also identify the cohomologies describing separately the deformations of the tensor product, the associator and the pentagonator. To obtain the above results, a cohomology theory for an arbitrary $K$-linear (unitary) pseudofunctor is introduced describing its purely pseudofunctorial deformations, and generalizing Yetter’s cohomology for semigroupal functors [@dY98]. The corresponding higher order obstructions will be considered in detail in a future paper.' address: 'Dept. Matemàtica Aplicada II, Universitat Politècnica de Catalunya' author: - Josep Elgueta bibliography: - 'cdtm2c1.bib' date: March 2002 title: 'Cohomology and deformation theory of monoidal 2-categories I' --- [^1] Introduction ============ This is the first of two papers where we intend to give a cohomological description of the infinitesimal deformations of a monoidal 2-category. Roughly speaking, a monoidal 2-category is a 2-category equipped with a binary operation, usually called the [*tensor product*]{}, defined at the three levels existing in any 2-category, i.e., objects, 1-morphisms and 2-morphisms, and which is associative and with a unit up to suitable 2-isomorphisms. Actually, in this paper we will consider the more general structure of a semigroupal 2-category , namely, a 2-category with a tensor product as above but which is only associative (up to a suitable 2-isomorphism) with no unit. More explicitly, we show that the first order (unitary) deformations of such an object  can be identified with the elements of some cohomology group associated to . The generalization to the case of monoidal categories and the question of the obstructions will be treated in a future paper. This work is an extension to the context of 2-categories of the theory developped by Crane and Yetter for semigroupal categories [@CY981] and by Yetter for braided monoidal categories [@dY98],[@dY01], which are in turn an extension to the context of (1-)categories of Gerstenhaber’s work on deformations of algebras [@mG63],[@mG64], later generalized to the case of Hopf algebras by Gerstenhaber and Schack [@GS90] (see also [@GS88],[@GS92]). These classical works should be viewed as the corresponding theories in the so-called [*0-dimensional algebra*]{} setting [@BD95], which is the algebra in the context of sets. The situation can be schematically represented as in the table below. This table is the $K$-linear version of the first two rows in the table of [*k-tuply monoidal n-categories*]{} of Baez and Dolan; see [@BD95], Table I. The $n$ here denotes the “dimensionality” of the algebraic framework we work with. So, dimension $n$ corresponds to work in the context of [*n-categories*]{}, a natural generalization of the notion of 2-category where we also have 3-morphisms between the 2-morphisms, and so on, until $n$-morphisms between $(n-1)$-morphisms. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ n=0 n=1 n=2 ----- --------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------------------ k=0 $\begin{array}{c} K\text{-vector} \\ \text{space} $\begin{array}{c} K\text{-linear} \\ \text{category} \end{array}$ $\begin{array}{c} K\text{-linear} \\ \text{2-category} \end{array}$ \end{array}$ k=1 $\begin{array}{c} K\text{-algebra} \\ $\begin{array}{c} $\begin{array}{c} K\text{-linear} \\ \text{monoidal 2-category} \end{array}$ {\mathbf (Gerstenhaber's\ work)} \end{array}$ K\text{-linear} \\ \text{monoidal category} \\ {\mathbf (Crane\text{-}Yetter's\ work)} \end{array}$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Notice that going from the top to the bottom row along the diagonal corresponds to taking the one object case (for example, a $K$-linear category of only one object is exactly the same thing as a $K$-algebra). For a more expanded explanation of this table, see the reference above. This idea of generalizing Gerstenhaber’s work to 1- and 2-categories comes from the important notion of [*categorification*]{}, which in the table corresponds to moving to the right. It first appears in the work by Crane and Frenkel [@CF94] on Hopf categories, although it seems it was already present in Grothendieck’s thought. A Hopf category is an analog of a Hopf algebra where the underlying $K$-vector space has been substituted by a special kind of $K$-linear category, usually called a 2-vector space over $K$. For a more precise definition, the reader is refered to Neuchl’s thesis [@mN97]. The basic idea behind the notion of categorification is that constructions from usual algebra can be translated to the level of categories and to higher levels, the [*n-categories*]{}, for $n\geq 1$, making one essential step: to substitute equations for isomorphisms. The price to pay is that it is necessary to simultaneously impose equations on these isomorphisms, which are the so-called [*coherence relations*]{}. These ideas have been developed by different authors, in particular, Crane and Yetter [@CY982] and in the more general language of n-categories, Baez and Dolan (see, for ex., [@BD98]). Apart from its own interest, our motivation for studying deformations of monoidal 2-categories has to be found in its potential applications to the construction of interesting four dimensional Topological Quantum Field Theories (briefly, TQFT’s). Indeed, in [@CF94], Crane and Frenkel suggest that Hopf categories may be used to construct four dimensional TQFT’s, in a similar way as three dimensional TQFT’s can be constructed from Hopf algebras (see, for ex., [@gK91],[@CFS94]). Now, it is well-known that three dimensional TQFT’s can also be obtained using the monoidal categories of representations of suitable Hopf algebras (see, for ex., [@TV92], [@dY94],[@BW96]). This clearly suggests the possibility that four dimensional TQFT’s could be obtained from the category of representations of suitable Hopf categories, which will be, going up in the categorification process, some kind of monoidal 2-categories (actually, Neuchl [@mN97] has proved that the 2-category of representations of a Hopf category is indeed monoidal). That idea has been made explicit by Mackaay [@mM99], who develops a method to construct invariants of piecewise linear four manifolds from a special kind of monoidal 2-categories he calls [*spherical 2-categories*]{}. His construction parallels that of Barrett and Westbury [@BW96] for three manifolds. This explains the interest of monoidal 2-categories in the construction of four dimensional TQFT’s. But, why are we interested in their deformations? The answer is again an expected analogy between the cases of dimension three and four. In dimension three, we can get a state sum invariant of a piecewise linear three-manifold using irreducible representations of an arbitrary semisimple Lie algebra. The method comes from the classical work of Ponzano-Regge [@PR68]. The problem is that the sum turns out to be infinite. Progress was made possible only when the corresponding quantum group was discovered, which is a deformation (as a [*braided bialgebra*]{}; see [@cK95]) of the classical universal envelopping algebra of the Lie algebra. Using the representations of the quantum group at a root of unity instead of those of the classical version, the state sum invariant becomes a convergent sum. That’s what Turaev and Viro do in their paper [@TV92]. The reader can also find more details, for example, in the book by Carter, Flath and Saito [@CFS95]. The hope is that a similar situation reproduces in dimension four. So, instead of having a Lie algebra or, equivalently, its universal envelopping algebra, which has a natural structure of a (trivially braided) Hopf algebra, we should now have a Hopf category, and instead of having the monoidal category of representations of the Lie algebra, we should have the monoidal 2-category of representations of the Hopf category. Via the reconstruction theorems of the Tannaka-Krein type, the deformations of the universal envelopping algebra of the Lie algebra correspond to deformations of its category of representations. Similarly, deformations of the Hopf category should correspond to deformations of its 2-category of representations. Therefore, we are indeed led in this way to consider the theory of deformations for monoidal 2-categories. In the above mentioned paper [@CF94], Crane and Frenkel already outline a method for constructing interesting Hopf categories out of the quantum groups and their canonical bases. A difficult point is to find the analog of the quantum groups in this new framework, which could be called [*2-quantum groups*]{} and which would correspond to non trivial deformations of these Hopf categories. Let’s say a few words about what it means to give a cohomological description, in the sense of Gerstenhaber, of the theory of deformations of a semigroupal 2-category. Although we will be thinking of this case, the situation is similar in all of the above mentioned settings. Given an arbitrary 2-category , it will be possible to define more than one semigroupal structure on it. So, we can think of a space X() whose points are in 1-1 correspondence with all such possible semigroupal structures on , up to a suitable notion of equivalence. The ultimate goal should be to have a description of such a [*moduli space*]{} X() in terms, for example, of a suitable parametrization of its points. However, this is difficult. The idea is then to focus the attention on one particular point in that space and to study the corresponding “tangent space”. That’s why we speak of [*infinitesimal*]{} deformations of the (reference) semigroupal 2-category. Clearly, the first point is how to formalize that idea of a tangent space, because a priori we have no differentiable manifold structure on X(). In the sequel, we will see how to do that. We will need to assume some $K$-linear structure on the 2-category, for some commutative unitary ring $K$, and to have some local $K$-algebra extending $K$, and over which the deformations will take place. In the classical algebra setting, this is accomplished by considering, instead of the original $K$-algebra $A$, its $K[[h]]$-linear extension $A[[h]]$ (see [@mG63]). According to Gerstenhaber’s foundational work, to give a cohomological description of such infinitesimal deformations amounts then to find a suitable cohomology $H^{\bullet}(\text\Cg)$ such that the so-called first order deformations (with respect to a formal deformation parameter) are classified, up to equivalence, by the elements of one of the cohomology groups $H^n(\text\Cg)$, for some $n$. But this is only the first point. According to Gerstenhaber, a nice cohomological description is required to further satisfy the property that the obstructions to extending such a first order deformation to higher order deformations or even to formal series deformations also live in some of the groups $H^m(\text\Cg)$. In the 0-dimensional setting of algebras, it turns out that the corresponding obstructions are described by a graded Lie algebra structure on the cochain complex governing the deformations [@mG64], and, after Gerstenhaber, this should be a basic principle of any obstruction theory. As mentioned before, however, in this paper we will not consider the question of higher order obstructions, whose treatment is defered to a future paper. Therefore, the goal of the present work is to just develop the first of the above points, i.e., to identify the first order deformations of a semigroupal 2-category with the cocycles of a suitable cohomology theory. An important point is how the infinitesimal deformations of a semigroupal 2-category are defined. In the classical algebra setting [@mG64],[@GS88], recall that the deformation consists of taking a new (deformed) product $\mu_h$ of the form $$\mu_h(a,a')=\mu(a,a')+\mu_1(a,a')h+\mu_2(a,a')h^2+\cdots$$ where $\mu:A\times A{\longrightarrow}A$ denotes the original (undeformed) product and the $\mu_i:A\times A{\longrightarrow}A$, $i\geq 1$, are suitable $K$-bilinear maps such that $\mu_h$ is indeed associative and with unit. In the category setting, this should correspond to considering a new (deformed) tensor product $\otimes_h$ between morphisms of the form $$f\otimes_h g=f\otimes g+(f\otimes_1 g)h+(f\otimes_2 g)h^2+\cdots$$ where $\otimes=\otimes_{(X,Y),(X',Y')}:{\mathcal{C}}\times{\mathcal{C}}((X,Y),(X',Y')){\longrightarrow}{\mathcal{C}}(X\otimes Y,X',\otimes Y')$ corresponds to the original tensor product and the $\otimes_i=(\otimes_i)_{(X,Y),(X',Y')}:{\mathcal{C}}\times{\mathcal{C}}((X,Y),(X',Y')){\longrightarrow}{\mathcal{C}}(X\otimes Y,X',\otimes Y')$, $i\geq 1$, are suitable $K$-bilinear functors. In the category setting, however, we should further consider possible deformations of the structural isomorphisms taking account of the associativity and unit character of the deformed tensor product, i.e., we should consider, for example, a new (deformed) associator $a_h$ of the form $$(a_h)_{X,Y,Z}=a_{X,Y,Z}+a^{(1)}_{X,Y,Z}h+a^{(2)}_{X,Y,Z}h^2+\cdots$$ for suitable morphisms $a^{(i)}_{X,Y,Z}:X\otimes(Y\otimes Z){\longrightarrow}(X\otimes Y)\otimes Z$. Now, in the definition of an infinitesimal deformation of a monoidal category ${\mathcal{C}}$, as given by Crane and Yetter [@CY981], the only structure susceptible of being deformed is that defined by these structural isomorphisms [^2]. In other words, the tensor product $\otimes:{\mathcal{C}}\times{\mathcal{C}}{\longrightarrow}{\mathcal{C}}$ is assumed to remain the same (except for a trivial linear extension). Apart from the fact that this clearly simplifies the theory, there is another reason that may induce to adopt this point of view. Indeed, in his milestone paper [@vD91], Drinfeld proved that the category of representations of the quantum group $U_h(\text\gg)$ associated to a simple Lie algebra , which corresponds to a certain deformation in the above generic sense of the category of representations of $U(\text\gg)$, is in fact equivalent to the category of representations of $U(\text\gg)[[h]]$ but with a non trivially deformed associator (see also [@cK95]). Hence, at least in this case, it is enough to consider those deformations where only the isomorphisms included in the monoidal structure are deformed, keeping the tensor product undeformed. In defining the infinitesimal deformations of a semigroupal 2-category we will adopt the same point of view as Crane and Yetter. So, an infinitesimal deformation of a semigroupal 2-category will be defined in such a way that the only things susceptible of deformation are the [*2-isomorphisms*]{} defining the semigroupal structure on the 2-category. Contrary to the case of monoidal categories, however, this involves many things. So, among the 2-isomorphisms susceptible of deformation, we can distinguish three groups: (1) the 2-isomorphisms included in the tensor product, coming from the weakening of the definition of the tensor product as a bifunctor, (2) the 2-isomorphisms included in the associator, and coming from the weakening of the naturality of the maps $a_{X,Y,Z}$, and (3) the 2-isomorphisms included in the so called pentagonator, coming from the weakening of the pentagon axiom on the associator. In a generic infinitesimal deformation, all of them will be deformed. The outline of the paper is as follows. In Section 2 we recall the basic definitions from bicategory theory, together with the corresponding strictification theorem (MacLane-Pare’s theorem). In Section 3 we generalize to (unitary) pseudofunctors Epstein’s coherence theorem for semigroupal functors [@dE66] and introduce the analog of Crane-Yetter’s “padding” composition operators [@CY981] in this setting. They are essential in the development of the theory. Section 4 is devoted to reviewing in detail the definition of a semigroupal 2-category, giving a formulation adapted to our purposes, and we also give an explicit definition of the corresponding notion of morphism, deduced from the notion of morphisms between tricategories as it appears in the paper by Gordon, Power and Street [@GPS95]. In Section 5, we give the precise definition of deformation of a semigroupal 2-category we will work with, together with the notion of equivalence of deformations. For later use, we also define in this section the notion of purely pseudofunctorial infinitesimal deformation of a pseudofunctor. In Section 6, and using results from Section 3, we develop a cohomology theory for the purely pseudofunctorial infinitesimal deformations of a pseudofunctor, which partially generalizes Yetter’s theory for monoidal functors [@dY98]. The cohomological description of the deformations of a semigroupal 2-category is then initiated in Section 7, where we consider the particular case of the [*pentagonator-deformations*]{}, i.e., those deformations where only the pentagonator is deformed, all the other structural 2-isomorphisms remaining undeformed. The next section is devoted to determine a cohomological description of the infinitesimal deformations involving both the tensor product and the associator. We do that in the special case where the deformations are [*unitary*]{}, i.e., such that the structural 2-isomorphisms $\otimes_0(X,Y)$ remain undeformed. They will be called [*unitary (tensorator,associator)-deformations*]{}. We also identify cohomologies describing the deformations separately of both structures. For the sake of simplicity, in this section we restrict ourselves to the case of a Gray semigroup. Finally, in Section 9 we show how the cohomologies in Sections 7 and 8 fit together to give a cohomology which describes the generic (unitary) deformations. Basic concepts from bicategory theory ===================================== Recall that a bicategory, also called lax or weak 2-category, and first defined by Bénabou [@jB67], can be obtained from a category after doing the following two steps: (1) enrich the sets of morphisms with the category of small categories in the sense of Kelly [@gK82], and (2) weaken the associativity and unit axioms on the composition by substituting 2-isomorphisms for the equations, with the consequent introduction of coherence relations on these 2-isomorphisms, as explained in the introduction. When we do that, we obtain the following definition. \[bicategory\] A bicategory  consists of: - A class $|\text\Cg|$ of objects or 0-cells. - For any ordered pair of objects $X,Y\in|\text\Cg|$, a small [*category*]{} $(X,Y)$. [The]{} objects of $(X,Y)$, denoted by $f:X\longrightarrow Y$, are called 1-morphisms or 1-cells, and its morphisms, denoted by $\tau:f\Longrightarrow f'$, are called 2-morphisms or 2-cells. Remark that, included in this data $(X,Y)$, there is a distinguished identity 2-morphism $1_f:f\Longrightarrow f$ for any 1-morphism $f:X\longrightarrow Y$, and an [*associative*]{} composition between 2-morphisms, called vertical composition. Given 1-morphisms $f,f',f'':X\longrightarrow Y$, the vertical composite of two 2-morphisms $\tau:f\Longrightarrow f'$ and $\tau':f'\Longrightarrow f''$ will be denoted by $\tau'\cdot\tau$. - For any ordered triple of objects $X,Y,Z\in|\text\Cg|$, a [*functor*]{} $$c_{X,Y,Z}:\text\Cg(X,Y)\times \text\Cg(Y,Z)\longrightarrow \text\Cg(X,Z)$$ [These]{} functors provide us not only the composition $c_{X,Y,Z}(f,g):=g\circ f$ of two 1-morphisms $f:X\longrightarrow Y$ and $g:Y\longrightarrow Z$, but also a second composition between 2-morphisms, called the horizontal composition, which involves three objects. If $f,f':X\longrightarrow Y$ and $g,g':Y\longrightarrow Z$, the horizontal composition $c_{X,Y,Z}(\tau,\eta)$ between the two 2-morphisms $\tau:f\Longrightarrow f'$ and $\eta:g\Longrightarrow g'$ will be denoted by $\eta\circ\tau$. In a general bicategory, this composition may be non associative. - For any object $X\in|\text\Cg|$, a distinguished 1-morphism $id_X\in|\text\Cg(X,X)|$. - For any objects $X,Y,Z,T\in|\text\Cg|$ and any composable 1-morphisms $f:X\longrightarrow Y$, $g:Y\longrightarrow Z$, $h:Z\longrightarrow T$, a 2-isomorphism $\alpha_{h,g,f}:h\circ (g\circ f)\Longrightarrow (h\circ g)\circ f$, called the associator or associativity constraint on $f,g,h$. - For any 1-morphism $f:X\longrightarrow Y$, two 2-isomorphisms $\lambda_f:id_Y\circ f\Longrightarrow f$ and $\rho_f:f\circ id_X\Longrightarrow f$, called the left and right unit constraints on $f$, respectively. Moreover, these data must satisfy the following axioms: 1. The $\alpha_{h,g,f}$ are natural in $f,g,h$ and the $\lambda_f$ and $\rho_f$ natural in $f$. 2. The associator $\alpha=\{\alpha_{h,g,f}\}$ is such that the following diagram commutes: $$\xymatrix{ k\circ(h\circ(g\circ f))\ar[d]_{\alpha_{k,h,g\circ f}} \ar[rr]^{1_k\circ \alpha_{h,g,f}} & & k\circ((h\circ g)\circ f)\ar[d]^{\alpha_{k,h\circ g,f}} \\ (k\circ h)\circ (g\circ f)\ar[dr]_{\alpha_{k\circ h,g,f}} & & (k\circ(h\circ g))\circ f \ar[dl]^{\alpha_{k,h,g}\circ 1_f} \\ & ((k\circ h)\circ g)\circ f }$$ 3. The left and right unit constraints $\lambda=\{\lambda_f\}$ and $\rho=\{\rho_f\}$ make commutative the following diagram: $$\xymatrix{ (g\circ id_Y)\circ f\ar[rr]^{\alpha_{g,id_Y,f}} \ar[dr]_{\rho_g\circ 1_f} & & g\circ(id_Y\circ f)\ar[dl]^{1_g\circ \lambda_f} \\ & g\circ f }$$ When all the associators $\alpha_{h,g,f}$ and left and right unit constraints $\lambda_f$, $\rho_f$ are identities, which in particular means that the composition of 1-morphisms is strictly associative and the identity 1-morphisms are strict units, we will speak of a 2-category. The reader should check that a bicategory  with only one object corresponds exactly to the notion of a monoidal category. If $X$ is the only object of , the monoidal category is $\text\Cg(X,X)$ with the composition functor as tensor product. This fact will be used repeatedly in what follows. We also leave to the reader to check that in a 2-category horizontal composition is strictly associative and that the identity 2-morphisms of the identity 1-morphisms act as strict units with respect to horizontal composition. Both facts will also be frequently used. Given two bicategories  and , their cartesian product is defined as the bicategory $\text\Bg\times\text\Cg$ such that $$\begin{aligned} & |\text\Bg\times\text\Cg|:=|\text\Bg|\times|\text\Cg| \\ &(\text\Bg\times\text\Cg)((X,Y),(X',Y')):=\text\Bg(X,X')\times\text\Cg(Y,Y') \\ &c_{(X,Y),(X',Y'),X'',Y'')}^{\text\Bg\times\text\Cg}:=(c_{X,X',X''}^{\text\Bg}\times c_{Y,Y',Y''}^{\text\Cg})\circ P_{23}\end{aligned}$$ with identity 1-morphisms $id_{(X,Y)}=(id_X,id_Y)$ and whose structural 2-isomorpisms $\alpha_{(f'',g''),(f',g'),(f,g)}$, $\lambda_{(f,g)}$ and $\rho_{(f,g)}$ are componentwise given by those of  and  ($P_{23}$ denotes the functor which permutes factors 2 and 3). The same construction obviously extends to a finite number of bicategories. We will mainly work with 2-categories. This means no loss of generality because of the following strictification theorem for bicategories, due to MacLane and Pare [@MP85] (see also [@GPS95], $\S 1.3$): \[MacLane-Pare\] Any bicategory is biequivalent (in the sense defined below) to a 2-category. Diagramatically, a 2-category differs from a category in that it has vertices (the objects) and edges (the 1-morphisms) but also faces between pairs of edges. In other words, while a category can be represented as a 1-dimensional cellular complex, a 2-category is a 2-dimensional cellular complex. As a consequence, when working with 2-categories, a generic diagram will be a three dimensional one, with a new “pasting” game where both vertical and horizontal compositions are combined. We will find some examples in the sequel. Another significant difference is that in 2-categories (and in bicategories in general), we have 1-isomorphisms (i.e., invertible 1-morphisms), but also [*equivalences*]{}, i.e., 1-morphisms which are invertible only up to a 2-isomorphism. This leads to the notion of equivalent objects in a 2-category, which is weaker than the notion of isomorphic objects. We will also need the corresponding notion of morphism between bicategories. There are in the literature various versions and names for this notion. Following Gray [@jG74] I will call them pseudofunctors, although our definition differs slightly from that of Gray. \[pseudofunctor\] If  and  are two bicategories, a [*pseudofunctor*]{} from  to  is any quadruple ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*,{\mathcal{F}}_0)$, where - $|{\mathcal{F}}|:|\text\Cg|\longrightarrow|\text\Dg|$ is an object map (the image $|{\mathcal{F}}|(X)$ of $X\in|\text\Cg|$ will be denoted by ${\mathcal{F}}(X)$); - ${\mathcal{F}}_*=\{{\mathcal{F}}_{X,Y}:\text\Cg(X,Y)\longrightarrow\text\Dg({\mathcal{F}}(X),{\mathcal{F}}(Y))\}$ is a collection of functors, indexed by ordered pairs of objects $X,Y\in|\text\Cg|$; - $\widehat{{\mathcal{F}}}_*=\{\widehat{{\mathcal{F}}}_{X,Y,Z}: c^{\text\Dg}_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)}\circ({\mathcal{F}}_{X,Y}\times{\mathcal{F}}_{Y,Z}) \Longrightarrow{\mathcal{F}}_{X,Z}\circ c^{\text\Cg}_{X,Y,Z}\}$ is a family of natural isomorphisms, indexed by triples of objects $X,Y,Z\in|\text\Cg|$. Explicitly, this corresponds to having, for all composable 1-morphisms $X\stackrel{f}{\longrightarrow} Y\stackrel{g}{\longrightarrow} Z$, a 2-isomorphism $$\widehat{{\mathcal{F}}}_{X,Y,Z}(g,f):{\mathcal{F}}_{Y,Z}(g)\circ{\mathcal{F}}_{X,Y}(f)\Longrightarrow{\mathcal{F}}_{X,Z}(g\circ f)$$ natural in $(f,g)$, and - ${\mathcal{F}}_0=\{{\mathcal{F}}_0(X):{\mathcal{F}}_{X,X}(id_X)\Longrightarrow id_{{\mathcal{F}}(X)}\}$ is a collection of 2-isomorphisms, indexed by objects $X\in|\text\Cg|$. [Moreover]{}, this data must satisfy the following conditions: 1. ([*hexagonal axiom*]{}) for all composable 1-morphisms $X\stackrel{f}{\longrightarrow} Y\stackrel{g}{\longrightarrow} Z\stackrel{h}{\longrightarrow} T$, it commutes $$\xymatrix{ {\mathcal{F}}(h)\circ({\mathcal{F}}(g)\circ{\mathcal{F}}(f))\ar[r]^{1_{{\mathcal{F}}(h)}\circ\widehat{{\mathcal{F}}}(g,f)} \ar[d]_{\alpha_{{\mathcal{F}}(h),{\mathcal{F}}(g),{\mathcal{F}}(f)}} & {\mathcal{F}}(h)\circ{\mathcal{F}}(g\circ f)\ar[r]^{\widehat{{\mathcal{F}}}(h,g\circ f)} & {\mathcal{F}}(h\circ (g\circ f))\ar[d]^{{\mathcal{F}}(\alpha_{h,g,f})} \\ ({\mathcal{F}}(h)\circ{\mathcal{F}}(g))\circ{\mathcal{F}}(f)\ar[r]_{\widehat{{\mathcal{F}}}(h,g)\circ 1_{{\mathcal{F}}(f)}} & {\mathcal{F}}(h\circ g)\circ{\mathcal{F}}(f)\ar[r]_{\widehat{{\mathcal{F}}}(h\circ g,f)} & {\mathcal{F}}((h\circ g)\circ f) }$$ 2. ([*triangular axioms*]{}) for any 1-morphism $f:X\longrightarrow Y$, the following diagrams commute: $$\xymatrix{ {\mathcal{F}}(f)\circ id_{{\mathcal{F}}(X)}\ar[dr]_{\rho_{{\mathcal{F}}(f)}} & {\mathcal{F}}(f)\circ{\mathcal{F}}(id_X) \ar[l]_{1_{{\mathcal{F}}(f)}\circ {\mathcal{F}}_0(X)} \ar[r]^{\widehat{{\mathcal{F}}}(f,id_X)} & {\mathcal{F}}(f\circ id_X)\ar[ld]^{{\mathcal{F}}(\rho_f)} \\ & {\mathcal{F}}(f) & \\ id_{{\mathcal{F}}(Y)}\circ{\mathcal{F}}(f)\ar[dr]_{\lambda_{{\mathcal{F}}(f)}} & {\mathcal{F}}(id_Y)\circ{\mathcal{F}}(f)\ar[l]_{{\mathcal{F}}_0(Y)\circ 1_{{\mathcal{F}}(f)}} \ar[r]^{\widehat{{\mathcal{F}}}(id_Y,f)} & {\mathcal{F}}(id_Y\circ f)\ar[ld]^{{\mathcal{F}}(\lambda_f)} \\ & {\mathcal{F}}(f) & }$$ (here, and from now on, we just write $\widehat{{\mathcal{F}}}(g,f)$ and ${\mathcal{F}}(f)$, the indexing objects being omitted for short). The $\widehat{{\mathcal{F}}}(g,f)$ and ${\mathcal{F}}_0(X)$, for all objects $X$ and composable 1-morphisms $f,g$, will be called the [*structural 2-isomorphisms of*]{} ${\mathcal{F}}$, and the whole set will be called the [*pseudofunctorial structure on*]{} ${\mathcal{F}}$. When they are all identities, which in particular means that the functors ${\mathcal{F}}_{X,Y}$ preserve the composition of 1-morphisms and the identity 1-morphisms, the pseudofunctor will be called a [*2-functor*]{}. When only the ${\mathcal{F}}_0(X)$ are identities, we will call it a [*unitary pseudofunctor*]{}. If no confusion arises, a pseudofunctor ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*,{\mathcal{F}}_0)$ from  to  will be denoted by ${\mathcal{F}}:\text\Cg\longrightarrow\text\Dg$ or simply by ${\mathcal{F}}$. The only difference between this definition and the one by Gray in [@jG74] is our assumption that all structural 2-morphisms $\widehat{{\mathcal{F}}}(g,f)$ and ${\mathcal{F}}_0(X)$ are actually 2-isomorphisms. From the above definition, it follows immediately that a pseudofunctor between one object bicategories amounts to a monoidal functor between the corresponding monoidal categories, as the reader should check. Given two pseudofunctors ${\mathcal{F}}:\text\Bg{\longrightarrow}\text\Cg$ and ${\mathcal{G}}:\text\Cg{\longrightarrow}\text\Dg$, the composite pseudofunctor ${\mathcal{G}}\circ{\mathcal{F}}$ is defined by $$\begin{aligned} &|{\mathcal{G}}\circ{\mathcal{F}}|=|{\mathcal{G}}|\circ|{\mathcal{F}}| \\ &({\mathcal{G}}\circ{\mathcal{F}})_{X,Y}={\mathcal{G}}_{{\mathcal{F}}(X),{\mathcal{F}}(Y)}\circ{\mathcal{F}}_{X,Y} \\ &(\widehat{{\mathcal{G}}\circ{\mathcal{F}}})(g,f)={\mathcal{G}}(\widehat{{\mathcal{F}}}(g,f))\cdot\widehat{{\mathcal{G}}}({\mathcal{F}}(g),{\mathcal{F}}(f)) \\ &({\mathcal{G}}\circ{\mathcal{F}})_0(X)={\mathcal{G}}_0({\mathcal{F}}(X))\cdot{\mathcal{G}}({\mathcal{F}}_0(X))\end{aligned}$$ The direct product of pseudofunctors can also be defined, whose source and target bicategories are the corresponding product bicategories. We leave to the reader to write out the explicit definition. Finally, let’s recall that a pseudofunctor ${\mathcal{F}}:\text\Cg{\longrightarrow}\text\Dg$ is called a [*biequivalence*]{} if for any object $Y\in|\text\Dg|$, there exists an object $X\in|\text\Cg|$ whose image ${\mathcal{F}}(X)$ is equivalent to $Y$, and for any pair of objects $X,X'\in|\text\Cg|$ the functor ${\mathcal{F}}_{X,X'}$ is an equivalence of categories. As in the case of categories, there is a notion of morphism between pseudofunctors, which I will call pseudonatural transformations. \[trans\_quasi\] Let  and  be two bicategories, and ${\mathcal{F}},{\mathcal{G}}:\text\Cg\longrightarrow\text\Dg$ two pseudofunctors. Then a [*pseudonatural transformation from*]{} ${\mathcal{F}}$ [*to*]{} ${\mathcal{G}}$ is any pair $\xi=(\xi_*,\widehat{\xi}_*)$, where - $\xi_*=\{\xi_X:{\mathcal{F}}(X)\longrightarrow{\mathcal{G}}(X)\}$ is a collection of 1-morphisms, indexed by objects $X\in|\text\Cg|$; - $\widehat{\xi}_*=\{\widehat{\xi}_{X,Y}:c^{\text\Dg}_{{\mathcal{F}}(X),{\mathcal{G}}(X),{\mathcal{G}}(Y)}(\xi_X,-)\circ {\mathcal{G}}_{X,Y}\Longrightarrow c^{\text\Dg}_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{G}}(Y)}(-,\xi_Y)\circ{\mathcal{F}}_{X,Y}\}$ is a family of natural isomorphisms, indexed by pairs of objects $X,Y\in|\text\Cg|$. Explicitly, this means having for any 1-morphism $f:X\longrightarrow Y$ a 2-isomorphism $\widehat{\xi}_{X,Y}(f):{\mathcal{G}}_{X,Y}(f)\circ\xi_X\Longrightarrow\xi_Y\circ{\mathcal{F}}_{X,Y}(f)$, natural in $f$. [Moreover]{}, this data must satisfy the conditions 1. for all composable 1-morphisms $X\stackrel{f}{\longrightarrow} Y\stackrel{g}{\longrightarrow} Z$, the following diagram commutes $$\xymatrix{ {\mathcal{G}}(g)\circ({\mathcal{G}}(f)\circ\xi_X)\ar[d]_{\alpha_{{\mathcal{G}}(g),{\mathcal{G}}(f),\xi_X}} \ar[r]^{1_{{\mathcal{G}}(g)}\circ\widehat{\xi}(f)} & {\mathcal{G}}(g)\circ(\xi_Y\circ{\mathcal{F}}(f)) \ar[r]^{\alpha_{{\mathcal{G}}(g),\xi_Y,{\mathcal{F}}(f)}} & ({\mathcal{G}}(g)\circ\xi_Y)\circ{\mathcal{F}}(f) \ar[r]^{\widehat{\xi}(g)\circ 1_{{\mathcal{F}}(f)}} & (\xi_Z\circ{\mathcal{F}}(g))\circ{\mathcal{F}}(f) \ar[d]^{\alpha^{-1}_{\xi_Z,{\mathcal{F}}(g),{\mathcal{F}}(f)}} \\ ({\mathcal{G}}(g)\circ{\mathcal{G}}(f))\circ\xi_X\ar[d]_{\widehat{{\mathcal{G}}}(g,f)\circ 1_{\xi_X}} & & & \xi_Z\circ({\mathcal{F}}(g)\circ{\mathcal{F}}(f))\ar[d]^{1_{\xi_Z}\circ\widehat{{\mathcal{F}}}(g,f)} \\ {\mathcal{G}}(g\circ f)\circ\xi_X\ar[rrr]^{\widehat{\xi}(g\circ f)} & & & \xi_Z\circ{\mathcal{F}}(g\circ f) }$$ 2. for all objects $X$, the following diagram commutes $$\xymatrix{ \xi_X\circ id_{{\mathcal{F}}(X)}\ar[r]^{\rho_{\xi_X}} & \xi_X \ar[r]^{\lambda^{-1}_{\xi_X}} & id_{{\mathcal{G}}(X)}\circ\xi_X \\ \xi_X\circ{\mathcal{F}}(id_X)\ar[u]^{1_{\xi_X}\circ{\mathcal{F}}_0(X)} & & {\mathcal{G}}(id_X)\circ\xi_X\ar[u]_{{\mathcal{G}}_0(X)\circ 1_{\xi_X}}\ar[ll]^{\widehat{\xi}(id_X)} }$$ (for short again, here and from now on, we will omit the indexing objects in $\widehat{\xi}_{X,Y}(f)$). When all the 2-isomorphisms $\widehat{\xi}(f)$ are identities, $\xi$ will be called a [*2-natural transformation*]{}. On the other hand, if all the $\xi_X$ are 1-isomorphisms (resp. equivalences), we will speak of a [*pseudonatural isomorphism*]{} (resp. [*pseudonatural equivalence*]{}). If no confusion arises, a pseudonatural transformation $\xi=(\xi_*,\widehat{\xi}_*)$ between two pseudofunctors ${\mathcal{F}}$ and ${\mathcal{G}}$ will be denoted by $\xi:{\mathcal{F}}\Longrightarrow{\mathcal{G}}$ or simply by $\xi$. As in the case of pseudofunctors, there is also a more general notion of pseudonatural transformation, where the 2-cells $\widehat{\xi}(f)$ are not assumed to be invertible. The reader should be familiar with the formulas for the vertical and horizontal compositions of pseudonatural transformations. Explicitly, given pseudofunctors ${\mathcal{F}},{\mathcal{G}},{\mathcal{H}}:\text\Cg\longrightarrow\text\Dg$, recall that the vertical composite of $\xi:{\mathcal{F}}\Longrightarrow{\mathcal{G}}$ and $\zeta:{\mathcal{G}}\Longrightarrow{\mathcal{H}}$, denoted $\zeta\cdot\xi$, is defined by $$\begin{aligned} &(\zeta\cdot\xi)_X=\zeta_X\circ\xi_X \\ &\widehat{\zeta\cdot\xi}(f)=\alpha^{\text\Dg}_{\zeta_Y,\xi_Y,{\mathcal{F}}(f)} \cdot(1_{\zeta_Y}\circ\widehat{\xi}(f))\cdot (\alpha^{\text\Dg})^{-1}_{\zeta_Y,{\mathcal{G}}(f),\xi_X}\cdot (\widehat{\zeta}(f)\circ 1_{\xi_X})\cdot\alpha^{\text\Dg}_{{\mathcal{H}}(f),\zeta_X,\xi_X}\end{aligned}$$ On the other hand, given pseudofunctors ${\mathcal{F}},{\mathcal{F}}':\text\Bg{\longrightarrow}\text\Cg$ and ${\mathcal{G}}:\text\Cg{\longrightarrow}\text\Dg$, and a pseudonatural transformation $\xi:{\mathcal{F}}\Longrightarrow{\mathcal{F}}'$, the horizontal composition $1_{id_{{\mathcal{G}}}}\circ\xi$ is defined by $$\begin{aligned} &(1_{id_{{\mathcal{G}}}}\circ\xi)_X={\mathcal{G}}_{{\mathcal{F}}(X),{\mathcal{F}}'(X)}(\xi_X) \\ &\widehat{1_{id_{{\mathcal{G}}}}\circ\xi}(f)=\widehat{{\mathcal{G}}}^{-1}(\xi_Y,{\mathcal{F}}(f))\cdot {\mathcal{G}}(\widehat{\xi}(f))\cdot\widehat{{\mathcal{G}}}({\mathcal{F}}'(f),\xi_X)\end{aligned}$$ Finally, the horizontal composition $\zeta\circ 1_{id_{{\mathcal{F}}}}$, where $\zeta:{\mathcal{G}}\Longrightarrow{\mathcal{G}}':\text\Cg{\longrightarrow}\text\Dg$ is any pseudonatural transformation and ${\mathcal{F}}:\text\Bg{\longrightarrow}\text\Cg$ any pseudofunctor, is given by $$\begin{aligned} &(\zeta\circ 1_{id_{{\mathcal{F}}}})_X=\zeta_{{\mathcal{F}}(X)} \\ &\widehat{\zeta\circ 1_{id_{{\mathcal{F}}}}}(f)=\widehat{\zeta}({\mathcal{F}}(f))\end{aligned}$$ Let’s finish this section by recalling that, in the context of bicategories, there is still a notion of morphism between two pseudonatural transformations, usually called a modification, and which has no analog in the category setting. \[modification\] Let  and  be two bicategories, ${\mathcal{F}},{\mathcal{G}}:\text\Cg\longrightarrow\text\Dg$ two pseudofunctors and $\xi,\zeta:{\mathcal{F}}\Longrightarrow{\mathcal{G}}$ two pseudonatural transformations. Then, a [*modification from*]{} $\xi$ [*to*]{} $\zeta$ is any family of 2-morphisms $\text\nn=\{\text\nn_X:\xi_X\Longrightarrow\zeta_X\}$, indexed by the objects of , such that for any 1-morphism $f:X\longrightarrow Y$ in , it holds $$\widehat{\zeta}(f)\cdot(1_{{\mathcal{G}}(f)}\circ\text\nn_X)=(\text\nn_Y\circ 1_{{\mathcal{F}}(f)}) \cdot\widehat{\xi}(f).$$ This condition expresses the fact that the 2-morphisms $\text\nn_X$ are natural in $X$. A modification from $\xi$ to $\zeta$ will be denoted by $\text\nn:\xi\Longrightarrow\zeta$ or simply by  if no confusion arises. A family of 2-morphisms as above which not necessarily satisfy the previous naturality condition will be called a [*pseudomodification from*]{} $\xi$ [*to*]{} $\zeta$. This more general notion will be needed later. Coherence and padding operators for unitary pseudofunctors ========================================================== Before giving the definition of a semigroupal 2-category and the corresponding notion of morphism, we consider in this section a coherence theorem for unitary pseudofunctors which generalizes to the many objects setting Epstein’s coherence theorem for semigroupal functors [@dE66]. Such a coherence result allows us to introduce the analog of Crane-Yetter’s “padding” composition operators [@CY981] in this setting. As the reader will realize later, these results are essential in what follows. So, they are first used in Section 6 to associate a cochain complex to a unitary pseudofunctor describing its purely pseudofunctorial infinitesimal deformations, and which is a key ingredient in the definition of the double complex of a $K$-linear Gray semigroup introduced in Section 8. The coherence result is also needed to prove that this double complex is indeed a double complex. Recall that in [@dE66], for any pair of semigroupal[^3] categories $({\mathcal{B}},\otimes,a)$ and $(\hat{{\mathcal{B}}},\hat{\otimes},\hat{a})$ and a semigroupal functor $(G,\widehat{G})$, the author defines the $G$-[*iterates*]{} (of multiplicity $n$, $n\geq 1$) as the set of all functors ${\mathcal{B}}^n\longrightarrow\hat{{\mathcal{B}}}$ that can be obtained as compositions of product functors $G^i:{\mathcal{B}}^i\longrightarrow\hat{{\mathcal{B}}}^i$ ($i\leq n$) and $j$-iterates ($j\leq n$) of the tensor products $\otimes$ and $\hat{\otimes}$ in ${\mathcal{B}}$ and $\hat{{\mathcal{B}}}$, respectively, which are functors ${\mathcal{B}}^j\longrightarrow{\mathcal{B}}$ or $\hat{{\mathcal{B}}}^j\longrightarrow\hat{{\mathcal{B}}}$. So, a generic $G$-iterate (of multiplicity $n$) will apply the object $(A_1,\ldots,A_n)$ of ${\mathcal{B}}^n$ to an object of $\hat{{\mathcal{B}}}$ of the form $$G(A_1\otimes\cdots\otimes A_{i_1})\hat{\otimes} G(A_{i_1+1}\otimes\cdots\otimes A_{i_2})\hat{\otimes}\cdots\hat{\otimes} G(A_{i_r}\otimes\cdots\otimes A_n),$$ with a suitable parenthesization of the $A_i$’s inside each group and of the $\hat{\otimes}$-factors that we omit because it will depend on the $\otimes$- and $\hat{\otimes}$-iterates used. Let ${\mathcal{C}}at(G,\otimes,\hat{\otimes})$ be the category whose objects are all these $G$-iterates and whose morphisms are all the natural transformations between them. Then, the structural natural isomorphisms $\widehat{G},a,\hat{a}$ define a subcategory ${\mathcal{C}}at(G,\otimes,\hat{\otimes},\widehat{G},a,\hat{a})$ with the same objects as ${\mathcal{C}}at(G,\otimes,\hat{\otimes})$ but whose morphisms are only those induced by these natural isomorphisms $\widehat{G},a,\hat{a}$, which are called the canonical ones. More precisely, a canonical morphism is any morphism obtained as a compositions of expansions of instances of $\widehat{G},a,\hat{a}$ or its inverses, where by an expansion of a morphism $f$ one means any morphism obtained from $f$ by tensorially multiplying it by identity morphisms. For example, a canonical morphism from $G((A\otimes B)\otimes(C\otimes D))$ to $G((A\otimes B)\otimes C)\hat{\otimes}G(D)$ is $$(\widehat{G}(A\otimes B,C)\hat{\otimes} id_{G(D)})\circ \hat{a}_{G(A\otimes B),G(C),G(D)}\circ (id_{G(A\otimes B} \hat{\otimes}\widehat{G}(C,D)^{-1})\circ\widehat{G}(A\otimes B,C\otimes D)^{-1}.$$ A priori, there are other canonical isomorphisms between the same two objects, as the reader should check. Epstein’s coherence theorem states that, when $\widehat{G},a,\hat{a}$ satisfy the appropiate coherence relations, for any two objects of ${\mathcal{C}}at(G,\otimes,\hat{\otimes},\widehat{G},a,\hat{a})$ there is at most one morphism (actually an isomorphism). Let’s consider now the case of a unitary pseudofunctor $(|{\mathcal{G}}|,{\mathcal{G}}_*,\widehat{{\mathcal{G}}}_*)$ between two bicategories , $\hat{\text\Bg}$. To emphasize the similarity between both situations, we present here the “conversion table”: $$\begin{aligned} - & \longleftrightarrow & |\text\Bg|=\{X,Y,Z,\ldots\} \\ - & \longleftrightarrow & |\hat{\text\Bg}|=\{U,V,W,\ldots\} \\ {\mathcal{B}}& \longleftrightarrow & \text\Bg_*=\{\text\Bg(X,Y)\}_{X,Y} \\ \hat{{\mathcal{B}}} & \longleftrightarrow & \hat{\text\Bg}_*=\{\hat{\text\Bg}(U,V)\}_{U,V} \\ \otimes & \longleftrightarrow & c_*=\{c_{X,Y,Z}\}_{X,Y,Z} \\ \hat{\otimes} & \longleftrightarrow & \hat{c}_*=\{\hat{c}_{U,V,W}\}_{U,V,W} \\ a=\{a_{A,B,C}\}_{A,B,C} & \longleftrightarrow & \alpha_*=\{\alpha(X,Y,Z,T)=\{\alpha_{h,g,f}\}_{h,g,f}\}_{X,Y,Z,T} \\ \hat{a}=\{\hat{a}_{D,E,F}\}_{D,E,F} & \longleftrightarrow & \hat{\alpha}_*= \{\hat{\alpha}(U,V,W,S)=\{\hat{\alpha}_{t,s,r}\}_{t,s,r}\}_{U,V,W,S} \\ - & \longleftrightarrow & |{\mathcal{G}}|:|\text\Bg|\longrightarrow|\hat{\text\Bg}| \\ G:{\mathcal{B}}\longrightarrow\hat{{\mathcal{B}}} & \longleftrightarrow & {\mathcal{G}}_*=\{{\mathcal{G}}_{X,Y}:\text\Bg(X,Y)\longrightarrow\hat{\text\Bg}({\mathcal{G}}(X),{\mathcal{G}}(Y))\}_{X,Y} \\ \widehat{G}=\{\widehat{G}(A,B)\}_{A,B} & \longleftrightarrow & \widehat{{\mathcal{G}}}_*= \{\widehat{{\mathcal{G}}}_{X,Y,Z}=\{\widehat{{\mathcal{G}}}(g,f)\}_{g,f}\}_{X,Y,Z}\end{aligned}$$ Looking at this table, we see that to go to the bicategory-pseudofunctor setting simply means substituting any thing in the left-hand side by a family of things of exactly the same type and indexed by objects of the appropriate bicategory. One can now proceed in the same way as Epstein does. The ${\mathcal{G}}$-iterates of multiplicity $n$, $n\geq 2$, will now be functors $\text\Bg(X_1,X_2)\times\text\Bg(X_2,X_3)\times\cdots\times \text\Bg(X_n,X_{n+1})\longrightarrow\hat{\text\Bg}({\mathcal{G}}(X_1),{\mathcal{G}}(X_{n+1}))$, indexed by a collection $X_1,\ldots,X_{n+1}$ of $n+1$ objects of , and obtained as compositions of product functors ${\mathcal{G}}_{X_i,X_{i+1}}\times{\mathcal{G}}_{X_{i+1},X_{i+2}}\times\cdots\times{\mathcal{G}}_{X_j,X_{j+1}}$ with suitable iterates of the composition functors $c_{X,Y,Z}$ and $\hat{c}_{U,V,W}$. So, a generic ${\mathcal{G}}$-iterate will apply the 1-morphisms $(f_n,\ldots,f_1)\in\text\Bg(X_1,X_2)\times\text\Bg(X_2,X_3) \times\cdots\times\text\Bg(X_n,X_{n+1})$ to a 1-morphism in $\hat{\text\Bg}({\mathcal{G}}(X_1),{\mathcal{G}}(X_{n+1}))$ of the form $${\mathcal{G}}(f_1\circ\cdots\circ f_{i_1})\circ{\mathcal{G}}(f_{i_1+1}\circ\cdots \circ f_{i_2})\circ\cdots\circ{\mathcal{G}}(f_{i_{r+1}}\circ\cdots\circ f_{n-1})\ \ \ \ \ \ \ \ (*)$$ with the appropriate parenthesization according to the used composition functors. Similarly, a canonical 2-morphism will be any 2-morphism (actually, a 2-isomorphism) obtained as a vertical composition of expansions of instances of the given $\widehat{{\mathcal{G}}}_{X,Y,Z},\alpha(X,Y,Z),\hat{\alpha}(U,V,W)$ or its inverses, for all $X,Y,Z\in|\text\Bg|$ and all $U,V,W\in|\hat{\text\Bg}|$, where expansion now means the horizontal composition with identity 2-morphisms. We can then consider the analogs of the above categories, namely, ${\mathcal{C}}at(|{\mathcal{G}}|,{\mathcal{G}}_*,c_*,\hat{c}_*)$ and $ {\mathcal{C}}at(|{\mathcal{G}}|,{\mathcal{G}}_*,c_*,\hat{c}_*,\widehat{{\mathcal{G}}}_*,\alpha_*,\hat{\alpha}_*). $ We have then the following generalization of Epstein’s theorem to unitary pseudofunctors: \[coherence\_pseudofunctor\] Let , $\hat{\text\Bg}$ be two bicategories and let $(|{\mathcal{G}}|,{\mathcal{G}}_*,,\widehat{{\mathcal{G}}}_*)$ be a unitary pseudofunctor between them. Then for any pair of objects of ${\mathcal{C}}at(|{\mathcal{G}}|,{\mathcal{G}}_*,c_*,\hat{c}_*,\widehat{{\mathcal{G}}}_*,\alpha_*,\hat{\alpha}_*)$ there is at most one morphism. Formally, the proof is the same as that of Epstein, but ignoring the permutations which appear in his paper because we do not consider commutativity constraints. The main difference is that we work simultaneously with various functors and natural isomorphisms. This coherence theorem already appears in a different formulation in [@GPS95], $\S 1.6$. The previous result allows us to introduce the analog of Crane-Yetter’s “padding” composition operators [@CY981] in the context of a unitary pseudofunctor ${\mathcal{G}}$ between two bicategories  and $\hat{\text\Bg}$. The main difference is that now we have a whole collection of such padding operators, indexed by pairs of objects of the target bicategory $\hat{\text\Bg}$. So, given two such objects $U,V$, the situation is that depicted in Fig. \[figurapadding\]. We have a sequence $\tau_1,\ldots,\tau_n$ of 2-morphisms in $\hat{\text\Bg}$ such that the source 1-morphism of $\tau_{i+1}$ is canonically 2-isomorphic to the target 1-morphism of $\tau_i$ (i.e., they are 2-isomorphic through a composition of expansions of the structural 2-isomorphisms coming from ${\mathcal{G}},\text\Bg,\hat{\text\Bg}$). Then, define $$\lceil\tau_n\cdot\tau_{n-1}\cdot\cdots\cdot\tau_1\rceil_{U,V}:= \beta_n\cdot\tau_n\cdot\beta_{n-1}\cdot\tau_{n-1}\cdot\cdots\cdot \beta_1\cdot\tau_1\cdot\beta_0,$$ where the $\beta_i$’s are the canonical 2-isomorphisms between the target of $\tau_i$ and the source of $\tau_{i+1}$, $\beta_0$ is the canonical 2-isomorphism whose source 1-morphism has no identity composition factors and it is completely right-parenthesized and free from images of composite morphisms under ${\mathcal{G}}$, and $\beta_n$ is the canonical 2-isomorphism whose target 1-morphism has no identity composition factors and it is completely left-parenthesized and free from compositions both of whose factors are images under ${\mathcal{G}}$. Note that these are the padding operators when one chooses as “references” the ${\mathcal{G}}$-iterates $c^{(n)}\circ{\mathcal{G}}^{(n)}$ and ${\mathcal{G}}\circ ^{(n)}c$, where $c^{(n)}$ denotes the appropriate iterate of the composition functors of $\hat{\text\Bg}$ for the resulting composition to be completely right-parenthesized, $^{(n)}c$ the same thing but using the composition functors of  and so that the resulting composition is completely left-parenthesized, and ${\mathcal{G}}^{(n)}$ denotes the appropriate ${\mathcal{G}}$-iterate (probably with some factor equal to an identity functor). Other choices of references are also possible. That the above 2-morphism is well defined is a consequence of the previous coherence theorem. Let ${\mathcal{G}}=(|{\mathcal{G}}|,{\mathcal{G}}_*,\widehat{{\mathcal{G}}}_*)$ be a unitary pseudofunctor between two bicategories  and $\hat{\text\Bg}$. Let $X,Y,Z,T$ be objects of  and let us consider 1-morphisms $X\stackrel{f}{\longrightarrow} Y\stackrel{g}{\longrightarrow}Z\stackrel{h}{\longrightarrow} T$. Taking $U={\mathcal{G}}(X)$ and $V={\mathcal{G}}(T)$, we have $ \lceil1_{{\mathcal{G}}(h)}\circ\widehat{{\mathcal{G}}}(g,f)\rceil_{{\mathcal{G}}(X),{\mathcal{G}}(T)}= {\mathcal{G}}(\alpha^{\text\Bg}_{h,g,f})\cdot \widehat{{\mathcal{G}}}(h,g\circ f)\cdot(1_{{\mathcal{G}}(h)}\circ\widehat{{\mathcal{G}}}(g,f)). $ Semigroupal 2-categories and their morphisms ============================================ From now on, and unless otherwise indicated, all bicategories will be assumed to be 2-categories. This assumption does not imply loss of generality due to MacLane-Pare’s strictification theorem for bicategories (see Theorem \[MacLane-Pare\]). The objects of our interest are the semigroupal 2-categories. Recall that semigroupal 2-category is a monoidal 2-category without the unit object for the tensor product, and hence without the structural 1- and 2-isomorphisms related to the unital structure. A standard reference on monoidal 2-categories is the paper by Kapranov-Voevodsky [@KV94]. In that paper, however, they give an unraveled definition which involves many data and an even greater number of axioms. To make things more intelligible, it is worth to point out that a semigroupal 2-category is just the categorification of the definition of a semigroupal category. This naturally leads to the following definition (except for the $K_5$ coherence condition on the pentagonator). A [*semigroupal 2-category*]{} consists of the following data [**SBDi**]{} and axiom [**SBA**]{}: SBD1 : A 2-category . SBD2 : A pseudofunctor $\otimes:\text\Cg\times\text\Cg{\longrightarrow}\text\Cg$, called the [*tensor product*]{}. SBD3 : A pseudonatural isomorphism $a:\otimes^{(3)}\Longrightarrow ^{(3)}\otimes: \text\Cg\times\text\Cg\times\text\Cg{\longrightarrow}\text\Cg$, called the [*associator*]{}, where $\otimes^{(3)}$ and $^{(3)}\otimes$ denote, respectively, the composite pseudofunctors $\otimes\circ(id_{\text\Cg}\times\otimes)$ and $\otimes\circ(\otimes\times id_{\text\Cg})$. SBD4 : An invertible modification $\pi:a^{(4)}\Rightarrow ^{(4)}a: \otimes^{(4)}\Longrightarrow ^{(4)}\otimes: \text\Cg\times\text\Cg\times\text\Cg\times\text\Cg{\longrightarrow}\text\Cg$, called the [*pentagonator*]{}, where $\otimes^{(4)}$ and $^{(4)}\otimes$ denote, respectively, the composite pseudofunctors $\otimes\circ(id_{\text\Cg}\times\otimes)\circ(id_{\text\Cg}\times id_{\text\Cg}\times\otimes)$ and $\otimes\circ(\otimes\times id_{\text\Cg})\circ(\otimes\times id_{\text\Cg}\times id_{\text\Cg})$, and $a^{(4)}$, $^{(4)}a$ are the pseudonatural isomorphisms $$\begin{aligned} a^{(4)}&=&(1_{\otimes}\circ(a\times 1_{id}))\cdot(a\circ 1_{id\times\otimes\times id})\cdot(1_{\otimes}\circ(1_{id}\times a)) \\ ^{(4)}a&=&(a\circ 1_{\otimes\times id\times id})\cdot(a\circ 1_{id\times id\times\otimes})\end{aligned}$$ (here, $id$ denotes the identity 2-functor of ). See Fig. \[figura\_pentagonador\] SBA : The data $(\otimes,a,\pi)$ is such that the equality in Fig. \[relacioK5\] holds (to simplify notation, the tensor product of objects or 1-morphisms is denoted by simple juxtaposition, and the identity 1-morphisms are represented by the corresponding objects; for more details about the notations in this Figure, see the next Proposition). This condition will be called the $K_5$ [*coherence relation*]{} (the name comes from the fact that the two pastings in Fig. \[relacioK5\] respresent together a realization of the $K_5$ Stasheff polytope; see [@jS63]). [A]{} semigroupal 2-category will be denoted by $(\text\Cg,\otimes,a,\pi)$ and the triple $(\otimes,a,\pi)$ will be called a [*semigroupal structure*]{} on the 2-category . For convenience in what follows, we give an explicit description of the structural 1- and 2-isomorphisms involved in the previous definition, as well as the whole list of equations they must satisfy. \[semigroupal\_2\_category\] Let  be a 2-category. Then, a semigroupal structure $(\otimes,a,\pi)$ on  consists of the following data: $|\otimes|$ : An object $X\otimes Y$ for any object $(X,Y)$ of $\text\Cg\times\text\Cg$. $\otimes$ : A collection of functors $\otimes_{(X,Y),(X',Y')}:\text\Cg(X,X')\times\text\Cg(Y,Y'){\longrightarrow}\text\Cg(X\otimes Y,X'\otimes Y')$ for all $(X,Y),(X',Y')$ objects of $\text\Cg\times\text\Cg$. As usual, the image of the 1-morphism $(f,g):(X,Y){\longrightarrow}(X',Y')$ and the 2-morphism $(\tau,\sigma):(f,g)\Longrightarrow(\tilde{f},\tilde{g}):(X,Y){\longrightarrow}(X',Y')$ by this functor $\otimes_{(X,Y),(X',Y')}$ will be denoted by $f\otimes g$ and $\tau\otimes\sigma$, respectively. $\mathbf{\widehat{\otimes}}$ : A collection of 2-isomorphisms $\widehat{\otimes}((f',g'),(f,g)):(f'\otimes g')\circ(f\otimes g)\Longrightarrow (f'\circ f)\otimes (g'\circ g)$, for all composable 1-morphisms $(f,g):(X,Y){\longrightarrow}(X',Y')$ and $(f',g'):(X',Y'){\longrightarrow}(X'',Y'')$ of $\text\Cg\times\text\Cg$. $\mathbf{\otimes_0}$ : A collection of 2-isomorphisms $\otimes_0(X,Y):id_X\otimes id_Y\Longrightarrow id_{X\otimes Y}$, for all objects $(X,Y)$ of $\text\Cg\times\text\Cg$. a : A collection of 1-isomorphisms $a_{X,Y,Z}:X\otimes(Y\otimes Z){\longrightarrow}(X\otimes Y)\otimes Z$, for all objects $(X,Y,Z)$ of $\text\Cg\times\text\Cg\times\text\Cg$. $\mathbf{\widehat{a}}$ : A collection of 2-isomorphisms $\widehat{a}(f,g,h):((f\otimes g)\otimes h)\circ a_{X,Y,Z}\Longrightarrow a_{X',Y',Z'}\circ (f\otimes(g\otimes h))$, for all 1-morphisms $(f,g,h):(X,Y,Z){\longrightarrow}(X',Y',Z')$ of $\text\Cg\times\text\Cg\times\text\Cg$. $\mathbf{\pi}$ : A collection of 2-isomorphisms $\pi_{X,Y,Z,T}:(a_{X,Y,Z}\otimes id_T)\circ a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})\Longrightarrow a_{X\otimes Y,Z,T}\circ a_{X,Y,Z\otimes T}$, for all objects $(X,Y,Z,T)$ of $\text\Cg\times\text\Cg\times\text\Cg\times\text\Cg$. [Moreover]{}, all the above 1- and 2-isomorphisms must satisfy the following equations: A$\mathbf{\widehat{\otimes}}$1 : For all 2-morphisms $(\tau,\sigma):(f,g)\Longrightarrow(\tilde{f},\tilde{g}):(X,Y){\longrightarrow}(X',Y')$ and $(\tau',\sigma'):(f',g')\Longrightarrow(\tilde{f}',\tilde{g}'):(X',Y'){\longrightarrow}(X'',Y'')$ of $\text\Cg\times\text\Cg$ $$((\tau'\circ\tau)\otimes(\sigma'\circ\sigma))\cdot \widehat{\otimes}((f',g'),(f,g))= \widehat{\otimes}((\tilde{f}',\tilde{g}'),(\tilde{f},\tilde{g}))\cdot ((\tau'\otimes\sigma')\circ(\tau\otimes\sigma))$$ A$\mathbf{\widehat{\otimes}}$2 : For all composable 1-morphisms $(X,Y)\stackrel{(f,g)}{{\longrightarrow}}(X',Y')\stackrel{(f',g')}{{\longrightarrow}}(X'',Y'') \stackrel{(f'',g'')}{{\longrightarrow}}(X''',Y''')$ of $\text\Cg\times\text\Cg$ $$\begin{aligned} \lefteqn{\widehat{\otimes}((f'',g''),(f'\circ f,g'\circ g))\cdot (1_{f''\otimes g''}\circ\widehat{\otimes}((f',g')(f,g)))=} \\ & & =\widehat{\otimes}((f''\circ f',g''\circ g'),(f,g))\cdot (\widehat{\otimes}((f'',g''),(f',g'))\circ 1_{f\otimes g})\end{aligned}$$ A$\mathbf{\widehat{\otimes}}$3 : For any 1-morphism $(f,g):(X,Y){\longrightarrow}(X',Y')$ of $\text\Cg\times\text\Cg$ $$\begin{aligned} &\widehat{\otimes}((id_{X'},id_{Y'}),(f,g))=\otimes_0(X',Y')\circ 1_{f\otimes g} \\ &\widehat{\otimes}((f,g),(id_X,id_Y))=1_{f\otimes g}\circ\otimes_0(X,Y)\end{aligned}$$ A$\mathbf{\widehat{a}}$1 : For all 2-morphisms $(\tau,\sigma,\eta):(f,g,h)\Longrightarrow(\tilde{f},\tilde{g},\tilde{h}):(X,Y,Z){\longrightarrow}(X',Y',Z')$ of $\text\Cg^3$ $$(1_{a_{X',Y',Z'}}\circ(\tau\otimes(\sigma\otimes\eta)))\cdot\widehat{a}(f,g,h)= \widehat{a}(\tilde{f},\tilde{g},\tilde{h})\cdot(((\tau\otimes\sigma)\otimes\eta) \circ 1_{a_{X,Y,Z}})$$ A$\mathbf{\widehat{a}}$2 : For all composable 1-morphisms $(X,Y,Z)\stackrel{(f,g,h)}{\longrightarrow}(X',Y',Z')\stackrel{(f',g',h')}{{\longrightarrow}}(X'',Y'',Z'')$ of $\text\Cg\times\text\Cg\times\text\Cg$ $$\begin{aligned} \widehat{a}(f'\circ f,g'\circ g,h'\circ h)&\cdot((\widehat{\otimes}((f',g'),(f,g))\otimes 1_{h'\circ h})\circ 1_{a_{X,Y,Z}}) \\ &\cdot(\widehat{\otimes}((f'\otimes g',h'),(f\otimes g,h))\circ 1_{a_{X,Y,Z}})= \\ &\ \ \ \ \ \ \ \ \ \ \ \ =(1_{a_{X'',Y'',Z''}}\circ(1_{f'\circ f}\otimes\widehat{\otimes}((g',h')(g,h)))) \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot(1_{a_{X',Y',Z'}}\circ \widehat{\otimes}((f',g'\otimes h'),(f,g\otimes h))) \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot(\widehat{a}(f',g',h')\circ 1_{f\otimes(g\otimes h)})\cdot(1_{(f'\otimes g')\otimes h'}\circ\widehat{a}(f,g,h))\end{aligned}$$ A$\mathbf{\widehat{a}}$3 : For all objects $(X,Y,Z)$ of $\text\Cg\times\text\Cg\times\text\Cg$ $$\begin{aligned} \widehat{a}(id_X,id_Y,id_Z)=&(1_{a_{X,Y,Z}}\circ(\otimes_0(X,Y\otimes Z) \cdot(1_{id_X}\otimes\otimes_0(Y,Z)))^{-1})\cdot \\ &\ \ \ \cdot((\otimes_0(X\otimes Y,Z)\cdot(\otimes_0(X,Y)\otimes 1_{id_Z}))\circ 1_{a_{X,Y,Z}})\end{aligned}$$ A$\pi$1 : For any 1-morphism $(f,g,h,k):(X,Y,Z,T){\longrightarrow}(X',Y',Z',T')$ of $\text\Cg^4$ $$\begin{aligned} (1_{a_{X'\otimes Y',Z',T'}}\circ\widehat{a}&(f,g,h\otimes k))\cdot(\widehat{a}(f\otimes g,h,k)\circ 1_{a_{X,Y,Z\otimes T}})\cdot(1_{((f\otimes g)\otimes h)\otimes k}\circ\pi_{X,Y,Z,T})= \\ &=(\pi_{X',Y',Z',T'}\circ 1_{f\otimes(g\otimes(h\otimes k))})\cdot \\ &\ \ \ \cdot(1_{a_{X',Y',Z'}\otimes id_{T'}\circ a_{X,Y\otimes Z,T}}\circ\widehat{\otimes}((id_{X'},a_{Y',Z',T'}),(f,g\otimes(h\otimes k)))^{-1})\cdot \\ &\ \ \ \cdot(1_{(a_{X',Y',Z'}\otimes id_{T'})\circ a_{X,Y\otimes Z,T}}\circ (1_f\otimes\widehat{a}(g,h,k)))\cdot \\ &\ \ \ \cdot(1_{(a_{X',Y',Z'}\otimes id_{T'})\circ a_{X,Y\otimes Z,T}}\circ \widehat{\otimes}((f,(g\otimes h)\otimes k),(id_X,a_{Y,Z,T})))\cdot \\ &\ \ \ \cdot(1_{a_{X',Y',Z'}\otimes id_{T'}}\circ \widehat{a}(f,g\otimes h,k)\circ 1_{id_X\otimes a_{Y,Z,T}})\cdot \\ &\ \ \ \cdot(\widehat{\otimes}((a_{X',Y',Z'},id_{T'}),(f\otimes(g\otimes h),k))^{-1} \circ 1_{a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})})\cdot \\ &\ \ \ \cdot((\widehat{a}(f,g,h)\otimes 1_k)\circ 1_{a_{X,Y\otimes Z,T} \circ(id_X\otimes a_{Y,Z,T})})\cdot \\ &\ \ \ \cdot(\widehat{\otimes}(((f\otimes g)\otimes h,k),(a_{X,Y,Z},id_T)) \circ 1_{a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})})\end{aligned}$$ A$\mathbf{\pi}$2 : For any object $(X,Y,Z,T,U)$ of $\text\Cg\times\text\Cg\times\text\Cg\times\text\Cg\times\text\Cg$ $$\begin{aligned} (\pi_{X\otimes Y,Z,T,U}&\circ 1_{a_{X,Y,Z\otimes(T\otimes Z)}})\cdot \\ &\cdot(1_{(a_{X\otimes Y,Z,T}\otimes id_U)\circ a_{X\otimes Y,Z\otimes T,U}}\circ\widehat{a}(id_X,id_Y,a_{Z,T,U})^{-1})\cdot \\ &\cdot(1_{a_{X\otimes Y,Z,T}\otimes id_U}\circ\pi_{X,Y,Z\otimes T,U}\circ 1_{id_X\otimes(id_Y\otimes a_{Z,T,U})})\cdot \\ &\cdot((\pi_{X,Y,Z,T}\tilde{\otimes} 1_{id_U})\circ 1_{a_{X,Y\otimes(Z\otimes T),U}\circ(id_X\otimes a_{Y,Z\otimes T,U})\circ(id_X\otimes(id_Y\otimes a_{Z,T,U}))})= \\ &\ \ \ \ \ \ \ =(1_{a_{(X\otimes Y)\otimes Z,T,U}}\circ\pi_{X,Y,Z,T\otimes U})\cdot \\ &\ \ \ \ \ \ \ \ \ \ \ \cdot (\widehat{a}(a_{X,Y,Z},id_T,id_U)\circ 1_{a_{X,Y\otimes Z,T\otimes U}\circ(id_X\otimes a_{Y,Z,T\otimes U})})\cdot \\ &\ \ \ \ \ \ \ \ \ \ \ \cdot (1_{(a_{X,Y,Z}\otimes id_T)\otimes id_U}\circ \pi_{X,Y\otimes Z,T,U}\circ 1_{id_X\otimes a_{Y,Z,T\otimes U}})\cdot \\ &\ \ \ \ \ \ \ \ \ \ \ \cdot (1_{((a_{X,Y,Z}\otimes id_T)\otimes id_U)\circ(a_{X,Y\otimes Z,T}\otimes id_U) \circ a_{X,(Y\otimes Z)\otimes T,U}}\circ(1_{id_X}\tilde{\otimes}\pi_{Y,Z,T,U}))\cdot \\ &\ \ \ \ \ \ \ \ \ \ \ \cdot (1_{((a_{X,Y,Z}\otimes id_T)\otimes id_U)\circ (a_{X,Y\otimes Z,T}\otimes id_U)}\circ\widehat{a}(id_X,a_{Y,Z,T},id_U)\circ \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \circ 1_{(id_X\otimes a_{Y,Z\otimes T,U})\circ(id_X\otimes(id_Y\otimes a_{Z,T,U}))})\end{aligned}$$ We will refer to the previous equations as the [*structural equations*]{} of a semigroupal 2-category. Notice that, in Equation (A$\pi$2), both terms $1_{id_X}\tilde{\otimes}\pi_{Y,Z,T,U}$ and $\pi_{X,Y,Z,T}\tilde{\otimes} 1_{id_U}$ denote pastings of the corresponding terms $1_{id_X}\otimes\pi_{Y,Z,T,U}$ and $\pi_{X,Y,Z,T}\otimes 1_{id_U}$. For example, the reader may check that the first term is given by $$\begin{aligned} 1_{id_X}\tilde{\otimes}\pi_{Y,Z,T,U}&=&\widehat{\otimes}((id_X,a_{Y\otimes Z,T,U}),(id_X,a_{Y,Z,T\otimes U}))^{-1}\cdot(1_{id_X}\otimes\pi_{Y,Z,T,U})\cdot \\ & &\cdot\widehat{\otimes}((id_X,a_{Y,Z,T}\otimes id_U),(id_X,a_{Y,Z\otimes T,U}\circ(id_Y\otimes a_{Z,T,U})))\cdot \\ & &\cdot(1_{id_X\otimes(a_{Y,Z,T}\otimes id_U)}\circ\widehat{\otimes}((id_X,a_{Y,Z\otimes T,U}),(id_X,id_Y\otimes a_{Z,T,U})))\end{aligned}$$ A similar expression gives us the pasting $\pi_{X,Y,Z,T}\tilde{\otimes} 1_{id_U}$. Equations $(A\widehat{\otimes}i)$ correspond to the naturality of the 2-isomorphisms $\widehat{\otimes}((f',g'),(f,g))$ and the two axioms on the pseudofunctorial structure which appear in the definition of pseudofunctor. Similarly, equations $(A\widehat{a}i)$ correspond to the naturality of the 2-isomorphisms $\widehat{a}(f,g,h)$ and the axioms appearing in the definition of pseudonatural transformation. On the other hand, Equation $(A\pi 1)$ corresponds to the naturality condition on the pentagonator $\pi_{X,Y,Z,T}$ in $(X,Y,Z,T)$, namely, $$\widehat{^{(4)}a}(f,g,h,k)\cdot(1_{((f\otimes g)\otimes h)\otimes k} \circ\pi_{X,Y,Z,T})=(\pi_{X',Y',Z',T'}\circ 1_{f\otimes(g\otimes(h\otimes k))}) \cdot\widehat{a^{(4)}}(f,g,h,k)$$ after making explicit the 2-isomorphisms $\widehat{^{(4)}a}(f,g,h,k)$ and $\widehat{a^{(4)}}(f,g,h,k)$ using the definitions in Section 2. Finally, Equation $(A\pi 2)$ is the algebraic expression of the $K_5$ coherence relation. A semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$ is called [*strict*]{} when all the above structural isomorphisms are identities (notice that this is not possible for an arbitrary tensor product $\otimes$; for example, it must satisfy that $X\otimes(Y\otimes Z)=(X\otimes Y)\otimes Z$, etc.). Let us remark that, apart from the structural 1- and 2-isomorphisms related to the unital structure that do not appear in our definition above, Kapranov-Voevodsky’s definition of a monoidal 2-category (see [@KV94]) includes a different collection of structural 2-isomorphisms. So, instead of our $\widehat{\otimes}((f',g'),(f,g)):(f'\otimes g')\circ(f\otimes g)\Longrightarrow (f'\circ f)\otimes (g'\circ g)$, they introduce the two sets of 2-isomorphisms $\otimes_{f,f',Y}:(f'\otimes Y)\circ (f\otimes Y)\Longrightarrow (f'\circ f)\otimes Y$ and $\otimes_{X,g,g'}:(X\otimes g')\circ(X\otimes g)\Longrightarrow X\otimes (g'\circ g)$ together with the basic 2-isomorphisms $\otimes_{f,g}:(f\otimes Y')\circ(X\otimes g)\Longrightarrow(X'\otimes g)\circ(f\otimes Y)$. Similarly, instead of our $\widehat{a}(f,g,h):((f\otimes g)\otimes h)\circ a_{X,Y,Z}\Longrightarrow a_{X',Y',Z'}\circ(f\otimes(g\otimes h))$, they use 2-isomorphisms $a_{f,Y,Z}:((f\otimes Y)\otimes Z)\circ a_{X,Y,Z}\Longrightarrow a_{X',Y',Z'}\circ(f\otimes(Y\otimes Z))$ and the similarly defined $a_{X,g,Z}$, $a_{X,Y,h}$. This obviously implies a different set of axioms. However, both formulations are equivalent, and correspond to the two possible ways of defining a “bipseudofunctor” directly as a pseudofunctor of two variables or as two collections of pseudofunctors of one variable. Although it is possible to work with Kapranov-Voevodsky’s 2-isomorphisms, the cohomological nature of the axioms is much more clear when working with those of the previous proposition. Let us further remark that the special case where the pseudofunctor $\otimes:\text\Cg^2{\longrightarrow}\text\Cg$ in our definition is [*cubical*]{} (see the definition below) corresponds, in the Kapranov-Voevodsky’s formulation, to the notion of a [*quasifunctor of 2-variables*]{} introduced by Gray in [@jG74], p.56. The equivalence between both notions, cubical pseudofunctor and quasifunctor of 2-variables, is in fact the content of Proposition I.4.8. in Gray’s book. We do not enter into the details of the equivalence, but let us mention that, in terms of our $\widehat{\otimes}((f',g'),(f,g))$, the above Kapranov-Voevodsky’s 2-isomorphisms $\otimes_{f,g}$ correspond to $$\otimes_{f,g}=\widehat{\otimes}((id_{X'},g),(f,id_Y))^{-1}\cdot \widehat{\otimes}((f,id_{Y'}),(id_X,g))$$ and conversely, our $\widehat{\otimes}((f',g'),(f,g))$ are given by $$\widehat{\otimes}((f',g'),(f,g))=1_{f'\otimes Y''}\circ(\otimes_{f,g'})^{-1}\circ 1_{X\otimes g}$$ The reader may also check that our structural equation $(A\widehat{a}2)$ exactly corresponds to the axiom Kapranov and Voevodsky denote by $(\rightarrow\otimes\rightarrow\otimes\bullet)$ (see Fig. \[cub\]) together with two more similar axioms. A fundamental fact in the theory of semigroupal 2-categories is the corresponding [*strictification theorem*]{}, due to Gordon-Power-Street [@GPS95]. In fact, they proved a much more general strictification theorem, valid for an arbitrary [*tricategory*]{} (i.e., the categorification of the notion of a bicategory). In the same way as a monoidal category just corresponds to a bicategory of only one object, a monoidal 2-category is just a tricategory of only one object [^4]. Now, contrary to the case of bicategories, not all tricategories are equivalent to the corresponding 3-categories (the reader may figure out the precise definition of such objects). Indeed, some of the structural 3-isomorphisms can not be strictified in general, i.e., made equal to identitites. In our case, this means that an arbitrary semigroupal 2-category is in some sense equivalent to a particular kind of semigroupal 2-categories, which, following Day and Street [@DS97], we will call [*Gray semigroups*]{} [^5], and which are not the strict semigroupal 2-categories. Since this theorem plays an essential role in what follows, allowing us to greatly simplify the theory, we review here the precise definitions. Let  be any 2-category. A pseudofunctor ${\mathcal{F}}:\text\Cg\times\text\Cg{\longrightarrow}\text\Cg$ is called [*cubical*]{} if its structural 2-isomorphisms $\widehat{{\mathcal{F}}}((f',g'),(f,g)):{\mathcal{F}}(f',g')\circ{\mathcal{F}}(f,g)\Longrightarrow {\mathcal{F}}(f'\circ f,g'\circ g)$ and ${\mathcal{F}}_0(X,Y):{\mathcal{F}}(id_X,id_Y)\Longrightarrow id_{{\mathcal{F}}(X,Y)}$ are such that: 1. The $\widehat{{\mathcal{F}}}((f',id_{Y'}),(f,g))$ and $\widehat{{\mathcal{F}}}((f,g'),(id_X,g))$ are all identity 2-morphisms. 2. ${\mathcal{F}}$ is a unitary pseudofunctor, i.e., for all $(X,Y)$, ${\mathcal{F}}_0(X,Y)=1_{id_{{\mathcal{F}}(X,Y)}}$. Notice that our definition here differs from that in [@GPS95], p.31, in that we explicitly require the pseudofunctor to be unitary. Indeed, although the authors say that this condition follows from the cubical condition of ${\mathcal{F}}$, it seems that this is not the case, and the assumption must be included in the definition [^6]. A [*cubical semigroupal 2-category*]{} is any semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$ such that the tensor product $\otimes$ is a cubical pseudofunctor. A cubical semigroupal 2-category will be called a [*Gray semigroup*]{} whenever its structural 2-isomorphisms included in the associator $a$ and the pentagonator $\pi$ are all identities. The analogous notions in the more general context of tricategories are respectively called [*cubical tricategories*]{} and [*Gray categories*]{} in [@GPS95]. A Gray semigroup will be simply denoted by $(\text\Cg,\otimes)$, the $a$ and $\pi$ being trivial. We leave to the reader to make explicit this definition. Notice that the set of structural 2-isomorphisms reduces in this case to the $\widehat{\otimes}((f',g'),(f,g))$, most of which are moreover trivial by the cubical condition. This is the reason a Gray semigroup is usually described in terms of Kapranov-Voevodsky’s 2-isomorphisms $\otimes_{f,g}$. It is worth to point out that not every cubical pseudofunctor $\otimes$ defines a structure of Gray semigroup on a 2-category. The fundamental strictification theorem for semigroupal 2-categories can now be stated as follows: ([@GPS95]) \[strictification\] Every semigroupal 2-category is equivalent (in a sense we do not make precise) to a Gray semigroup. After reading the next section, where the notion of a morphism between semigroupal 2-categories is defined, the reader may figure out by himself the sense in which this equivalence should be understood. Let us finish this section by giving the corresponding notion of morphisms between semigroupal 2-categories, which will be needed in the next section in order to define equivalence of deformations. In the case of Gray semigroups, this definition appears, for example, in [@DS97], Def.2 (in fact, they define morphism between Gray monoids). Our definition below follows from the general definition of morphism between tricategories which appears in [@GPS95] when restricted to the one object case (and forgetting the unital structure). Let $(\text\Cg,\otimes,a,\pi)$ and $(\text\Cg',\otimes',a',\pi')$ be semigroupal 2-categories. A [*semigroupal pseudofunctor*]{} from  to $\text\Cg'$ is a pseudofunctor ${\mathcal{F}}:\text\Cg{\longrightarrow}\text\Cg'$ together with the following data [**SPDi**]{} and axiom [**SPA**]{}: SPD1 : A pseudonatural isomorphism $\psi:\otimes'\circ ({\mathcal{F}}\times{\mathcal{F}})\Longrightarrow {\mathcal{F}}\circ\otimes:\text\Cg\times\text\Cg\to\text\Cg'$. SPD2 : An invertible modification $\omega:(1_{{\mathcal{F}}}\circ a)\cdot(\psi\circ 1_{id_{\text\Cg}\times\otimes})\cdot(1_{\otimes'}\circ(1_{{\mathcal{F}}}\times\psi))\Rightarrow(\psi\circ 1_{\otimes\times id_{\text\Cg}})\cdot(1_{\otimes'}\circ(\psi\times 1_{{\mathcal{F}}}))\cdot(a'\circ 1_{{\mathcal{F}}\times{\mathcal{F}}\times{\mathcal{F}}})$ (see Fig. \[omega\_figura\]) SPA : The pair $(\psi,\omega)$ is such that the equation in Fig. \[semigrupal\_figura\] holds (to simplify notation, the tensor product of objects and 1-morphisms is again denoted by simple juxtaposition and the identity 1-morphisms are represented by the corresponding objects; furthermore, the action of the pseudofunctor on objects, 1-morphisms or 2-morphisms is indicated by the symbols $[-]$, so that, for example, $\psi_{XY,Z}([Z][T])$ denotes the 1-morphism $\psi_{X\otimes Y,Z}\otimes'({\mathcal{F}}(id_Z)\otimes'{\mathcal{F}}(id_T))$. For more details about the notations appearing in this Figure, see the next Proposition). [A]{} semigroupal pseudofunctor will be denoted by the triple $({\mathcal{F}},\psi,\omega)$ and the pair $(\psi,\omega)$ will be called a [*semigroupal structure on*]{} ${\mathcal{F}}$. Observe that the above definition indeed corresponds to categorifying the definition of a semigroupal functor between semigroupal categories: the axiom on the semigroupal structure is substituted for the modification $\omega$, which in turn must satisfy the additional coherence relation (SPA). A more explicit description of a semigroupal structure $(\psi,\omega)$ on ${\mathcal{F}}$, with the whole list of equations on the structural 1- and 2-isomorphisms, is as follows: \[estructura\_semigrupal\] Let $(\text\Cg,\otimes,a,\pi)$ and $(\text\Cg',\otimes',a',\pi')$ be semigroupal 2-categories and ${\mathcal{F}}:\text\Cg{\longrightarrow}\text\Cg'$ a pseudofunctor. Then, a semigroupal structure $(\psi,\omega)$ on ${\mathcal{F}}:\text\Cg{\longrightarrow}\text\Cg'$ consists of: $\mathbf{\psi}$ : A collection of 1-isomorphisms $\psi_{X,Y}:{\mathcal{F}}(X)\otimes'{\mathcal{F}}(Y){\longrightarrow}{\mathcal{F}}(X\otimes Y)$ for all objects $(X,Y)$ of $\text\Cg\times\text\Cg$. $\mathbf{\widehat{\psi}}$ : A collection of 2-isomorphisms $\widehat{\psi}(f,g):{\mathcal{F}}(f\otimes g)\circ\psi_{X,Y}\Longrightarrow\psi_{X',Y'}\circ({\mathcal{F}}(f)\otimes'{\mathcal{F}}(g))$ for all 1-morphisms $(f,g):(X,Y){\longrightarrow}(X',Y')$ of $\text\Cg\times\text\Cg$. $\mathbf{\omega}$ : A collection of 2-isomorphisms $\omega_{X,Y,Z}:{\mathcal{F}}(a_{X,Y,Z})\circ\psi_{X,Y\otimes Z}\circ(id_{{\mathcal{F}}(X)}\otimes'\psi_{Y,Z})\Longrightarrow\psi_{X\otimes Y,Z}\circ(\psi_{X,Y}\otimes' id_{{\mathcal{F}}(Z)})\circ a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)}$ for all objects $(X,Y,Z)$ of $\text\Cg\times\text\Cg\times\text\Cg$. [Moreover]{}, these data must satisfy the following equations: A$\mathbf{\widehat{\psi}}$1 : For all 2-morphisms $(\tau,\sigma):(f,g)\Longrightarrow(\tilde{f},\tilde{g}):(X,Y){\longrightarrow}(X',Y')$ of $\text\Cg\times\text\Cg$ $$(1_{\psi_{X',Y'}}\circ({\mathcal{F}}(\tau)\otimes'{\mathcal{F}}(\sigma)))\cdot\widehat{\psi}(f,g)= \widehat{\psi}(\tilde{f},\tilde{g})\cdot({\mathcal{F}}(\tau\otimes\sigma)\circ 1_{\psi_{X,Y}})$$ A$\mathbf{\widehat{\psi}}$2 : For all composable 1-morphisms $(X,Y)\stackrel{(f,g)}{{\longrightarrow}}(X',Y')\stackrel{(f',g')}{{\longrightarrow}}(X'',Y'')$ of $\text\Cg\times\text\Cg$ $$\begin{aligned} \lefteqn{\widehat{\psi}(f'\circ f,g'\circ g)\cdot([{\mathcal{F}}(\widehat{\otimes}((f',g'),(f,g)))\cdot\widehat{{\mathcal{F}}}(f'\otimes g',f\otimes g)]\circ 1_{\psi_{X,Y}})=} \\ & &=(1_{\psi_{X'',Y''}}\circ[(\widehat{{\mathcal{F}}}(f',f)\otimes'\widehat{{\mathcal{F}}}(g',g))\cdot \widehat{\otimes}'(({\mathcal{F}}(f'),{\mathcal{F}}(g')),({\mathcal{F}}(f),{\mathcal{F}}(g)))])\cdot \\ & & \ \ \ \ \cdot(\widehat{\psi}(f',g')\circ 1_{{\mathcal{F}}(f)\otimes'{\mathcal{F}}(g)})\cdot(1_{{\mathcal{F}}(f'\otimes g')}\circ\widehat{\psi}(f,g))\end{aligned}$$ A$\mathbf{\widehat{\psi}}$3 : For all objects $(X,Y)$ of $\text\Cg\times\text\Cg$ $$\begin{aligned} \widehat{\psi}(id_X,id_Y)=(1_{\psi_{X,Y}}&\circ [\otimes'_0({\mathcal{F}}(X),{\mathcal{F}}(Y))\cdot({\mathcal{F}}_0(X)\otimes'{\mathcal{F}}_0(Y))])^{-1}\cdot \\ &\cdot([{\mathcal{F}}_0(X\otimes Y)\cdot{\mathcal{F}}(\otimes_0(X,Y))]\circ 1_{\psi_{X,Y}})\end{aligned}$$ A$\mathbf{\omega}$1 : For all 1-morphisms $(f,g,h):(X,Y,Z){\longrightarrow}(X',Y',Z')$ of $\text\Cg\times\text\Cg\times\text\Cg$ $$\begin{aligned} (\omega_{X',Y',Z'}&\circ 1_{{\mathcal{F}}(f)\otimes'({\mathcal{F}}(g)\otimes'{\mathcal{F}}(h))})\cdot \\ &\cdot(1_{{\mathcal{F}}(a_{X',Y',Z'})\circ\psi_{X',Y'\otimes Z'}}\circ\widehat{\otimes}'((id_{{\mathcal{F}}(X')},\psi_{Y',Z'}),({\mathcal{F}}(f),{\mathcal{F}}(g)\otimes'{\mathcal{F}}(h)))^{-1})\cdot \\ &\cdot(1_{{\mathcal{F}}(a_{X',Y',Z'})\circ\psi_{X',Y'\otimes Z'}}\circ(1_{{\mathcal{F}}(f)}\otimes'\widehat{\psi}(g,h)))\cdot \\ &\cdot(1_{{\mathcal{F}}(a_{X',Y',Z'})\circ\psi_{X',Y'\otimes Z'}}\circ\widehat{\otimes}'(({\mathcal{F}}(f),{\mathcal{F}}(g\otimes h)),(id_{{\mathcal{F}}(X)},\psi_{Y,Z})))\cdot \\ &\cdot(1_{{\mathcal{F}}(a_{X',Y',Z'})}\circ\widehat{\psi}(f,g\otimes h) \circ 1_{id_{{\mathcal{F}}(X)}\otimes'\psi_{Y,Z}})\cdot \\ &\cdot(\widehat{{\mathcal{F}}}(a_{X',Y',Z'},f\otimes(g\otimes h))^{-1}\circ 1_{\psi_{X,Y\otimes Z}}\circ 1_{id_{{\mathcal{F}}(X)}\otimes'\psi_{Y,Z}})\cdot \\ &\cdot({\mathcal{F}}(\widehat{a}(f,g,h))\circ 1_{\psi_{X,Y\otimes Z}}\circ 1_{id_{{\mathcal{F}}(X)}\otimes'\psi_{Y,Z}})\cdot \\ &\cdot(\widehat{{\mathcal{F}}}((f\otimes g)\otimes h,a_{X,Y,Z})\circ 1_{\psi_{X,Y\otimes Z}}\circ 1_{id_{{\mathcal{F}}(X)}\otimes'\psi_{Y,Z}})= \\ &\ \ \ \ \ =(1_{\psi_{X'\otimes Y',Z'}}\circ 1_{\psi_{X',Y'}\otimes' id_{{\mathcal{F}}(Z')}}\circ\widehat{a}'({\mathcal{F}}(f),{\mathcal{F}}(g),{\mathcal{F}}(h)))\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{\psi_{X'\otimes Y',Z'}}\circ \widehat{\otimes}'((\psi_{X',Y'},id_{{\mathcal{F}}(Z')}),({\mathcal{F}}(f)\otimes'{\mathcal{F}}(g),{\mathcal{F}}(h)))^{-1} \circ 1_{a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)}})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{\psi_{X'\otimes Y',Z'}}\circ(\widehat{\psi}(f,g)\otimes' 1_{{\mathcal{F}}(h)})\circ 1_{a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)}})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{\psi_{X'\otimes Y',Z'}}\circ \widehat{\otimes}'(({\mathcal{F}}(f\otimes g),{\mathcal{F}}(h)),(\psi_{X,Y},id_{{\mathcal{F}}(Z)}))\circ 1_{a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)}})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(\widehat{\psi}(f\otimes g,h)\circ 1_{(\psi_{X,Y}\otimes' id_{{\mathcal{F}}(Z)})\circ a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)}})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{{\mathcal{F}}((f\otimes g)\otimes h)}\circ\omega_{X,Y,Z})\end{aligned}$$ A$\mathbf{\omega}$2 : For all objects $(X,Y,Z,T)$ of $\text\Cg\times\text\Cg\times\text\Cg\times\text\Cg$ $$\begin{aligned} (1&_{\psi_{(X\otimes Y)\otimes Z,T}\circ(\psi_{X\otimes Y,Z}\otimes'{\mathcal{F}}(id_T))}\circ\widehat{a}'(\psi_{X,Y},{\mathcal{F}}(id_Z),{\mathcal{F}}(id_T))^{-1}\circ 1_{a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)\otimes'{\mathcal{F}}(T)}})\cdot \\ &\cdot(\tilde{\omega}_{X\otimes Y,Z,T}\circ 1_{(\psi_{X,Y}\otimes'({\mathcal{F}}(id_Z)\otimes'{\mathcal{F}}(id_T)))\circ a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)\otimes'{\mathcal{F}}(T)}})\cdot \\ &\cdot(1_{{\mathcal{F}}(a_{X\otimes Y,Z,T})\circ\psi_{X\otimes Y,Z\otimes T}}\circ\chi_{X,Y,Z,T}\circ 1_{a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z)\otimes'{\mathcal{F}}(T)}})\cdot \\ &\cdot(1_{{\mathcal{F}}(a_{X\otimes Y,Z,T})\circ\psi_{X\otimes Y,Z\otimes T}\circ (\psi_{X,Y}\otimes'{\mathcal{F}}(id_{Z\otimes T}))}\circ\widehat{a}'({\mathcal{F}}(id_X),{\mathcal{F}}(id_Y),\psi_{Z,T}))^{-1}\cdot \\ &\cdot(1_{{\mathcal{F}}(a_{X\otimes Y,Z,T})}\circ\tilde{\omega}_{X,Y,Z\otimes T}\circ 1_{{\mathcal{F}}(id_X)\otimes'({\mathcal{F}}(id_Y)\otimes'\psi_{Z,T})})\cdot \\ &\cdot(\tilde{{\mathcal{F}}}(\pi_{X,Y,Z,T})\circ 1_{\psi_{X,Y\otimes(Z\otimes T)}\circ({\mathcal{F}}(id_X) \otimes'\psi_{Y,Z\otimes T})\circ({\mathcal{F}}(id_X)\otimes'({\mathcal{F}}(id_Y)\otimes'\psi_{Z,T}))})= \\ &\ \ \ \ \ =(1_{\psi_{(X\otimes Y)\otimes Z,T}\circ(\psi_{X\otimes Y,Z}\otimes'{\mathcal{F}}(id_T))\circ((\psi_{X,Y}\otimes'{\mathcal{F}}(id_Z))\otimes'{\mathcal{F}}(id_T))} \circ\tilde{\pi}'_{{\mathcal{F}}(X),{\mathcal{F}}(Y),{\mathcal{F}}(Z),{\mathcal{F}}(T)})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{\psi_{(X\otimes Y)\otimes Z,T}}\circ (\tilde{\omega}_{X,Y,Z}\tilde{\otimes} 1_{{\mathcal{F}}(id_T)})\circ 1_{a'_{{\mathcal{F}}(X),{\mathcal{F}}(Y)\otimes'{\mathcal{F}}(Z),{\mathcal{F}}(T)}\circ({\mathcal{F}}(id_X)\otimes' a'_{{\mathcal{F}}(Y),{\mathcal{F}}(Z),{\mathcal{F}}(T)})})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(\widehat{\psi}(a_{X,Y,Z},id_T)\circ 1_{(\psi_{X,Y\otimes Z}\otimes'[id_T])\circ(([id_X]\otimes'\psi_{Y,Z})\otimes'[id_T])\circ a'_{[X],[Y]\otimes'[Z],[T]}\circ([id_X]\otimes' a'_{[Y],[Z],[T]})})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{[a_{X,Y,Z}\otimes id_T]\circ\psi_{X\otimes(Y\otimes Z),T} \circ(\psi_{X,Y\otimes Z}\otimes'[id_T])}\circ\widehat{a}'([id_X],\psi_{Y,Z},[id_T])^{-1} \circ 1_{[id_X]\otimes' a'_{[Y],[Z],[T]}})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{{\mathcal{F}}(a_{X,Y,Z}\otimes id_T)}\circ\tilde{\omega}_{X,Y\otimes Z,T}\circ 1_{({\mathcal{F}}(id_X)\otimes'(\psi_{Y,Z}\otimes'{\mathcal{F}}(id_T)))\circ({\mathcal{F}}(id_X)\otimes' a'_{{\mathcal{F}}(Y),{\mathcal{F}}(Z),{\mathcal{F}}(T)})})\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{{\mathcal{F}}(a_{X,Y,Z}\otimes id_T) \circ{\mathcal{F}}(a_{X,Y\otimes Z,T})\circ\psi_{X,(Y\otimes Z)\otimes T}}\circ (1_{{\mathcal{F}}(id_X)}\tilde{\otimes}\tilde{\omega}_{Y,Z,T}))\cdot \\ &\ \ \ \ \ \ \ \ \ \cdot(1_{[a_{X,Y,Z}\otimes id_T] \circ[a_{X,Y\otimes Z,T}]}\circ\widehat{\psi}(id_X,a_{Y,Z,T})\circ 1_{([id_X]\otimes'\psi_{Y,Z\otimes T})\circ([id_X]\otimes'([id_Y]\otimes'\psi_{Z,T}))})\end{aligned}$$ (to simplify notation, ${\mathcal{F}}(-)$ is denoted in some places by $[-]$). [In]{} the last equation (A$\omega$2), the term $\chi_{X,Y,Z,T}$ denotes the 2-isomorphism $$\begin{aligned} \chi_{X,Y,Z,T}=&\widehat{\otimes}(({\mathcal{F}}(id_{X\otimes Y}),\psi_{Z,T}), (\psi_{X,Y},{\mathcal{F}}(id_Z)\otimes'{\mathcal{F}}(id_T)))^{-1}\cdot(({\mathcal{F}}(\otimes_0(X,Y))\circ 1)\otimes' 1)\cdot \\ &\cdot(\widehat{\psi}(id_X,id_Y)^{-1}\otimes'\widehat{\psi}(id_Z,id_T))\cdot (1\otimes'({\mathcal{F}}(\otimes_0(Z,T)^{-1})\circ 1))\cdot \\ &\cdot\widehat{\otimes}'((\psi_{X,Y},{\mathcal{F}}(id_{Z\otimes T})),({\mathcal{F}}(id_X)\otimes'{\mathcal{F}}(id_Y),\psi_{Z,T}))\end{aligned}$$ Notice that, in the particular case of a Gray semigroup, they just reduce to Kapranov-Voevodsky’s 2-isomorphisms $\otimes_{\psi_{X,Y},\psi_{Z,T}}$, as it appears in Day-Street’s definition mentioned above [@DS97]. Observe also that all the terms $\tilde{\omega}_{X,Y,Z}$, $\tilde{\pi}_{X,Y,Z,T}$, $\tilde{{\mathcal{F}}}(\pi_{X,Y,Z,T})$, $\tilde{\omega}_{X,Y,Z}\tilde{\otimes}' 1_{{\mathcal{F}}(id_T)}$ $1_{{\mathcal{F}}(id_X)}\tilde{\otimes}'\tilde{\omega}_{Y,Z,T}$ are pastings of the corresponding 2-isomorphisms with the appropriate structural 2-isomorphisms from $\widehat{{\mathcal{F}}}$, ${\mathcal{F}}_0$ and $\widehat{\otimes}'$. We leave to the reader to find out the explicit formulas. The Proposition again follows from the definitions in Section 2. In particular, Equation (A$\omega$1) corresponds to the naturality condition on $\omega_{X,Y,Z}$ in the object $(X,Y,Z)$, and (A$\omega$2) is the algebraic expression of the coherence relation (SPA). Later on, we will need this Proposition in the very special case where $\text\Cg'=\text\Cg$ (but with different semigroupal structures $(\otimes,a,\pi)$ and $(\otimes',a',\pi')$) and ${\mathcal{F}}=id_{\text\Cg}$, the identity 2-functor of . Deformations of pseudofunctors and semigroupal 2-categories =========================================================== In this section we formalize the idea outlined in the introduction, i.e., we “linearize” the problem of deforming a semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$. To do that, we will need to assume that  has some $K$-linear structure, for some commutative ring with unit $K$. Before that, however, we introduce the notion of a purely pseudofunctorial infinitesimal deformation of a ($K$-linear) pseudofunctor, a notion which appears later in Section 8 when we study the deformations of the tensor product in a semigroupal 2-category. The corresponding notions of equivalent deformations are also introduced, and they are made explicit in the case of first order deformations for later use. Recall that a category ${\mathcal{C}}$ is called $K$-linear when all its hom-sets ${\mathcal{C}}(A,B)$, $A,B\in|{\mathcal{C}}|$, are $K$-modules and all composition maps are $K$-bilinear. On the other hand, a $K$-linear functor between two $K$-linear categories ${\mathcal{C}}, {\mathcal{D}}$ is any functor $F:{\mathcal{C}}\longrightarrow{\mathcal{D}}$ such that all maps $F_{A,B}:{\mathcal{C}}(A,B)\longrightarrow{\mathcal{D}}(F(A),F(B))$, $A,B\in|{\mathcal{C}}|$, are $K$-linear. The analogous definitions for 2-categories are as follows. Let $K$ a commutative ring with unit. A $K$-[*linear 2-category*]{} is a 2-cateogry  such that all its hom-categories $(X,Y)$ are $K$-linear, and all the composition functors $c_{X,Y,Z}:\text\Cg(X,Y)\times \text\Cg(X,Z)\longrightarrow \text\Cg(X,Z)$ are $K$-bilinear. Given $K$-linear 2-categories  and , a $K$-[*linear pseudofunctor*]{} between them is any pseudofunctor ${\mathcal{F}}:\text\Cg\longrightarrow \text\Dg$ such that all functors ${\mathcal{F}}_{X,Y}:\text\Cg(X,Y){\longrightarrow}\text\Dg({\mathcal{F}}(X),{\mathcal{F}}(Y))$, $X,Y\in|\text\Cg|$, are $K$-linear. Notice that, according to this definition, we only have a structure of $K$-module on the sets of 2-morphisms. This will mean that, in our definition of deformation below, all structural 1-morphisms will remain undeformed, and the only thing susceptible to be deformed will be the 2-morphisms. The following result brings together some easy facts about $K$-linear 2-categories and pseudofunctors whose proof is left to the reader. \[prop\_K\_linear\] Let ,  and  be $K$-linear 2-categories, and ${\mathcal{F}},{\mathcal{F}}':\text\Bg{\longrightarrow}\text\Cg$ and ${\mathcal{G}}:\text\Cg{\longrightarrow}\text\Dg$ $K$-linear pseudofunctors. Then: \(i) The product 2-category $\text\Bg\times\text\Cg$ is $K$-linear. \(ii) The composition pseudofunctor ${\mathcal{G}}\circ{\mathcal{F}}:\text\Bg{\longrightarrow}\text\Dg$ is $K$-linear. \(iii) For any $\xi,\zeta:{\mathcal{F}}\Longrightarrow{\mathcal{F}}'$, the set ${\rm Mod}(\xi,\zeta)$ (resp. ${\rm PseudMod}(\xi,\zeta)$) of modifications (resp. pseudomodifications) between $\xi$ and $\zeta$ is a $K$-vector space. Our main objects of interest are the $K$-linear semigroupal 2-categories, defined as follows: A $K$-[*linear semigroupal 2-category*]{} is a semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$ such that both  and $\otimes$ are $K$-linear. Fundamental for the definitions of infinitesimal deformation given later are the notions of $R$-linear extension for $K$-linear (semigroupal) 2-categories and $K$-linear pseudofunctors, for any $K$-algebra $R$. As the reader will see, in the first case it provides us with the necessary “tangent space” at the point of $X(\text\Cg)$ defined by the semigroupal structure in question. \[extensio\_categoria\] Let  be a $K$-linear 2-category. Given a $K$-algebra $R$, the $R$-[*linear extension of*]{}  is the $R$-linear 2-category $\text\Cg^0_R$ defined as follows: (1) its objects and 1-morphisms are the same as in , (2) its sets of 2-morphisms are given by $(\text\Cg^0_R(X,Y))(f,f'):=(\text\Cg(X,Y))(f,f')\otimes_K R$, (3) the vertical composition is defined by $(\tau\otimes r)\cdot(\tilde{\tau}\otimes\tilde{r}):=(\tau\cdot\tilde{\tau})\otimes(r\tilde{r})$ and by linear extension, (4) the composition functors $(c_{X,Y,Z})_R\equiv \circ_R:\text\Cg^0_R(X,Y)\times\text\Cg^0_R(Y,Z){\longrightarrow}\text\Cg^0_R(X,Z)$ are defined on 1-morphisms as in  and on 2-morphisms by $(\eta\otimes s)\circ_R(\tau\otimes r):=(\eta\circ\tau)\otimes(rs)$ and by linear extension, and (5) the identity 1-morphisms are the same as in $\text\Cg$. The reader may easily check that these data indeed define an $R$-linear 2-category. The reason to add the zero superscript in $\text\Cg^0_R$ will be soon understood. If $K$ is a topological ring and $R$ is an -adically complete local $K$-algebra (for example, $R=K[[h]]$), it can be defined the -adically complete $R$-linear extension of . This extension is the starting point for the definition of the -adically complete infinitesimal deformations. In this work, however, we are mainly interested in the non topological case, and we leave to the reader to figure out the corresponding definitions in this topological setting. We are specially interested in the case $R=K[\epsilon]/<\epsilon^{n+1}>$. The corresponding $R$-linear extension will be denoted by $\text\Cg^0_{(n)}$. In this case, a generic 2-morphism $\tau_{\epsilon}:f\Longrightarrow f':X\longrightarrow Y$ in the linear extension can be written in the form $$\label{tau_n} \tau_{\epsilon}=\tau_0+\tau_1\epsilon+\cdots+\tau_n\epsilon^n$$ where $\tau_0,\ldots,\tau_n\in\text\Cg(X,Y)(f,f')$. The above definition is part of a functor of extension of scalars for $K$-linear 2-categories, a fact which allows us to further introduce the required $R$-linear extension of a $K$-linear pseudofunctor. \[extensio\] Let ,  be two $K$-linear 2-categories and $R$ a $K$-algebra. Then, any $K$-linear pseudofunctor ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*,{\mathcal{F}}_0)$ from  to  extends to an $R$-linear pseudofunctor ${\mathcal{F}}^0_R=(|{\mathcal{F}}|^0_R,({\mathcal{F}}_*)^0_R,(\widehat{{\mathcal{F}}}_*)^0_R,({\mathcal{F}}_0)^0_R)$ from $\text\Cg^0_R$ to $\text\Dg^0_R$. Furthermore, if $\xi=(\xi_*,\widehat{\xi}_*)$ is a pseudonatural transformation between two $K$-linear pseudofunctors ${\mathcal{F}}$ and ${\mathcal{G}}$, it extends to a pseudonatural transformation $\xi^0_R=((\xi_*)^0_R,(\widehat{\xi}_*)^0_R))$ between the $R$-linear extensions ${\mathcal{F}}^0_R$ and ${\mathcal{G}}^0_R$, and the same thing for modifications between pseudonatural transformations. Take $|{\mathcal{F}}|^0_R=|{\mathcal{F}}|$, and for any pair of objects $X,Y\in|\text\Cg|$, define the functor $({\mathcal{F}}_{X,Y})^0_R$ as follows: $({\mathcal{F}}_{X,Y})^0_R(f)={\mathcal{F}}_{X,Y}(f)$ for all 1-morphisms $f:X{\longrightarrow}Y$, and on 2-morphisms, take $({\mathcal{F}}_{X,Y})^0_R(\tau\otimes r)={\mathcal{F}}_{X,Y}(\tau)\otimes r$ and extend by linearity. Finally, define a pseudofunctorial structure on ${\mathcal{F}}^0_R$ by taking $(\widehat{{\mathcal{F}}})^0_R(g,f)=\widehat{{\mathcal{F}}}(g,f)\otimes 1$ and $({\mathcal{F}}_0)^0_R(X)={\mathcal{F}}_0(X)\otimes 1$ for all objects $X,Y,Z$ and 1-morphisms $f,g$. The rest of the proposition is proved similarly and is left to the reader. As a by-product, we obtain the notion of $R$-linear extension for $K$-linear semigroupal 2-categories. Indeed, we have: \[extensio\_semigrupal\] Let $(\text\Cg,\otimes,a,\pi)$ be a $K$-linear semigroupal 2-category. Then, for any $K$-algebra $R$, the extension $\text\Cg^0_R$ inherits a structure $(\otimes^0_R,a^0_R,\pi^0_R)$ of $R$-linear semigroupal 2-category. Indeed, to give a semigroupal structure on a 2-category means to give a pseudofunctor, a pseudonatural isomorphism and a modification, and all of them can be extended according to the previous Proposition. We leave to the reader to check that this extensions satisfy the appropriate axioms. We can now define the corresponding notions of infinitesimal deformation. Let us begin with the case of a $K$-linear pseudofunctor. According to Proposition \[extensio\], given such a pseudofunctor ${\mathcal{F}}:\text\Cg{\longrightarrow}\text\Dg$, we have a “copy” of it ${\mathcal{F}}_R^0:\text\Cg_R^0{\longrightarrow}\text\Dg^0_R$ in the “category of $R$-linear pseudofunctors”. The reason to consider such a copy is that the infinitesimal deformations of ${\mathcal{F}}$ will actually be, strictly speaking, deformations of that copy, for some local $K$-algebra $R$. The copy itself will be called the [*null deformation of*]{} ${\mathcal{F}}$ [*over*]{} $R$. A generic deformation is then defined as follows. Let ,  be two $K$-linear 2-categories, and ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*,{\mathcal{F}}_0)$ a $K$-linear pseudofunctor between them. Given a local $K$-algebra $R$, a [*purely pseudofunctorial infinitesimal deformation of*]{} ${\mathcal{F}}$ [*over*]{} $R$ is the pair $(|{\mathcal{F}}|^0_R,({\mathcal{F}}_*)^0_R)$ of Proposition \[extensio\] equipped with a pseudofunctorial structure $((\widehat{{\mathcal{F}}}_*)_R,({\mathcal{F}}_0)_R)$ which reduces mod.  to that of the null deformation. When $R=K[\epsilon]/<\epsilon^{n+1}>$, the corresponding deformations are called [*purely pseudofunctorial*]{} $n^{th}$-[*order deformations of*]{} ${\mathcal{F}}$. The terms “purely pseudofunctorial” in this definition refer to the fact that the only deformed thing is the pseudofunctorial structure of ${\mathcal{F}}_R^0$, the source and target 2-categories remaining undeformed, in the sense that they are simply substituted for the corresponding $R$-linear extensions. For example, it is easy to see that to give a purely pseudofunctorial $n^{th}$-order deformation of ${\mathcal{F}}$ simply amounts to give new families of 2-isomorphisms of the form $$\begin{aligned} &\widehat{{\mathcal{F}}}_{\epsilon}(g,f)=\widehat{{\mathcal{F}}}(g,f)+\widehat{{\mathcal{F}}}^{(1)}(g,f)\epsilon+\cdots+ \widehat{{\mathcal{F}}}^{(n)}(g,f)\epsilon^n \\ &({\mathcal{F}}_0)_{\epsilon}(X)={\mathcal{F}}_0(X)+{\mathcal{F}}_0^{(1)}(X)\epsilon+\cdots+{\mathcal{F}}_0^{(n)}(X)\epsilon^n\end{aligned}$$ where $\widehat{{\mathcal{F}}}^{(i)}(g,f):{\mathcal{F}}(g)\circ {\mathcal{F}}(f)\Longrightarrow{\mathcal{F}}(g\circ f)$ and ${\mathcal{F}}_0^{(i)}(X):{\mathcal{F}}(id_X)\Longrightarrow id_{{\mathcal{F}}(X)}$, for all $i=1,\ldots,n$, are suitable 2-morphisms in  such that the above 2-isomorphisms indeed define a pseudofunctorial structure on the pair $(|{\mathcal{F}}|_R^0,({\mathcal{F}}_*)_R^0)$. To emphasize that, a purely pseudofunctorial $n^{th}$-order deformation will be denoted by the pair $(\{\widehat{{\mathcal{F}}}^{(i)}\}_{i=1,\ldots,n},\{{\mathcal{F}}_0^{(i)}\}_{i=1,\ldots,n})$. In particular, when all these 2-morphisms are zero, we recover the null deformation ${\mathcal{F}}^0_R$. We are only interested in the equivalence classes of such purely pseudofunctorial infinitesimal deformations, two such deformations being considered equivalent in the following sense: Let ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*,{\mathcal{F}}_0)$ be a $K$-linear pseudofunctor. Then, given two purely pseudofunctorial infinitesimal deformations ${\mathcal{F}}_R=((\widehat{{\mathcal{F}}}_*)_R,({\mathcal{F}}_0)_R)$ and ${\mathcal{F}}'_R=((\widehat{{\mathcal{F}}}_*)'_R,({\mathcal{F}}_0)'_R)$, they are called [*equivalent*]{} if there exists a pseudonatural isomorphism $\xi:{\mathcal{F}}_R\Longrightarrow{\mathcal{F}}'_R$ such that 1. $\xi_X=id_{{\mathcal{F}}(X)}$ for all objects $X\in|\text\Cg|$, and 2. $\widehat{\xi}(f)=1_{{\mathcal{F}}(f)}\ \ (\text{mod. \mm})$, for all 1-morphisms $f$ of . [For]{} later use, let us make explicit what this definition means in the case of first order deformations. \[equivalent\_pseudo\] Two purely pseudofunctorial first order deformations ${\mathcal{F}}_{\epsilon}$, ${\mathcal{F}}'_{\epsilon}$ of a $K$-linear pseudofunctor ${\mathcal{F}}:\text\Cg{\longrightarrow}\text\Dg$, defined by 2-morphisms $(\widehat{{\mathcal{F}}}^{(1)}(g,f),{\mathcal{F}}_0^{(1)}(X))$ and $((\widehat{{\mathcal{F}}}^{(1)})'(g,f),({\mathcal{F}}_0^{(1)})'(X))$, respectively, are equivalent if and only if there exists 2-morphisms $\widehat{\xi}^{(1)}(f):{\mathcal{F}}(f)\Longrightarrow{\mathcal{F}}(f)$, for all 1-morphisms $f$ of , satisfying the following conditions: 1. They are natural in $f$. 2. For all composable 1-morphisms $X\stackrel{f}{{\longrightarrow}}Y\stackrel{g}{{\longrightarrow}}Z$, it holds $$\begin{aligned} (\widehat{{\mathcal{F}}}^{(1)})'(g,f)-\widehat{{\mathcal{F}}}^{(1)}(g,f)=\widehat{{\mathcal{F}}}&(g,f) \cdot(1_{{\mathcal{F}}(g)}\circ\widehat{\xi}^{(1)}(f))-\widehat{\xi}^{(1)}(g\circ f)\cdot\widehat{{\mathcal{F}}}(g,f)+ \\ &+\widehat{{\mathcal{F}}}(g,f)\cdot (\widehat{\xi}^{(1)}(g)\circ 1_{{\mathcal{F}}(f)})\end{aligned}$$ 3. For all objects $X$ of , it holds $$({\mathcal{F}}_0^{(1)})'(X)-{\mathcal{F}}_0^{(1)}(X)={\mathcal{F}}_0(X)\cdot\widehat{\xi}^{(1)}(id_X)$$ Indeed, let us go back to the definition of pseudonatural transformation (see Definition \[trans\_quasi\]) and take ${\mathcal{F}}={\mathcal{F}}_{\epsilon}$, ${\mathcal{G}}={\mathcal{F}}_{\epsilon}'$, and $\xi$ defined by $\xi_X=id_{{\mathcal{F}}(X)}$ and $\widehat{\xi}(f)=1_{{\mathcal{F}}(f)}+\widehat{\xi}^{(1)}(f)\epsilon$. The conditions above follow then by writing out the first order terms in $\epsilon$ in every condition satisfied by $\xi$. Let us consider now the deformations of a $K$-linear semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$. As in the case of a pseudofunctor, the first thing we need is its “copy” $(\text\Cg^0_R,\otimes^0_R,a^0_R,\pi^0_R)$ in the “category of $R$-linear semigroupal 2-categories” (see Corollary \[extensio\_semigrupal\]). It will be called the [*null deformation of*]{} $(\text\Cg,\otimes,a,\pi)$ [*over*]{} $R$. A generic infinitesimal deformation is then a deformation of that copy. More precisely: \[deformacio\_2\_categoria\_semigrupal\] Let $(\text\Cg,\otimes,a,\pi)$ be a $K$-linear semigroupal 2-category, and let $R$ be a local $K$-algebra, with maximal ideal . An [*infinitesimal deformation of*]{} $(\text\Cg,\otimes,a,\pi)$ [*over*]{} $R$ is the $R$-linear extension 2-category $\text\Cg^0_R$ equipped with a semigroupal structure $(\otimes_R,a_R,\pi_R)$ which reduces mod.  to that of the null deformation. More explicitly, $(\otimes_R,a_R,\pi_R)$ must be such that: (1) $\otimes_R$ only differs from $\otimes^0_R$ in the pseudofunctorial structure, (2) the 1-isomorphisms $(a_R)_{X,Y,Z}$ coincide with those of $a_R^0$, and (3) all structural 2-isomorphisms $\widehat{\otimes}_R((f',g'),(f,g)),(\otimes_0)_R(X,Y), \widehat{a}_R(f,g,h),(\pi_R)_{X,Y,Z,T}$ reduce mod.  to those of the null deformation. When $R=K[\epsilon]/<\epsilon^{n+1}>$, the corresponding infinitesimal deformations are called $n^{th}$-[*order deformations of*]{} . Since the associator as well as the left and right unit constraints on the composition of 1-morphisms (which we are assuming trivial) both live in the “deformable world” of 2-morphisms, it would be possible to modify the above definitions of infinitesimal deformation in such a way that also the bicategory structure of  is deformed. In the case of a pseudofunctor, this will lead us to the notion of a not necessarily purely pseudofunctorial deformation. However, we will not proceed in this direction, and we will assume that all bicategories, the undeformed as well as the deformed ones, are always 2-categories. For example, according to the above definition, an arbitrary $n^{th}$-order deformation of $(\text\Cg,\otimes,a,\pi)$ amounts to a new set of structural 2-isomorphism $\widehat{\otimes}_{\epsilon},(\otimes_0)_{\epsilon},\widehat{a}_{\epsilon},\pi_{\epsilon}$ of the form in Eq.(\[tau\_n\]) with the zero order term equal to the original 2-isomorphism, i.e., $$\begin{aligned} &\widehat{\otimes}_{\epsilon}((f',g'),(f,g))=\widehat{\otimes}((f',g'),(f,g))+ \widehat{\otimes}^{(1)}((f',g'),(f,g))\epsilon+\cdots+\widehat{\otimes}^{(n)}((f',g'),(f,g)) \epsilon^n \\ &(\otimes_0)_{\epsilon}(X,Y)=\otimes_0(X,Y)+\otimes_0^{(1)}(X,Y)\epsilon+ \cdots+ \otimes_0^{(n)}(X,Y)\epsilon^n \\ &\widehat{a}_{\epsilon}(f,g,h)=\widehat{a}(f,g,h)+\widehat{a}^{(1)}(f,g,h)\epsilon+\cdots+ \widehat{a}^{(n)}(f,g,h)\epsilon^n \\ &(\pi_{\epsilon})_{X,Y,Z,T}=\pi_{X,Y,Z,T}+ (\pi^{(1)})_{X,Y,Z,T}\epsilon+\cdots+(\pi^{(n)})_{X,Y,Z,T}\epsilon^n\end{aligned}$$ where $\widehat{\otimes}^{(i)}((f',g'),(f,g))$, $\otimes_0^{(i)}(X,Y)$, $\widehat{a}^{(i)}(f,g,h)$ and $(\pi^{(i)})_{X,Y,Z,T}$, for all $i=1,\ldots,n$, are suitable 2-morphisms in  with the same source and target 1-morphisms as the corresponding undeformed 2-isomorphisms and such that the whole set of new 2-isomorphisms satisfy all the necessary equations to define a semigroupal structure on $\text\Cg^0_{(n)}$. Such a $n^{th}$-order deformation will be denoted by $(\{\widehat{\otimes}^{(i)}\}_i, \{\otimes_0^{(i)}\}_i, \{\widehat{a}^{(i)}\}_i,\{\pi^{(i)}\}_i)$. In particular, when all these 2-morphisms are zero we again recover the null deformation. As in the case of pseudofunctors, we are only interested in the equivalence classes of infinitesimal deformations. Let $(\text\Cg,\otimes,a,\pi)$ be a $K$-linear semigroupal 2-category. Two infinitesimal deformations over $R$ $(\otimes_R,a_R,\pi_R)$ and $(\otimes'_R,a'_R,\pi'_R)$ are called [*equivalent*]{} if the identity 2-functor $id_{\text\Cg^0_R}:(\text\Cg^0_R,\otimes_R,a_R,\pi_R){\longrightarrow}(\text\Cg^0_R,\otimes'_R,a'_R,\pi'_R)$ admits a semigroupal structure $(\psi,\omega)$ such that: 1. $\psi_{X,Y}=id_{X\otimes Y}$ for all objects $X,Y$; 2. $\widehat{\psi}(f,g)=1_{f\otimes g}\ \ (\text{mod. \mm})$, for all 1-morphisms $f,g$, and 3. $\omega_{X,Y,Z}=(\otimes_0(X\otimes Y,Z)^{-1}\circ 1_{a_{X,Y,Z}})\cdot(1_{a_{X,Y,Z}}\circ\otimes_0(X,Y\otimes Z))\ \ (\text{mod. \mm})$, for all objects $X,Y,Z$. The deformations will be called $\omega$-[*equivalent*]{} when there exists a semigroupal structure $(\psi,\omega)$ satisfying the first and third conditions above and such that $\widehat{\psi}(f,g)=1_{f\otimes g}$ for all $f,g$ (not only mod. ). Similarly, the deformations will be called $\psi$-[*equivalent*]{} when there exists a semigroupal structure $(\psi,\omega)$ satisfying the first and second conditions above and such that $\omega_{X,Y,Z}=(\otimes_0(X\otimes Y,Z)^{-1}\circ 1_{a_{X,Y,Z}})\cdot(1_{a_{X,Y,Z}}\circ\otimes_0(X,Y\otimes Z))$ for all $X,Y,Z$ (not only mod. ). Let us also make explicit for its later use what this definition means in the case of first order deformations. To simplify equations, however, let us assume, without loss of generality by Theorem \[strictification\], that the undeformed tensor product $\otimes$ is unitary, i.e., that all 2-isomorphisms $\otimes_0(X,Y)$ are identities (see Definition \[pseudofunctor\]). Notice, however, that the deformed tensor product may no longer be unitary. \[equivalents\] Let $(\text\Cg,\otimes,a,\pi)$ be a $K$-linear semigroupal 2-category, with $\otimes$ a unitary tensor product. Let’s consider two first order deformations defined by 2-morphisms $(\widehat{\otimes}^{(1)},\otimes_0^{(1)},\widehat{a}^{(1)},\pi^{(1)})$ and $((\widehat{\otimes}^{(1)})',(\otimes_0^{(1)})',(\widehat{a}^{(1)})',(\pi^{(1)})')$. Then, they are equivalent if and only there exists 2-morphisms $\widehat{\psi}^{(1)}(f,g):f\otimes g\Longrightarrow f\otimes g$ and $(\omega^{(1)})_{X,Y,Z}:a_{X,Y,Z}\Longrightarrow a_{X,Y,Z}$, for all objects $X,Y,Z$ and 1-morphisms $f,g$ of , such that the following equations hold: E$\widehat{\psi}$1 : For all 2-morphisms $(\tau,\sigma):(f,g)\Longrightarrow(\tilde{f},\tilde{g})$ of $\text\Cg^2$ $$(\tau\otimes\sigma)\cdot\widehat{\psi}^{(1)}(f,g)= \widehat{\psi}^{(1)}(\tilde{f},\tilde{g})\cdot(\tau\otimes\sigma)$$ E$\widehat{\psi}$2 : For all composable 1-morphisms $(X,Y)\stackrel{(f,g)}{{\longrightarrow}}(X',Y')\stackrel{(f',g')}{{\longrightarrow}}(X'',Y'')$ of $\text\Cg^2$ $$\begin{aligned} \widehat{\psi}^{(1)}(f'\circ f,g'\circ g)\cdot&\widehat{\otimes}((f',g'),(f,g)))+ \widehat{\otimes}^{(1)}(f',g'),(f,g))= \\ &=(\widehat{\otimes}^{(1)})'((f',g'),(f,g))+\widehat{\otimes}((f',g'),(f,g)) \cdot(\widehat{\psi}^{(1)}(f',g')\circ 1_{f\otimes g})+ \\ &\ \ \ \ + \widehat{\otimes}((f',g'),(f,g))\cdot(1_{f'\otimes g'}\circ\widehat{\psi}^{(1)}(f,g))\end{aligned}$$ E$\widehat{\psi}$3 : For all objects $(X,Y)$ of $\text\Cg\times\text\Cg$ $$\widehat{\psi}^{(1)}(id_X,id_Y)=\otimes_0^{(1)}(X,Y)- (\otimes_0^{(1)})'(X,Y)$$ E$\omega$1 : For all 1-morphisms $(f,g,h):(X,Y,Z){\longrightarrow}(X',Y',Z')$ of $\text\Cg\times\text\Cg\times\text\Cg$ $$\begin{aligned} (\widehat{a}^{(1)})'(f,g,h)-&\widehat{a}(f,g,h)\cdot((\widehat{\otimes}^{(1)})'((id_{X'\otimes Y'},id_{Z'}),(f\otimes g,h))\circ 1_{a_{X,Y,Z}})+ \\ & +\widehat{a}(f,g,h)\cdot((\widehat{\psi}^{(1)}(f,g)\otimes 1_h)\circ 1_{a_{X,Y,Z}})+ \\ &+\widehat{a}(f,g,h)\cdot((\widehat{\otimes}^{(1)})'((f\otimes g,h),(id_{X\otimes Y},id_Z))\circ 1_{a_{X,Y,Z}})+\ \\ &+\widehat{a}(f,g,h)\cdot(\widehat{\psi}^{(1)}(f\otimes g,h)\circ 1_{a_{X,Y,Z}})+ \\ &+\widehat{a}(f,g,h)\cdot(1_{(f\otimes g)\otimes h}\circ (\omega^{(1)})_{X,Y,Z})= \\ &\ \ \ \ =((\omega^{(1)})_{X',Y',Z'}\circ 1_{f\otimes(g\otimes h)})\cdot\widehat{a}(f,g,h)- \\ &\ \ \ \ \ \ \ -(1_{a_{X',Y',Z'}}\circ(\widehat{\otimes}^{(1)})'((id_{X',}id_{Y'\otimes Z'}),(f,g\otimes h)))\cdot\widehat{a}(f,g,h)+ \\ &\ \ \ \ \ \ \ +(1_{a_{X',Y',Z'}}\circ(1_f\otimes\widehat{\psi}^{(1)}(g,h))) \cdot\widehat{a}(f,g,h)+ \\ &\ \ \ \ \ \ \ + (1_{a_{X',Y',Z'}}\circ(\widehat{\otimes}^{(1)})'((f,g\otimes h), (id_X,id_{Y\otimes Z})))\cdot\widehat{a}(f,g,h)+ \\ &\ \ \ \ \ \ \ + (1_{a_{X',Y',Z'}}\circ\widehat{\psi}^{(1)}(f,g\otimes h))\cdot\widehat{a}(f,g,h)+\widehat{a}^{(1)}(f,g,h)\end{aligned}$$ E$\omega$2 : For all objects $(X,Y,Z,T)$ of $\text\Cg^4$ $$\begin{aligned} (\pi^{(1)})'&_{X,Y,Z,T}+ \\ &+\pi_{X,Y,Z,T}\cdot((\otimes_0^{(1)})'((X\otimes Y)\otimes Z,T)\circ 1_{(a_{X,Y,Z}\otimes id_T)\circ a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})})+ \\ &+\pi_{X,Y,Z,T}\cdot(((\omega^{(1)})_{X,Y,Z}\otimes 1_{id_T})\circ 1_{a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})})+ \\ &+ \pi_{X,Y,Z,T}\cdot(1_{a_{X,Y,Z}\otimes id_T}\circ(\otimes_0^{(1)})'(X\otimes(Y\otimes Z),T) \circ 1_{a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})})+ \\ &+\pi_{X,Y,Z,T}\cdot (\widehat{\psi}^{(1)}(a_{X,Y,Z},id_T) \circ 1_{a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})})- \\ &-\pi_{X,Y,Z,T}\cdot (1_{a_{X,Y,Z}\otimes id_T}\circ(\widehat{a}^{(1)})'(id_X,id_{Y\otimes Z},id_T)\circ 1_{id_X\otimes a_{Y,Z,T}})+ \\ &+\pi_{X,Y,Z,T}\cdot (1_{a_{X,Y,Z}\otimes id_T}\circ(\omega^{(1)})_{X,Y\otimes Z,T}\circ 1_{id_X\otimes a_{Y,Z,T}})+ \\ &+\pi_{X,Y,Z,T}\cdot(1_{(a_{X,Y,Z}\otimes id_T)\circ a_{X,Y\otimes Z,T}}\circ(\otimes_0^{(1)})'(X,(Y\otimes Z)\otimes T)\circ 1_{id_X\otimes a_{Y,Z,T}})+ \\ &+\pi_{X,Y,Z,T}\cdot(1_{(a_{X,Y,Z}\otimes id_T)\circ a_{X,Y\otimes Z,T}}\circ(1_{id_X}\otimes(\omega^{(1)})_{Y,Z,T}))+ \\ &+ \pi_{X,Y,Z,T}\cdot(1_{(a_{X,Y,Z}\otimes id_T)\circ a_{X,Y\otimes Z,T}\circ(id_X\otimes a_{Y,Z,T})}\circ(\otimes_0^{(1)})'(X,Y\otimes (Z\otimes T)))+ \\ &+\pi_{X,Y,Z,T}\cdot(1_{(a_{X,Y,Z}\otimes id_T)\circ a_{X,Y\otimes Z,T}}\circ\widehat{\psi}^{(1)}(id_X,a_{Y,Z,T}))= \\ &\ \ \ \ = -(\widehat{a}^{(1)}(id_{X\otimes Y},id_Z,id_T)\circ 1_{a_{X,Y,Z\otimes T}})\cdot\pi_{X,Y,Z,T}+ \\ &\ \ \ \ \ \ +((\omega^{(1)})_{X\otimes Y,Z,T}\circ 1_{a_{X,Y,Z\otimes T}}) \cdot\pi_{X,Y,Z,T}- \\ &\ \ \ \ \ \ -(1_{a_{X\otimes Y,Z,T}}\circ \widehat{\otimes}^{(1)}((id_{X\otimes Y},id_{Z\otimes T}),(id_{X\otimes Y},id_{Z\otimes T})) \circ 1_{a_{X,Y,Z\otimes T}})\cdot\pi_{X,Y,Z,T}- \\ &\ \ \ \ \ \ -(1_{a_{X\otimes Y,Z,T}}\circ (\widehat{\psi}^{(1)}(id_X,id_Y)\otimes 1_{id_{Z\otimes T}}) \circ 1_{a_{X,Y,Z\otimes T}})\cdot\pi_{X,Y,Z,T}+ \\ &\ \ \ \ \ \ +(1_{a_{X\otimes Y,Z,T}}\circ (1_{id_{X\otimes Y}}\otimes\widehat{\psi}^{(1)}(id_Z,id_T)) \circ 1_{a_{X,Y,Z\otimes T}})\cdot\pi_{X,Y,Z,T}+ \\ &\ \ \ \ \ \ +(1_{a_{X\otimes Y,Z,T}}\circ (\widehat{\otimes}^{(1)})'((id_{X\otimes Y},id_{Z\otimes T}),(id_{X\otimes Y},id_{Z\otimes T})) \circ 1_{a_{X,Y,Z\otimes T}})\cdot\pi_{X,Y,Z,T}- \\ &\ \ \ \ \ \ -(1_{a_{X\otimes Y,Z,T}}\circ(\widehat{a}^{(1)})'(id_X,id_Y,id_{Z\otimes T}))\cdot\pi_{X,Y,Z,T}+ \\ &\ \ \ \ \ \ + (1_{a_{X\otimes Y,Z,T}}\circ(\omega^{(1)})_{X,Y,Z\otimes T})\cdot\pi_{X,Y,Z,T}+ \\ &\ \ \ \ \ \ +(\pi^{(1)})_{X,Y,Z,T}\end{aligned}$$ The proof is a long but straightforward computation of the first order term in each of the conditions in Proposition \[estructura\_semigrupal\] when ${\mathcal{F}}$ is taken equal to the identity 2-functor $id_{\text\Cg^0_{(1)}}$, the semigroupal structures $(\otimes,a,\pi)$ and $(\otimes',a',\pi')$ are those of the first order deformations, and the $\psi$ and $\omega$ are of the form $$\begin{aligned} &\psi_{X,Y}=id_{X\otimes Y} \\ &\widehat{\psi}(f,g)=1_{f\otimes g}+\widehat{\psi}^{(1)}(f,g)\epsilon \\ &\omega_{X,Y,Z}=1_{a_{X,Y,Z}}+\omega^{(1)}_{X,Y,Z}\epsilon\end{aligned}$$ Notice that the zero order term of $\omega$ is trivial because we are assuming $\otimes$ is unitary. Cohomology of a unitary pseudofunctor ===================================== This section contains preliminary results that will be used in Section 8 to construct the cochain complex which describes the simultaneous deformations of both the tensor product and the associator in a $K$-linear semigroupal 2-category. More explicitly, we associate a cohomology to an arbitrary $K$-linear unitary pseudofunctor and prove that this cohomology describes its purely pseudofunctorial infinitesimal deformations in the sense of Gerstenhaber. The main idea is to use the fact mentioned in Section 2 that a pseudofunctor between one object bicategories corresponds to the notion of a monoidal functor. In this sense, our results generalize the cohomology theory for monoidal functors described by Yetter in [@dY98]. Let us remark that our restriction to the case of unitary pseudofunctors implies no loss of generality for our purposes, because, as indicated before, the results obtained here will just be used to study the deformations of the tensor product and the associator in a $K$-linear semigroupal 2-category. Now, by Theorem \[strictification\], the undeformed semigroupal 2-category may be assumed to be a Gray semigroup and, hence, such that the original tensor product is indeed a unitary pseudofunctor. Let us consider a $K$-linear unitary pseudofunctor ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*)$ between $K$-linear 2-categories  and . Since we assume  is a 2-category, for each $n\geq 2$ and each ordered family $X_0,\ldots,X_n$ of $n+1$ objects of , we have a uniquely induced composition functor $c^{\text\Cg}_{X_0,\ldots,X_n}:\text\Cg(X_{n-1},X_n)\times\text \Cg(X_{n-2},X_{n-1})\times\cdots\times\text\Cg(X_0,X_1)\longrightarrow\text\Cg(X_0,X_n)$, obtained by applying the appropriate elementary composition functors $c^{\text\Cg}_{X,Y,Z}$ in any order [^7]. In the same way, we have the induced composition functors $c^{\text\Dg}_{{\mathcal{F}}(X_0),\ldots,{\mathcal{F}}(X_n)}$ for all $X_0,\ldots,X_n$. Then, given $X_0,\ldots,X_n$, let’s consider the functors $C{\mathcal{F}}_{X_0,\ldots,X_n},{\mathcal{F}}C_{X_0,\ldots,X_n}:\text\Cg(X_{n-1},X_n)\times\text \Cg(X_{n-2},X_{n-1})\times\cdots\times\text\Cg(X_0,X_1){\longrightarrow}\text\Dg({\mathcal{F}}(X_0),{\mathcal{F}}(X_n))$ defined by $$\begin{aligned} C{\mathcal{F}}_{X_0,\ldots,X_n}&:=c^{\text\Dg}_{{\mathcal{F}}(X_0),\ldots,{\mathcal{F}}(X_n)} \circ({\mathcal{F}}_{X_{n-1},X_n}\times{\mathcal{F}}_{X_{n-2},X_{n-1}}\times\cdots\times{\mathcal{F}}_{X_0,X_1}) \\ {\mathcal{F}}C_{X_0,\ldots,X_n}&:={\mathcal{F}}_{X_0,X_n}\circ c^{\text\Cg}_{X_0,\ldots,X_n}\end{aligned}$$ When $n=1$, let $C{\mathcal{F}}_{X_0,X_1}:={\mathcal{F}}_{X_0,X_1}=:{\mathcal{F}}C_{X_0,X_1}$. We now define the vector spaces $X^n({\mathcal{F}})$ of the cochain complex we are looking for as follows: $$X^n({\mathcal{F}}):=\left\{ \begin{array}{ll} \prod_{X_0,\ldots,X_n\in|\text\Cg| } \text{Nat}(C{\mathcal{F}}_{X_0,\ldots,X_n},{\mathcal{F}}C_{X_0,\ldots,X_n}) & n\geq 1 \\ 0 & \text{otherwise} \end{array}\right.$$ Notice that they are indeed vector spaces over $K$ because we are assuming that the target 2-category  is $K$-linear. According to this definition, a generic element $\phi\in X^n({\mathcal{F}})$, $n\geq 1$, is of the form $\phi=(\phi_{X_0,\ldots,X_n})_{X_0,\ldots,X_n}$, with $\phi_{X_0,\ldots,X_n}=\{\phi(f_0,\ldots,f_{n-1})\ |\ f_i\in|\text\Cg(X_{n-i-1},X_{n-i})|,\ i=0,\ldots,n-1\}$ and $$\phi(f_0,\ldots,f_{n-1}):{\mathcal{F}}(f_0)\circ{\mathcal{F}}(f_1)\circ\cdots\circ{\mathcal{F}}(f_{n-1})\Longrightarrow {\mathcal{F}}(f_0\circ f_1\circ\cdots\circ f_{n-1})$$ a 2-morphism natural in $(f_0,\ldots,f_{n-1})$. On the other hand, the “padding” composition operators introduced in Section 3 allows us to define coboundary maps $\delta:X^{n-1}({\mathcal{F}})\longrightarrow X^n({\mathcal{F}})$, for all $n\geq 2$, in the usual way. So, if $\phi\in X^{n-1}({\mathcal{F}})$, $\delta\phi\in X^n({\mathcal{F}})$ is given by $$\begin{aligned} (\delta\phi)(f_0,f_1,\ldots,f_{n-1})=&\lceil 1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1}) \rceil_{{\mathcal{F}}(X_0),{\mathcal{F}}(X_n)}+ \\ &+\sum_{i=1}^{n-1}(-1)^i\lceil\phi(f_0,\ldots,f_{i-1}\circ f_i,\ldots,f_{n-1})\rceil_{{\mathcal{F}}(X_0),{\mathcal{F}}(X_n)}+ \\ &+(-1)^n\lceil\phi(f_0,\ldots,f_{n-2})\circ 1_{{\mathcal{F}}(f_{n-1})}\rceil_{{\mathcal{F}}(X_0),{\mathcal{F}}(X_n)}\end{aligned}$$ for all $f_i\in|\text\Cg(X_{n-i-1},X_{n-i})|$, $i=0,\ldots,n-1$. \[complex\] For any $K$-linear unitary pseudofunctor ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*)$, the pair $(X^{\bullet}({\mathcal{F}}),\delta)$ is a cochain complex. The map $\delta$ is clearly linear in $\phi$. On the other hand, naturality of $\phi(f_1,\ldots,f_{n-1})$ in $(f_1,\ldots,f_{n-1})$ easily implies the naturality of each summand of $\delta(\phi)(f_0,\ldots,f_{n-1})$ in $(f_0,\ldots,f_{n-1})$. For example, making explicit the padding, the first term is $$\lceil 1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1}) \rceil_{{\mathcal{F}}(X_0),{\mathcal{F}}(X_n)}=\widehat{{\mathcal{F}}}(f_0,f_1\circ\ldots\circ f_{n-1})\cdot (1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1}))$$ Now, for any 2-morphism $(\tau_0,\ldots,\tau_{n-1}):(f_0,\ldots,f_{n-1})\Longrightarrow(f'_0,\ldots,f'_{n-1})$, the naturality of $\widehat{{\mathcal{F}}}$ and $\phi$ in its arguments and the interchange law gives that $$\begin{aligned} {\mathcal{F}}(\tau_0\circ\ldots\circ&\tau_{n-1})\cdot\lceil 1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1}) \rceil_{{\mathcal{F}}(X_0),{\mathcal{F}}(X_n)}= \\ &={\mathcal{F}}(\tau_0\circ\ldots\circ\tau_{n-1})\cdot \widehat{{\mathcal{F}}}(f_0,f_1\circ\ldots\circ f_{n-1})\cdot (1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1})) \\ &=\widehat{{\mathcal{F}}}(f'_0,f'_1\circ\ldots\circ f'_{n-1})\cdot ({\mathcal{F}}(\tau_0)\circ{\mathcal{F}}(\tau_1\circ\ldots\circ\tau_{n-1}))\cdot (1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1})) \\ &=\widehat{{\mathcal{F}}}(f'_0,f'_1\circ\ldots\circ f'_{n-1})\cdot ({\mathcal{F}}(\tau_0)\circ[{\mathcal{F}}(\tau_1\circ\ldots\circ\tau_{n-1})\cdot \phi(f_1,\ldots,f_{n-1})]) \\ &=\widehat{{\mathcal{F}}}(f'_0,f'_1\circ\ldots\circ f'_{n-1})\cdot ({\mathcal{F}}(\tau_0)\circ[\phi(f'_1,\ldots,f'_{n-1})\cdot({\mathcal{F}}(\tau_1)\circ\ldots\circ{\mathcal{F}}(\tau_{n-1}))]) \\ &=\widehat{{\mathcal{F}}}(f'_0,f'_1\circ\ldots\circ f'_{n-1})\cdot (1_{{\mathcal{F}}(f'_0)}\circ\phi(f'_1,\ldots,f'_{n-1}))\cdot ({\mathcal{F}}(\tau_0)\circ{\mathcal{F}}(\tau_1)\circ\ldots\circ{\mathcal{F}}(\tau_{n-1})) \\ &=\lceil 1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1}) \rceil_{{\mathcal{F}}(X_0),{\mathcal{F}}(X_n)}\cdot ({\mathcal{F}}(\tau_0)\circ{\mathcal{F}}(\tau_1)\circ\ldots\circ{\mathcal{F}}(\tau_{n-1}))\end{aligned}$$ The other terms are similarly worked. Finally, to prove that $\delta^2=0$, notice first that, in computing $\delta^2(\phi)$, one can initially forget the padding operators and take them into account at the end of the computation. For example, the first term of $\delta^2(\phi)$ reads $$\begin{aligned} \delta(\lceil 1_{{\mathcal{F}}}\circ\phi\rceil)(f_0,\ldots,f_n)&= \lceil 1_{{\mathcal{F}}(f_0)}\circ\lceil 1_{{\mathcal{F}}(f_1)}\circ\phi(f_2,\ldots,f_n)\rceil\rceil \\ &\ \ \ \ \ \ \sum_{i=1}^n(-1)^i\lceil\lceil (1_{{\mathcal{F}}}\circ\phi) (f_0,\ldots,f_{i-1}\circ f_i,\ldots,f_n)\rceil\rceil \\ &\ \ \ \ \ \ (-1)^{n+1}\lceil\lceil 1_{{\mathcal{F}}(f_0)}\circ\phi(f_1,\ldots,f_{n-1})\rceil\circ 1_{{\mathcal{F}}(f_n)}\rceil\end{aligned}$$ Now, it is easy to check that the horizontal compositions of the 2-morphisms $1_{{\mathcal{F}}(f_0)}$ and $1_{{\mathcal{F}}(f_n)}$ in the first and last terms commute with the padding. Our assertion follows then from the obvious fact that taking a padding of a padding is the same as doing nothing. So, let’s provisionally forget the extra padding operators in the computation of $\delta^2(\phi)$ and use the same argument which shows the $\delta$ in the bar resolution satisfies $\delta^2=0$ to deduce that the terms formally cancel out each other. Reinserting now the padding operators in each summand of this formal expression, corresponding terms still cancel out each other because, by the coherence theorem, their paddings will also coincide. This complex will be called the [*purely pseudofunctorial deformation complex of*]{} ${\mathcal{F}}$, and the corresponding cohomology will be denoted by $H^{\bullet}({\mathcal{F}})$. Notice that the dependence of this cohomology on the structural 2-isomorphisms $\widehat{{\mathcal{F}}}_*$ of ${\mathcal{F}}$ is entirely encoded in the padding operators involved in the definition of $\delta$. Let us suppose that both $\Cg$ and  have only one object. Let us denote by $X$ the only object of $\Cg$, so that the ${\mathcal{F}}(X)$ will be the only object of . If we denote the (unique) composition functor $c_{X,X,X}:\Cg(X,X)\times\Cg(X,X){\longrightarrow}\Cg(X,X)$ in $\Cg$ by $\otimes^{\Cg}$ and in the same way denote by $\otimes^{\Dg}$ the (unique) composition functor in , the purely pseudofunctorial deformation complex of ${\mathcal{F}}$ clearly reduces to $$X^n({\mathcal{F}}):=\left\{ \begin{array}{ll} {\rm Nat}((\otimes^{\Dg})^n\circ{\mathcal{F}}^n,{\mathcal{F}}\circ(\otimes^{\Cg})^n) & n\geq 1 \\ 0 & {\rm otherwise} \end{array}\right.$$ which is exactly the cochain complex associated by Yetter [@dY98] to a semigroupal functor. We have then the following generalization of Yetter’s result [@dY98]: The equivalences classes of purely pseudofunctorial first order deformations of a $K$-linear unitary pseudofunctor ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*)$ are in bijection with the elements of $H^2({\mathcal{F}})$. Let us consider 2-isomorphisms $\widehat{{\mathcal{F}}}_{\epsilon}(g,f)=\widehat{{\mathcal{F}}}(g,f)+\widehat{{\mathcal{F}}}^{(1)}(g,f)\epsilon$ and $({\mathcal{F}}_0)_{\epsilon}(X)=1_{id_{{\mathcal{F}}(X)}}+{\mathcal{F}}_0^{(1)}(X)\epsilon$, with $\widehat{{\mathcal{F}}}^{(1)}(g,f):{\mathcal{F}}(g)\circ{\mathcal{F}}(f)\Longrightarrow{\mathcal{F}}(g\circ f)$ and ${\mathcal{F}}_0^{(1)}:{\mathcal{F}}(id_X)\Longrightarrow id_{{\mathcal{F}}(X)}$. We want to find the necessary and sufficient conditions on the $\widehat{{\mathcal{F}}}^{(1)}(g,f)$ and the ${\mathcal{F}}_0^{(1)}(X,Y)$ for these 2-isomorphisms to define a purely pseudofunctorial first order deformation of ${\mathcal{F}}$. Let us first observe the following, which in particular shows that such a first order deformation of a unitary pseudofunctor ${\mathcal{F}}$ is completely determined by the 2-morphisms $\widehat{{\mathcal{F}}}^{(1)}(g,f)$ [^8]: Let ${\mathcal{F}}=(|{\mathcal{F}}|,{\mathcal{F}}_*,\widehat{{\mathcal{F}}}_*)$ be a $K$-linear unitary pseudofunctor between $K$-linear 2-categories  and , and let’s consider a purely pseudofunctorial first order deformation given by 2-morphisms $\widehat{{\mathcal{F}}}^{(1)}(g,f)$ and ${\mathcal{F}}_0^{(1)}(X)$ (the deformation need not be unitary). Then, for all objects $X$ of , we have: \(i) $\widehat{{\mathcal{F}}}(id_X,id_X)=1_{id_{{\mathcal{F}}(X)}}$. \(ii) ${\mathcal{F}}_0^{(1)}(X)=\widehat{{\mathcal{F}}}^{(1)}(id_X,id_X)$. \(i) For any pseudofunctor between 2-categories, it directly follows from the axioms that $\widehat{{\mathcal{F}}}(f,id_X)=1_{{\mathcal{F}}(f)}\circ {\mathcal{F}}_0(X)$, for all 1-morphisms $f$. In particular, this is true when $f=id_X$. Now, if ${\mathcal{F}}$ is unitary, we have ${\mathcal{F}}(id_X)=id_{{\mathcal{F}}(X)}$, and since in any 2-category identity 2-morphisms of an identity 1-morphism are units with respect to horizontal composition, we get $\widehat{{\mathcal{F}}}(id_X,id_X)={\mathcal{F}}_0(X)=1_{id_{{\mathcal{F}}(X)}}$. \(ii) The same argument as before shows that $({\mathcal{F}}_0)_{\epsilon}(X)=\widehat{{\mathcal{F}}}_{\epsilon}(id_X,id_X)$. Notice that, although ${\mathcal{F}}_{\epsilon}$ is no longer unitary, it still holds that ${\mathcal{F}}_{\epsilon}(id_X)=id_{{\mathcal{F}}_{\epsilon}(X)}$, which is the only thing needed to show the previous equality. The desired result follows then by taking the first order terms in $\epsilon$. Let us now prove the proposition. According to the lemma and the definition of a pseudofunctor, the 2-morphisms $\widehat{{\mathcal{F}}}^{(1)}(g,f)$ and ${\mathcal{F}}_0^{(1)}(X)$ above define a purely pseudofunctorial first order deformation of ${\mathcal{F}}$ if and only if: (1) ${\mathcal{F}}_0^{(1)}(X)=\widehat{{\mathcal{F}}}^{(1)}(id_X,id_X)$, and (2) the $\widehat{{\mathcal{F}}}^{(1)}(g,f)$ are such that $\widehat{{\mathcal{F}}}_{\epsilon}(g,f)$ is natural in $(g,f)$ and satisfies the hexagonal and triangular axioms in Definition \[pseudofunctor\]. Since $\widehat{{\mathcal{F}}}(g,f)$ is natural in $g,f$ by hypothesis, naturality in $g,f$ of $\widehat{{\mathcal{F}}}_{\epsilon}(g,f)$ amounts to the naturality of $\widehat{{\mathcal{F}}}^{(1)}(g,f)$ in $g,f$. Hence the $\widehat{{\mathcal{F}}}^{(1)}(g,f)$ define an element $\widehat{{\mathcal{F}}}^{(1)}\in X^2({\mathcal{F}})$. On the other hand, the hexagonal axiom on ${\mathcal{F}}_{\epsilon}$ gives the following condition on $\widehat{{\mathcal{F}}}^{(1)}$: for all composable morphisms $X\stackrel{f}{\longrightarrow}Y\stackrel{g}{\longrightarrow}Z\stackrel{h}{\longrightarrow}T$ $$\begin{aligned} \widehat{{\mathcal{F}}}&(h,g\circ f)\cdot(1_{{\mathcal{F}}(h)}\circ\widehat{{\mathcal{F}}}^{(1)}(g,f))+ \widehat{{\mathcal{F}}}^{(1)}(h,g\circ f)\cdot(1_{{\mathcal{F}}(h)}\circ \widehat{{\mathcal{F}}}(g,f))= \\ &= \widehat{{\mathcal{F}}}^{(1)}(h\circ g,f)\cdot(\widehat{{\mathcal{F}}}(h,g)\circ 1_{{\mathcal{F}}(f)})+ \widehat{{\mathcal{F}}}(h\circ g,f)\cdot(\widehat{{\mathcal{F}}}^{(1)}(h,g)\circ 1_{{\mathcal{F}}(f)})\end{aligned}$$ It is easily seen that this condition exactly corresponds to the fact that $\delta(\widehat{{\mathcal{F}}}^{(1)})=0$. Hence, $\widehat{{\mathcal{F}}}^{(1)}$ is a 2-cocycle of the complex $X^{\bullet}({\mathcal{F}})$. As regards the triangular axioms, notice that they imply no additional conditions on $\widehat{{\mathcal{F}}}^{(1)}$. For example, since ${\mathcal{F}}_0^{(1)}(X)=\widehat{{\mathcal{F}}}^{(1)}(id_X,id_X)$, the first of these triangular axioms gives the condition $$\widehat{{\mathcal{F}}}^{(1)}(f,id_X)=1_{{\mathcal{F}}(f)}\circ\widehat{{\mathcal{F}}}^{(1)}(id_X,id_X)$$ for all 1-morphisms $f:X{\longrightarrow}Y$. Now, this condition is nothing more than the condition $\delta(\widehat{{\mathcal{F}}}^{(1)})(f,id_X,id_X)=0$, as the reader may easily check. Suppose now that the 2-morphisms $(\widehat{{\mathcal{F}}}^{(1)})'(g,f)$ define another purely pseudofunctorial first order deformation of ${\mathcal{F}}$ equivalent to the previous one. We need to show that $\widehat{{\mathcal{F}}}^{(1)}$ and $(\widehat{{\mathcal{F}}}^{(1)})'$ are cohomologous 2-cocycles. Now, from Proposition \[equivalent\_pseudo\] and by definition of $\delta$, it follows immediately that both deformations are equivalent if and only if there exists $\widehat{\xi}^{(1)}\in X^1({\mathcal{F}})$ such that $$(\widehat{{\mathcal{F}}}^{(1)})'-\widehat{{\mathcal{F}}}^{(1)}=\delta(\widehat{\xi}^{(1)}),$$ as required. Let’s remark that the third condition in Proposition \[equivalent\_pseudo\] is again superfluous. Indeed, just take $f=g=id_X$ in the second condition and use the previous lemma to conclude that $$({\mathcal{F}}_0^{(1)})'(X)-{\mathcal{F}}_0^{(1)}(X)=(\widehat{{\mathcal{F}}}^{(1)})'(id_X,id_X)- \widehat{{\mathcal{F}}}^{(1)}(id_X,id_X)=\widehat{\xi}^{(1)}(id_X)={\mathcal{F}}_0(X)\cdot\widehat{\xi}^{(1)}(id_X)$$ Cohomology theory for the deformations of the pentagonator ========================================================== In this section we initiate the study of the infinitesimal deformations of a $K$-linear semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$. Notice first of all that, according to Definition \[deformacio\_2\_categoria\_semigrupal\], in a generic infinitesimal deformation of  all structural 2-isomorphisms $\widehat{\otimes}((f',g'),(f,g))$, $\otimes_0(X,Y)$, $\widehat{a}(f,g,h)$, $\pi_{X,Y,Z,T}$ will be deformed. Now, instead of treating directly such a generic deformation from the outset, we will proceed in three steps. So, in this section we consider those infinitesimal deformations where only the pentagonator is deformed, the tensor product and the associator remaining undeformed. They will be called infinitesimal [*pentagonator-deformations*]{}. In the following section we will treat the case where both the tensor product and the associator are simultaneously deformed, although under the assumption that the tensor product remains unitary, even after the deformation. These deformations will be called infinitesimal [*unitary (tensorator,associator)-deformations*]{}. We will obtain in this way two different cohomologies that separately describe the deformations of both parts of the semigroupal structure. Section 9 is devoted to see how both cohomologies fit together in a global cohomology describing the generic infinitesimal unitary deformations. Let us consider an arbitrary $K$-linear semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$. Recall from Section 4 that, given the data $(\text\Cg,\otimes,a)$, a pentagonator $\pi$ is defined as a modification between two induced pseudonatural transformations $a^{(4)},^{(4)}a:\otimes^{(4)}\Longrightarrow ^{(4)}\otimes$ which satisfies the $K_5$ coherence relation. More generally, given the pseudofunctor $\otimes:\text\Cg\times\text\Cg{\longrightarrow}\text\Cg$, we can consider the induced pseudofunctors $\otimes^{(n)},^{(n)}\otimes:\text\Cg\times\stackrel{n)}{\cdots}\times\text\Cg{\longrightarrow}\text\Cg$, $n\geq 3$, defined by $$\begin{aligned} &\otimes^{(n)}=\otimes\circ(id_{\text\Cg}\times\otimes)\circ(id_{\text\Cg}\times id_{\text\Cg}\times\otimes)\circ\cdots\circ(id_{\text\Cg}\times\stackrel{n-2)}{\cdots}\times id_{\text\Cg}\times\otimes) \\ &^{(n)}\otimes=\otimes\circ(\otimes\times id_{\text\Cg})\circ(\otimes\times id_{\text\Cg}\times id_{\text\Cg})\circ\cdots\circ(\otimes\times id_{\text\Cg}\times\stackrel{n-2)}{\cdots}\times id_{\text\Cg})\end{aligned}$$ These are just two examples of a lot of induced tensor products of multiplicity $n$. In the same way, we can generalize the induced pseudonatural transformations $a^{(4)},^{(4)}a$ to suitable pseudonatural transformations $a^{(n)},^{(n)}a:\otimes^{(n)}\Longrightarrow ^{(n)}\otimes$, for all $n\geq 4$. Here, we also have many possible choices, because there are many possibles $a$[*-paths*]{} (i.e., paths constructed as compositions of expansions of instances of the 1-isomorphisms $a_{X,Y,Z}$) from the completely right-parenthesized object $\otimes^{(n)}(X_1,\ldots,X_n)$ to the completely left-parenthesized one $^{(n)}\otimes(X_1,\ldots,X_n)$. In the case $n=4$, the 1-isomorphisms of $a^{(4)}$ and $^{(4)}a$ are defined by taking the [*extremal paths*]{}, i.e., those characterized by the fact that, in each step, always the most internal parenthesis or the most external parenthesis, respectively, is moved. This leads us to introduce the following generalization. Given 1-isomorphisms $a_{(X,Y,Z)}:X\otimes(Y\otimes Z){\longrightarrow}(X\otimes Y)\otimes Z$ for all objects $(X,Y,Z)$ of $\text\Cg^3$, let $a^{(n)}_{(X_1,\ldots,X_n)}$, $^{(n)}a_{(X_1,\ldots,X_n)}$, $n\geq 4$, be the $a$-paths from $\otimes^{(n)}(X_1,\ldots,X_n)$ to $^{(n)}\otimes(X_1,\ldots,X_n)$ induced by the $a_{X,Y,Z}$ and corresponding to always moving the most internal parenthesis and the most external parenthesis, respectively. It is possible to give a more explicit description of these 1-isomorphisms. Indeed, the objects $\otimes^{(n)}(X_1,\ldots,X_n)$ and $^{(n)}\otimes(X_1,\ldots,X_n)$ can be graphically represented as in Fig.\[objectes\]. Then, the $a$-path $a^{(n)}_{X_1,\dots,X_n}$ corresponds to moving to the left all the legs associated to the objects $X_2,\ldots,X_{n-1}$, starting with $X_{n-1}$ and so on until $X_2$, while the path $^{(n)}a_{X_1,\dots,X_n}$ corresponds to doing the same thing but starting with $X_2$ and so on until $X_{n-1}$. Using this graphical presentation, we obtain the following description of both paths: \[na\_an\] For any $n\geq 4$ and any objects $X_1,\ldots,X_n$, we have $$\begin{aligned} a^{(n)}_{X_1,\dots,X_n}=&\left(\prod_{i=3}^n(\cdots((a_{X_1,^{(i-2)}\otimes (X_2,\ldots,X_{i-1}),X_i}\otimes id_{X_{i+1}})\otimes id_{X_{i+2}})\cdots)\otimes id_{X_n}\right)\circ \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \circ(id_{X_1}\hat{\otimes} a^{(n-1)}_{X_2,\ldots,X_n}) \\ ^{(n)}a_{X_1,\dots,X_n}&=\prod_{i=2}^{n-1}a_{^{(n-i)}\otimes(X_1,\ldots,X_{n-i}),X_{n-i+1}, \otimes^{(i-1)}(X_{n-i+2},\ldots,X_n)}\end{aligned}$$ (the product denotes composition of 1-morphisms and the symbol $\hat{\otimes}$ in the term $id_{X_1}\hat{\otimes} a^{(n-1)}_{X_2,\ldots,X_n}$ is intended to mean the composition of the tensor multiplications of $id_{X_1}$ by each one of the composition factors defining $a^{(n-1)}_{X_2,\ldots,X_n}$). For example, when $n=5$, the reader may easily check that one recovers the $a$-paths that appear in Fig. \[relacioK5\] defining the common boundary of the polytope. By definition, $a^{(n)},^{(n)}a:\otimes^{(n)}\Longrightarrow ^{(n)}\otimes$ are the pseudonatural isomorphisms (induced by $a$) whose 1-isomorphisms are precisely the above 1-morphisms $a^{(n)}_{X_1,\ldots,X_n},^{(n)}a_{X_1,\ldots,X_n}$. So, from the formulas defining the vertical composition of 2-morphisms and the horizontal compositions of the form $\xi\circ 1_{{\mathcal{F}}}$ (see Section 2), it is clear that $^{(n)}a$ is the pseudonatural isomorphism given by the pasting $$^{(n)}a=\prod_{i=2}^{n-1}(a\circ 1_{^{(n-i)}\otimes\times id_{\Cg}\times\otimes^{(i-1)}})$$ the product here denoting vertical composition of pseudonatural transformations. The formula giving the pasting that defines $a^{(n)}$ is a bit more complicated and is omitted because it is not relevant in what follows. In Fig. \[figura\_pastings\_a\_5\], however, both pastings are explicitly represented in the case $n=5$. This defines the pseudonatural isomorphisms $a^{(n)},^{(n)}a$ for all $n\geq 4$. When $n=1,2,3$, let us take $\otimes^{(2)}=^{(2)}\otimes=\otimes$, $\otimes^{(1)}=^{(1)}\otimes=id_{\text\Cg}$ and define $$\begin{aligned} &a^{(3)}=^{(3)}a=a \\ &a^{(2)}=^{(2)}a=1_{\otimes} \\ &a^{(1)}=^{(1)}a=1_{id_{\text\Cg}}\end{aligned}$$ We can now define the cochain complex we are looking for. So, for all $n\in{\mathbf N}$, let’s denote by $\widetilde{X}^n_{pent}(\text\Cg)$ the following vector spaces over $K$: $$\widetilde{X}^n_{pent}(\text\Cg)=\left\{ \begin{array}{ll} \text{PseudMod}(a^{(n+1)},^{(n+1)}a) & \text{if}\ n\geq 0 \\ 0 & \text{otherwise} \end{array} \right.$$ where $\text{PseudMod}(a^{(n+1)},^{(n+1)}a)$ denotes the set of pseudomodifications from $a^{(n+1)}$ to $^{(n+1)}a$ (see Definition \[modification\]). They are indeed vector spaces over $K$ because of Proposition \[prop\_K\_linear\]. As for the coboundary operator $\delta_{pent}:\widetilde{X}^{n-1}_{pent}(\text\Cg){\longrightarrow}\widetilde{X}^n_{pent}(\text\Cg)$, we would like to take the usual formula, i.e., $$(\delta_{pent}(\text\nn))_{X_0,\ldots,X_n}\approx 1_{id_{X_0}}\otimes\text\nn_{X_1,\ldots,X_n}+ \sum_{i=1}^n(-1)^i\text\nn_{X_0,\ldots,X_{i-1}\otimes X_i,\ldots,X_n}+ (-1)^{n+1}\text\nn_{X_0,\ldots,X_{n-1}}\otimes 1_{id_{X_n}}$$ But the 2-morphisms corresponding to each of the terms in this sum are not 2-cells from $a^{(n+1)}_{X_0,\ldots,X_n}$ to $^{(n+1)}a_{X_0,\ldots,X_n}$, as required. If $\sigma:f\Longrightarrow f'$ denotes any one of these 2-morphisms, the situation is like that in Fig. \[figura\_extensio\]. Note, however, that $f,f'$ are always $a$-paths, because  is a modification from $a^{(n)}$ to $^{(n)}a$. The claim is that, once more, there exist suitable analogs of Crane-Yetter’s “padding” operators in this 2-dimensional setting of pastings that give sense to the previous definition. Behind these new “padding” operators there is again a coherence theorem, which in this case can be stated as follows: Let $(\text\Cg,\otimes,a,\pi)$ be a semigroupal 2-category, and let $U,V$ be any two objects of both obtained as a certain tensor product of the objects $X_1,\ldots,X_n$. Then, given any two $a$-paths from $U$ to $V$, there is a unique 2-isomorphism between them constructed as a pasting of instances of the structural 2-isomorphisms of  and identity 2-morphisms (of expansions of instances) of the structural 1-isomorphisms. Although this is a particular consequence of the strictification theorem for semigroupal 2-categories (see Theorem \[strictification\]), let’s give a direct and somewhat more appealing argument using the Stasheff polyhedra [@jS63]. Recall that in [@jS63], the author introduces, for each $n\geq 2$, a polyhedron $K_n$ whose vertices are in bijection with all possible parenthesizations of a word $x_1x_2\cdots x_n$ of length $n$ and whose edges all correspond to moves of the type $-(--){\longrightarrow}(--)-$, where $-$ stands for a letter or a block of letters. Stasheff shows that $K_n$ is homeomorphic to the $(n-2)$-dimensional ball $D^{n-2}$. In particular, $K_5$ is a homeomorphic image of the 3-ball whose faces (six pentagons and three quadrilaterals) are those represented in Fig. \[relacioK5\]. On the other hand, notice that the $(n-3)$-dimensional faces of $K_n$ constituting its boundary $\partial K_n$ correspond to all meaningful ways of inserting one pair of parentheses $x_1x_2\cdots(x_k\cdots x_{k+s-1})\cdots x_n$, where $2\leq s\leq n-1$ and $1\leq k\leq n-s+1$ (in particular, $K_n$ has $n(n-1)/2-1$ such faces). Since the next insertion of parentheses must be either within the block $(x_k\cdots x_{k+s-1})$ or treating this block as a unit, this face can be thought of as a homeomorphic image of $K_{n-s+1}\times K_s$. All these faces are not disjoint, but intersect along their boundaries in such a way that the “edges” so formed correspond to inserting two pairs of parentheses in the word $x_1\cdots x_n$. This allows one to construct the $K_n$, for all $n\geq 2$, by induction: $K_2$ is a point, and given $K_2,\ldots,K_{n-1}$, the next one $K_n$ is defined as the cone on $\partial K_n$, where $\partial K_n$ is a quotient of the form $$\partial K_n:=\left(\coprod_{s,k}(K_{n+s-1}\times K_s)_k\right)/\sim$$ with $2\leq s\leq n-1$ and $1\leq k\leq n-s+1$. MacLane’s classical coherence theorem for semigroupal categories (see, for example, [@sM98]) is nothing more that an algebraic interpretation of the fact that the 2-dimensional skeleton of $K_n$, for all $n\geq 4$, is a union of homeomorphic copies of $K_4$ or $K_3\times K_3$ (see [@SS93]), a copy of $K_4$ corresponding to an instance of the Stasheff pentagon axiom and a copy of $K_3\times K_3$ corresponding to an instance of a naturality square of $a_{X,Y,Z}$ applied to a morphism which is itself some $a_{X',Y',Z'}$. In the same way, the above coherence result we want to prove is a consequence of the following fact about these polyhedra: For all $n\geq 5$, the 3-dimensional skeleton of $K_n$ is a union of homeomorphic copies of $K_5$, $K_3\times K_4$ or $K_3\times K_3\times K_3$. Indeed, the 3-cells of $K_n$ correspond to all ways of inserting $n-5$ pairs of parenthesis in the word $x_1\cdots x_n$. Now, as we have seen before, the insertion of the first pair gives an $(n-3)$-cell of $K_n$ homeomorphic to a suitable product $K_{s'_1}\times K_{s'_2}$. Similarly, the insertion of the second pair corresponds to an $(n-4)$-cell of $K_n$ homeomorphic to some product $K_{s''_1}\times K_{s''_2}\times K_{s''_3}$, because it is obtained by substituting one of the previous factors $K_{s'_i}$ for one of its faces, etc. We conclude that all 3-cells of $K_n$ will be homeomorphic images of a suitable product $K_{s_1}\times\cdots\times K_{s_{n-4}}$. Furthermore, since $K_{s_i}$ is of dimension $s_i-2$, it must be $(s_1-2)+\cdots+(s_{n-4}-2)=3$. This, together with the fact that $s_i\geq 2$ for all $i=1,\ldots,n-3$, implies that at most three of the $s_i$ can be greater than 2. In other words, any 3-cell of $K_n$ is homeomorphic to a product of $n-7$ copies of $K_2$ by $K_3\times K_3\times K_3$, by $K_2\times K_3\times K_4$ or by $K_2\times K_2\times K_5$. To prove the proposition using this lemma, let’s consider the $(\otimes,a,\pi)$-[*realization*]{} of $K_n$ associated to the objects $X_1,\ldots,X_n$, defined as follows: (1) as vertices, it has all possible tensor products of $X_1,\ldots,X_n$, with all possible parenthesizations, (2) as edges, it has expansions of instances of the structural 1-isomorphisms $a_{X,Y,Z}$, and (3) as 2-faces, instances of the structural 2-isomorphisms $\widehat{\otimes}((f',g'),(f,g))$, $\otimes_0(X,Y)$, $\widehat{a}(f,g,h)$ and $\pi_{X,Y,Z,T}$. Observe that, here, the $X,Y,Z,T$ are all objects obtained as tensor products of the $X_1,\ldots,X_n$, and that the $f,g,h,f',g'$ are all identity 1-morphisms or instances of the 1-isomorphisms $a_{X,Y,Z}$. Any a-path from $U$ to $V$ is then a path in this realization of $K_n$, and the 2-isomorphisms mentioned in the proposition between two such paths correspond to 2-faces between them in this realization. Now, two such 2-faces are equal whenever the 3-cell diagram they define is commutative. But, according to the previous lemma, any 3-cell in $K_n$ is a union of 3-cells of the types $K_5$, $K_3\times K_4$ and $K_3\times K_3\times K_3$. The proof of the proposition then finishes by checking that the 3-dimensional diagrams corresponding to these three possible types of 3-cells are just realizations of $K_5$, pentagonal prisms corresponding to the naturality of the pentagonator in any one of the variables and instances of the cube in Fig. \[cub\], all of them commutative by hypothesis. This unique 2-isomorphism will be called the [*canonical*]{} 2-isomorphism between both $a$-paths. Using them, we can extend any $\sigma:f\Longrightarrow f'$ above to a 2-morphism from $a^{(n+1)}_{X_0,\ldots,X_n}$ to $^{(n+1)}a_{X_0,\ldots,X_n}$ as follows. Since the source and target objects $Y$, $Y'$ of $f$ and $f'$ are canonically isomorphic to the reference objects $\otimes^{(n+1)}(X_0,\ldots,X_n)$ and $^{(n+1)}\otimes(X_0,\ldots,X_n)$, we can choose $a$-paths $g:\otimes^{(n+1)}(X_0,\ldots,X_n){\longrightarrow}Y$ and $g':Y'{\longrightarrow}^{(n+1)}\otimes(X_0,\ldots,X_n)$, represented in Fig. \[figura\_extensio\] by dashed arrows. Now, by the previous coherence theorem, there are unique canonical 2-isomorphisms $\gamma_{g,g'}:a^{(n+1)}_{X_0,\ldots,X_n}\Longrightarrow g'\circ f\circ g$ and $\gamma'_{g,g'}:g'\circ f\circ g\Longrightarrow ^{(n+1)}a_{X_0,\ldots,X_n}$. The desired extension of $\sigma$ is then the pasting $$\lceil\lceil\sigma\rceil\rceil_{g,g'}:= \gamma'_{g,g'}\cdot(1_{g'}\circ\sigma\circ 1_g)\cdot\gamma_{g,g'}.$$ Since the 2-morphism $\sigma$ will not generally be a pasting of the structural 2-isomorphisms of , this extension may a priori depend on the paths $g,g'$. The next result shows that this is not the case. In the above notations, the extension $\lceil\lceil\sigma\rceil\rceil_{g,g'}$ is independent of the chosen canonical 1-isomorphisms $g,g'$. In particular, there is a unique extension of $\sigma$ by canonical 2-isomorphisms. Let $\hat{g},\hat{g}'$ be any other choice. Then, we have the two extensions $$\begin{aligned} \lceil\lceil\sigma\rceil\rceil_{g,g'}&= \gamma'_{g,g'}\cdot(1_{g'}\circ\sigma\circ 1_g)\cdot\gamma_{g,g'} \\ \lceil\lceil\sigma\rceil\rceil_{\hat{g},\hat{g}'}&= \gamma'_{\hat{g},\hat{g}'}\cdot(1_{\hat{g}'}\circ\sigma\circ 1_{\hat{g}}) \cdot\gamma_{\hat{g},\hat{g}'}\end{aligned}$$ Now, since $g,\hat{g}$ are both canonical 1-isomorphisms between the same vertices, coherence theorem implies that there exists a unique canonical 2-isomorphism $\tau:g\Longrightarrow \hat{g}$. By the same reason, we also have a unique canonical 2-isomorphism $\tau':g'\Longrightarrow \hat{g}'$. Hence, the pastings $$\begin{aligned} &(\tau'\circ 1_f\circ\tau)\cdot\gamma_{g,g'} \\ &\gamma'_{g,g'}\cdot((\tau')^{-1}\circ 1_{f'}\circ\tau^{-1})\end{aligned}$$ define canonical 2-isomorphisms from $a^{(n+1)}_{X_0,\ldots,X_n}$ to $\hat{g}'\circ f\circ\hat{g}$ and from $\hat{g}'\circ f'\circ\hat{g}$ to $^{(n+1)}a_{X_0,\ldots,X_n}$, respectively. By unicity, we must have $$\begin{aligned} \gamma_{\hat{g},\hat{g}'}&=(\tau'\circ 1_f\circ\tau)\cdot\gamma_{g,g'} \\ \gamma'_{\hat{g},\hat{g}'}&=\gamma'_{g,g'}\cdot((\tau')^{-1}\circ 1_{f'}\circ\tau^{-1})\end{aligned}$$ Hence, applying the interchange law, we obtain that $$\begin{aligned} \lceil\lceil\sigma\rceil\rceil_{\hat{g},\hat{g}'}&= \gamma'_{g,g'}\cdot((\tau')^{-1}\circ 1_{f'}\circ\tau^{-1}) \cdot(1_{\hat{g}'}\circ\sigma\circ 1_{\hat{g}})\cdot(\tau'\circ 1_f\circ\tau) \cdot\gamma_{g,g'} \\ &=\gamma'_{g,g'}\cdot(1_{g'}\circ\sigma\circ 1_g)\cdot\gamma_{g,g'} \\ &=\lceil\lceil\sigma\rceil\rceil_{g,g'}.\end{aligned}$$ Let us consider a 2-morphism $\sigma:f\Longrightarrow f'$, where $f,f'$ are some $a$-paths between suitable parenthesizations of the tensor product of $X_0,\ldots,X_n$. Then, there exists a unique extension of $\sigma$ by canonical 2-isomorphisms to a 2-morphism between the reference $a$-paths $a^{(n+1)}_{X_0,\ldots,X_n}$ and $^{(n+1)}a_{X_0,\ldots,X_n}$. Let us denote by $\lceil\lceil\sigma\rceil\rceil$ this unique extension of $\sigma$ by canonical 2-isomorphisms. The $\lceil\lceil-\rceil\rceil$ are, then, the analogs of the “padding” operators in this 2-dimensional setting (for the chosen reference 1-morphisms). Notice that they should be strictly denoted by $\lceil\lceil-\rceil\rceil_{X_0,\ldots,X_n}$, because there is such an operator for every ordered set of objects $(X_0,\ldots,X_n)$. We can now define the coboundary operator $\delta_{pent}:\widetilde{X}^{n-1}_{pent}(\Cg){\longrightarrow}\widetilde{X}^n_{pent}(\Cg)$ by $$\begin{aligned} (\delta_{pent}(\text\nn))_{X_0,\ldots,X_n}=\lceil\lceil &1_{id_{X_0}}\otimes\text\nn_{X_1,\ldots,X_n}\rceil\rceil \\ &+\sum_{i=1}^n(-1)^{i}\lceil\lceil\text\nn_{X_0,\ldots,X_{i-1}\otimes X_i,\ldots,X_n}\rceil\rceil+(-1)^{n+1}\lceil\lceil\text\nn_{X_0,\ldots,X_{n-1}}\otimes 1_{id_{X_n}}\rceil\rceil.\end{aligned}$$ Using similar arguments to those made to prove Proposition \[complex\], it can be shown that For any $K$-linear semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$, the pair $(\widetilde{X}^{\bullet}_{pent}(\text\Cg),\delta_{pent})$ is a cochain complex. This complex will be called the [*general pentagonator-deformation complex of*]{} $(\text\Cg,\otimes,a,\pi)$, and the corresponding cohomology groups will be denoted by $\widetilde{H}^{\bullet}_{pent}(\text\Cg)$, the semigroupal structure $(\otimes,a,\pi)$ being omitted for the sake of simplicity. Note that the dependence on the pentagonator $\pi$ comes exclusively through the “padding” operators $\lceil\lceil-\rceil\rceil$. Although this complex and its cohomology will be relevant in the sequel, it is not the right complex describing the infinitesimal pentagonator-deformations. Indeed, we need to take the following subcomplex: The vector subspaces ${\rm Mod}(a^{(n)},^{(n)}a)\subset\widetilde{X}^{n-1}_{pent}(\text\Cg)$ define a subcomplex of the general pentagonator-deformation complex of . We only need to see that the naturality of the $\text\nn_{X_1,\ldots,X_n}$ in $(X_1,\ldots,X_n)$ implies that of the $(\delta_{pent}(\text\nn))_{X_0,\ldots,X_n}$ in $(X_0,\ldots,X_n)$. Let us consider for example the first term $\lceil\lceil 1_{id_{X_0}}\otimes\text\nn_{X_1,\ldots,X_n}\rceil\rceil$. The naturality in $(X_1,\ldots,X_n)$ of $\text\nn_{X_1,\ldots,X_n}$ implies that of $1_{id_{X_0}}\otimes\text\nn_{X_1,\ldots,X_n}$ in $(X_0,\ldots,X_n)$. So, the situation is like that in Fig. \[naturalitat\], with two cylinders, one inside the other. We already know that the inner one commutes, and we want to see that the same is true for the outer one. Let’s think of this outer cylinder without the inner one as being decomposed in its upper and lower halves. Each one of these halves is itself a cylinder. Now, both bases of any one of these cylinders will correspond to [*modifications*]{} between $a^{(n+1)}$ or $^{(n+1)}a$ and some other induced pseudonatural isomorphism. Indeed, they are nothing more than the canonical 2-isomorphisms of the previous coherence theorem between the corresponding $a$-paths. But these 2-isomorphisms are pastings of 2-isomorphisms all natural in $(X_0,\ldots,X_n)$. It then follows that both halves also commute. This subcomplex will be denoted by $X^{\bullet}_{pent}(\text\Cg)$ and called the [*pentagonator-deformation complex of*]{} . If we denote its cohomology by $H_{pent}^{\bullet}(\text\Cg)$, we have the following: For any $K$-linear semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$, the $\omega$-equivalence classes of its first order pentagonator-deformations are in bijection with the elements of $H^3_{pent}(\text\Cg)$. Let’s consider 2-isomorphisms of the form $$\begin{aligned} \widehat{\otimes}_{\epsilon}((f',g'),(f,g))&=\widehat{\otimes}((f',g'),(f,g)) \\ (\otimes_0)_{\epsilon}(X,Y)&=(\otimes_0)(X,Y) \\ \widehat{a}_{\epsilon}(f,g,h)&=\widehat{a}(f,g,h) \\ (\pi_{\epsilon})_{X,Y,Z,T}&=\pi_{X,Y,Z,T}+(\pi^{(1)})_{X,Y,Z,T}\epsilon\end{aligned}$$ From Proposition \[semigroupal\_2\_category\], it is easy to check that they define a semigroupal structure on $\text\Cg^0_{(1)}$ (hence, a first order pentagonator-deformation of $\text\Cg$) if and only if the following conditions are satisfied (each condition is identified by the structural equation it comes from): A$\pi$1 : The $\pi^{(1)}_{X,Y,Z,T}$ are natural in $(X,Y,Z,T)$, i.e., they define an element $\pi^{(1)}\in X^3_{pent}(\text\Cg)$. A$\pi$2 : $\delta_{pent}(\pi^{(1)})=0$. The remaining structural equations are clearly superfluous in this case. This proves that first order pentagonator-deformations of  indeed correspond to 3-cocylces of $X^{\bullet}_{pen}(\text\Cg)$. Let’s suppose now that two 3-cocycles $\pi^{(1)}$ and $(\pi^{(1)})'$ define $\omega$-equivalent first order pentagonator-deformations. We need to see that they are cohomologous cocycles. Indeed, from Proposition \[equivalents\], it easily follows that both deformations are $\omega$-equivalent if and only if there exists 2-morphisms $(\omega^{(1)})_{X,Y,Z}:a_{X,Y,Z}\Longrightarrow a_{X,Y,Z}$, hence, and element $\omega^{(1)}\in\widetilde{X}^2_{pent}(\text\Cg)$, such that (each condition is again identified with the correspondiong condition in Proposition \[equivalents\] it comes from) E$\omega$1 : The $(\omega^{(1)})_{X,Y,Z}$ are natural in $(X,Y,Z)$, i.e., $\omega^{(1)}\in X^2_{pent}(\text\Cg)$. E$\omega$2 : $(\pi^{(1)})'-\pi^{(1)}=-\delta_{pent}(\omega^{(1)})$ (the remaining conditions $(E\widehat{\psi}1)-(E\widehat{\psi}3)$ in Proposition \[equivalents\] are clearly empty in this case). Hence, both 3-cocycles are indeed cohomologous. Cohomology theory for the unitary deformations of the tensor product and the associator ======================================================================================= As already indicated in the previous section, in this section we give a cohomological description of the infinitesimal unitary[^9] (tensorator,associator)-deformations. To do that, we will make the simplifying assumption that the undeformed semigroupal 2-category is actually a Gray semigroup, since otherwise the theory becomes extremely cumbersome. This means, however, no loss of generality because of Theorem \[strictification\] The situation we will encounter for these deformations closely resembles the cohomology theory discovered by Gerstenhaber and Schack[@GS90] to describe the infinitesimal deformations of a bialgebra, and later extended by Crane and Yetter [@CY981] to the case of a bitensor category (i.e., the categorification of a bialgebra). So, we associate a double complex to any $K$-linear Gray semigroup and prove that the second cohomology group of the corresponding total complex provides us with the desired description of the simultaneous first order unitary deformations of both the tensor product and the associator. As we will see, the role played by the multiplication and comultiplication in the bialgebra case corresponds in our case to the tensor product and composition of 1-morphisms. Furthermore, from this double complex we will easily get cohomologies describing the (unitary) deformations of the tensor product and the associator separately. Roughly, they are respectively related to the rows and the columns of the double complex of $(\text\Cg,\otimes)$, in much the same way as in the classical bialgebra case. Let $(\text\Cg,\otimes)$ be a $K$-linear Gray semigroup. In particular, $\otimes:\text\Cg\times\text\Cg{\longrightarrow}\text\Cg$ is a cubical pseudofunctor. Recall, however, that not all cubical pseudofunctors $\otimes$ will provide the 2-category  with the structure of a Gray semigroup. More explicitly, the non-trivial 2-isomorphisms $\widehat{\otimes}((f',g'),(f,g))$ must additionally satisfy the equation $$\begin{aligned} (\widehat{\otimes}((f',g'),(f,g))\otimes 1_{h'\circ h})\cdot\widehat{\otimes}((f'\otimes g',h')&,(f\otimes g,h))= \\ &=(1_{f'\circ f}\otimes\widehat{\otimes}((g',h'),(g,h)))\cdot\widehat{\otimes}((f',g'\otimes h'),(f,g\otimes h))\end{aligned}$$ coming from the structural condition $(A\widehat{a}2)$ in Proposition \[semigroupal\_2\_category\] when the associator is trivial (the reader may easily check that the remaining structural equations $(A\widehat{a}3)$ and $(A\pi 1)-(A\pi 2)$ give no additional conditions on $\otimes$). Recall from the previous section that, for all $n\geq 1$, we introduced pseudofunctors $\otimes^{(n)},^{(n)}\otimes:\text\Cg\times\stackrel{n)}{\cdots}\times\text\Cg{\longrightarrow}\text\Cg$. Then, because of the above additional equation on $\otimes$, in a Gray semigroup we have the following: \[o\_n\] Let $(\text\Cg,\otimes)$ be a Gray semigroup. Then, for all $n\geq 1$, we have the equality of pseudofunctors $$^{(n)}\otimes=\otimes^{(n)}:=\otimes(n).$$ Moreover, $\otimes(n)$ is unitary For $n=1,2$ it is obvious. Let’s consider the case $n\geq 3$. Since the structural isomorphisms $a_{X,Y,Z}$, $\widehat{a}(f,g,h)$ and $\otimes_0(X,Y)$ are identities, it is clear that we only need to prove that $$\widehat{^{(n)}\otimes}((f'_1,\ldots,f'_n),(f_1,\ldots,f_n))= \widehat{\otimes^{(n)}}((f'_1,\ldots,f'_n),(f_1,\ldots,f_n))$$ The proof is by induction on $n$. The case $n=3$ is nothing more than the previously mentioned additional equation, as the reader may easily check. Let $n>3$. By definition of $^{(n)}\otimes$ and using the induction hypothesis, we have $$\begin{aligned} \widehat{^{(n)}\otimes}((f'_1,\ldots,f'_n),(f_1,\ldots,f_n))&= (\widehat{\otimes}((f'_1,f'_2),(f_1,f_2))\otimes 1_{(f'_3\circ f_3)\otimes\cdots\otimes(f'_n\circ f_n)})\cdot \\ &\ \ \ \ \ \ \cdot\widehat{^{(n-1)}\otimes} ((f'_1\otimes f'_2,\ldots,f'_n),(f_1\otimes f_2,\ldots,f_n)) \\ &=(\widehat{\otimes}((f'_1,f'_2),(f_1,f_2))\otimes 1_{(f'_3\circ f_3)\otimes\cdots\otimes(f'_n\circ f_n)})\cdot \\ &\ \ \ \ \ \ \cdot\widehat{\otimes^{(n-1)}} ((f'_1\otimes f'_2,\ldots,f'_n),(f_1\otimes f_2,\ldots,f_n))\end{aligned}$$ Now, from the definition of $\otimes^{(n-1)}$ and using the equality $(\tau'\cdot\tau)\otimes(\sigma'\cdot\sigma)= (\tau'\otimes\sigma')\cdot(\tau\otimes\sigma)$, it follows that $$\begin{aligned} \widehat{\otimes^{(n-1)}} ((f'_1\otimes f'_2,\ldots,f'_n),(f_1\otimes f_2,\ldots,f_n))&= (1_{(f'_1\otimes f'_2)\circ (f_1\otimes f_2)}\otimes \widehat{\otimes^{(n-2)}}((f'_3,\ldots,f'_n),(f_3,\ldots,f_n)))\cdot \\ &\ \ \ \cdot\widehat{\otimes} ((f'_1\otimes f'_2,f'_3\otimes\cdots\otimes f'_n), (f_1\otimes f_2,f_3\otimes\cdots\otimes f_n))\end{aligned}$$ Therefore, we have $$\begin{aligned} \widehat{^{(n)}\otimes}((f'_1\otimes f'_2,\ldots,f'_n),(f_1\otimes f_2,\ldots,f_n))&= (\widehat{\otimes}((f'_1,f'_2),(f_1,f_2))\otimes 1_{(f'_3\circ f_3)\otimes\cdots\otimes(f'_n\circ f_n)})\cdot \\ &\ \ \ \cdot(1_{(f'_1\otimes f'_2)\circ (f_1\otimes f_2)}\otimes \widehat{\otimes^{(n-2)}}((f'_3,\ldots,f'_n),(f_3,\ldots,f_n)))\cdot \\ &\ \ \ \cdot\widehat{\otimes} ((f'_1\otimes f'_2,f'_3\otimes\cdots\otimes f'_n), (f_1\otimes f_2,f_3\otimes\cdots\otimes f_n)) \\ &=(1_{(f'_1\circ f_1)\otimes(f'_2\circ f_2)}\otimes \widehat{\otimes^{(n-2)}}((f'_3,\ldots,f'_n),(f_3,\ldots,f_n)))\cdot \\ &\ \ \ \cdot(\widehat{\otimes}((f'_1,f'_2),(f_1,f_2))\otimes 1_{(f'_3\otimes\cdots\otimes f'_n)\circ(f_3\otimes\cdots\otimes f_n)}) \cdot \\ &\ \ \ \cdot\widehat{\otimes} ((f'_1\otimes f'_2,f'_3\otimes\cdots\otimes f'_n), (f_1\otimes f_2,f_3\otimes\cdots\otimes f_n))\end{aligned}$$ The proof finishes by applying $(A\widehat{a}2)$ to the last two factors. Since all the pseudofunctors $\otimes(n)$ are unitary, we have for each of them the corresponding cochain complex $X^{\bullet}(\otimes(n))$, $n\geq 1$, describing their purely pseudofunctorial deformations (see Section 6). More precisely, if $m\leq 0$, $X^m(\otimes(n))=0$, while for all $m\geq 1$, it is $$X^m(\otimes(n))=\prod_{(X^0_1,\ldots,X^0_n),\ldots,(X^m_1,\ldots,X^m_n)\in|\text\Cg^n|} \text{Nat}(C\otimes(n)_{(X^0_i),\ldots,(X^m_i)},\otimes(n)C_{(X^0_i),\ldots,(X^m_i)})$$ To simplify, we write $(X^j_i)$ instead of $(X^j_1,\ldots,X^j_n)$. Here, $C\otimes(n)_{(X^0_i),\ldots,(X^m_i)}$ and $\otimes(n)C_{(X^0_i),\ldots,(X^m_i)}$ denote the functors $$\begin{aligned} \text\Cg^n((X^{m-1}_1,\ldots,X^{m-1}_n),(X^m_1,\ldots,X^m_n))\times \cdots\times\text\Cg^n&((X^0_1,\ldots,X^0_n),(X^1_1,\ldots,X^1_n)){\longrightarrow}\\ &{\longrightarrow}\text\Cg(X^0_1\otimes\cdots\otimes X^0_n,X^m_1\otimes\cdots\otimes X^m_n)\end{aligned}$$ which apply the composable 1-morphisms $$(f^1_i,\ldots,f^n_i):(X^{m-1-i}_1,\ldots,X^{m-1-i}_n){\longrightarrow}(X^{m-i}_1,\ldots,X^{m-i}_n),\ \ \ \ \ \ i=0,\ldots,m-1$$ to $(f^1_0\otimes\cdots\otimes f^m_0)\circ\cdots\circ(f_{m-1}^1\otimes\cdots\otimes f_{m-1}^n)$ and $(f_0^1\circ\cdots\circ f_{m-1}^1)\otimes\cdots\otimes(f_0^n\circ\cdots\circ f_{m-1}^n)$, respectively. Hence, a generic element $\phi\in X^m(\otimes(n))$ is a collection of 2-morphisms $$\begin{aligned} \phi((f_0^1,\ldots,f_0^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n)): (f^1_0\otimes\cdots&\otimes f^n_0)\circ\cdots\circ(f_{m-1}^1\otimes\cdots\otimes f_{m-1}^n)\Longrightarrow \\ &\Longrightarrow(f_0^1\circ\cdots\circ f_{m-1}^1)\otimes\cdots\otimes(f_0^n\circ\cdots\circ f_{m-1}^n)\end{aligned}$$ natural in the $(f_i^1,\ldots,f_i^n)$. In particular, notice that the structural 2-isomorphisms $\widehat{\otimes}((f',g'),(f,g))$ define an element $\widehat{\otimes}\in X^2(\otimes(2))$, while the $\widehat{a}(f,g,h)$ define an element $\widehat{a}\in X^1(\otimes(3))$ (the trivial one in the case of a Gray semigroup). Instead of $X^m(\otimes(n))$, let’s use the more suggestive notation $X^{m-1,n-1}(\text\Cg,\otimes)$, or just $X^{m-1,n-1}(\text\Cg)$, for these $K$-vector spaces (the change of indices is for later convenience). They can be arranged as in Figure \[figura\_complex\_doble\], with $m-1\geq 0$ and $n-1\geq 0$ being the row and column index, respectively. Since the elements $\phi\in X^{m-1,0}(\text\Cg)$ are of the form $\phi(f_0,\ldots,f_{m-1}):f_0\circ\cdots\circ f_{m-1}\Longrightarrow f_0\circ\cdots\circ f_{m-1}$, while those $\phi\in X^{0,n}(\text\Cg)$ are of the form $\phi(f^0,\ldots,f^n):f^0\otimes\cdots\otimes f^n\Longrightarrow f^0\otimes\cdots\otimes f^n$, we can think of the rows and columns as related to the composition and the tensor product, respectively, of 1-morphisms. Arranged in this way, each row corresponds to the cochain complexes $X^{\bullet}(\otimes(n))$, the coboundary operators $\delta_h:X^{m-1,n-1}(\text\Cg){\longrightarrow}X^{m,n-1}(\text\Cg)$, $m\geq 1$, being those defined in the previous section. Namely, if $\phi\in X^{m-1,n-1}(\text\Cg)$, then $$\begin{aligned} (\delta_h(\phi))((f^1_0,\ldots,&f_0^n),\ldots,(f^1_m,\ldots,f_m^n))= \lceil 1_{f_0^1\otimes\cdots\otimes f_0^n}\circ \phi((f^1_1,\ldots,f_1^n),\ldots,(f^1_m,\ldots,f^n_m)\rceil \\ &+\sum_{i=1}^{m}(-1)^i\lceil\phi((f^1_0,\ldots,f_0^n),\ldots,(f^1_{i-1}\circ f^1_i,\ldots,f_{i-1}^n\circ f_i^n),\ldots,(f^1_{m},\ldots,f^n_{m}))\rceil+ \\ &+(-1)^{m+1}\lceil\phi((f^1_0,\ldots,f_0^n),\ldots,(f^1_{m-1},\ldots,f_{m-1}^n))\circ 1_{f^1_{m}\otimes\cdots\otimes f^n_{m}}\rceil\end{aligned}$$ The claim is that it is possible to define vertical coboundary maps $\delta_v:X^{m-1,n-1}(\text\Cg){\longrightarrow}X^{m-1,n}(\text\Cg)$, for all $m\geq 1$, making each column a cochain complex and in such a way that the whole set of vector spaces and maps define a double complex. Indeed, if $\phi\in X^{m-1,n-1}(\text\Cg)$, let’s define $$\begin{aligned} (\delta_v(\phi))((f^0_0,\ldots,&f_0^n),\ldots,(f^0_{m-1},\ldots,f_{m-1}^n))= \lceil 1_{f_0^0\circ\cdots\circ f_{m-1}^0}\otimes \phi((f^1_0,\ldots,f_0^n),\ldots,(f^1_{m-1},\ldots,f^n_{m-1})\rceil \\ &+\sum_{i=1}^n(-1)^i\lceil\phi((f^0_0,\ldots,f_0^{i-1}\otimes f_0^i,\ldots,f_0^n),\ldots,(f^0_{m-1},\ldots,f_{m-1}^{i-1}\otimes f_{m-1}^i,\ldots,f^n_{m-1}))\rceil+ \\ &+(-1)^{n+1}\lceil\phi((f^0_0,\ldots,f_0^{n-1}),\ldots, (f^0_{m-1},\ldots,f_{m-1}^{n-1}))\otimes 1_{f^n_0\circ\cdots\circ f^n_{m-1}}\rceil\end{aligned}$$ Using arguments similar to those used to prove previous results of the same kind, one shows the following (once more, the coherence theorem for unitary pseudofunctors takes account of the padding operators): For all $m\geq 1$, the pair $(X^{m,\bullet}(\text\Cg),\delta_v)$ is a cochain complex. Actually, as indicated before, we have the following stronger result, which is fundamental in our theory: The $K$-vector spaces $X^{\bullet,\bullet}(\text\Cg)$ together with the above maps $\delta_h:X^{\bullet,\bullet}(\text\Cg){\longrightarrow}X^{\bullet+1,\bullet}(\text\Cg)$ and $\delta_v:X^{\bullet,\bullet}(\text\Cg){\longrightarrow}X^{\bullet,\bullet+1}(\text\Cg)$ define a double complex. It remains to prove that both coboundary maps $\delta_h$ and $\delta_v$ commute. Let’s consider an element $\phi\in X^{m-2,n-1}(\text\Cg)$, $m\geq 2,n\geq 1$, with $\phi=\{\phi((f_1^1,\ldots,f_1^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n))\}$. Then, the reader may easily check that $$\begin{aligned} (\delta_h&(\delta_v(\phi))((f_0^0,\ldots,f_0^n),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^n))= \\ &=\lceil 1_{f_0^0\otimes\cdots\otimes f_0^n}\circ\lceil 1_{f_1^0\circ\cdots\circ f_{m-1}^0} \otimes\phi((f_1^1,\ldots,f_1^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n))\rceil\rceil \\ &\ \ +\sum_{i=1}^n(-1)^i\lceil 1_{f_0^0\otimes\cdots\otimes f_0^n}\circ\lceil\phi((f_1^0,\ldots,f_1^{i-1}\otimes f_1^i,\ldots,f_1^n),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{i-1}\otimes f_{m-1}^i,\ldots,f_{m-1}^n))\rceil\rceil \\ &\ \ +(-1)^{n+1}\lceil 1_{f_0^0\otimes\cdots\otimes f_0^n}\circ\lceil\phi((f_1^0,\ldots,f_1^{n-1}),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{n-1})) \otimes 1_{f_1^n\circ\cdots\circ f_{m-1}^n}\rceil\rceil \\ &\ \ +\sum_{i=1}^{m-1}(-1)^i \lceil 1_{f_0^0\circ\cdots\circ f_{m-1}^0}\otimes \phi((f_0^1,\ldots,f_o^n),\ldots,(f_{i-1}^1\circ f_i^1,\ldots,f_{i-1}^n\circ f_i^n),\ldots, (f_{m-1}^1,\ldots,f_{m-1}^n))\rceil \\ &\ \ +\sum_{i=1}^{m-1}\sum_{j=1}^n(-1)^{i+j}\lceil \phi((f_0^0,\ldots,f_0^{j-1}\otimes f_0^j,\ldots,f_0^n),\ldots,(f_{i-1}^0\circ f_i^0,\ldots \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots,(f_{i-1}^{j-1}\circ f_i^{j-1}) \otimes(f_{i-1}^j\circ f_i^j), \ldots,f_{i-1}^n\circ f_i^n),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{j-1} \otimes f_{m-1}^j,\ldots,f_{m-1}^n))\rceil \\ &\ \ +\sum_{i=1}^{m-1}(-1)^{i+n+1} \lceil\phi((f_0^0,\ldots,f_0^{n-1}),\ldots,(f_{i-1}^0\circ f_i^0,\ldots, f_{i-1}^{n-1}\circ f_i^{n-1}),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{n-1}))\otimes 1_{f_0^n\circ\cdots\circ f_{m-1}^n}\rceil \\ &\ \ +(-1)^m\lceil\lceil 1_{f_0^0\circ\cdots\circ f_{m-2}^0}\otimes \phi((f_0^1,\ldots,f_0^n),\ldots,(f_{m-2}^1,\ldots,f_{m-2}^n))\rceil \circ 1_{f_{m-1}^0\otimes\cdots\otimes f_{m-1}^n}\rceil \\ &\ \ +\sum_{i=1}^n(-1)^{m+i} \lceil\lceil\phi((f_0^0,\ldots,f_0^{i-1}\otimes f_0^i,\ldots,f_o^n),\ldots, (f_{m-2}^0,\ldots,f_{m-2}^{i-1}\otimes f_{m-2}^i,\ldots,f_{m-2}^n))\rceil\circ 1_{f_{m-1}^0\otimes\cdots\otimes f_{m-1}^n}\rceil \\ &\ \ +(-1)^{m+n+1}\lceil\lceil \phi((f_0^0,\ldots,f_0^{n-1}),\ldots,(f_{m-2}^0,\ldots,f_{m-2}^{n-1})) \otimes 1_{f_0^n\circ\cdots\circ f_{m-2}^n}\rceil \circ 1_{f_{m-1}^0\otimes\cdots\otimes f_{m-1}^n}\rceil\end{aligned}$$ while $$\begin{aligned} (\delta_v&(\delta_h(\phi))((f_0^0,\ldots,f_0^n),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^n))= \\ &=\lceil 1_{f_0^0\circ\cdots\circ f_{m-1}^0}\circ\lceil 1_{f_0^1\otimes\cdots \otimes f_0^n} \otimes\phi((f_1^1,\ldots,f_1^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n))\rceil\rceil \\ &\ \ +\sum_{i=1}^{m-1}(-1)^i \lceil 1_{f_0^0\circ\cdots\circ f_{m-1}^0}\otimes\lceil \phi((f_0^1,\ldots,f_o^n),\ldots,(f_{i-1}^1\circ f_i^1,\ldots,f_{i-1}^n\circ f_i^n),\ldots, (f_{m-1}^1,\ldots,f_{m-1}^n))\rceil\rceil \\ &\ \ +(-1)^m\lceil 1_{f_0^0\circ\cdots\circ f_{m-1}^0}\otimes\lceil \phi((f_0^1,\ldots,f_0^n),\ldots,(f_{m-2}^1,\ldots,f_{m-2}^n)) \circ 1_{f_{m-1}^1\otimes\cdots\otimes f_{m-1}^n}\rceil\rceil \\ &\ \ +\sum_{i=1}^n(-1)^i\lceil 1_{f_0^0\otimes\cdots\otimes f_0^n}\circ\phi(f_1^0,\ldots,f_1^{i-1}\otimes f_1^i,\ldots,f_1^n),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{i-1}\otimes f_{m-1}^i,\ldots,f_{m-1}^n))\rceil \\ &\ \ +\sum_{i=1}^n\sum_{j=1}^{m-1}(-1)^{i+j}\lceil \phi((f_0^0,\ldots,f_0^{i-1}\otimes f_0^i,\ldots,f_0^n),\ldots,(f_{j-1}^0\circ f_j^0,\ldots \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots,\ldots,(f_{j-1}^{i-1}\otimes f_{j-1}^i) \circ(f_j^{i-1}\otimes f_j^i), \ldots,f_{j-1}^n\circ f_j^n),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{i-1} \otimes f_{m-1}^i,\ldots,f_{m-1}^n))\rceil \\ &\ \ +\sum_{i=1}^n(-1)^{m+i} \lceil\phi((f_0^0,\ldots,f_0^{i-1}\otimes f_0^i,\ldots,f_o^n),\ldots, (f_{m-2}^0,\ldots,f_{m-2}^{i-1}\otimes f_{m-2}^i,\ldots,f_{m-2}^n))\circ 1_{f_{m-1}^0\otimes\cdots\otimes f_{m-1}^n}\rceil \\ &\ \ +(-1)^{n+1}\lceil\lceil 1_{f_0^0\otimes\cdots\otimes f_0^{n-1}}\circ\phi((f_1^0,\ldots,f_1^{n-1}),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{n-1}))\rceil \otimes 1_{f_0^n\circ\cdots\circ f_{m-1}^n}\rceil \\ &\ \ +\sum_{i=1}^{m-1}(-1)^{i+n+1} \lceil\lceil\phi((f_0^0,\ldots,f_0^{n-1}),\ldots,(f_{i-1}^0\circ f_i^0,\ldots, f_{i-1}^{n-1}\circ f_i^{n-1}),\ldots,(f_{m-1}^0,\ldots,f_{m-1}^{n-1}))\rceil\otimes 1_{f_0^n\circ\cdots\circ f_{m-1}^n}\rceil \\ &\ \ +(-1)^{m+n+1}\lceil\lceil \phi((f_0^0,\ldots,f_0^{n-1}),\ldots,(f_{m-2}^0,\ldots,f_{m-2}^{n-1})) \circ 1_{f_{m-1}^0\otimes\cdots\otimes f_{m-1}^{n-1}}\rceil \otimes 1_{f_0^n\circ\cdots\circ f_{m-1}^n}\rceil\end{aligned}$$ Notice that in both expressions there are nine terms. Now, recall that taking the padding $\lceil-\rceil$ of some 2-morphism $(-)$ simply means to take a vertical composition of $(-)$ with the appropriate expansions of the 2-isomorphisms $\widehat{\otimes}(-,-)$. It then follows by the interchange law that $$\lceil 1_{f_0^0\otimes\cdots\otimes f_0^n}\circ\lceil-\rceil\rceil=\lceil\lceil 1_{f_0^0\otimes\cdots\otimes f_0^n}\circ(-)\rceil\rceil=\lceil 1_{f_0^0\otimes\cdots\otimes f_0^n}\circ(-)\rceil$$ This proves the equality between the second term in the expression of $\delta_h(\delta_v(\phi))$ and the fourth term in the expression of $\delta_v(\delta_h(\phi))$. The same argument shows the equality between the eighth and sixth terms in the first and second expression, respectively. On the other hand, we can also conclude that the first term in $\delta_h(\delta_v(\phi))$ is the padding of $$1_{f_0^0\otimes\cdots\otimes f_0^n} \circ(1_{f_1^0\circ\cdots\circ f_{m-1}^0} \otimes\phi((f_1^1,\ldots,f_1^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n)))$$ But by the naturality of the $\widehat{\otimes}(-,-)$ (equation $(A\widehat{\otimes}1)$), this is the same as $$\widehat{\otimes}(-,-)^{-1}\cdot(1_{f_0^0\circ\cdots\circ f_{m-1}^0}\otimes (1_{f_0^1\otimes\cdots\otimes f_0^n}\circ\phi((f_1^1,\ldots,f_1^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n)))) \cdot\widehat{\otimes(-,-)}$$ whose padding clearly coincides with that of $$1_{f_0^0\circ\cdots\circ f_{m-1}^0}\otimes (1_{f_0^1\otimes\cdots\otimes f_0^n}\circ\phi((f_1^1,\ldots,f_1^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n)))$$ Using now that $(\tau'\cdot\tau)\otimes(\sigma'\cdot\sigma)=(\tau'\otimes\sigma')\cdot(\tau\otimes\sigma)$, we obtain that both first terms also coincide. Similar arguments can be made to show the equality between both last terms and between the terms: third and seventh, fourth and second, sixth and eighth and seventh and third in the first and second expressions, respectively. Hence, it only remains to prove the equality between both fifth terms in each expression, and this easily follows from the naturality of $\phi$ in its arguments applied to the 2-morphism $$1_{f_0^0\otimes\cdots\otimes f_0^n}\circ\cdots\circ(1_{f_{i-1}^0\circ f_i^0}\otimes\cdots\otimes\widehat{\otimes}((f_{j-1}^{i-1},f_{j-1}^i),(f_j^{i-1},f_j^i)) \otimes\cdots\otimes 1_{f_{j-1}^n\circ f_j^n})\circ\cdots\circ 1_{f_{m-1}^0\otimes\cdots\otimes f_{m-1}^n}$$ This double complex $(X^{\bullet,\bullet}(\text\Cg),\delta_h,\delta_v)$ will be called the [*extended double complex*]{} of the Gray semigroup $(\text\Cg,\otimes)$. We are actually interested in the double complex obtained after deleting the bottom row $X^{m,0}(\text\Cg)$. It will be called the [*double complex*]{} of $(\text\Cg,\otimes)$. Furthermore, for our purposes, we need to take a subcomplex of this double complex. This is related to the fact that we only consider infinitesimal [*unitary*]{} deformations. An element $\phi\in X^{m-1,n-1}(\text\Cg)$ will be called [*special*]{} if whenever $(f_i^1,\ldots,f_i^n)=(id,\ldots,id)$ for some $i\in\{0,\ldots,m-1\}$, it holds $$\phi((f_0^1,\ldots,f_0^n),\ldots,(f_{m-1}^1,\ldots,f_{m-1}^n))=0$$ The set of $\phi\in X^{m-1,n-1}(\text\Cg)$, $m,n\geq 1$, which are special clearly define a vector subspace, which will be denoted by $X_s^{m-1,n-1}(\text\Cg)$. We have then the following: The vector subspaces $X_s^{m,n}(\text\Cg)$, $m,n\geq 0$, define a subcomplex $X_s^{\bullet,\bullet}(\text\Cg)$ of the extended double complex of $(\text\Cg,\otimes)$. We need to see that both coboundary operators $\delta_h$ and $\delta_v$ preserve the special elements. Indeed, let $\phi\in X_s^{m-1,n-1}(\text\Cg)$. Then, from the above expression of $(\delta_h(\phi))$, it is clear that when $((f_i^1,\ldots,f_i^n)=(id,\ldots,id)$ for some $i\in\{0,\ldots,m-1\}$, all terms are zero except the $(i+1)^{th}$ and the $(i+2)^{th}$ terms, which are equal but of opposite sign (recall that, $\otimes$ being unitary, the tensor product of identity 1-morphisms is always an identity 1-morphism, and that the identity 2-morphism of an identity 1-morphism is a unit with respect to horizontal composition). Hence, $\delta_h(\phi)$ is special. On the other hand, if $\phi$ is special and $((f_i^1,\ldots,f_i^n)=(id,\ldots,id)$ for some $i\in\{0,\ldots,m-1\}$, all terms in $\delta_v(\phi)$ are clearly zero, so that $\delta_v(\phi)$ is also special. The double complex defined by the vector subspaces $X_s^{m,n}(\text\Cg)$, $m,n\geq 0$, and the corresponding restrictions of both $\delta_h$ and $\delta_v$ will be called the [*special extended double complex*]{} of $(\text\Cg,\otimes)$, or just the [*special double complex*]{} of $(\text\Cg,\otimes)$, when the bottom row is deleted. Let $X^{\bullet}_{tens,ass}(\text\Cg)$ denote the total complex associated to the special double complex $(X_s^{\bullet,\bullet}(\text\Cg),\delta_h,\delta_v)$ of $(\text\Cg,\otimes)$. It will be called the [*unitary (tensorator,associator)-deformation complex*]{} of $(\text\Cg,\otimes)$. By definition, it is the cochain complex with vector spaces $$X^q_{tens,ass}(\text\Cg)=\bigoplus_{\begin{array}{c}m+n=q \\ m\geq 0,n\geq 1 \end{array}} X_s^{m,n}(\text\Cg),\ \ \ \ \ q\geq 1$$ and coboundary operators $\delta_{tens,ass}:X^q_{tens,ass}(\text\Cg){\longrightarrow}X^{q+1}_{tens,ass}(\text\Cg)$ given by $$\delta_{tens,ass}=\bigoplus_{\begin{array}{c} m+n=q \\ m\geq 0,n\geq 1 \end{array}} ((-1)^n\delta_h+\delta_v)\ \ \ \ \ \ q\geq 1$$ The corresponding cohomology groups will be denoted by $H^{\bullet}_{tens,ass}(\text\Cg)$. The reason we choose the above name for this total complex is the following theorem: \[tensorator,associator-deformations\] If $(\text\Cg,\otimes)$ is a $K$-linear Gray semigroup, the $\psi$-equivalence classes of its first order unitary (tensorator,associator)-deformations are in bijection with the elements of the group $H^2_{tens,ass}(\text\Cg)$. Let’s consider 2-isomorphisms $$\begin{aligned} \widehat{\otimes}_{\epsilon}((f',g'),(f,g))&=\widehat{\otimes}((f',g'),(f,g))+ \widehat{\otimes}^{(1)}((f',g'),(f,g))\epsilon \\ (\otimes_0)_{\epsilon}(X,Y)&=1_{id_{X\otimes Y}}\\ \widehat{a}_{\epsilon}(f,g,h)&=1_{f\otimes g\otimes h}+\widehat{a}^{(1)}(f,g,h)\epsilon \\ (\pi_{\epsilon})_{X,Y,Z,T}&=1_{id_{X\otimes Y\otimes Z\otimes T}}\end{aligned}$$ Substituting these 2-isomorphisms in the structural equations in Proposition \[semigroupal\_2\_category\] and computing the first order term in $\epsilon$, it turns out that they define a first order unitary (tensorator,associator)-deformation of  if and only if the 2-morphisms $\widehat{\otimes}^{(1)}((f',g'),(f,g))$, $\widehat{a}^{(1)}(f,g,h)$ satisfy the following conditions: A$\widehat{\otimes}$1 : The $\widehat{\otimes}^{(1)}((f',g'),(f,g))$ are natural in $(f',g'),(f,g)$, i.e., they define an element $$\widehat{\otimes}^{(1)}\in X^{1,1}(\text\Cg)=X^2(\otimes(2))$$ A$\widehat{\otimes}$2 : For all composable 1-morphisms $(f'',g''),(f',g'),(f,g)$ $$\begin{aligned} \widehat{\otimes}&^{(1)}((f'',g''),(f'\circ f,g'\circ g)) \cdot(1_{f''\otimes g''}\circ\widehat{\otimes}((f',g'),(f,g)))+ \\ \widehat{\otimes}&((f'',g''),(f'\circ f,g'\circ g)) \cdot(1_{f''\otimes g''}\circ\widehat{\otimes}^{(1)}((f',g'),(f,g)))= \\ &\ \ \ \ \ \ \ \ \ \ \ \ =\widehat{\otimes}^{(1)}((f''\circ f',g''\circ g'),(f,g)) \cdot(\widehat{\otimes}((f'',g''),(f',g'))\circ 1_{f\otimes g})+ \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \widehat{\otimes}((f''\circ f',g''\circ g'),(f,g)) \cdot(\widehat{\otimes}^{(1)}((f'',g''),(f',g'))\circ 1_{f\otimes g})\end{aligned}$$ It is easy to check that this is exactly the condition $\delta_h(\widehat{\otimes}^{(1)})=0$. A$\widehat{\otimes}$3 : For all 1-morphisms $(f,g):(X,Y){\longrightarrow}(X',Y')$, it holds $$\widehat{\otimes}^{(1)}((id_{X'},id_{Y'}),(f,g))= \widehat{\otimes}^{(1)}((f,g),(id_X,id_Y))=0$$ i.e., $\widehat{\otimes}^{(1)}\in X^{1,1}_s(\text\Cg)\subset X^2_{tens,ass}(\text\Cg)$. A$\widehat{a}$1 : The $\widehat{a}^{(1)}(f,g,h)$ are natural in $(f,g,h)$, i.e., they define an element $$\widehat{a}^{(1)}\in X^{0,2}(\text\Cg)=X^1(\otimes(3))$$ A$\widehat{a}$2 : For all 1-morphisms $(f,g,h)$ $$\begin{aligned} (\widehat{\otimes}&^{(1)}((f',g'),(f,g))\otimes 1_{h'\circ h})\cdot\widehat{\otimes}((f'\otimes g',h'),(f\otimes g,h))+ \\ &(\widehat{\otimes}((f',g'),(f,g))\otimes 1_{h'\circ h})\cdot\widehat{\otimes}^{(1)}((f'\otimes g',h'),(f\otimes g,h)) \\ &\widehat{a}^{(1)}(f'\circ f,g'\circ g,h'\circ h)\cdot(\widehat{\otimes}((f',g'),(f,g))\otimes 1_{h'\circ h})\cdot\widehat{\otimes}((f'\otimes g',h'),(f\otimes g,h))= \\ &\ \ \ \ \ \ \ \ \ \ =(1_{f'\circ f}\otimes\widehat{\otimes}^{(1)}((g',h')(g,h)))\cdot \widehat{\otimes}((f',g'\otimes h'),(f,g\otimes h))+ \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ (1_{f'\circ f}\otimes\widehat{\otimes}((g',h')(g,h)))\cdot \widehat{\otimes}^{(1)}((f',g'\otimes h'),(f,g\otimes h))+ \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ (1_{f'\circ f}\otimes\widehat{\otimes}((g',h')(g,h)))\cdot \widehat{\otimes}((f',g'\otimes h'),(f,g\otimes h))\cdot (\widehat{a}^{(1)}(f',g',h')\circ 1_{f\otimes(g\otimes h)})+ \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ (1_{f'\circ f}\otimes\widehat{\otimes}((g',h')(g,h)))\cdot \widehat{\otimes}((f',g'\otimes h'),(f,g\otimes h))\cdot (1_{(f'\otimes g')\otimes h'}\circ\widehat{a}^{(1)}(f,g,h))\end{aligned}$$ It is easy to check that this is exactly the condition $\delta_v(\widehat{\otimes}^{(1)})+\delta_h(\widehat{a}^{(1)})=0$ A$\widehat{a}$3 : For all objects $(X,Y,Z)$, it holds $$\widehat{a}^{(1)}(id_X,id_Y,id_Z)=0$$ i.e., $\widehat{a}^{(1)}\in X^{0,2}_s(\text\Cg)\subset X^2_{tens,ass}(\text\Cg)$. A$\pi$1 : For all 1-morphisms $(f,g,h,k)$ $$\begin{aligned} \widehat{a}&^{(1)}(f,g,h\otimes k)+\widehat{a}^{(1)}(f\otimes g,h,k)= \\ &\ \ \ \ \ \ =-\widehat{\otimes}^{(1)} ((id_{X'},id_{Y'\otimes Z'\otimes T'}),(f,g\otimes h\otimes k)) +1_f\otimes \widehat{a}^{(1)}(g,h,k) \\ &\ \ \ \ \ \ \ \ \ \ +\widehat{\otimes}^{(1)} ((f,g\otimes h\otimes k),(id_{X},id_{Y\otimes Z\otimes T})) +\widehat{a}^{(1)}(f,g\otimes h,k) \\ &\ \ \ \ \ \ \ \ \ \ -\widehat{\otimes}^{(1)} ((id_{X'\otimes Y'\otimes Z'},id_{T'}),(f\otimes g\otimes h,k)) +\widehat{a}^{(1)}(f,g,h)\otimes 1_k \\ &\ \ \ \ \ \ \ \ \ \ +\widehat{\otimes}^{(1)} ((f\otimes g\otimes h,k),(id_{X\otimes Y\otimes Z},id_T))\end{aligned}$$ Since the terms in $\widehat{\otimes}^{(1)}$ are all zero by condition $(A\widehat{\otimes}3)$, this exactly corresponds to the condition $\delta_v(\widehat{a}^{(1)})=0$. As the reader may easily check, the structural equation $(A\pi 2)$ gives no additional conditions for this kind of deformation. Now, $(A\widehat{\otimes}1),(A\widehat{\otimes}3)$ together with $(A\widehat{a}1),(A\widehat{a}3)$ say that $(\widehat{a}^{(1)},\widehat{\otimes}^{(1)})\in X^2_{tens,ass}(\text\Cg)$. On the other hand, we have $$\delta_{tens,ass}(\widehat{a}^{(1)},\widehat{\otimes}^{(1)})=(\delta_v(\widehat{a}^{(1)}), \delta_h(\widehat{a}^{(1)})+\delta_v(\widehat{\otimes}^{(1)}), -\delta_h(\widehat{\otimes}^{(1)}))$$ Hence, $(A\widehat{\otimes}2)$, $(A\widehat{a}2)$ and $(A\pi 1)$ together say that $(\widehat{a}^{(1)},\widehat{\otimes}^{(1)})$ is a 2-cocycle. Let us now suppose that $((\widehat{a}^{(1)})',(\widehat{\otimes}^{(1)})')$ is another 2-cocycle defining a $\psi$-equivalent first order unitary (tensorator,associator)-deformation of . We need to see that both 2-cocycles are actually cohomologous. Indeed, from Proposition \[equivalents\], it immediately follows that they are $\psi$-equivalent deformations if and only if there exists 2-morphisms $\widehat{\psi}^{(1)}(f,g):f\otimes g\Longrightarrow f\otimes g$ such that E$\widehat{\psi}$1 : The $\widehat{\psi}^{(1)}(f,g)$ are natural in $(f,g)$, i.e., they define an element $\widehat{\psi}^{(1)}\in X^{0,1}(\text\Cg)$ E$\widehat{\psi}$2 : $(\widehat{\otimes}^{(1)})'-\widehat{\otimes}^{(1)}=-\delta_h(\widehat{\psi}^{(1)})$. E$\widehat{\psi}$3 : $\widehat{\psi}^{(1)}$ is special. E$\omega$1 : $(\widehat{a}^{(1)})'-\widehat{a}^{(1)}=\delta_v(\widehat{\psi}^{(1)})$. (in this case, the condition coming from equation $(E\omega 2)$ is empty). The first and third conditions together say that $\widehat{\psi}^{(1)}\in X^{0,1}_s(\text\Cg)=X^1_{tens,ass}(\text\Cg)$, while the second and fourth express nothing more than the fact that $$((\widehat{a}^{(1)})',(\widehat{\otimes}^{(1)})')- (\widehat{a}^{(1)},\widehat{\otimes}^{(1)})=\delta_{tens,ass}(\widehat{\psi}^{(1)})$$ as required. With the above results, it is easy to obtain a cochain complex whose cohomology describes the infinitesimal [*associator-deformations*]{} of $(\text\Cg,\otimes)$, i.e., those deformations where only the associator is deformed. Indeed, such a deformation is given by 2-isomorphisms of the form $$\begin{aligned} \widehat{\otimes}_{\epsilon}((f',g'),(f,g))&=\widehat{\otimes}((f',g'),(f,g)) \\ (\otimes_0)_{\epsilon}(X,Y)&=1_{id_{X\otimes Y}} \\ \widehat{a}_{\epsilon}(f,g,h)&=1_{f\otimes g\otimes h}+\widehat{a}^{(1)}(f,g,h)\epsilon \\ (\pi_{\epsilon})_{X,Y,Z,T}&=1_{id_{X\otimes Y\otimes Z\otimes T}}\end{aligned}$$ According to the proof of the previous theorem, they define a first-order associator-deformation of  if and only if the $\widehat{a}^{(1)}(f,g,h)$ define an element $\widehat{a}^{(1)}\in X_s^{0,2}(\text\Cg)$ which moreover satisfies that $$\delta_v(\widehat{a}^{(1)})=0=\delta_h(\widehat{a}^{(1)})$$ The first equality just says that $\widehat{a}^{(1)}$ is a 2-cocycle of the cochain complex $(X^{0,\bullet}_s(\text\Cg),\delta_v)$, while the second one serves to define a subcomplex of this complex. More explicitly, let’s define $$X^n_{ass}(\text\Cg):=\text{Ker}(\delta_h:X^{0,n}_s(\text\Cg){\longrightarrow}X^{1,n}_s(\text\Cg))\ \ \ \ \ \ n\geq 0$$ Since $X^{\bullet,\bullet}_s(\text\Cg)$ is a double complex, its horizontal coboundary map $\delta_h$ is a morphism of complexes. But the kernel of a morphism of complexes is a subcomplex. Hence, the subspaces $X^n_{ass}(\text\Cg)$, $n\geq 0$, together with the corresponding restriction of $\delta_v$, which will be denoted by $\delta_{ass}$, define a cochain complex. Let us call it the [*associator-deformation complex*]{} of the Gray semigroup $(\text\Cg,\otimes)$. Then, if $H^{\bullet}_{ass}(\text\Cg)$ denote the corresponding cohomology groups, the following result is an immediate consequence of the previous theorem: If $(\text\Cg,\otimes)$ is a $K$-linear Gray semigroup, the $\psi$-equivalence classes of its first order associator-deformations are in bijection with the elements of $H^2_{ass}(\text\Cg)$. Notice that the $\widehat{\psi}^{(1)}\in X^{0,2}_s(\text\Cg)$ must be such that $\delta_h(\widehat{\psi}^{(1)})=-(\widehat{\otimes}^{(1)})'+\widehat{\otimes}^{(1)}=0$, i.e., $\widehat{\psi}^{(1)}\in X^1_{ass}(\text\Cg)$, as required. In a similar way, we can easily get a cochain complex describing the infinitesimal [*unitary tensorator-deformations*]{} of $\text\Cg$, i.e., those deformations where only the tensor product is deformed, and in such a way that it remains unitary. Indeed, such a deformation is given by 2-isomorphisms of the form $$\begin{aligned} \widehat{\otimes}_{\epsilon}((f',g'),(f,g))&=\widehat{\otimes}((f',g'),(f,g))+ \widehat{\otimes}^{(1)}((f',g'),(f,g))\epsilon \\ (\otimes_0)_{\epsilon}(X,Y)&=1_{id_{X\otimes Y}} \\ \widehat{a}_{\epsilon}(f,g,h)&=1_{f\otimes g\otimes h}\\ (\pi_{\epsilon})_{X,Y,Z,T}&=1_{id_{X\otimes Y\otimes Z\otimes T}}\end{aligned}$$ Again going back to the proof of Theorem \[tensorator,associator-deformations\], it immediately follows that they define a first-order unitary tensorator-deformation of  if and only if the $\widehat{\otimes}^{(1)}((f',g'),(f,g))$ define an element $\widehat{\otimes}^{(1)}\in X_s^{1,1}(\text\Cg)$ which moreover satisfies that $$\delta_h(\widehat{\otimes}^{(1)})=0=\delta_v(\widehat{\otimes}^{(1)})$$ Now, recall that $X_s^{n,1}(\text\Cg)=X_s^{n+1}(\otimes)$, the special $n$-cochains of the purely pseudofunctorial deformation complex of the (unitary) pseudofunctor $\otimes$ (see Section 6). Then, the first equality just says that $\widehat{\otimes}^{(1)}$ is a 2-cocycle of this cochain complex $X^{\bullet}_s(\otimes)$, while the second one serves to define a subcomplex of this complex. More explicitly, let’s define $$X^n_{tens}(\text\Cg):=\text{Ker}(\delta_v:X^{n+1}_s(\otimes){\longrightarrow}X^{n,1}_s(\text\Cg))\ \ \ \ \ \ n\geq 0$$ The same argument as before shows that these subspaces define a subcomplex. Let us call it the [*unitary tensorator-deformation complex*]{} of the Gray semigroup $(\text\Cg,\otimes)$. Then, if $H^{\bullet}_{tens}(\text\Cg)$ denote the corresponding cohomology groups, the following result is an immediate consequence of the previous theorem: If $(\text\Cg,\otimes)$ is a $K$-linear Gray semigroup, the $\psi$-equivalence classes of its first order unitary tensorator-deformations are in bijection with the elements of $H^2_{tens}(\text\Cg)$. Cohomology theory for the generic unitary deformations ====================================================== In Sections 7 and 8 we have constructed complexes $X^{\bullet}_{pent}(\text\Cg)$ and $X^{\bullet}_{tens,ass}(\text\Cg)$ whose cohomologies separately describe the infinitesimal unitary deformations of the pentagonator, on the one hand, and the tensor product and the associator, on the other, of a $K$-linear Gray semigroup $(\text\Cg,\otimes)$ (actually, of an arbitrary $K$-linear semigroupal 2-category in the case of the deformations of the pentagonator). The goal of this section is to obtain a cohomology describing the simultaneous unitary deformations of both sets of structural 2-isomorphisms. To do that, we will need to go back to the bigger cochain complex $\widetilde{X}^{\bullet}_{pent}(\text\Cg)\supseteq X^{\bullet}_{pent}(\text\Cg)$ introduced in Section 7. The appropriate cohomology for the generic unitary deformations turns out to be the total complex of a modified version of the double complex of $(\text\Cg,\otimes)$ introduced in the previous section. The modification consists of the substitution of the first column $X^{1,\bullet}(\text\Cg)$ in this double complex by a suitable cone of that column and the general pentagonator-deformation complex $\widetilde{X}^{\bullet}_{pent}(\text\Cg)$. Recall that given two cochain complexes $(A^{\bullet},\delta_A)$ and $(B^{\bullet},\delta_B)$ and a morphism of complexes $\varphi:(A^{\bullet},\delta_A){\longrightarrow}(B^{\bullet},\delta_B)$, the cone of $A^{\bullet}$ and $B^{\bullet}$ over $\varphi$ is the cochain complex $(C^{\bullet}_{\varphi}(A^{\bullet},B^{\bullet}),\delta_C)$ defined by $$C^n_{\varphi}(A^{\bullet},B^{\bullet})=B^n\oplus A^{n+1}$$ and $$\delta_C(b,a)=(\delta_B(b)+\varphi(a),-\delta_A(a)),\ \ \ \ (b,a)\in B^n\oplus A^{n+1}.$$ The minus sign in $\delta_A$ is to ensure that $\delta_C\circ\delta_C=0$. For more details see, for example, Weibel[@cW94]. Let $(\text\Cg,\otimes)$ be a $K$-linear Gray semigroup, and let’s consider the general pentagonator-deformation complex $\widetilde{X}^{\bullet}_{pent}(\text\Cg)$ defined in Section 7. The associator $a$ of  being trivial, this complex reduces to $$\widetilde{X}^{n-1}_{pent}(\text\Cg)=\text{PseudMod}({\mathbf 1}_n,{\mathbf 1}_n),\ \ \ \ \ n\geq 1$$ where ${\mathbf 1}_n$ is the pseudonatural isomorphism ${\mathbf 1}_n:\otimes(n)\Longrightarrow\otimes(n)$ whose structural 1- and 2-isomorphisms are all identities, while the coboundary is given by $$(\delta_{pent}(\text\nn))_{X_0,\ldots,X_n}=1_{id_{X_0}}\otimes\text\nn_{X_1,\ldots,X_n}+ \sum_{i=1}^n(-1)^{i}\text\nn_{X_0,\ldots,X_{i-1} \otimes X_i,\ldots,X_n}+(-1)^{n+1}\text\nn_{X_0,\ldots,X_{n-1}}\otimes 1_{id_{X_n}}$$ (in this case, the padding operators are superfluous). Notice that a generic element $\text\nn\in\widetilde{X}^{n-1}_{pent}(\text\Cg)$, $n\geq 1$, is just a collection of 2-morphisms $\text\nn_{X_1,\ldots,X_n}:id_{X_1\otimes\cdots\otimes X_n}\Longrightarrow id_{X_1\otimes\cdots\otimes X_n}$, for all objects $(X_1,\ldots,X_n)$ of $\text\Cg^n$. Let us consider, on the other hand, the complex $(X^{0,\bullet}(\text\Cg),\delta_v)$ corresponding to the first column of the extended double complex of $(\text\Cg,\otimes)$. We can define $K$-linear maps $\varphi:\widetilde{X}^{n-1}_{pent}(\text\Cg){\longrightarrow}X^{0,n-1}(\text\Cg)$, $n\geq 1$ by $$(\varphi(\text\nn))(f_1,\ldots,f_n)=\text\nn_{X'_1,\ldots,X'_n}\circ 1_{f_1\otimes\cdots\otimes f_n}-1_{f_1\otimes\ldots\otimes f_n}\circ\text\nn_{X_1,\ldots,X_n}$$ for all 1-morphisms $(f_1,\ldots,f_n):(X_1,\ldots,X_n){\longrightarrow}(X'_1,\ldots,X'_n)$ of $\text\Cg^n$. The above maps $\varphi_{\bullet}:\widetilde{X}^{\bullet}_{pent}(\text\Cg){\longrightarrow}X^{0,\bullet}(\text\Cg)$ define a morphism of cochain complexes. Let $\text\nn\in\widetilde{X}^{n-1}_{pent}(\text\Cg)$. Then, we have $$\begin{aligned} (\delta_v(\varphi(\text\nn)))(f_0,\ldots,f_n)&=1_{f_0}\otimes (\varphi(\text\nn))(f_1,\ldots,f_n)+ \\ &\ \ \ +\sum_{i=1}^n(-1)^i(\varphi(\text\nn)) (f_0,\ldots,f_{i-1}\otimes f_i,\ldots,f_n)+ \\ &\ \ \ + (-1)^{n+1}(\varphi(\text\nn))(f_0,\ldots,f_{n-1})\otimes 1_{f_n} \\ &=-1_{f_0}\otimes(1_{f_1\otimes\cdots\otimes f_n} \circ\text\nn_{X_1,\ldots,X_n})+ \\ &\ \ \ +1_{f_0}\otimes(\text\nn_{X'_1,\ldots,X'_n}\circ 1_{f_1\otimes\cdots\otimes f_n})- \\ &\ \ \ -\sum_{i=1}^n(-1)^i1_{f_0\otimes\cdots\otimes f_n} \circ\text\nn_{X_0,\ldots,X_{i-1}\otimes X_i,\ldots,X_n}+ \\ &\ \ \ + \sum_{i=1}^n(-1)^i\text\nn_{X'_0,\ldots,X'_{i-1}\otimes X'_i,\ldots,X'_n}\circ 1_{f_0\otimes\cdots\otimes f_n}- \\ &\ \ \ -(-1)^{n+1}(1_{f_0\otimes\cdots\otimes f_{n-1}} \circ\text\nn_{X_0,\ldots,X_{n-1}})\otimes 1_{f_n}+ \\ &\ \ \ +(-1)^{n+1}(\text\nn_{X'_0,\ldots,X'_{n-1}}\circ 1_{f_0\otimes\cdots\otimes f_{n-1}})\otimes 1_{f_n}\end{aligned}$$ (the reader may easily check that the padding operators appearing in the definition of $\delta_v$ are indeed trivial in this case). Now, since  is a Gray semigroup, it is $\widehat{\otimes}((f_0,f_1\otimes\cdots\otimes f_n),(id_{X_0},id_{X_1\otimes\cdots\otimes X_n}))=\widehat{\otimes}((id_{X_0},id_{X_1\otimes\cdots\otimes X_n}),(f_0,f_1\otimes\cdots\otimes f_n))=1_{f_0\otimes\cdots\otimes f_n}$. Therefore, using Equation $(A\widehat{\otimes}1)$, we get $$\begin{aligned} &1_{f_0}\otimes(1_{f_1\otimes\cdots\otimes f_n} \circ\text\nn_{X_1,\ldots,X_n})=1_{f_0\otimes\cdots\otimes f_n} \circ(1_{id_{X_0}}\otimes\text\nn_{X_1,\ldots,X_n}) \\ &1_{f_0}\otimes(\text\nn_{X'_1,\ldots,X'_n}\circ 1_{f_1\otimes\cdots\otimes f_n})=(1_{id_{X_0}}\otimes\text\nn_{X'_1,\ldots,X'_n})\circ 1_{f_0\otimes\cdots\otimes f_n}\end{aligned}$$ The last two terms are treated similarly. It follows immediately that $\delta_v(\varphi(\text\nn))=\varphi(\delta_{pent}(\text\nn))$, as required. Notice that the pentagonator-deformation subcomplex $X^{\bullet}_{pent}(\text\Cg)\subseteq\widetilde{X}^{\bullet}_{pent}(\text\Cg)$ is nothing more that the kernel of this morphism $\varphi$. Associated to this cochain map, there is the corresponding cone complex, which will be denoted by $(X^{\bullet}_{pent,ass}(\text\Cg),\delta_{pent,ass})$. By definition $$X^n_{pent,ass}(\text\Cg)=X^{0,n}(\text\Cg)\oplus \widetilde{X}^{n+1}_{pent}(\text\Cg),\ \ \ \ \ \ \ n\geq 0$$ with coboundary map $\delta_{pent,ass}:X^n_{pent,ass}(\text\Cg){\longrightarrow}X^{n+1}_{pent,ass}(\text\Cg)$ given by $$\delta_{pent,ass}(\phi,\text\nn)=(\delta_v(\phi)+\varphi(\text\nn),-\delta_{pent}(\text\nn))$$ for all $(\phi,\text\nn)\in X^n_{pent,ass}(\text\Cg)$. We have the following modified version of the extended double complex of $(\text\Cg,\otimes)$: Let’s substitute the first column $(X^{0,\bullet}(\text\Cg),\delta_v)$ of the extended double complex $(X^{\bullet,\bullet}(\text\Cg),\delta_h,\delta_v)$ of $(\text\Cg,\otimes)$ for the previous cone complex $(X^{\bullet}_{pent,ass}(\text\Cg),\delta_{pent,ass})$, and the coboundary maps $\delta_h:X^{0,n}(\text\Cg){\longrightarrow}X^{1,n}(\text\Cg)$, $n\geq 0$, for the maps $\delta'_h:X^n_{pent,ass}(\text\Cg){\longrightarrow}X^{1,n}(\text\Cg)$ given by the projection to $X^{0,n}(\text\Cg)$ followed by $\delta_h$. Then, the resulting collection of $K$-vector spaces and linear maps is a double complex (see Fig.  \[figura\_complex\_doble\_modificat\]). By the way the $\delta'_h$ are defined, it is clear that the new rows are cochain complexes. So, we only need to see that the squares on the left of Fig???? still commute. Let $(\phi,\text\nn)\in X^{n-1}_{pent,ass}(\text\Cg)$. Since the coboundary maps $\delta_h,\delta_v$ commute, we have $$\begin{aligned} \delta'_h(\delta_{pen,ass}(\phi,\text\nn))&= \delta'_h(\delta_v(\phi)+\varphi(\text\nn),-\delta_{pent}(\text\nn)) \\ &=\delta_h(\delta_v(\phi))+\delta_h(\varphi(\text\nn)) \\ &=\delta_v(\delta_h(\phi))+\delta_h(\varphi(\text\nn)) \\ &=\delta_v(\delta'_h(\phi,\text\nn))+\delta_h(\varphi(\text\nn))\end{aligned}$$ Therefore, the proof reduces to show that the term $\delta_h(\varphi(\text\nn))$ is zero for all $\text\nn\in\widetilde{X}^n_{pent}(\text\Cg)$. Now, by definition of $\delta_h$ and $\varphi$, we have $$\begin{aligned} \delta_h(\varphi(\text\nn))((f'_0,\ldots,f'_n),(f_0,\ldots,f_n))&= \lceil 1_{f'_0\otimes\cdots\otimes f'_n}\circ(\varphi(\text\nn))(f_0,\ldots,f_n)\rceil \\ &\ \ \ \ \ \ \ \ \ -\lceil(\varphi(\text\nn))(f'_0\circ f_0,\ldots,f'_n\circ f_n)\rceil \\ &\ \ \ \ \ \ \ \ \ +\lceil(\varphi(\text\nn))(f'_0,\ldots,f'_n)\circ 1_{f_0\otimes\cdots\otimes f_n}\rceil \\ &=-\lceil 1_{f'_0\otimes\cdots\otimes f'_n}\circ (1_{f_0\otimes\cdots\otimes f_n}\circ\text\nn_{X_0,\ldots,X_n})\rceil \\ &\ \ \ \ \ \ \ \ \ +\lceil 1_{f'_0\otimes\cdots\otimes f'_n}\circ(\text\nn_{X'_0,\ldots,X'_n}\circ 1_{f_0\otimes\cdots\otimes f_n})\rceil \\ &\ \ \ \ \ \ \ \ \ +\lceil 1_{(f'_0\circ f_0)\otimes\cdots\otimes (f'_n\circ f_n)} \circ\text\nn_{X_0,\ldots,X_n}\rceil \\ &\ \ \ \ \ \ \ \ \ -\text\nn_{X''_0,\ldots,X''_n}\circ 1_{(f'_0\circ f_0)\otimes\cdots\otimes (f'_n\circ f_n)}\rceil \\ &\ \ \ \ \ \ \ \ \ -\lceil (1_{f'_0\otimes\cdots\otimes f'_n}\circ\text\nn_{X'_0,\ldots,X'_n})\circ 1_{f_0\otimes\cdots\otimes f_n}\rceil \\ &\ \ \ \ \ \ \ \ \ +\lceil(\text\nn_{X''_0,\ldots,X''_n}\circ 1_{f'_0\otimes\cdots\otimes f'_n})\circ 1_{f_0\otimes\cdots\otimes f_n}\rceil\end{aligned}$$ Since  is a 2-category, the second and fifth terms clearly cancel out each other. On the other hand, using again that  is a 2-category and the interchange law and the naturality of the 2-morphisms $\widehat{\otimes(n+1)}((f'_0,\ldots,f'_n),(f_0,\ldots,f_n))$ in $(f'_0,\ldots,f'_n),(f_0,\ldots,f_n)$, we have $$\begin{aligned} \lceil 1_{f'_0\otimes\cdots\otimes f'_n}\circ&(1_{f_0\otimes\cdots\otimes f_n}\circ\text\nn_{X_0,\ldots,X_n})\rceil= \\ &=\widehat{\otimes(n+1)}((f'_0,\ldots,f'_n),(f_0,\ldots,f_n))\cdot (1_{(f'_0\otimes\cdots\otimes f'_n)\circ(f_0\otimes\cdots\otimes f_n)}\circ\text\nn_{X_0,\ldots,X_n}) \\ &=(\widehat{\otimes(n+1)}((f'_0,\ldots,f'_n),(f_0,\ldots,f_n))\circ 1_{id_{X_0\otimes\cdots\otimes X_n}})\cdot (1_{(f'_0\otimes\cdots\otimes f'_n)\circ(f_0\otimes\cdots\otimes f_n)}\circ\text\nn_{X_0,\ldots,X_n}) \\ &=(\widehat{\otimes(n+1)}((f'_0,\ldots,f'_n),(f_0,\ldots,f_n))\cdot 1_{(f'_0\otimes\cdots\otimes f'_n)\circ(f_0\otimes\cdots\otimes f_n)})\circ \text\nn_{X_0,\ldots,X_n} \\ &=(1_{(f'_0\circ f_0)\otimes\cdots\otimes (f'_n\circ f_n)}\cdot \widehat{\otimes(n+1)}((f'_0,\ldots,f'_n),(f_0,\ldots,f_n)))\circ\text\nn_{X_0,\ldots,X_n} \\ &=(1_{(f'_0\circ f_0)\otimes\cdots\otimes (f'_n\circ f_n)}\circ\text\nn_{X_0,\ldots,X_n}) \cdot\widehat{\otimes(n+1)}((f'_0,\ldots,f'_n),(f_0,\ldots,f_n))) \\ &=\lceil 1_{(f'_0\circ f_0)\otimes\cdots\otimes (f'_n\circ f_n)} \circ\text\nn_{X_0,\ldots,X_n} \rceil\end{aligned}$$ Therefore, the first and third term above also cancel out each other. The same thing can be shown similarly for the fourth and sixth terms. This new double complex will be called the [*modified extended double complex*]{} of $(\text\Cg,\otimes)$, and denoted by $X^{\bullet,\bullet}_{mod}(\text\Cg)$. So $$X^{m,n}_{mod}(\text\Cg)=\left\{ \begin{array}{ll} X^{0,n}(\text\Cg)\oplus\widetilde{X}^{n+1}_{pent}(\text\Cg) & \text{if}\ m=0,n\geq 0 \\ X^{m,n}(\text\Cg) & \text{if}\ m\geq 1,n\geq 0 \end{array}\right.$$ For short, the corresponding horizontal and vertical coboundary maps will be denoted by $\delta'_h$ and $\delta'_v$, respectively. But remind that $\delta'_v=\delta_v$ except for the first column, where $\delta'_v=\delta_{pent,ass}$, and that $\delta'_h=\delta_h$ except for $m=0$, where it is the projection to the first component followed by $\delta_h$. Let us proceed now as in Section 8 and consider the double complex obtained from $X^{\bullet,\bullet}_{mod}(\text\Cg)$ after deleting the first row. It will be the [*modified double complex*]{} of $(\text\Cg,\otimes)$. Furthermore, let’s take the subcomplex $X^{\bullet,\bullet}_{mod,s}(\text\Cg)$ of this modified double complex corresponding to the special elements. These are defined in the same way as in Section 8 for all $m\geq 1,n\geq 1$, while $(\phi,\text\nn)\in X^{0,n}_{mod}(\text\Cg)$, $n\geq 1$, is called special whenever $\phi$ is special. We leave to the reader to check that special elements are preserved by the coboundary maps $\delta'_h,\delta'_v$, so that $X^{\bullet,\bullet}_{mod,s}(\text\Cg)$ is indeed a double complex (the [*modified special double complex*]{} of $(\text\Cg,\otimes)$). Then, we can consider the associated total complex, which will be denoted by $X^{\bullet}_{unit}(\text\Cg)$, and called the [*unitary deformation complex*]{} of the Gray semigroup $(\text\Cg,\otimes)$. By definition $$X^q_{unit}(\text\Cg)=\bigoplus_{\begin{array}{c} m+n=q \\ m\geq 0,n\geq 1 \end{array}}X^{m,n}_{mod,s}(\text\Cg), \ \ \ \ \ q\geq 1$$ and the coboundary operator $\delta_{unit}:X^q_{unit}(\text\Cg){\longrightarrow}X^{q+1}_{unit}(\text\Cg)$ is $$\delta_{unit}=\bigoplus_{\begin{array}{c} m+n=q \\ m\geq 0,n\geq 1 \end{array}} ((-1)^n\delta'_h+\delta'_v),\ \ \ \ \ q\geq 1$$ If $H^{\bullet}_{unit}(\text\Cg)$ denote its cohomology groups, we have then the following final theorem, which says that this is the right cochain complex describing the generic unitary deformations: Given a $K$-linear Gray semigroup $(\text\Cg,\otimes)$, the equivalence classes of its first order unitary deformations are in bijection with the elements of $H^2_{unit}(\text\Cg)$. Let’s consider 2-isomorphisms of the form $$\begin{aligned} \widehat{\otimes}_{\epsilon}((f',g'),(f,g))&=\widehat{\otimes}((f',g'),(f,g))+ \widehat{\otimes}^{(1)}((f',g'),(f,g)) \\ (\otimes_0)_{\epsilon}(X,Y)&=1_{id_{X\otimes Y}} \\ \widehat{a}_{\epsilon}(f,g,h)&=1_{f\otimes g\otimes h}+\widehat{a}^{(1)}(f,g,h)\epsilon \\ (\pi_{\epsilon})_{X,Y,Z,T}&=1_{id_{X\otimes Y\otimes Z\otimes T}}+(\pi^{(1)})_{X,Y,Z,T}\epsilon\end{aligned}$$ with $\widehat{\otimes}^{(1)}((f',g'),(f,g)):(f'\otimes g')\circ (f\otimes g)\Longrightarrow (f'\circ f)\otimes (g'\circ g)$, $\widehat{a}^{(1)}(f,g,h):f\otimes g\otimes h\Longrightarrow f\otimes g\otimes h$ and $\pi^{(1)}_{X,Y,Z,T}:id_{X\otimes Y\otimes Z\otimes T}\Longrightarrow id_{X\otimes Y\otimes Z\otimes T}$. In particular, the $\pi^{(1)}_{X,Y,Z,T}$ clearly define an element $$\pi^{(1)}\in\widetilde{X}^3_{pent}(\text\Cg)\subseteq X^{0,2}_s(\text\Cg)\oplus \widetilde{X}^3_{pent}(\text\Cg)=X^{0,2}_{mod,s}(\text\Cg)\subseteq X^2_{unit}(\text\Cg)$$ Then, applying Proposition \[estructura\_semigrupal\], it follows that the above 2-isomorphisms define a semigroupal structure on $\text\Cg_{(1)}^0$ (hence, a first order unitary deformation of $\text\Cg$) if and only if: A$\widehat{\otimes}$1 : The $\widehat{\otimes}^{(1)}((f',g'),(f,g))$ define an element $\widehat{\otimes}^{(1)}\in X^{1,1}(\text\Cg)=X^{1,1}_{mod}(\text\Cg)$. A$\widehat{\otimes}$2 : $\widehat{\otimes}^{(1)}$ is such that $\delta_h(\widehat{\otimes}^{(1)})=0$. A$\widehat{\otimes}$3 : $\widehat{\otimes}^{(1)}$ is special. A$\widehat{a}$1 : The $\widehat{a}^{(1)}(f,g,h)$ define an element $\widehat{a}^{(1)}\in X^{0,2}(\text\Cg)\subset X^{0,2}(\text\Cg)\oplus\widetilde{X}^3_{pent}(\text\Cg)=X^{0,2}_{mod}(\text\Cg)$. A$\widehat{a}$2 : $\widehat{a}^{(1)}$ and $\widehat{\otimes}^{(1)}$ are such that $\delta_h(\widehat{a}^{(1)})+\delta_v(\widehat{\otimes}^{(1)})=0$. A$\widehat{a}$3 : $\widehat{a}^{(1)}$ is special. A$\pi$1 : $\widehat{a}^{(1)}$ and $\pi^{(1)}$ are such that $\delta_v(\widehat{a}^{(1)})+\varphi(\pi^{(1)})=0$. A$\pi$2 : $\pi^{(1)}$ is such that $\delta_{pent}(\pi^{(1)})=0$. Now, $\pi^{(1)}\in\widetilde{X}^3_{pent}(\text\Cg)$ together with $(A\widehat{\otimes} 1)$, $(A\widehat{\otimes} 3)$ and $(A\widehat{a} 1)$, $(A\widehat{a} 3)$ say that $$((\widehat{a}^{(1)},\pi^{(1)}),\widehat{\otimes}^{(1)})\in (X^{0,2}_s(\text\Cg)\oplus\widetilde{X}^3_{pent}(\text\Cg))\oplus X^{1,1}_s(\text\Cg)=X^2_{unit}(\text\Cg)$$ On the other hand, we have $$\delta_{unit}((\widehat{a}^{(1)},\pi^{(1)}),\widehat{\otimes}^{(1)})= ((\delta_v(\widehat{a}^{(1)})+\varphi(\pi^{(1)}),-\delta_{pent}(\pi^{(1)})), \delta_h(\widehat{a}^{(1)})+\delta_v(\widehat{\otimes}^{(1)}), -\delta_h(\widehat{\otimes}^{(1)}))$$ Hence, $(A\pi 1)$, $(A\pi 2)$, $(A\widehat{a}2)$ and $(A\widehat{\otimes}2)$ together exactly say that $((\widehat{a}^{(1)},\pi^{(1)}),\widehat{\otimes}^{(1)})$ is a 2-cocycle. Let’s consider $(((\widehat{a}^{(1)})',(\pi^{(1)})'),(\widehat{\otimes}^{(1)})')$ another 2-cocycle defining an equivalent first order unitary deformation of . Then, by Proposition \[equivalents\] applied to our situation, there exists 2-morphisms $\widehat{\psi}^{(1)}(f,g):f\otimes g\Longrightarrow f\otimes g$ and $(\omega^{(1)})_{X,Y,Z}:id_{X\otimes Y\otimes Z}\Longrightarrow id_{X\otimes Y\otimes Z}$ (in particular, $\omega^{(1)}\in\widetilde{X}^2_{pent}(\text\Cg)$) such that E$\widehat{\psi}$1 : The $\widehat{\psi}^{(1)}(f,g)$ define an element $\widehat{\psi}^{(1)}\in X^{0,1}(\text\Cg)$. E$\widehat{\psi}$2 : $(\widehat{\otimes}^{(1)})'-\widehat{\otimes}^{(1)}=-\delta_h(\widehat{\psi}^{(1)})$. E$\widehat{\psi}$3 : $\widehat{\psi}^{(1)}$ is special. E$\omega$1 : $(\widehat{a}^{(1)})'-\widehat{a}^{(1)}=\delta_v(\widehat{\psi}^{(1)})+\varphi(\omega^{(1)})$ E$\omega$2 : $(\pi^{(1)})'-\pi^{(1)}=-\delta_{pent}(\omega^{(1)})$ The first and third conditions together say that $(\widehat{\psi}^{(1)},\omega^{(1)})\in X^{0,1}_s(\text\Cg)\oplus\widetilde{X}^2_{pent}(\text\Cg)= X^{0,1}_{mod,s}(\text\Cg)=X^1_{unit}(\text\Cg)$. On the other hand, the reader may check that $$\delta_{unit}(\widehat{\psi}^{(1)},\omega^{(1)})= ((\delta_v(\widehat{\psi}^{(1)})+\varphi(\omega^{(1)}),-\delta_{pent}(\omega^{(1)})), -\delta_h(\widehat{\psi}^{(1)}))$$ so that the remaining conditions just say that $$(((\widehat{a}^{(1)})',(\pi^{(1)})'),(\widehat{\otimes}^{(1)})')- ((\widehat{a}^{(1)},\pi^{(1)}),\widehat{\otimes}^{(1)})= \delta_{unit}(\widehat{\psi}^{(1)},\omega^{(1)})$$ Hence, both 2-cocycles are cohomologous, as required. Concluding remarks ================== The present work is not intended to give the complete picture of the theory of infinitesimal deformations of a monoidal 2-category. Indeed, various points still remain for future work. Among them, let us mention the following: 1 : There is the all-important question of the higher-order obstructions. This turned out to be the most difficult point in the cohomological deformation theory for monoidal categories first initiated by Crane and Yetter [@CY981] and further developped by Yetter [@dY01]. We guess that our cohomological description also fits nicely into the general picture established by Gerstenhaber for a good deformation theory [@mG64]. But as already mentioned, this is left for a future work. 2 : As stated at the beginning of this work, the ultimate goal should be to get a cohomological description of the infinitesimal deformations of a [*monoidal*]{} 2-category. Hence, it also deserves further work the question of how to take into account the additional unital structure in the whole theory. In the case of monoidal categories, Yetter [@dY01] has shown that the deformations of this additional structure are already determined by those of the semigroupal structure. It seems possible that the same situation reproduces in our framework. 3 : Finally, in the present work we have restricted our attention to those infinitesimal deformations of the semigroupal 2-category $(\text\Cg,\otimes,a,\pi)$ such that the bicategory structure of $\text\Cg$ remains undeformed. But, as pointed out previously, this structure can also be deformed. As regards this point, notice that the elements $\phi\in X^{2,0}(\text\Cg)$ of our extended double complex are of the form $$\phi(h,g,f):h\circ g\circ f\Longrightarrow h\circ g\circ f$$ (we are thinking of a Gray semigroup, so that parenthesis are not needed here). This suggests that the possible deformations of the bicategory structure of  may be related to such elements $\phi\in X^{2,0}(\text\Cg)$. In this sense, the situation can once more resemble that encountered in the deformation theory of a bialgebra. Indeed, it can be shown (see [@GS92],[@sS92],[@SS93]) that the right cochain complex describing the deformations of a bialgebra as a [*quasibialgebra*]{} (i.e., coassociative only up to conjugation) is precisely that associated to the double complex obtained after adding the bottom row of the full Gerstenhaber-Schack complex, which had to be deleted to study the deformations in the bialgebra setting. In our case, the weakening of the coassociativity condition should correspond to the weakening of the 2-category condition $\alpha_{h,g,f}=1_{h\circ g\circ f}$. It seems possible, then, that taking into account the deleted bottom row in our double complex is just the only step needed to consider these more general deformations of . Another important point not addressed in this paper, and which we are currently working on, is the question of examples. In particular, a simple example of a $K$-linear Gray semigroup where our theory can be applied is that introduced by Mackaay [@mM00], denoted by [**N**]{}$(G,H,K^*)$, and associated to a pair of finite groups $G,H$ (with $H$ an abelian group); it includes as a special case the (semistrict version of) the monoidal 2-category of 2-vector spaces. More interesting examples, however, are expected to come from the 2-categories of representations of the Hopf categories associated to quantum groups, whose construction was sketched by Crane and Frenkel [@CF94]. Finally, let us finish by mentioning the interest our work may have for homotopy theory. Indeed, since Grothendieck [@aG83], it was suspected that homotopy $n$-types were somewhat equivalent to certain algebraic structures called [*weak n-groupoids*]{}, which should be a particular kind of [*weak n-categories*]{} characterized by the fact that all morphisms are invertible up to suitable equivalence. Recently, Tamsamani [@zT99] realized this idea, giving a precise definition of a weak $n$-groupoid, for any non-negative integer $n$, together with a suitable notion of equivalence, and showing that the equivalence classes of weak $n$-groupoids bijectively correspond to homotopy classes of $n$-anticonnected CW-complexes. Since weak 3-groupoids with one object should correspond to a special type of semigroupal 2-categories, it naturally raises the question about the meaning our cohomology theory has in this topological setting. Such a relation between certain monoidal 2-categories and homotopy 3-types has recently been discussed by Mackaay [@mM00], who conjectures that the classification of [*semi-weak*]{} monoidal 2-category structures on the above mentioned 2-category [**N**]{}$(G,H,K^*)$ boils down to the classification up to homotopy equivalence of connected 3-anticonnected ($>$1-simple) CW-complexes $X$ with $\pi_1(X)=G$, $\pi_2(X)=H$ and $\pi_3(X)=K^*$. Via Postnikov’s theory, this leds him to conjecture a bijection between the equivalence classes of semi-weak monoidal 2-category structures on [**N**]{}$(G,H,K^*)$ and pairs of cohomology classes $\alpha\in H^3(B_G,H)$ and $\beta\in H^4(W(\alpha),K^*)$ ($W(\alpha)$ denotes a certain CW-complex induced by $\alpha$; see [@mM00]). [[*Acknowledgements.*]{}]{} First of all I want to thank Louis Crane (my coadvisor together with Sebastià Xambó) for suggesting me the topic of this work for my PhD, and Louis Crane and David Yetter for the helpful discussions with them. I also want to thank the Department of Mathematics of Kansas State University (KSU) and Louis Crane (and his wife) in particular for their hospitality during my stay in Manhattan, where I started this work. Finally, I want to thank the Departament d’Universitats, Recerca i Societat de la Informació (DURSI) of the Generalitat the Catalunya and the Universitat Politècnica de Catalunya (UPC) for their financial support which enabled me to visit KSU. [^1]: [^2]: As shown by Yetter [@dY01], it is enough to only consider deformations of the associator, since they already induce a deformation of the unital structure. [^3]: The term semigroupal applied to categories means a category with a tensor product and a coherent associator, but without unit constraints; when applied to functors, it means a functor $F$ between semigroupal (or monoidal) categories together with a coherent natural isomorphism $\widehat{F}:\otimes\circ(F\times F)\Longrightarrow F\circ\otimes$, but without the isomorphism $F_0:F(I)\longrightarrow I$. Note also that Epstein actually considers semigroupal categories equipped with a symmetry. [^4]: Strictly speaking, tricategories of one object correspond to the more general notion of a monoidal [*bicategory*]{}. [^5]: Actually, they use the name [*Gray monoid*]{}, because they consider monoidal 2-categories. [^6]: We would like to thank J. Power and R. Street for the emails interchanged about this point. [^7]: Here, we think of the elementary composition functors $c^{\text\Cg}_{X,Y,Z}$ as defined on the product category $\text\Cg(Y,Z)\times\text\Cg(X,Y)$, instead of $\text\Cg(X,Y)\times\text\Cg(Y,Z)$. Hence, they differ from those appearing in Definition \[bicategory\] by a permutation functor. [^8]: This result should be viewed as an analog of Yetter’s result that a semigroupal deformation of a monoidal functor uniquely extends to a monoidal deformation. See [@dY01], Theorem 17.2. [^9]: Recall that the term [*unitary*]{} applied to an infinitesimal deformation always means that the deformed tensor product is supposed to be still unitary, i.e., such that the 2-isomorphisms $\otimes_0(X,Y)$ remain trivial even after the deformation. At the moment of writing, the author doesn’t know how to take into account the non trivial deformations of these 2-isomorphisms.

--- abstract: 'We experimentally probe complex bio-photonic architecture of microstructures on the transparent insect wings by a simple, non-invasive, real time optical technique. A stable and reproducible far-field diffraction pattern in transmission was observed using collimated cw and broadband fs pulses. A quantitative analysis of the observed diffraction pattern unveiled a new form of long-range semi-periodic order of the microstructures over $mm$ scale. These observations agree well with Fourier analysis of SEM images of the wing taken at various length scales. We propose a simple quantitative model based on optical diffraction by an array of non-overlapping microstructures with minimal disorder which supports our experimental observations. Two different applications of our techniques are demonstrated. First, by scanning the laser beam across the wing sample we observed a rotation of the original diffraction profile which gives direct signature of organizational symmetry of microstructures. Second, we report the first optical detection of reorganization in the photonic architecture on the Drosophila wings by various genetic mutations. These results have potentials for design and development of diffractive optical components for applications and identifying routes to genetic control of biomemetic devices.' address: - '$^{a}$Femtosecond Laser Laboratory, Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Manauli-140306, Punjab, India.' - '$^{b}$Department of Biological Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Manauli-140306, Punjab, India.' author: - 'Pramod Kumar$^{a\dagger}$, Danish Shamoon$^{a}$, Dhirendra P. Singh$^{a}$, Sudip Mandal$^{b}$, Kamal P. 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Introduction ============ Nature has developed a remarkable variety of photonic structures in various insect wings [@pet2003; @bar2011; @Laszlo2011]. The cooperation of structural heterogeneities (regularity and irregularity) [@Shuichi2005; @Poya11] in these natural bio-photonic architectures at optical wavelength scale interact with light in a specific way to produce various optical effects such as reflection [@Hafiz2006], interference [@yosh2008], diffraction [@Feng2011] fluorescence [@Eloise2011], iridescence [@step2009], and polarization sensitivity [@shinya2007]. Compared to the equivalent man-made optical devices, the biophotonic structures often possess greater complexity and could outperform their functions in some cases [@bar2011; @kesong2011]. Due to the presence of multiple length scales and diversity in their design the optical behavior of such arrangements is still not fully understood. While the optical effects in the non-transparent wings of the butterfly and beetles have been well studied [@Laszlo2011; @Poya11; @shinya2007; @Shao2010], the transparent wings of many insects (Drosophila, bees, dragonfly) have attracted much less attention. Previous experiments on thin transparent wings observed various interference colours under white light illumination and quantified their transmission and reflection spectra [@Valerie2009; @Ekaterina2011]. Recently, the transparent wings of firefly have been exploited in designing and optimizing optical components such as anti-reflection elements in laser diodes [@annick2013]. High resolution techniques like scanning electron microscopy (SEM) and atomic force microscopy have resolved structural complexity of transparent wings (of dragonfly) at micro and nano-scale [@Hooper2006]. These studies revealed that the wing surface is decorated with a large number of micro-structures [@Laszlo2011; @Shao2010; @Ekaterina2011; @heeso2010]. These microstructures were known to provide anti-wetting, self-cleaning and aerodynamic properties to the wing surface [@wanger1996; @ghir1974; @sane2003]. Recently, modulation of friction and adhesion on these microstuctures was also observed [@ashima]. However, the long-range organizing principles and symmetry of the complex photonic architectue in transparent wings remain unexplored. A knowledge of their organization would be fundamental to understand how these systems have been naturally optimized to coherently manipulate light for various functions [@Mathias2010; @Shinya2004]. Using high resolution images from SEM/TEM to extract structural order over entire wing surface is not feasible and may even be misleading. Therefore, an efficient and quantitative approach to explore correlations in spatial architecture over the entire length scale of the wing is desired. One attractive possibility is to exploit high sensitivity of optical method that combines right spectral properties with high spatial coherence such as transmitted diffraction pattern [@Feng2011; @kumar2013]. Remarkably, our technique is very sensitive and efficient to extract *in situ* structural organization and symmetry of microstructures in a single-shot manner. We demonstrate that variability in the diffraction pattern is directly correlated with arrangements in the spatial distribution of the scales that cover the wing membranes. This optical imaging technique opens new ways for the non-invasive study and classification of different forms of irregularity in structural patterns. The investigation of natural microstructures lead to the knowledge that can be applied to design biomemetic materials with designer photonic properties with reduced engineering effort and cost of their fabrication [@bar2011; @Poya11]. The aim of this paper is to address the following questions: Is there any long range ($mm$ scale) order and organizational symmetry in the array of microstructures on the transparent insect wing? Can we exploit complex diffraction pattern to quantitatively unveil new features in the arrangement of microstructure array? Answering these questions are crucial to understand design principles and multi-functional role of transparent wings [@Hooper2006; @Shinya2004; @ashima]. The paper is organized as follows. In section $2$, we describe our experimental setup. In section $3$, we report experimental observations of the diffraction pattern using various lasers. We also give a theoretical interpretation of these results using SEM analysis of the samples. In section $4$ and section $5$ we demonstrate applications of the optical techniques by measuring correlated diffraction pattern at various scales in the wing and quantifying the role of genetic mutations on the photonic architecture of the transparent wings of the Drosophila. Experimental set-up {#section-setup} =================== A schematic diagram of our experimental set-up and its actual picture is shown in Fig. \[fig1\]. In our set-up a collimated laser beam passes through a wing sample that is mounted on a $xyz$ micrometer translation stage. The transmitted laser intensity was captured through the wings by a digital camera and analyzed. We used both monochromatic cw lasers at two visible wavelengths in red and green ($\lambda=532~nm$ & $632~nm$) and femtosecond pulses centered in near IR range $800~nm$. These wavelengths are chosen to match the transparency window of the our wing sample. The typical intensity transmission coefficient of our insect wing is around $60 \%$ at these wavelengths. The $1/e^2$ full-waist of the collimated laser beams was around $1mm$ which is much smaller than the typical wing size $>1 cm$. The input beam profile of these lasers is shown in Fig. \[fig2\]. The far-field diffraction pattern was captured on a white screen fixed at $D=20.5~cm$ from the wing sample. We have observed that the diffraction pattern is fully developed after few $cm$ from the wing and it simply diverges thereafter due to geometrical effect. It is worth mentioning that with this simple set-up, no preparation of the wing sample is required. In fact, it can be used for in vivo non-destructive imaging of the wing with the insect alive. The laser powers were very low and no sign of optical damage was seen on the wing surface. To validate the sensitivity of our optical technique we also performed SEM images of the wing surfaces. In addition, we scanned the laser beam across the wing sample and the resulting laser diffraction was recorded in a single-shot manner to probe local variation in the wing structure. We also demonstrate how our optical technique can detect structural reorganization of microstructure array in genetically mutated wings of the Drosophila. Results and discussion {#section-results} ====================== Observation and analysis of complex diffraction pattern {#subsection-diffraction} ------------------------------------------------------- ![ Experimental measurement of complex diffraction pattern. Left column (a, d, g) incident beam profiles for pulsed fs ($800nm$, 10 fs, 2nJ@78MHz) laser, cw $\lambda=532nm$ green solid-state laser and a cw red diode laser ($\lambda=632nm$) from top to bottom, respectively. Middle column (b, e, h) diffraction pattern on a screen using beam profiles in the left. Scale bar is $3 cm$. Right column (c,f,i) intensity cuts along x axis for the corresponding diffraction pattern in the left. []{data-label="fig2"}](Pfig2.pdf){width="80.00000%"} Remarkably, a collimated laser beam ($1/e^2$ full waist around $1mm$) formed a stable and characteristic diffraction pattern after passing through the transparent wing-sample. The laser powers are around $5-200 mW$ which is below their damage thresholds. The far-field diffraction pattern was observed for (i) broadband femtosecond laser pulses centered at $800nm$ (top row in Fig. \[fig2\]), (ii) a cw $532~nm$ green laser, and (iii) a cw $632~nm$ red laser (bottom row in Fig. \[fig2\]). Note that the intensity profile of the diffraction pattern was recorded on a calibrated screen for all the cases. For nearly Gaussian input beams, the observed intensity pattern $I(x,y)$ exhibited a bright central spot and up to two distinct higher order maxima in the form of curved lobes. These lobes are symmetrically located on both sides of the bright central spot. The femtosecond pulse is used to show the robustness of the diffraction pattern under broadband coherent source in the IR range. We have computed the corresponding spatial frequencies along x and y axes, $k_x=(2\pi/{\lambda D}) x $ and $k_y=(2\pi/{\lambda D}) y$, respectively. Here D is the screen to wing distance and x, y are distances measured on the screen from the central spot. The position of the first lobe in the case of cw lasers correspond to spatial frequency of around $0.5 \times 10^{-6} radian/meter$. These spatial frequencies agree well with our theoretical analysis, as shown later. The corresponding intensity-cuts of the diffraction patterns along x axis (right columns in Fig. \[fig2\]) also confirm these values. ![ Laser diffraction from a single microstructure and background. (a) Schematic of the set-up. A triplet lens objective was used to create $<5 \mu m$ spot diameter with the green laser beam. The wing was mounted on a micrometer translation stage. (a-e) Various diffraction patterns of the background and microstructure when the micro-spot was scanned along the long axis of the wing. The pictures were taken on a calibrated screen at $5~cm$ away. []{data-label="fig3"}](Pfig4.pdf){width="70.00000%"} To understand the formation of the diffraction pattern we performed SEM images of the wing surface (Fig. \[fig4\]). The high resolution SEM images showed that the wing surface is decorated with a large number of non overlapping microstructures. The flat background of the wing exhibits nano-grain like features which are typically smaller than the wavelength of the light. These microstructures were elongated having typical length and width around $5-8 \mu m$ and $1-2 \mu m$, respectively. Note that their dimensions are comparable to the used laser wavelengths, these act as efficient “photonic elements” that diffract light. To demonstrate how the observed far-field diffraction pattern emerges by a large number of microstructures (typically $10^{4-6}$ in beam waist of $1.0~mm$ ) we computed Fast Fourier transforms (FFT) of the SEM images at various scales. The SEM images contained total $N \times M$ pixels and it was sampled with the spatial resolution $\Delta x= L/M $ and $\Delta y= L/N$ along x and y axes, respectively. The complimentary Fourier domain then had spatial frequencies $k_{x}= m \Delta k_x$ and $k_{y}= n \Delta k_{y}$ where $m,n$ are integers and $\Delta k_{x,y}=2\pi/L$ determines resolution of the Fourier domain [@dav2003; @mas2003]. As we increased the area of SEM image to incorporate more number of microstructures (Fig. \[fig4\]) the corresponding FFT showed an emergence of the higher order lobes. The corresponding spatial frequency was around $0.5 \times 10^{-6} radian/meter$ that agrees quantitatively with the location of the experimental one. One can define a quasi-periodic function $\Lambda(x, y)$ that determines the average spacing between microstructures. As one can see in the SEM image that the quasi-period $\Lambda(x, y)$ is a function of position rather than a constant. Therefore, the input beam would be diffracted by an angle $\Lambda (x)\sin\Theta_{x}=m\lambda$ [@nan2013]. The corresponding average period between the microstructures along the x-axis was around $12~\mu m$ that also agrees well with the SEM image analysis. To further prove that the diffraction lobes are due to quasiperiodic organization of microstructure array, we performed experiment using tightly focused laser beam. Using a high numerical aperture (NA 0.2) triplet-lens objective we generated a micro-spot of full width $<5 \mu m$. Note that this spot size is comparable with the typical size of a single microstructure. We recorded the diffraction pattern on a screen kept about $5~cm$ away for two different cases when the laser spot is on the microstructrue and when the laser spot is on the background, i.e., between the two microstructures. We observed that these diffraction patterns are qualitatively different compared to the case when we used $1mm$ collimated beam. The FFT of high resolution SEM image with single microstructure also produced a similar complex diffraction pattern that matched with the experimental observations (see Fig. 3 and 4). This demonstrated that the observed diffraction is a result of quasi-periodicity in the array of microstructures. In the following we develop a theoretical model to explain our experimental results. Theoretical understanding of the experimental results {#subsection-theory} ----------------------------------------------------- We present a simple quantitative model to provide further insight into the experiment by generating the far-field diffraction from a two-dimensional array of non-overlapping microstructures. Each micro-structure was modeled by an amplitude transmission function $t(x,y)$. The total transmission $T$ of the wing was then due to $N \times M $ total number of microstructures arranged in a lattice with average distance between them being $d$. Although, a similar jitter model was previously proposed to explain spectral properties of an array of identical nonoverlapping grains [@mah1992; @hen1978; @mar1979; @ru1993]. However, we introduce multiple disorders to model the laser diffraction through complex wing surface. We generated a two-dimensional total brightness distribution T(x,y) by creating square cells, each containing one microstructure, arranged in a rectangular array of size $N \times M$. The distance between two microstructures in the square lattice was $d$ along the two orthogonal directions. In the case of perfectly ordered array of identical microstructure, one can write, $$T(x, y) = t(x,y) \star \sum_{n=0}^{N-1} \sum_{m=0}^{M-1} \ \delta(x -n d) \delta (y - m d),$$ where star denotes convolution and $\delta(x)$ is a Dirac delta function centered at $x=0$. To model structural organization of microstructures on the wing surface, we added disorder in the shape of each microstruture, their position and orientation. The resulting stochastic two-dimensional brightness function is given by, $$T(x, y) = \sum_{n=0}^{N-1} \sum_{m=0}^{M-1} \ t(R - \alpha_{mn} - \xi(n) - \eta_{m,n} ),$$ where $\alpha_{m,n}$ is the position vector of each microstructure, $\xi$ defines a two component stochastic vector quantifying the deviation of their position from their nominal center and $\eta_{m,n}$ is a set of variables determining the shape distortions. The stochastic variables are delta correlated in position as, $$\left\langle \xi_{x,y} (n) \xi_{x,y}(n') \right\rangle = A_{x,y} \delta(n-n'),$$ where $A_{x,y}$ denote the amplitude of the noise along x and y directions. The Fourier transform of the transmission is given by $T(k_x, k_y) = FT(T(x,y))$ where $k_{x,y}$ denote spatial frequencies [@yama2012]. The average normalized far-field diffraction pattern is given by, $$I(x, y) = C \left| T(k_x,k_y) \right|^{2}$$ where $C$ is a proportionality constant. Note that the above equation is usually used for monochromatic incident field and it can be extended to a broadband pulse of width $\Delta\lambda$ that could lead to further detail of quasi-periodicity [@liu1997; @yama2012]. We simulated the above model in Matlab by generating the transmission function $t(x,y)$ in the form of a curved ellipse of about $2~\mu m$ width and about $8~\mu m$ length that is comparable to the dimensions observed in the SEM image. Each microstructure consisted of $16 \times 16$ pixels where each pixel defined $750~nm$ resolution. The total number of microstructures in our simulations were more than $10^3$. We verified that in the case of a perfect square grid of identical microstructures (Eq. 1), the simulated diffraction pattern consisted of ordered spots typical of regular diffraction grating (Fig. \[fig5\]a, d). To model our experiment we generated complex aperture functions by adding various kinds of disorders in the shape, orientation and position of each element. In Fig. \[fig5\]b, we show the disordered array which is produced by adding noise on the shape and orientation of $t(x,y)$. The variance in the fluctuations in the length and width was $8 \pm 4.5~\mu m$ and $2 \pm 0.75~\mu m$, respectively and their orientation was allowed to fluctuate randomly by $\pm 30^\circ$. The chosen parameters lie in the range seen in the SEM images. The corresponding far-field diffraction pattern for this organization showed that higher than second order spots vanished but their organization was on the square grid. Finally, to create the various symmetries in the form of rings we randomly selected several patches and added a global rotation by less than $\pm 15 ^\circ$ in order to create a wavy pattern on the grid structure. In this case, various symmetries overlapped and produced ring-like pattern similar to the experiment (see Fig. \[fig5\]c). Note that the location of the first order peak correspond to spatial frequency $0.5\times 10^{-6}$ $radian/m$ close to the ones seen in the experiment as well as in the Fourier analysis of the SEM images. Though the minimally stochastic model uses many approximations, it can reproduce key signatures of the experiment quite well. Rotations of the diffraction pattern at various length scales {#subsection-rotation} ============================================================= We can probe local variations in the structural arrangement across the wing by simply scanning the laser beam. This is possible since the $1mm$ beam spot was much smaller than the wing size $>1cm$. To demonstrate this, we recorded the diffraction pattern at various scales by scanning the laser spot along the wing length (Fig. 6). A rotation of the original diffraction pattern was observed for both the pulsed and cw lasers. This behavior directly reflects the local symmetry and its spatial correlation along the entire wing surface. It should be mentioned that one can easily vary the spot size to further probe local and the global organizations in a single shot manner. The observed rotation in the diffraction pattern suggests a systematic rotation in the arrangement of the hooks without much change in their density and shape. To verify this, we recorded SEM images of various portions of the wing and numerically computed FFT of the images (Fig. 6(k-o)). The SEM-FFT analysis also produced a similar rotation in the diffraction as shown in Fig. 6. This confirmed our optical observation. Therefore, our technique is very attractive and efficient to reveal the complex arrangements of millions of these photonic elements on the wing surface. We have also observed similar results in insect wings of the Drosophila which suggest a generic nature of the reported phenomenon. The functional significance of these rotations and their development aspects requires further experimentations. Genetic control of the photonic architecture {#section-application} ============================================= We report the first optical measurements of reorganization in mictrostructure array due to the genetic mutations on the transparent wings of the Drosophila melanogaster. The Drosophila wings are ideal system for this kind of study since development, structure and function of the wings have been studied extensively and the roles of several genetic mutants are well characterized. For our analyses we selected two different mutants, Cyo and vg, that either produces curly wings or generates small stumpy wing rudiments, respectively. The wild type unmutated wings were kept as controls (reference). Although the developmental dynamics of Drosophila wings along with the implications of these mutants on the structural and functional aspect have been explored, much less is known about how these mutations affect organization of the microstructures on the wing surface. Using diffraction pattern in transmission offers us a unique optical technique to make a quantitative comparison among wings produced by various mutations. First, we recorded the diffraction pattern from a normal fly-wing as our control. This clearly showed a ring shaped pattern revealing average periodicity of around $20~\mu m$. Compared to the control wing, in Cyo wings, the diffraction ring became smaller with a weak second order lobe appeared. This suggests that the average periodicity increased and their organization became more ordered compared to the normal wings. In contrast, in vg wings the diffraction pattern was speckle-like without any higher order maxima and minima. This means that in this case both the symmetry and average periodicity is completely absent. Note that if such an information is attempted by SEM imaging, it would be very tedious and inefficient process. This is clearly illustrated in Fig. 7 where SEM images of various mutants are recorded and corresponding FFT is computed. The good matching between computational and experimental patterns again confirmed the implications of this technique. The spatial coherence of the laser and sensitivity of the diffraction pattern offer unique advantage over other methods. This experimental evidence of global correlation and its genetic control could be potentially useful to understand how one can manipulate genes to control the natural photonic architecture as well as for other potential biological applications in understanding structure-function relation in the genetic pathways. Conclusion and outlook ====================== In summary, we show that the diffraction pattern through the transparent insect wings is correlated with the spatial organization of the microstructures at various length scales. We demonstrate that the microstructures on the transparent wings possess a long-range quasi-periodic order and characteristic organizational symmetry as unveiled by an appearance of the stable and robust diffraction pattern. These observations are in quantitative agreement with a Fourier analysis of high resolution SEM images of the wing surface. The existence of average periodicity was also supported by observations of diffraction pattern of single microstructure and background using tightly focused laser beam. Furthermore, we proposed a simple quantitative model to explain our observations that showed the existence of minimal disorder in the microstructure organization. Two different applications of our optical technique were demonstrated. First, by scanning the laser beam across the wing surface, a rotation of the original diffraction pattern was observed that demonstrated symmetry in the spatial organization of microstructures. Second, we reported first optical measurements on how various genetic mutations reorganize the biophotonic architecture. The proposed optical technique is potentially attractive to quantify natural photonic architecture on a large variety of transparent insect wings in a single-shot manner. These tools would be crucial to understand design principles of photonic crystals with potential for biomimetic applications that may lead to novel optical devices [@bar2011; @Mathias2010; @gust2009] Acknowledgement =============== We thank the DST and IISER Mohali for supporting this research. Pramod Kumar acknowledges postdoctoral fellowship from IISER Mohali. The invaluable help of Mrs. B. Basoya and Dr. Vinod Kumar are hereby grateful acknowledged. KPS acknowledges DST for financial support.

--- abstract: 'Mortality rates are often disaggregated by different attributes, such as sex, state, education, religion or ethnicity. Forecasting mortality rates at the national and sub-national levels plays an important role in making social policies associated with the national and sub-national levels. However, base forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider the problem of reconciling mortality rate forecasts from the viewpoint of grouped time-series forecasting methods [@HAA+11]. A bottom-up method and an optimal combination method are applied to produce point forecasts of infant mortality rates that are aggregated appropriately across the different levels of a hierarchy. We extend these two methods by considering the reconciliation of interval forecasts through a bootstrap procedure. Using the regional infant mortality rates in Australia, we investigate the one-step-ahead to 20-step-ahead point and interval forecast accuracies among the independent and these two grouped time-series forecasting methods. The proposed methods are shown to be useful for reconciling point and interval forecasts of demographic rates at the national and sub-national levels, and would be beneficial for government policy decisions regarding the allocations of current and future resources at both the national and sub-national levels.' bibliography: - 'hts.bib' --- Reconciling forecasts of infant mortality rates at national and sub-national levels: Grouped time-series methods Han Lin Shang [^1]\ Australian National University **Keywords** bottom-up forecasts – hierarchical forecasting – optimal combination – reconciling forecasts – Australian infant mortality rates Introduction ============ The infant mortality rate is a useful indicator of a country’s level of health or development and it is a component of the physical quality of life index. In some societies, sex-specific infant mortality may reveal gender inequalities. For instance, many Asian countries are known to have a preference for sons, which has stimulated research into gender bias, such as in India [@GMH00], Bangladesh [@RD93], China [@CB96], the Republic of Korea [@PC95], and sub-Saharan Africa [@FK14]. Anomalous female infant mortality is a sign of gender stratification, and as such is in need of detailed investigation by social and medical scientists. As a part of the United Nations Millennium Development Goals, the infant mortality rate has been widely studied by official statistical agencies worldwide, including the United Nations Statistics Division (<http://unstats.un.org/unsd/demographic/products/vitstats>), the United Nations International Children’s Emergency Fund (<http://www.unicef.org>), the World Health Organization (<http://www.who.int/whosis/mort/en>), the World Bank ([data.worldbank.org/indicator/SP.DYN.IMRT.IN](data.worldbank.org/indicator/SP.DYN.IMRT.IN)), as well as demographic and medical research communities. For example, [@FC06] investigated the gender imbalance in infant mortality in a cross-national study consisting of many developing nations, while [@DCV+08] studied the rise and fall of excess male infant mortality in a cross-national study consisting of many developed nations. Furthermore, [@ABO+06] studied the differences in infant mortality between rural and urban areas in Australia. It is not only important to analyze infant mortality by state and examine variations across different states, but also important to analyze the infant mortality rate by sex and examine the hypothesis whether or not the female infant mortality rate will continue to be higher than the male infant mortality rate. With both aggregated and disaggregated historical time series, we aim to model and forecast sex-specific infant mortality rates at national and sub-national levels. When these data are forecast independently without any constraint, we often confront the account balancing problem, where the forecasts at the sub-national level may not add up to the forecasts at the national level. This is known as forecast reconciliation, which has long been studied by [@SCM42] and further studied by [@Weale88] and [@SW09], in the context of balancing the national economic account. Here, we extend this forecast reconciliation from economics to demography. The reconciliation methods proposed will not only enhance interpretation of mortality forecasts, but can also improve forecast accuracy as it obeys a group structure. Any improvements in the forecast accuracy of mortality would be beneficial for governments, in particular for determining age of retirements; annuity providers and corporate pension funds for allocating pension benefits at the national and sub-national levels. To the best of our knowledge, there is little or no work on reconciling forecasts of infant mortality rates at the different levels of a hierarchy, where infant mortality rates can be disaggregated by sex and state. We consider a bottom-up method and an optimal combination method of [@HAA+11], and extend these methods to model rates instead of counts. These methods do not only produce point forecasts for infant mortality rates at the national and sub-national levels, but also the point forecasts at the sub-national level sum up to the forecasts at the national level. As a result, the point forecasts and the original time series both preserve the group structure. The main contribution of this paper is to put forward a bootstrap procedure for constructing prediction intervals for the bottom-up and optimal combination methods, since forecast uncertainty can never be overlooked. When we observe multiple time series that are correlated, we often confront the so-called grouped time series. Grouped time series are typically time series organized in a hierarchical structure based on different attributes, such as sex, state, education, religion or ethnicity. For example, [@AAH09] disaggregate the Australian tourism demand by states. Tourism demand within each state is then disaggregated into different zones. Tourism demand within each zone is further divided into different regions. In demographic forecasting, the infant mortality rates in Australia can first be disaggregated by sex. Within each sex, mortality rates can then be further disaggregated by the different Australian states. The first example is referred to as a hierarchical time series, in which the order of disaggregation is unique. The second example, which will be studied here, is called a grouped time series. Grouped time series can be thought of as hierarchical time series without a unique hierarchical structure. In other words, the infant mortality rates in Australia can also be first disaggregated by state and then by sex. Existing approaches to hierarchical/grouped time-series forecasting in econometrics and statistics usually consider a top-down method, bottom-up method, middle-out method or an optimal combination method. A top-down method predicts the aggregated series at the top level and then disaggregates the forecasts based on historical or forecast proportions [see for example, @GS90]. The bottom-up method involves forecasting each of the disaggregated series at the lowest level of the hierarchy and then using simple aggregation to obtain forecasts at the higher levels of the hierarchy [see for example, @Kahn98]. In practice, it is common to combine both methods, where forecasts are obtained for each series at an intermediate level of the hierarchy, before aggregating them to the series at the top level and disaggregating them to the series at the bottom level. This method is referred to as the middle-out method. [@HAA+11] and [@HLW16] proposed an optimal combination method, where base forecasts are obtained independently for all series at all levels of the hierarchy and then a linear regression model is used with an ordinary least squares (OLS) or a generalized least squares (GLS) estimator to optimally combine and reconcile these forecasts. They showed that the revised forecasts do not only add up across the hierarchy, but they are also unbiased and have minimum variance amongst all combined forecasts under some simple assumptions [@HAA+11]. To the best of our knowledge, these four hierarchical time-series methods are only applicable to counts not rates. Among the four hierarchical time-series forecasting methods, the top-down and middle-out methods are not suitable for analyzing grouped time series because of the non-unique structure of the hierarchy. In Section \[sec:2\], we first revisit a bottom-up and an optimal combination method to produce point forecasts of infant mortality rates, and then propose a bootstrap method to reconcile interval forecasts. Using the Australian infant mortality rates described in Section \[sec:3\], we investigate the one-step-ahead to 20-step-ahead point and interval forecast accuracies in Sections \[sec:4\] and \[sec:5\], respectively. Conclusions are given in Section \[sec:6\], along with some reflections on how the methods developed here might be further extended. In the Appendix, we present some details on maximum entropy bootstrapping. Some grouped time-series forecasting methods {#sec:2} ============================================ Notation -------- For ease of explanation, we will introduce the notation using the Australian data example (see Section \[sec:31\] for more details). The generalization to other contexts should be apparent. The Australian data follow a multilevel geographical hierarchy coupled with a sex grouping variable. The geographical hierarchy is shown in Fig. \[fig:1\], where Australia is split into eight regions. =\[minimum size = 8mm\] =\[sibling distance=16mm,font=\] =\[sibling distance=22mm,font=\] =\[sibling distance=40mm,font=\] child [node\[circle,draw\] [R1]{} child [node\[circle,draw\] [F]{}]{} child [node\[circle,draw\] [M]{}]{} ]{} child [node\[circle,draw\] [R2]{} child [node\[circle,draw\] [F]{}]{} child [node\[circle,draw\] [M]{}]{} ]{} child [node [$\cdots$]{}edge from parent\[draw=none\]]{} child [node\[circle,draw\] [R8]{} child[node\[circle,draw\] [F]{}]{} child[node\[circle,draw\] [M]{}]{} ]{}; Let $\bm{C}_t = \left[C_t, \bm{C}^{\top}_{1,t}, \dots, \bm{C}^{\top}_{K,t}\right]^{\top}$, where $C_t$ is the total of all series at time $t=1,2,\dots,n$, $\bm{C}_{k,t}$ represents the vector of all observations at level $k$ at time $t$ and $^{\top}$ symbolizes the matrix transpose. As shown in Fig. \[fig:1\], counts at higher levels can be obtained by summing the series below. $$\begin{aligned} C_{t}& = C_{\text{R1},t} + C_{\text{R2},t} + \cdots + C_{\text{R8},t},\\ C_{\text{R1},t} &= C_{\text{R1}*\text{F},t}+C_{\text{R1}*\text{M},t}.\end{aligned}$$ Alternatively, we can also express the hierarchy using a matrix notation [see @AAH09]. Note that $$\bm{C}_t = \bm{S}\times\bm{C}_{K,t},$$ where $\bm{S}$ is a “summing" matrix of order $m\times m_K$, $m$ represents the total number of series ($1+2+8+16=27$ for the hierarchy in Fig. \[fig:1\]) and $m_K$ represents the total number of bottom-level series. The summing matrix $\bm{S}$, which delineates how the bottom-level series are aggregated, is consistent with the group structure. For modeling mortality *counts*, we can express the hierarchy in Fig. \[fig:1\] as =0.1cm $$\underbrace{ \left[ \begin{array}{l} C_{\text{T},t} \\ C_{\textcolor{red}{\text{F},t}} \\ C_{\textcolor{red}{\text{M},t}} \\ C_{\textcolor{a0}{\text{R1*T},t}} \\ C_{\textcolor{a0}{\text{R2*T},t}} \\ \vdots \\ C_{\textcolor{a0}{\text{R8*T},t}} \\ C_{\textcolor{purple}{\text{R1*F},t}} \\ C_{\textcolor{purple}{\text{R1*M},t}} \\ C_{\textcolor{purple}{\text{R2*F},t}} \\ C_{\textcolor{purple}{\text{R2*M},t}} \\ \vdots \\ C_{\textcolor{purple}{\text{R8*F},t}} \\ C_{\textcolor{purple}{\text{R8*M},t}} \\ \end{array} \right]}_{\bm{C}_t} = \underbrace{\left[ \begin{array}{ccccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 & 1 \\ \textcolor{red}{1} & \textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{0} & \cdots & \textcolor{red}{1} & \textcolor{red}{0} \\ \textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{0} & \textcolor{red}{1} & \cdots & \textcolor{red}{0} & \textcolor{red}{1} \\ \textcolor{a0}{1} & \textcolor{a0}{1} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \cdots & \textcolor{a0}{0} & \textcolor{a0}{0} \\ \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{1} & \textcolor{a0}{1} & \textcolor{a0}{0} & \textcolor{a0}{0} & \cdots & \textcolor{a0}{0} & \textcolor{a0}{0} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \cdots & \textcolor{a0}{1} & \textcolor{a0}{1} \\ \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \textcolor{purple}{0} & \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{1} & \textcolor{purple}{0}\\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{1} \\ \end{array} \right]}_{\bm{S}_t} \underbrace{\left[ \begin{array}{l} C_{\text{R1*F},t} \\ C_{\text{R1*M},t} \\ C_{\text{R2*F},t} \\ C_{\text{R2*M},t} \\ \vdots \\ C_{\text{R8*F},t} \\ C_{\text{R8*M},t} \\ \end{array} \right]}_{\bm{C}_{K,t}}.$$ For modeling mortality *rates*, we can express the hierarchy in Fig. \[fig:1\] as =0.1cm $$\underbrace{ \left[ \begin{array}{l} R_{\text{T},t} \\ R_{\textcolor{red}{\text{F},t}} \\ R_{\textcolor{red}{\text{M},t}} \\ R_{\textcolor{a0}{\text{R1*T},t}} \\ R_{\textcolor{a0}{\text{R2*T},t}} \\ \vdots \\ R_{\textcolor{a0}{\text{R8*T},t}} \\ R_{\textcolor{purple}{\text{R1*F},t}} \\ R_{\textcolor{purple}{\text{R1*M},t}} \\ R_{\textcolor{purple}{\text{R2*F},t}} \\ R_{\textcolor{purple}{\text{R2*M},t}} \\ \vdots \\ R_{\textcolor{purple}{\text{R8*F},t}} \\ R_{\textcolor{purple}{\text{R8*M},t}} \\ \end{array} \right]}_{\bm{R}_t} = \underbrace{\left[ \begin{array}{ccccccccccc} \frac{E_{\text{R1*F},t}}{E_{\text{T},t}} & \frac{E_{\text{R1*M},t}}{E_{\text{T},t}} & \frac{E_{\text{R2*F},t}}{E_{\text{T},t}} & \frac{E_{\text{R2*M},t}}{E_{\text{T},t}} & \frac{E_{\text{R3*F},t}}{E_{\text{T},t}} & \frac{E_{\text{R3*M},t}}{E_{\text{T},t}} & \cdots & \frac{E_{\text{R8*F},t}}{E_{\text{T},t}} & \frac{E_{\text{R8*M},t}}{E_{\text{T},t}} \\ \textcolor{red}{\frac{E_{\text{R1*F},t}}{E_{\text{F},t}}} & \textcolor{red}{0} & \textcolor{red}{\frac{E_{\text{R2*F},t}}{E_{\text{F},t}}} & \textcolor{red}{0} & \textcolor{red}{\frac{E_{\text{R3*F},t}}{E_{\text{F},t}}} & \textcolor{red}{0} & \cdots & \textcolor{red}{\frac{E_{\text{R8*F},t}}{E_{\text{F},t}}} & \textcolor{red}{0} \\ \textcolor{red}{0} & \textcolor{red}{\frac{E_{\text{R1*M},t}}{E_{\text{M},t}}} & \textcolor{red}{0} & \textcolor{red}{\frac{E_{\text{R2*M},t}}{E_{\text{M},t}}} & \textcolor{red}{0} & \textcolor{red}{\frac{E_{\text{R3*M},t}}{E_{\text{M},t}}} & \cdots & \textcolor{red}{0} & \textcolor{red}{\frac{E_{\text{R8*M},t}}{E_{\text{M},t}}} \\ \textcolor{a0}{\frac{E_{\text{R1*F},t}}{E_{\text{R1*T},t}}} & \textcolor{a0}{\frac{E_{\text{R1*M},t}}{E_{\text{R1*T},t}}} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \cdots & \textcolor{a0}{0} & \textcolor{a0}{0} \\ \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{\frac{E_{\text{R2*F},t}}{E_{\text{R2*T},t}}} & \textcolor{a0}{\frac{E_{\text{R2*M},t}}{E_{\text{R2*T},t}}} & \textcolor{a0}{0} & \textcolor{a0}{0} & \cdots & \textcolor{a0}{0} & \textcolor{a0}{0} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \textcolor{a0}{0} & \cdots & \textcolor{a0}{\frac{E_{\text{R8*F},t}}{E_{\text{R8*T},t}}} & \textcolor{a0}{\frac{E_{\text{R8*M},t}}{E_{\text{R8*T},t}}} \\ \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \textcolor{purple}{0} & \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{1} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{0} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{1} & \textcolor{purple}{0}\\ \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \textcolor{purple}{0} & \cdots & \textcolor{purple}{0} & \textcolor{purple}{1} \\ \end{array} \right]}_{\bm{S}_t} \underbrace{\left[ \begin{array}{l} R_{\text{R1*F},t} \\ R_{\text{R1*M},t} \\ R_{\text{R2*F},t} \\ R_{\text{R2*M},t} \\ \vdots \\ R_{\text{R8*F},t} \\ R_{\text{R8*M},t} \\ \end{array} \right]}_{\bm{R}_{K,t}},$$ where $\textit{E}_{\text{R1}* \text{F},t}/\textit{E}_{\text{T},t}$ represents the ratio between the exposure-to-risk for female series in Region 1 and the exposure-to-risk for total series in entire Australia at time $t$, and $R_{\text{R1}*\text{F},t} = \textit{D}_{\text{R1}*\text{F},t}/\textit{E}_{\text{R1}*\text{F},t}$ represents the mortality rate given by the ratio between the number of deaths and exposure-to-risk for female series in region 1 at time $t$, for $t=1,\dots,n$. Based on the information available up to and including time $n$, we are interested in computing forecasts for each series at each level, giving $m$ base forecasts for the forecasting period $n+h,\dots,n+w$, where $h$ represents the forecast horizon and $w\geq h$ represents the last year of the forecasting period. We denote - $\widehat{R}_{\text{T},n+h}$ as the $h$-step-ahead base forecast of Series Total in the forecasting period, - $\widehat{R}_{\text{R1},n+h}$ as the $h$-step-ahead forecast of the series Region 1, and - $\widehat{R}_{\text{R1}*\text{F},n+h}$ as the $h$-step-ahead forecast of the female series in Region 1. These base forecasts can be obtained for each series in the hierarchy using a suitable forecasting method, such as the automatic autoregressive integrated moving average (ARIMA) [@HK08] implemented here. They are then combined in such ways to produce final forecasts for the whole hierarchy that aggregate in a manner which is consistent with the structure of the hierarchy. We refer to these as revised forecasts and denote them as $\overline{R}_{\text{T},n+h}$ and $\overline{\bm{R}}_{k,n+h}$ for level $k=1,\dots,K$. In the following sections, we describe two ways of combining the base forecasts in order to obtain revised forecasts. These two methods were originally proposed for modeling counts, here we extend these methods for modeling rates. Bottom-up method ---------------- One of the commonly used methods for hierarchical/grouped time-series forecasting is the bottom-up method [e.g., @Kinney71; @DM92; @ZT00]. This method involves first generating base forecasts for each series at the bottom level of the hierarchy and then aggregating these upwards to produce revised forecasts for the whole hierarchy. As an example, let us consider the hierarchy of Fig. \[fig:1\]. We first generate $h$-step-ahead base forecasts for the bottom-level series, namely $\widehat{R}_{\text{R}_1*\text{F},n+h},\widehat{R}_{\text{R}_1*\text{M},n+h}, \widehat{R}_{\text{R}_2*\text{F},n+h},\widehat{R}_{\text{R}_2*\text{M},n+h}, \cdots,$ $\widehat{R}_{\text{R}_8*\text{F},n+h},\widehat{R}_{\text{R}_8*\text{M},n+h}$. Aggregating these up the hierarchy, we get $h$-step-ahead forecasts for the rest of series, as stated below. - $\overline{R}_{\text{F},n+h} = \frac{E_{\text{R1*F},n+h}}{E_{\text{T},n+h}} \times \widehat{R}_{\text{R1*F},n+h}+\frac{E_{\text{R2*F},n+h}}{E_{\text{T},n+h}}\times\widehat{R}_{\text{R2*F},n+h}+\cdots + \frac{E_{\text{R8*F},n+h}}{E_{\text{T},n+h}}\times\widehat{R}_{\text{R8*F},n+h},$ - $\overline{R}_{\text{M},n+h} = \frac{E_{\text{R1*M},n+h}}{E_{\text{T},n+h}} \times \widehat{R}_{\text{R1*M},n+h}+\frac{E_{\text{R2*M},n+h}}{E_{\text{T},n+h}}\times\widehat{R}_{\text{R2*M},n+h}+\cdots + \frac{E_{\text{R8*M},n+h}}{E_{\text{T},n+h}}\times\widehat{R}_{\text{R8*M},n+h}, \text{and}$ - $\overline{R}_{n+h} = \frac{E_{\text{F},n+h}}{E_{\text{T},n+h}}\times\overline{R}_{\text{F},n+h}+\frac{E_{\text{M},n+h}}{E_{\text{T},n+h}}\times\overline{R}_{\text{M},n+h}$, where $\overline{R}_{\text{F},n+h}$ and $\overline{R}_{\text{M},n+h}$ represent reconciled forecasts. The revised forecasts for the bottom-level series are the same as the base forecasts in the bottom-up method (i.e., $\overline{R}_{\text{R1*F},n+h}=\widehat{R}_{\text{R1*F},n+h}$). The bottom-up method can also be expressed by the summing matrix and we write $$\overline{\bm{R}}_{n+h} = \bm{S} \times \widehat{\bm{R}}_{K,n+h},$$ where $\overline{\bm{R}}_{n+h}=\left[\overline{R}_{n+h},\overline{\bm{R}}_{1,n+h}^{\top},\dots,\overline{\bm{R}}_{K,n+h}^{\top}\right]^{\top}$ represents the revised forecasts for the whole hierarchy and $\widehat{\bm{R}}_{K,n+h}$ represents the bottom-level forecasts. The bottom-up method has an agreeable feature in that no information is lost due to aggregation, and it performs well when the signal-to-noise ratio is strong at the bottom-level series. On the other hand, it may lead to inaccurate forecasts of the top-level series, when there are many missing or noisy data at the bottom level [see for example, @SW79; @STM88]. Optimal combination ------------------- This method involves first producing base forecasts independently for each time series at each level of a hierarchy. As these base forecasts are independently generated, they will not be ‘aggregate consistent’ (i.e., they will not sum appropriately according to the group structure). The optimal combination method optimally combines the base forecasts through linear regression by generating a set of revised forecasts that are as close as possible to the base forecasts but that also aggregate consistently within the group. The essence is derived from the representation of $h$-step-ahead base forecasts for the entire hierarchy by linear regression. That is, $$\widehat{\bm{R}}_{n+h} = \bm{S}\times \bm{\beta}_{n+h} + \bm{\varepsilon}_{n+h},$$ where $\widehat{\bm{R}}_{n+h}$ is a vector of the $h$-step-ahead base forecasts for the entire hierarchy, stacked in the same hierarchical order as for original data matrix $\bm{R}_t$ for $t=1,\dots,n$; $\bm{\beta}_{n+h}=\text{E}[\bm{R}_{K,n+h}|\bm{R}_1,\dots,\bm{R}_n]$ is the unknown mean of the base forecasts of the bottom level $K$; and $\bm{\varepsilon}_{n+h}$ represents the estimation errors in the regression, which has zero mean and unknown covariance matrix $\bm{\Sigma}_h$. Given the base forecasts approximately satisfy the group aggregation structure (which should occur for any reasonable set of forecasts), the errors approximately satisfy the same aggregation structure as the data. That is, $$\bm{\varepsilon}_{n+h} \approx \bm{S}\times\bm{\varepsilon}_{K,n+h}, \label{eq:OLS}$$ where $\bm{\varepsilon}_{K,n+h}$ represents the forecast errors in the bottom level. Under this assumption, @HAA+11 [Theorem 1] show that the best linear unbiased estimator for $\bm{\beta}_{n+h}$ is $$\widehat{\bm{\beta}}_{n+h} = \left(\bm{S}^{\top}\bm{\Sigma}_h^{+}\bm{S}\right)^{-1}\bm{S}^{\top}\bm{\Sigma}_h^+\widehat{\bm{R}}_{n+h},$$ where $\bm{\Sigma}_h^+$ denotes the Moore-Penrose generalized inverse of $\bm{\Sigma}_h$. The revised forecasts are then given by $$\overline{\bm{R}}_{n+h} = \bm{S}\times\widehat{\bm{\beta}}_{n+h}.$$ The revised forecasts are unbiased, since $\bm{S}\left(\bm{S}^{\top}\bm{\Sigma}_h^{+}\bm{S}\right)^{-1}\bm{S}^{\top}\bm{\Sigma}_h^+=\bm{I}_m$ where $\bm{I}_m$ denotes $(m\times m)$ identity matrix and $m$ represents the total number of series; the revised forecasts have minimum variances $\text{Var}[\overline{\bm{R}}_{n+h}|\bm{R}_1,\dots,\bm{R}_n] = \bm{S}\left(\bm{S}^{\top}\bm{\Sigma}_h^+\bm{S}\right)^{-1}\bm{S}^{\top}$. Under the assumption given in Eq. , the estimation problem reduces from GLS to OLS, thus it is ideal for handling large-dimensional covariance structures. Even if the aggregation errors do not satisfy this assumption, the OLS solution will still be a consistent way of reconciling the base forecasts [@HLW16]. On the other hand, it is possible that assumption  becomes less and less adequate, in particular for a longer and longer forecast horizon. [@HLW16] proposed a GLS estimator, where the elements of $\bm{\Sigma}_h^+$ are set to the inverse of the variances of the base forecasts, $\text{Var}(y_{n+1}-\widehat{y}_{n+1|n})$. Note that we use the one-step-ahead forecast variances, not the $h$-step-ahead forecast variances. This is because the one-step-ahead forecast variances are readily available as the residual variances for each of the base forecasting models. We assume that these are approximately proportional to the $h$-step-ahead forecast variances, which is true for almost all standard time series forecasting models [see e.g., @HKO+08]. Univariate time-series forecasting method ----------------------------------------- For each series given in Table \[tab:3\], we consider a univariate time-series forecasting method, namely the automatic ARIMA method. This univariate time-series forecasting method is able to model non-stationary time series containing a stochastic trend component. As the yearly mortality data do not contain seasonality, the ARIMA has the general form: $$\left(1-\phi_1 B - \cdots - \phi_pB^p\right)\left(1-B\right)^dx_t = \gamma + \left(1+\theta_1B + \cdots + \theta_qB^q\right)w_t,$$ where $\gamma$ represents the intercept, $\left(\phi_1,\cdots,\phi_p\right)$ represent the coefficients associated with the autoregressive component, $\left(\theta_1,\cdots,\theta_q\right)$ represent the coefficients associated with the moving average component, $B$ denotes the backshift operator, and $d$ denotes the order of integration. We use the automatic algorithm of [@HK08] to choose the optimal orders of autoregressive $p$, moving average $q$ and difference order $d$. $d$ is selected based on successive Kwiatkowski-Phillips-Schmidt-Shin (KPSS) unit-root test [@KPSS92]. KPSS tests are used for testing the null hypothesis that an observable time series is stationary around a deterministic trend. We first test the original time series for a unit root; if the test result is significant, then we test the differenced time series for a unit root. The procedure continues until we obtain our first insignificant result. Having determined $d$, the orders of $p$ and $q$ are selected based on the optimal Akaike information criterion (AIC) with a correction for small sample sizes [@Akaike74]. Having identified the optimal ARIMA model, maximum likelihood method can then be used to estimate the parameters. Note that instead of ARIMA, other univariate time-series forecasting methods (such as exponential smoothing models [@HKO+08]) or multivariate time-series forecasting methods (such as vector autoregressive models [@Lutkepohl06]) can be used. However, as the effort in comparing forecast accuracy obtained from these models might distract too much from our emphasis on forecast reconciliation, we do not address these other methods in this paper. Instead, we save discussion of these other models for future research. Prediction interval construction {#sec:2.4} -------------------------------- To construct a prediction interval, we consider a combination of the maximum entropy bootstrap proposed by [@Vinod04] and a parametric bootstrap. The parametric bootstrap captures the forecast uncertainty in the underlying time-series extrapolation models. In contrast to nonparametric bootstrap, the parametric bootstrap method is comparably fast to compute when the group structure contains many sub-national series; and it also enjoys an optimal convergence rate when the underlying parametric model assumptions are satisfied. These assumptions include: the order of ARIMA model is selected correctly and the parameters are estimated correctly. [@Kilian01] pointed out that the adverse consequences of bootstrapping an over-parameterized model are much less severe than those of bootstrapping an under-parameterized model, and suggested the optimal order selection be based on the AIC rather than the Bayesian Information Criterion. By using the AIC, the parametric bootstrap algorithm conditions on the lag order estimates from the original time series as though they were the true lag orders. In other words, the parametric bootstrapping ignores the sampling uncertainty associated with the lag order estimates and may lead to erroneous inferences [see @CL11 Chapter 8 for examples when parametric bootstrap is invalid]. As a possible remedy, the maximum entropy bootstrap generates a set of bootstrap samples from the original time series. From bootstrapped samples, the optimal orders selected are allowed to be different and do not necessarily condition on the lag order estimates from the original time series. Instead, the maximum entropy bootstrap re-estimates the lag order in each bootstrap sample. The maximum entropy bootstrap possesses several advantages: (1) stationarity is not required; (2) the bootstrap technique computes the ranks of a time series; since the ranks of observations are invariant under a large class of monotone transformations, this invariance property yields robustness of rank-based statistics against outliers and other distributional departures; (3) bootstrap samples satisfy the ergodic theorem, central limit theorem and mean-preserving constraint; (4) it is suitable for panel time series, where the cross covariance of the original time series is reasonably well preserved. The methodology and an algorithm of the maximum entropy bootstrap are described in [@VL09]. In the Appendix, we have briefly outlined the maximum entropy bootstrap algorithm. Computationally, the `eboot.pdata.fra`e  function in the *meboot* package [@VL09] in [R]{} language [@Team13] was utilized for producing bootstrap samples for all the time series at different levels of a hierarchy. These bootstrap samples are capable of mimicking the correlation within and between original multiple time series. For each bootstrapped time series, we then fitted an optimal ARIMA model (see Section \[sec:4\]). Assuming the fitted ARIMA model is correct, future sample paths of mortality rates and exposure-to-risk are separately simulated. As for the two grouped time-series forecasting methods, those simulated forecasts are reconciled through the summing matrix. With a set of the bootstrapped forecasts, we can assess the forecast uncertainty by constructing the prediction intervals using corresponding $\alpha/2$ and $1-\alpha/2$ quantiles, at a specified nominal coverage probability denoted by $1-\alpha$. By averaging the prediction intervals over all bootstrapped samples, we obtained an averaged prediction interval. For a reasonably large level of significance $\alpha$, such as $\alpha=0.2$, averaging prediction intervals works well as we estimate the center distribution of the quantiles. Data sets {#sec:3} ========= Australian infant mortality rates {#sec:31} --------------------------------- We apply the bottom-up and optimal combination methods to model and forecast infant mortality rates across the different sexes and states in Australia. For each series, we have yearly observations on the infant mortality rates from 1933 to 2003. This data set was obtained from the Australian Social Science Data Archive (<http://www.assda.edu.au/>) and is also publicly available in the *addb* package [@Hyndman10] in the [R]{} language. The structure of the hierarchy is displayed in Table \[tab:3\]. At the top level, we have the total infant mortality rates for Australia. At Level 1, we can split these total rates by sex, although we note the possibility of splitting the total rates by region. At Level 2, the total rates are disaggregated by eight different regions of Australia: New South Wales (NSW), Victoria (VIC), Queensland (QLD), South Australia (SA), Western Australia (WA), Tasmania (TAS), the Australian Capital Territory and the Overseas Territories (ACTOT), and the Northern Territory (NT). At the bottom level, the total rates are disaggregated by different regions of Australia for each sex. This gives 16 series at the bottom level and 27 series in total. 0.38in Level Number of series -------------------- -- ------------------ Australia 1 Sex 2 State 8 Sex $\times$ State 16 Total 27 : Hierarchy of Australian infant mortality rates.[]{data-label="tab:3"} Figure \[fig:3\] shows a few selected series of the infant mortality rates disaggregated by sex, state, and sex and state. As an illustration, based on the data from 1933 to 1983, we apply the bottom-up method to forecast infant mortality rates from 1984 to 2003. The forecasts indicate a continuing decline in infant mortality rates, due largely to improved health services. Moreover, the male infant mortality rates are slightly higher than the female infant mortality rates in Australia. This confirms the early findings of [@DCV+08] and [@Pongou13], and it can be explained by environmental causes and also by sex differences in genetic structure and biological makeup, with boys being biologically weaker and more susceptible to diseases and premature death. ![Infant mortality rates can be disaggregated by sex in Level 1, region in Level 2, and sex and region in Level 3. For clarity of presentation, we plot only two regions in Level 2, and two sexes of region 1 in Level 3. Based on the data from 1933 to 1983, the bottom-up method is used to produce 20-steps-ahead forecasts from 1984 to 2003 across different levels of the hierarchy. The thicker line(s) represent(s) the historical data, while the thinner line(s) represent(s) the forecasts.[]{data-label="fig:3"}](infantforecast){width="\textwidth"} Results of the point forecasts {#sec:4} ============================== Point forecast evaluation ------------------------- A rolling window analysis of a time-series model is commonly used to assess model and parameter stabilities over time. It assesses the constancy of a model’s parameter by computing parameter estimates and their forecasts over a rolling window of a fixed size through the sample [see @ZW06 Chapter 9 for details]. Using the first 51 observations from 1933 to 1983 in the Australian infant mortality rates, we produce one to 20-step-ahead point forecasts. Through a rolling windows approach, we re-estimate the parameters in the univariate time-series forecasting models using the first 52 observations from 1933 to 1984. Forecasts from the estimated models are then produced for one to 19-step-ahead. We iterate this process by increasing the sample size by one year until reaching the end of the data period in 2003. This process produces 20 one-step-ahead forecasts, 19 two-step-ahead forecasts, 18 three-step-ahead forecasts, etc., and one 20-step-ahead forecast. We compare these forecasts with the holdout samples to determine the out-of-sample point forecast accuracy. To evaluate the point forecast accuracy, we use the mean absolute forecast error (MAFE) and root mean squared forecast error (RMSFE), which are the absolute and squared percentage errors averaged across years in the forecasting period. As two measures of accuracy, the MAFE and RMSFE show the average difference between estimated and actual populations, regardless of whether the individual estimates were too high or too low. As a measure of bias, the mean forecast error (MFE) shows the average of errors. For each series $j$, they can be defined as $$\begin{aligned} \text{MFE}_{j}(h) &= \frac{1}{(21-h)}\sum^{n+(20-h)}_{\omega=n}(R_{\omega+h,j}-\widehat{R}_{\omega+h,j}), \\ \text{MAFE}_{j}(h) &= \frac{1}{(21-h)}\sum^{n+(20-h)}_{\omega=n}\Big|R_{\omega+h,j}-\widehat{R}_{\omega+h,j}\Big|,\quad \text{and} \\ \text{RMSFE}_{j}(h) &= \sqrt{\frac{1}{(21-h)}\sum^{n+(20-h)}_{\omega=n}\left(R_{\omega+h,j}-\widehat{R}_{\omega+h,j}\right)^2}, \end{aligned}$$ where $n$ denotes the sample size used for the fitting period for $h=1,2,\dots,20$. By averaging MFE$_{j}(h)$, MAFE$_{j}(h)$ and RMSFE$_j(h)$ across the number of series within each level of a hierarchy, we obtain an overall assessment of the bias and point forecast accuracy for each level and horizon within a hierarchy, denoted by MFE($h$), MAFE$(h)$ and RMSFE$(h)$. They are defined as $$\begin{aligned} \text{MFE}(h) & = \frac{1}{m_k}\sum^{m_k}_{j=1}\text{MFE}_j(h), \\ \text{MAFE}(h) &= \frac{1}{m_k}\sum^{m_k}_{j=1}\text{MAFE}_j(h), \quad \text{and} \\ \text{RMSFE}(h) &= \frac{1}{m_k}\sum^{m_k}_{j=1}\text{RMSFE}_j(h), \end{aligned}$$ where $m_k$ denotes the number of series at the $k^{\text{th}}$ level of the hierarchy, for $k=1,\dots,K$. Point forecast accuracy of Australian infant mortality rates ------------------------------------------------------------ In Tables \[tab:AUS\_MFE\],  \[tab:AUS\_MAFE\] and \[tab:AUS\_RMSFE\], we present the MFE$(h)$, MAFE$(h)$ and RMSFE$(h)$ for each level of the hierarchy using the bottom-up and optimal combination methods, and a base forecasting method (i.e., without reconciling forecasts). For ease of comparison, we highlight in bold the method that performs the best for each level of the hierarchy and each forecast horizon, defined as the method with the smallest MFE($h$), MAFE$(h)$ and RMSFE$(h)$. 0.108in [@lrrrr|rrrr@]{}\ & Total & Sex & Region & Sex$\times$ Region & Total & Sex & Region & Sex$\times$ Region\ $h$ & &\ 1 & -0.004 & -0.005 & -0.037 & -0.065 & -0.013 & -0.013 & -0.065 & -0.065\ 2 & -0.005 & -0.006 & -0.052 & -0.081 & -0.016 & -0.016 & -0.081 & -0.081\ 3 & -0.015 & -0.016 & -0.074 & -0.106 & -0.030 & -0.030 & -0.106 & -0.106\ 4 & -0.021 & -0.022 & -0.095 & -0.126 & -0.039 & -0.039 & -0.125 & -0.126\ 5 & -0.028 & -0.028 & -0.114 & -0.147 & -0.050 & -0.049 & -0.147 & -0.147\ 6 & -0.039 & -0.040 & -0.140 & -0.175 & -0.065 & -0.065 & -0.175 & -0.175\ 7 & -0.049 & -0.050 & -0.164 & -0.199 & -0.079 & -0.078 & -0.199 & -0.199\ 8 & -0.063 & -0.064 & -0.191 & -0.229 & -0.097 & -0.097 & -0.230 & -0.229\ 9 & -0.072 & -0.073 & -0.215 & -0.255 & -0.110 & -0.110 & -0.256 & -0.255\ 10 & -0.085 & -0.086 & -0.242 & -0.283 & -0.127 & -0.127 & -0.284 & -0.283\ 11 & -0.090 & -0.091 & -0.259 & -0.304 & -0.136 & -0.136 & -0.305 & -0.304\ 12 & -0.090 & -0.091 & -0.270 & -0.319 & -0.141 & -0.141 & -0.321 & -0.319\ 13 & -0.089 & -0.090 & -0.284 & -0.338 & -0.147 & -0.146 & -0.340 & -0.338\ 14 & -0.085 & -0.087 & -0.296 & -0.356 & -0.154 & -0.153 & -0.358 & -0.356\ 15 & -0.080 & -0.082 & -0.311 & -0.371 & -0.158 & -0.157 & -0.373 & -0.371\ 16 & -0.067 & -0.070 & -0.326 & -0.386 & -0.156 & -0.155 & -0.389 & -0.386\ 17 & -0.069 & -0.072 & -0.370 & -0.424 & -0.172 & -0.170 & -0.428 & -0.424\ 18 & -0.060 & -0.064 & -0.415 & -0.426 & -0.167 & -0.166 & -0.430 & -0.426\ 19 & & & -0.404 & -0.435 & -0.167 & -0.164 & -0.442 & -0.435\ 20 & -0.058 & & -0.415 & -0.456 & -0.190 & -0.185 & -0.470 & -0.456\ Mean & -0.055 & -0.056 & -0.234 & -0.274 & -0.111 & -0.110 & -0.276 & -0.274\ Median & -0.061 & -0.064 & -0.250 & -0.294 & -0.132 & -0.131 & -0.295 & -0.294\ $h$ & &\ 1 & 0.010 & 0.011 & & & & & -0.025 & -0.025\ 2 & 0.011 & 0.011 & & & & & -0.035 & -0.035\ 3 & & & & & -0.015 & -0.015 & -0.053 & -0.053\ 4 & & & & & -0.021 & -0.021 & -0.066 & -0.066\ 5 & & & & & -0.029 & -0.029 & -0.081 & -0.081\ 6 & & & & & -0.042 & -0.042 & -0.102 & -0.102\ 7 & & & & & -0.053 & -0.053 & -0.118 & -0.119\ 8 & & & & & -0.068 & -0.068 & -0.142 & -0.142\ 9 & & & & & -0.079 & -0.079 & &\ 10 & & & -0.182 & -0.181 & -0.093 & -0.092 & &\ 11 & & & -0.196 & -0.194 & -0.099 & -0.098 & &\ 12 & & & -0.203 & -0.202 & -0.100 & -0.099 & &\ 13 & & & -0.211 & -0.210 & -0.100 & -0.100 & &\ 14 & & & -0.218 & -0.216 & -0.101 & -0.100 & &\ 15 & & & -0.224 & -0.223 & -0.099 & -0.098 & &\ 16 & & & -0.228 & -0.226 & -0.089 & -0.088 & &\ 17 & & & -0.254 & -0.252 & -0.096 & -0.095 & &\ 18 & & & -0.247 & -0.248 & -0.086 & -0.086 & &\ 19 & -0.043 & -0.044 & -0.240 & -0.241 & -0.076 & -0.076 & &\ 20 & & & -0.253 & -0.248 & -0.090 & -0.088 & &\ Mean & & & -0.161 & -0.160 & -0.067 & -0.067 & &\ Median & & & -0.189 & -0.188 & -0.083 & -0.083 & &\ 0.12in [@lrrrr|rrrr@]{}\ & Total & Sex & Region & Sex$\times$ Region & Total & Sex & Region & Sex$\times$ Region\ $h$ & &\ 1 & 0.037 & 0.039 & 0.097 & 0.140 & 0.040 & 0.041 & 0.118 & 0.140\ 2 & & & 0.104 & 0.152 & & 0.041 & 0.128 & 0.152\ 3 & & & 0.120 & 0.168 & 0.050 & 0.052 & 0.148 & 0.168\ 4 & & 0.059 & 0.132 & 0.182 & 0.063 & 0.063 & 0.163 & 0.182\ 5 & & & 0.153 & 0.202 & 0.073 & 0.073 & 0.186 & 0.202\ 6 & 0.078 & 0.078 & 0.179 & 0.226 & 0.083 & 0.083 & 0.212 & 0.226\ 7 & 0.076 & 0.079 & 0.192 & 0.239 & 0.091 & 0.091 & 0.225 & 0.239\ 8 & 0.084 & 0.085 & 0.214 & 0.260 & 0.106 & 0.106 & 0.251 & 0.260\ 9 & 0.085 & 0.087 & 0.231 & 0.278 & 0.113 & 0.114 & 0.269 & 0.278\ 10 & 0.089 & 0.091 & 0.251 & 0.299 & 0.127 & 0.127 & 0.295 & 0.299\ 11 & 0.090 & 0.091 & 0.264 & 0.315 & 0.136 & 0.136 & 0.312 & 0.315\ 12 & 0.090 & 0.091 & 0.274 & 0.329 & 0.141 & 0.141 & 0.325 & 0.329\ 13 & 0.089 & 0.090 & 0.289 & 0.349 & 0.147 & 0.146 & 0.345 & 0.349\ 14 & 0.085 & 0.087 & 0.301 & 0.366 & 0.154 & 0.153 & 0.363 & 0.366\ 15 & 0.080 & 0.082 & 0.316 & 0.381 & 0.158 & 0.157 & 0.379 & 0.381\ 16 & 0.067 & 0.070 & 0.331 & 0.396 & 0.156 & 0.155 & 0.392 & 0.396\ 17 & 0.069 & 0.072 & 0.370 & 0.434 & 0.172 & 0.170 & 0.430 & 0.434\ 18 & 0.060 & 0.066 & 0.418 & 0.445 & 0.167 & 0.166 & 0.441 & 0.445\ 19 & & & 0.411 & 0.465 & 0.167 & 0.164 & 0.463 & 0.465\ 20 & 0.058 & 0.052 & 0.432 & 0.495 & 0.190 & 0.185 & 0.499 & 0.495\ Mean & 0.069 & 0.070 & 0.254 & 0.306 & 0.119 & 0.118 & 0.297 & 0.306\ Median & 0.073 & 0.075 & 0.257 & 0.307 & 0.132 & 0.131 & 0.304 & 0.307\ $h$ & &\ 1 & & & 0.093 & 0.124 & 0.036 & & &\ 2 & 0.043 & 0.045 & 0.103 & 0.132 & & 0.041 & &\ 3 & 0.047 & 0.048 & 0.120 & 0.144 & 0.045 & 0.046 & &\ 4 & 0.058 & & 0.130 & 0.153 & 0.057 & & &\ 5 & 0.068 & 0.067 & 0.145 & 0.168 & 0.065 & & &\ 6 & & & 0.166 & 0.184 & 0.076 & 0.075 & &\ 7 & & & 0.174 & 0.188 & 0.076 & 0.076 & &\ 8 & & & 0.185 & 0.200 & 0.084 & 0.084 & &\ 9 & & & 0.194 & 0.208 & 0.088 & 0.088 & &\ 10 & & & 0.206 & 0.217 & 0.094 & 0.094 & &\ 11 & & & 0.215 & 0.223 & 0.099 & 0.098 & &\ 12 & & & 0.220 & 0.232 & 0.100 & 0.099 & &\ 13 & & & 0.235 & 0.246 & 0.100 & 0.100 & &\ 14 & & & 0.241 & 0.256 & 0.101 & 0.100 & &\ 15 & & & 0.260 & 0.270 & 0.099 & 0.098 & &\ 16 & & & 0.271 & 0.281 & 0.089 & 0.088 & &\ 17 & & & 0.302 & 0.315 & 0.096 & 0.095 & &\ 18 & & & 0.325 & 0.331 & 0.086 & 0.086 & &\ 19 & 0.043 & 0.044 & 0.332 & 0.339 & 0.076 & 0.076 & &\ 20 & & & 0.344 & 0.345 & 0.090 & 0.088 & &\ Mean & & & 0.213 & 0.228 & 0.080 & 0.080 & &\ Median & & & 0.211 & 0.220 & 0.087 & 0.087 & &\ 0.12in [@lcccc|cccc@]{}\ & Total & Sex & Region & Sex$\times$ Region & Total & Sex & Region & Sex$\times$ Region\ $h$ & &\ 1 & 0.050 & 0.052 & 0.159 & 0.254 & 0.050 & 0.052 & 0.210 & 0.254\ 2 & & 0.052 & 0.179 & 0.270 & 0.049 & & 0.227 & 0.270\ 3 & & & 0.213 & 0.304 & 0.062 & 0.064 & 0.269 & 0.304\ 4 & & 0.071 & 0.251 & 0.326 & 0.074 & 0.075 & 0.293 & 0.326\ 5 & 0.075 & 0.078 & 0.274 & 0.359 & 0.085 & 0.087 & 0.331 & 0.359\ 6 & 0.087 & 0.089 & 0.323 & 0.400 & 0.099 & 0.100 & 0.374 & 0.400\ 7 & 0.089 & 0.093 & 0.335 & 0.422 & 0.107 & 0.108 & 0.395 & 0.422\ 8 & 0.099 & 0.103 & 0.364 & 0.469 & 0.123 & 0.124 & 0.439 & 0.469\ 9 & 0.098 & 0.103 & 0.404 & 0.506 & 0.130 & 0.131 & 0.475 & 0.506\ 10 & 0.103 & 0.109 & 0.445 & 0.543 & 0.142 & 0.144 & 0.517 & 0.543\ 11 & 0.100 & 0.105 & 0.483 & 0.585 & 0.148 & 0.149 & 0.555 & 0.585\ 12 & 0.101 & 0.106 & 0.498 & 0.610 & 0.153 & 0.155 & 0.575 & 0.610\ 13 & 0.096 & 0.103 & 0.529 & 0.643 & 0.154 & 0.155 & 0.610 & 0.643\ 14 & 0.088 & 0.093 & 0.557 & 0.688 & 0.160 & 0.160 & 0.649 & 0.688\ 15 & 0.089 & 0.095 & 0.598 & 0.706 & 0.164 & 0.164 & 0.682 & 0.706\ 16 & 0.071 & 0.080 & 0.644 & 0.745 & 0.157 & 0.157 & 0.727 & 0.745\ 17 & 0.074 & 0.083 & 0.716 & 0.813 & 0.173 & 0.174 & 0.806 & 0.813\ 18 & 0.071 & 0.082 & 0.831 & 0.844 & 0.168 & 0.169 & 0.835 & 0.844\ 19 & & & 0.843 & 0.869 & 0.167 & 0.167 & 0.859 & 0.869\ 20 & 0.058 & & 0.888 & 0.901 & 0.190 & 0.186 & 0.902 & 0.901\ Mean & 0.078 & 0.082 & 0.477 & 0.563 & 0.128 & 0.129 & 0.536 & 0.563\ Median & 0.081 & 0.086 & 0.464 & 0.564 & 0.145 & 0.146 & 0.536 & 0.564\ $h$ & &\ 1 & 0.049 & 0.053 & 0.149 & 0.199 & & & &\ 2 & 0.055 & 0.059 & 0.155 & 0.204 & 0.049 & 0.052 & &\ 3 & 0.057 & 0.061 & 0.197 & 0.235 & 0.057 & 0.059 & &\ 4 & & 0.072 & 0.216 & 0.252 & 0.067 & & &\ 5 & & & 0.245 & 0.274 & 0.076 & 0.078 & &\ 6 & & & 0.279 & 0.306 & 0.085 & 0.087 & &\ 7 & & & 0.286 & 0.317 & 0.089 & 0.091 & &\ 8 & & & 0.321 & 0.353 & 0.099 & 0.101 & &\ 9 & & & 0.346 & 0.378 & 0.100 & 0.103 & &\ 10 & & & 0.379 & 0.403 & 0.107 & 0.109 & &\ 11 & & & 0.407 & 0.435 & 0.107 & 0.109 & &\ 12 & & & 0.418 & 0.449 & 0.109 & 0.110 & &\ 13 & & & 0.441 & 0.472 & 0.105 & 0.107 & &\ 14 & & & 0.464 & 0.498 & 0.104 & 0.105 & &\ 15 & & & 0.492 & 0.513 & 0.104 & 0.105 & &\ 16 & & & 0.523 & 0.539 & 0.090 & 0.091 & &\ 17 & & & 0.589 & 0.596 & 0.097 & 0.098 & &\ 18 & & & 0.607 & 0.618 & 0.089 & 0.091 & &\ 19 & 0.043 & 0.046 & 0.616 & 0.624 & 0.076 & 0.076 & &\ 20 & & 0.057 & 0.646 & 0.651 & 0.090 & 0.088 & &\ Mean & & & 0.389 & 0.416 & 0.087 & 0.089 & &\ Median & & & 0.393 & 0.419 & 0.090 & 0.091 & &\ Based on the MFE$(h)$, the optimal combination methods generally outperform the base and bottom-up forecasting methods. In the top level and Level 1, the optimal combination (OLS) method has smaller forecast bias than the optimal combination (GLS) method at all horizons, with exceptions of $h=1$ and $h=2$. At Level 2 and the bottom level, the forecasts obtained from the optimal combination (OLS) method have smaller forecast bias than the optimal combination (GLS) method at the shorter forecast horizons from $h=1$ to $h=9$, but less so at the longer forecast horizons. Based on the MAFE$(h)$ and RMSFE$(h)$, the optimal combination methods generally outperform the base and bottom-up forecasting methods. In the top level and Level 1, the optimal combination (OLS) method has smaller forecast errors than the optimal combination (GLS) method at the medium to long forecast horizons, but less so at the shorter forecast horizons. At Level 2 and the bottom level, the forecasts obtained from the optimal combination (GLS) method outperforms the optimal combination (OLS) method for every forecast horizon. Averaging across all levels of a hierarchy, the point forecasts obtained from the optimal combination (GLS) method are the most accurate in all methods investigated, and the method produces reconciled forecasts that obey a grouped structure. To highlight the discrepancy in point forecasts between the base forecasts and optimal combination forecasts, we present a diagnostic plot showing the 20-step-ahead forecasts obtained from these two methods. As an illustration, since the base forecasts provide a foundation for the reconciled forecasts obtained from the optimal combination (OLS) method, the diagnostic plot allows us to visualize the forecasts that are similar or different between the two methods. As shown in Fig. \[fig:diag\_1\], there are almost no difference between the two methods at the top level and Level 1. At Level 2, there is only a difference for the NT region. At the bottom level, the largest differences for both sexes are ACTOT and NT regions. ![Diagnostic plot of the 20-step-ahead forecasts at each level of the hierarchy for the Australian infant mortality between the base and optimal combination forecasts.[]{data-label="fig:diag_1"}](ind_rescale){width="\textwidth"} Influence of the $S$ matrix on point forecast accuracy ------------------------------------------------------ The potential improvement in point forecast accuracy in the reconciliation methods relies crucially on the accurate forecast of the $S$ matrix. Recall that the $S$ matrix includes ratios of forecast exposure-at-risk. To forecast these exposure-at-risk, we again use the automatic ARIMA method to model and forecast exposure-at-risk at the logarithmic scale. By taking the exponential back-transformation, forecast exposure-at-risk in the original scale is obtained. In Tables \[tab:MAFE\_BU\], \[tab:MAFE\_OLS\] and \[tab:MAFE\_GLS\], we compare the MAFE among the reconciliation methods with forecast and holdout $S$ matrices. 0.36cm -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- $h$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ 1 0.0406 0.1175 2 0.0407 0.1277 3 0.0504 0.0515 0.1483 4 0.0629 0.0627 5 0.0728 0.0727 0.1862 6 0.0831 0.0831 0.2123 7 0.0908 0.0907 0.2254 8 0.1059 0.1058 0.2514 9 0.1126 0.1145 10 0.1270 0.1266 0.2953 11 0.1364 0.1360 0.3125 12 0.1414 0.1411 0.3252 13 0.1467 0.1464 0.3454 14 0.1536 0.1527 0.3627 15 0.1576 0.1566 0.3787 16 0.1560 0.1547 0.3915 17 0.1720 0.1703 0.4302 18 0.1673 0.1659 0.4413 19 0.1670 0.1643 0.4632 20 0.1898 0.1847 0.4988 Mean 0.1187 0.1181 0.2972 Median 0.1317 0.1313 0.3039 -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- : A comparison of MAFE ($\times 100$) between the bottom-up method with forecast $S$ and actual $S$ matrices.[]{data-label="tab:MAFE_BU"} 0.36cm -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- $h$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ 1 0.0377 0.0927 2 0.0438 0.0456 0.1028 3 0.0472 0.0488 0.1198 4 0.0579 0.0595 5 0.0673 6 0.0717 0.0713 7 0.0672 0.0675 0.1747 8 0.0695 0.0706 9 0.0674 0.0709 0.1945 10 0.0681 0.0686 0.2065 11 0.0683 0.0685 0.2149 12 0.0682 0.0678 0.2204 13 0.0672 0.0669 0.2354 14 0.0662 0.0656 0.2412 15 0.0632 0.0626 0.2596 16 0.0516 0.0542 0.2707 17 0.0573 0.0569 0.3019 18 0.0499 0.0508 0.3253 19 0.0429 0.0442 0.3325 20 0.0518 0.0515 0.3438 Mean 0.0589 0.0598 0.2131 Median 0.0647 0.0641 0.2107 -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- : A comparison of MAFE $(\times 100)$ between the optimal combination method (the OLS estimator) with forecast $S$ and actual $S$ matrices.[]{data-label="tab:MAFE_OLS"} 0.36cm -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- $h$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ 1 0.0363 0.0367 2 0.0406 0.0410 0.0936 3 0.0462 0.1089 4 0.0569 5 0.0649 0.0651 0.1313 6 0.0756 0.0752 7 0.0761 0.0765 0.1569 8 0.0840 0.0839 0.1726 9 0.0879 0.0880 0.1822 10 0.0939 0.0941 0.1961 11 0.0986 0.0981 0.2032 12 0.0995 0.0992 0.2071 13 0.1004 0.1001 0.2166 14 0.1007 0.0999 0.2210 15 0.0990 0.0982 0.2262 16 0.0888 0.0880 0.2264 17 0.0957 0.0947 0.2498 18 0.0865 0.0864 0.2614 19 0.0764 0.0760 0.2580 20 0.0899 0.0875 0.2776 Mean 0.0798 0.0797 0.1872 Median 0.0872 0.0869 0.1996 -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- : A comparison of MAFE $(\times 100)$ between the optimal combination method (the GLS estimator) with forecast $S$ and actual $S$ matrices.[]{data-label="tab:MAFE_GLS"} At the top two levels, more accurate point forecasts can be obtained by using the holdout $S$ matrix. At the bottom two levels, there are comparably smaller differences in point forecast accuracy between the forecast and actual $S$ matrices. Results of the interval forecasts {#sec:5} ================================= As described in Section \[sec:2.4\], we constructed pointwise prediction intervals using the maximum entropy and parametric bootstrap methods. The maximum entropy bootstrap method generates bootstrap samples that preserve the correlation in the original time series, whereas the parametric bootstrap method generates bootstrap forecasts for each bootstrap sample. Based on these bootstrap forecasts, we assess the variability of point forecasts by constructing prediction intervals based on quantiles. By averaging over all bootstrap prediction intervals, we obtain the averaged prediction intervals. For a reasonably large level of significance $\alpha$, such as $\alpha=0.2$, averaging prediction intervals works well as we estimate the center distribution of the quantiles. Due to heavy computational cost, there are 100 bootstrap samples obtained by a maximum entropy bootstrap. Within each bootstrap sample, the number of parametric bootstrap forecasts is 100. Figure \[fig:interval\] shows the 80% pointwise averaged prediction intervals of the direct 20-steps-ahead Australian infant mortality rate forecasts for a few selected series at each level of the hierarchy from 1984 to 2003. At the top level, there seems to be a larger difference in interval forecasts between the base forecasting and two grouped time-series methods. From the middle to bottom levels, the interval forecasts are very similar between the three methods. For the optimal combination (GLS) method, the construction of prediction interval is hindered by the difficulty in measuring forecast uncertainty associated with $\bm{\Sigma}^+$, and thus we leave this for future research. ![Based on the Australian infant mortality data from 1933 to 1983, we produce 20-steps-ahead prediction intervals for years 1984 to 2003, at the nominal coverage probability of 80%. For ease of presentation, we show the 80% prediction intervals for a few selected series at each level of the hierarchy.[]{data-label="fig:interval"}](interval_forecast_h20_total){width="\textwidth"} Interval forecast evaluation ---------------------------- At a nominal level of 80%, prediction intervals are constructed by taking corresponding quantiles, where the lower bound is denoted by $\widehat{L}_{\omega+h,j}$ and the upper bound is denoted by $\widehat{U}_{\omega+h,j}$ for $j=1,\dots,m$ and $m$ representing the total number of series in a hierarchy. With a pointwise prediction interval and its corresponding holdout data point in the forecasting period, we can assess interval forecast accuracy by the interval score of [@GR07], defined as $$\begin{aligned} S_{\alpha,j}\left(\widehat{L}_{\omega+h,j}, \widehat{U}_{\omega+h,j}, Y_{\omega+h,j}\right) = \left(\widehat{U}_{\omega+h,j} - \widehat{L}_{\omega+h,j}\right) &+ \frac{2}{\alpha}\left(\widehat{L}_{\omega+h,j} - Y_{\omega+h,j}\right) \mathds{1}\left\{Y_{\omega+h,j} < \widehat{L}_{\omega+h,j}\right\} \\ &+\frac{2}{\alpha}\left(Y_{\omega+h,j} - \widehat{U}_{\omega+h,j}\right)\mathds{1}\left\{Y_{\omega+h,j}>\widehat{U}_{\omega+h,j}\right\},\end{aligned}$$ where $Y_{\omega+h,j}$ represents the holdout samples in the forecasting period for the series $j$, and $\mathds{1}\{\cdot\}$ is a binary indicator function. This interval score combines the halfwidth of the prediction intervals with the coverage probability difference between the nominal and empirical coverage probabilities. Intuitively, a forecaster is rewarded for narrow prediction intervals, but a penalty is incurred, the size of which depends on the level of significance $\alpha$, if the holdout samples lie outside the prediction intervals. For each series $j$ at each forecast horizon, we obtain $$\overline{S}_{\alpha,j}(h) = \frac{1}{(21-h)}\sum^{n+(20-h)}_{\omega=n}S_{\alpha,j}\left(\widehat{L}_{\omega+h,j}, \widehat{U}_{\omega+h,j}, Y_{\omega+h,j}\right), \qquad h=1,\dots,20,$$ where $S_{\alpha,j}\left(\widehat{L}_{\omega+h,j}, \widehat{U}_{\omega+h,j}, Y_{\omega+h,j}\right)$ denotes the interval score at each level of the hierarchy for the holdout samples in the forecasting period. By averaging the interval score $\overline{S}_{\alpha,j}(h)$ across the number of series within each level of a hierarchy, we obtain an overall assessment of the interval forecast accuracy for each level within a hierarchy. The mean interval score is then defined by $$\overline{S}_{\alpha,k}(h) = \frac{1}{m_k}\sum^{m_k}_{j=1}\overline{S}_{\alpha,j}(h),$$ where $m_k$ denotes the number of series at the $k^{\text{th}}$ level of the hierarchy, for $k=1,\dots,K$. Interval forecast accuracy of Australian infant mortality --------------------------------------------------------- In Table \[tab:interval\_score\], we present the mean interval scores for the one-step-ahead to 20-step-ahead forecasts at each level of the hierarchy between the three methods. For ease of comparison, we highlight in bold the method that performs the best for each level of the hierarchy and each forecast horizon, based on the smallest $\overline{S}_{\alpha, k}(h)$. 0.095in [@ lcccc|cccc@]{}\ & Total & Sex & Region & Sex$\times$ Region & Total & Sex & Region & Sex$\times$ Region\ $h$ & &\ 1 & $\textBF{0.17}$ & $\textBF{0.18}$ & $\textBF{0.55}$ & $0.77$ & $0.24$ & $0.22$ & $0.63$ & $0.76$\ 2 & $\textBF{0.17}$ & $\textBF{0.21}$ & $\textBF{0.59}$ & $0.81$ & $0.27$ & $0.24$ & $0.70$ & $0.81$\ 3 & $\textBF{0.21}$ & $\textBF{0.26}$ & $\textBF{0.65}$ & $0.88$ & $0.36$ & $0.31$ & $0.78$ & $0.88$\ 4 & $\textBF{0.23}$ & $\textBF{0.28}$ & $\textBF{0.69}$ & $0.91$ & $0.46$ & $0.38$ & $0.82$ & $0.91$\ 5 & $\textBF{0.26}$ & $\textBF{0.32}$ & $\textBF{0.74}$ & $0.96$ & $0.54$ & $0.45$ & $0.89$ & $0.96$\ 6 & $\textBF{0.31}$ & $\textBF{0.35}$ & $\textBF{0.80}$ & $1.02$ & $0.65$ & $0.53$ & $1.01$ & $1.02$\ 7 & $\textBF{0.34}$ & $\textBF{0.37}$ & $\textBF{0.81}$ & $1.08$ & $0.73$ & $0.60$ & $1.10$ & $1.08$\ 8 & $\textBF{0.37}$ & $\textBF{0.39}$ & $\textBF{0.87}$ & $1.18$ & $0.85$ & $0.68$ & $1.22$ & $1.18$\ 9 & $\textBF{0.35}$ & $\textBF{0.38}$ & $\textBF{0.93}$ & $1.28$ & $0.94$ & $0.74$ & $1.32$ & $1.28$\ 10 & $\textBF{0.37}$ & $\textBF{0.40}$ & $\textBF{1.02}$ & $1.36$ & $1.05$ & $0.85$ & $1.43$ & $1.36$\ 11 & $\textBF{0.35}$ & $\textBF{0.38}$ & $\textBF{1.07}$ & $1.47$ & $1.12$ & $0.90$ & $1.52$ & $1.49$\ 12 & $\textBF{0.36}$ & $\textBF{0.38}$ & $\textBF{1.07}$ & $1.52$ & $1.16$ & $0.92$ & $1.58$ & $1.52$\ 13 & $\textBF{0.29}$ & $\textBF{0.34}$ & $\textBF{1.13}$ & $1.62$ & $1.20$ & $0.95$ & $1.69$ & $1.62$\ 14 & $\textBF{0.26}$ & $\textBF{0.29}$ & $\textBF{1.16}$ & $1.74$ & $1.25$ & $0.99$ & $1.80$ & $1.73$\ 15 & $\textBF{0.30}$ & $\textBF{0.31}$ & $\textBF{1.17}$ & $1.84$ & $1.30$ & $1.02$ & $1.97$ & $1.82$\ 16 & $\textBF{0.24}$ & $\textBF{0.24}$ & $\textBF{1.26}$ & $1.89$ & $1.27$ & $0.99$ & $2.12$ & $1.88$\ 17 & $\textBF{0.24}$ & $\textBF{0.24}$ & $\textBF{1.46}$ & $2.05$ & $1.40$ & $1.11$ & $2.33$ & $2.06$\ 18 & $\textBF{0.25}$ & $\textBF{0.25}$ & $\textBF{1.78}$ & $2.19$ & $1.39$ & $1.09$ & $2.48$ & $2.18$\ 19 & $\textBF{0.24}$ & $\textBF{0.24}$ & $\textBF{1.96}$ & $\textBF{2.19}$ & $1.42$ & $1.11$ & $2.46$ & $2.20$\ 20 & $\textBF{0.25}$ & $\textBF{0.24}$ & $\textBF{2.19}$ & $2.36$ & $1.62$ & $1.29$ & $2.59$ & $\textBF{2.28}$\ Mean & $\textBF{0.28}$ & $\textBF{0.30}$ & $\textBF{1.10}$ & $1.46$ & $0.96$ & $0.77$ & $1.52$ & $1.45$\ Median & $\textBF{0.26}$ & $\textBF{0.30}$ & $\textBF{1.04}$ & $1.42$ & $1.09$ & $0.87$ & $1.48$ & $1.42$\ $h$ &\ 1 & $0.27$ & $0.21$ & $0.60$ & $\textBF{0.72}$\ 2 & $0.28$ & $0.23$ & $0.68$ & $\textBF{0.76}$\ 3 & $0.37$ & $0.30$ & $0.76$ & $\textBF{0.83}$\ 4 & $0.48$ & $0.37$ & $0.82$ & $\textBF{0.86}$\ 5 & $0.55$ & $0.44$ & $0.91$ & $\textBF{0.92}$\ 6 & $0.66$ & $0.54$ & $1.05$ & $\textBF{0.97}$\ 7 & $0.74$ & $0.59$ & $1.15$ & $\textBF{1.02}$\ 8 & $0.87$ & $0.69$ & $1.27$ & $\textBF{1.11}$\ 9 & $0.95$ & $0.75$ & $1.37$ & $\textBF{1.20}$\ 10 & $1.06$ & $0.85$ & $1.49$ & $\textBF{1.27}$\ 11 & $1.13$ & $0.89$ & $1.59$ & $\textBF{1.36}$\ 12 & $1.16$ & $0.90$ & $1.64$ & $\textBF{1.41}$\ 13 & $1.19$ & $0.92$ & $1.73$ & $\textBF{1.50}$\ 14 & $1.22$ & $0.95$ & $1.83$ & $\textBF{1.58}$\ 15 & $1.24$ & $0.96$ & $2.00$ & $\textBF{1.67}$\ 16 & $1.21$ & $0.91$ & $2.16$ & $\textBF{1.71}$\ 17 & $1.31$ & $1.01$ & $2.44$ & $\textBF{1.88}$\ 18 & $1.31$ & $1.00$ & $2.66$ & $\textBF{2.09}$\ 19 & $1.34$ & $1.01$ & $2.78$ & $\textBF{2.19}$\ 20 & $1.56$ & $1.22$ & $2.99$ & $2.40$\ Mean & $0.94$ & $0.74$ & $1.59$ & $\textBF{1.37}$\ Median & $1.09$ & $0.87$ & $1.54$ & $\textBF{1.32}$\ \ Based on the overall interval forecast accuracy $\overline{S}_{\alpha,k}(h)$, the base forecasting method gives the most accurate interval forecasts at the top three levels, but the optimal combination method demonstrates the best interval forecast accuracy for the bottom-level series. Averaged over all levels of the hierarchy, the base forecasting method outperforms the two grouped time-series methods in terms of mean interval scores. A possible explanation for the inferior interval accuracy of the grouped time-series forecasting methods is that they require the accurate forecasts of the $S$ matrix consisting of the forecast exposure-to-risk, which may introduce additional forecast uncertainty. However, from a viewpoint of forecast interpretation, the grouped time series methods produce interval forecasts that obey a grouped time-series structure. Due to the limited space, although not shown in the paper, the grouped time-series forecasting methods can improve interval forecast accuracy in another data set, namely the Japanese data set [@JMD15]. When the forecasts of the exposure-to-risk are accurate, the reconciliation of interval mortality forecasts are more accurate than the base interval forecasts. These results can be obtained upon request from the author. Influence of the $S$ matrix on interval forecast accuracy --------------------------------------------------------- The potential improvement in interval forecast accuracy in the reconciliation methods relies crucially on the accurate forecast of the $S$ matrix. Recall that the $S$ matrix includes ratios of forecast exposure-at-risk. To obtain bootstrap forecasts of these exposure-at-risk, we use the parametric bootstrap and maximum entropy bootstrap methods to simulate future samples of the exposure-at-risk at the logarithmic scale. By taking the exponential back-transformation, bootstrap forecasts of exposure-at-risk in the original scale are obtained. In Tables \[tab:interval\_score\_BU\] and \[tab:interval\_score\_optim\], we compare the interval score among the reconciliation methods with forecast and holdout $S$ matrices. 0.44cm -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- $h$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ 1 0.24 2 0.27 0.24 3 0.36 0.31 4 0.46 0.38 0.83 5 0.54 0.45 0.92 6 0.65 0.53 1.06 7 0.73 0.60 1.16 8 0.85 0.69 1.29 1.18 9 0.94 0.74 1.39 10 1.05 0.85 1.52 1.36 11 1.12 0.90 1.63 1.49 12 1.16 0.92 1.69 1.53 13 1.20 0.95 1.83 14 1.25 0.99 1.95 1.74 15 1.30 1.02 2.14 16 1.27 0.99 2.29 1.88 17 1.40 1.11 2.52 2.06 18 1.39 1.09 2.68 2.18 19 1.42 1.11 2.70 2.20 20 1.62 1.29 2.83 2.32 Mean 0.96 0.77 1.63 Median 1.09 0.87 1.57 1.42 -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- : A comparison of interval score $(\times 100)$ between the bottom-up method with forecast $S$ and actual $S$ matrices.[]{data-label="tab:interval_score_BU"} 0.44cm -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- $h$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ $\widehat{S}_{n+h|n}$ $S_{n+h}$ 1 0.23 0.61 0.73 2 0.28 0.69 3 0.37 0.78 0.84 4 0.48 0.85 0.87 5 0.55 0.44 0.94 6 0.66 0.54 1.09 0.98 7 0.74 0.59 1.19 1.03 8 0.87 0.69 1.33 1.12 9 0.95 0.75 1.44 1.21 10 1.06 0.85 1.58 1.28 11 1.13 0.89 1.67 12 1.16 0.90 1.75 13 1.19 0.92 1.88 14 1.22 0.95 1.99 15 1.24 0.96 2.19 16 1.21 0.91 2.34 1.71 17 1.31 1.01 2.62 1.88 18 1.31 1.00 2.84 2.09 19 1.34 1.01 2.94 2.19 20 1.56 1.22 3.17 2.40 Mean 0.94 0.74 1.70 Median 1.09 0.87 1.63 -------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- ----------------------- ----------- : A comparison of interval score $(\times 100)$ between the optimal combination method with forecast $S$ and actual $S$ matrices.[]{data-label="tab:interval_score_optim"} At the top two levels, more accurate interval forecasts can be obtained by using the holdout $S$ matrix. At the Region level, the forecast $S$ matrix gives a smaller interval score than the holdout $S$ matrix. This rather surprising result may due to the forecast uncertainty associated with the mortality rates. At the bottom level, there is no difference in terms of interval forecast accuracy between the forecast and actual $S$ matrices. Conclusions {#sec:6} =========== This article adapts a bottom-up method and an optimal combination method for modeling and forecasting grouped time series of infant mortality rates. The bottom-up method models and forecasts time series at the bottom level and then aggregates to the top level using the summing matrix. The optimal combination method optimally combines the base forecasts through linear regression by generating a set of revised forecasts that are as close as possible to the base forecasts but that also aggregate consistently within the group. Under a mild assumption, regression coefficient can be estimated by either OLS or GLS estimator. Using the regional infant mortality rates in Australia, we implemented these two grouped time-series forecasting methods that reconcile forecasts across different levels of a hierarchy. Furthermore, we compared the one-step-ahead to 20-step-ahead point forecast accuracy, and found that the optimal combination method has the smallest overall forecast error in the Australian data set considered. Through the maximum entropy and parametric bootstrap methods, we present a means of constructing pointwise prediction intervals for grouped time series. The maximum entropy bootstrap is capable of mimicking the correlation within and between the original time series. For each bootstrapped time series, we can then fit an optimal ARIMA model and generate forecasts; from these forecasts the corresponding prediction intervals are obtained. Averaging over all prediction intervals, we obtain averaged prediction intervals to evaluate forecast uncertainty associated with the point forecasts. In the Australian data set, we found that the base forecasting method gives the best overall interval forecast accuracy, but the two grouped time-series forecasting methods produce interval forecasts that obey a group structure and thus ease of interpretation. It is noteworthy that the accuracy of the reconciliation methods crucially depends on the forecast accuracy of the summing matrix. Although the forecast $S$ matrix does not differ much from the holdout $S$ matrix, the reconciliation methods enjoy improved forecast accuracy with the holdout $S$ matrix at the top and middle levels, but less so at the bottom level. There are several ways in which this study could be further extended and we briefly outline five of these. First, the methods are proposed from a frequentist viewpoint, and they can be compared with a hierarchical Bayesian method. Secondly, the methodology can be applied to cause-specific mortality, considered in [@ML97], [@GK08] and [@GS15]. Thirdly, the methodology can be applied to other demographic data, such as population size. Fourthly, forecasts can also be obtained by multivariate time-series forecasting methods, such as vector autoregressive models, in order to take into account possible correlations between and within multiple time series. Finally, the idea of grouped time series can be extended to functional time series [see @SH16], where each series is a time series of functions, such as age-specific demographic rates. This work provides a natural foundation for such extensions. Appendix: Maximum entropy bootstrap algorithm An overview of the maximum entropy bootstrap algorithm is provided for generating a random realization of a univariate time series $x_t$. Consult [@Vinod04] for more details and an example. In the maximum entropy bootstrap algorithm, 1. Sort the original data in increasing order to create order statistics $x_{(t)}$ and store the ordering index vector. 2. Compute intermediate points $z_t = \frac{x_{(t)}+x_{(t+1)}}{2}$ for $t=1,\dots,n-1$ from the order statistics. 3. Compute the trimmed mean, denoted by $m_{\text{trim}}$ of deviations $x_t - x_{t-1}$ among our consecutive observations. Compute the lower limit for the left tail as $z_0 = x_{(1)} - m_{\text{trim}}$ and the upper limit for the right tail as $z_n = x_{(n)}+m_{\text{trim}}$. These limits become the limiting intermediate points. 4. Compute the mean of the maximum entropy density within each interval such that the “mean-preserving constraint" is satisfied. Interval means are denoted as $m_t$. The means for the first and last intervals have simpler formulas: $$\left\{ \begin{array}{l} m_1=0.75 x_{(1)}+0.25 x_{(2)} \\ m_k = 0.25x_{(k-1)} + 0.5 x_{(k)} + 0.25 x_{(k+1)}, \qquad k=2,\dots,n \\ m_n = 0.25 x_{(n-1)} + 0.75 x_{(n)} \end{array} \right.$$ 5. Generate random numbers from Uniform$[0,1]$, compute sample quantiles of the maximum entropy density at those points and sort them. 6. Re-order the sorted sample quantiles by using the ordering index of Step 1. This recovers the time dependence relationships of the originally observed data. 7. Repeat Steps 2 to 6 several times. [^1]: Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, Acton ACT 2601, Australia; Telephone number: +61(2) 6125 0535; Fax number: +61(2) 6125 0087; Email address: [email protected]

--- abstract: 'We show that a right-angled Artin group, defined by a graph $\Gamma$ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if $\Gamma$ is biconnected. Further, we compute JSJ–decompositions of 1–ended right-angled Artin groups over infinite cyclic subgroups.' address: | Department of Mathematical Sciences\ University of Arkansas\ Fayetteville, AR 72701 author: - Matt Clay bibliography: - '../bibliography.bib' title: 'When does a right-angled Artin group split over ${\mathbb{Z}}$?' --- Introduction {#sec:intro} ============ Given a finite simplicial graph $\Gamma$, the right-angled Artin group (RAAG) $A(\Gamma)$ is the group with generating set $\Gamma^0$, the vertices of $\Gamma$, and with relations $[v,w] = 1$ whenever vertices $v$ and $w$ span an edge in $\Gamma$. That is: $$A(\Gamma) = {\langle}\, \Gamma^0 \mid [v,w] = 1 \, \forall v,w \in \Gamma^0 \mbox{ that span an edge in } \Gamma \, {\rangle}$$ Right-angled Artin groups, simple to define, are at the focal point of many recent developments in low-dimensional topology and geometric group theory. This is in part due to the richness of their subgroups, in part due to their interpretation as an interpolation between free groups and free abelian groups and also in part due to the frequency at which they arise as subgroups of geometrically defined groups. Recent work of Agol, Wise and Haglund in regards to the Virtual Haken Conjecture show a deep relationship between 3–manifold groups and right-angled Artin groups [@ar:Agol13; @ar:HW08; @ar:HW12; @un:Wise; @bk:Wise12]. One of the results of this paper computes JSJ–decompositions for 1–ended right-angled Artin groups. This decomposition is a special type of graph of groups decomposition over infinite cyclic subgroups, generalizing to the setting of finitely presented groups a tool from the theory of 3–manifolds. So to begin, we are first concerned with understanding when a right-angled Artin group splits over an infinite cyclic subgroup. Recall, a group $G$ *splits* over a subgroup $Z$ if $G$ can be decomposed as an amalgamated free product $G = A\ast_Z B$ with $A \neq Z \neq B$ or as an HNN-extension $G = A\ast_Z$. Suppose $\Gamma$ is a finite simplicial graph. A subgraph $\Gamma_1 \subseteq \Gamma$ is *induced* if two vertices of $\Gamma_1$ span an edge in $\Gamma_1$ whenever they span an edge in $\Gamma$. If $\Gamma_1 \subseteq \Gamma$ is a induced subgraph, then the natural map induced by subgraph inclusion $A(\Gamma_1) \to A(\Gamma)$ is injective. A vertex $v \in \Gamma^0$ is a *cut vertex* if the induced subgraph spanned by the vertices $\Gamma^0 - \{v\}$ has more connected components than $\Gamma$. A graph $\Gamma$ is *biconnected* if for each vertex $v \in \Gamma^0$, the induced subgraph spanned by the vertices $\Gamma^0 - \{v\}$ is connected. In other words, $\Gamma$ is biconnected if $\Gamma$ is connected and does not contain a cut vertex. Note, $K_2$, the complete graph on two vertices, is biconnected. \[rem:cut vertex\] There is an obvious sufficient condition for a right-angled Artin group to split over a subgroup isomorphic to ${\mathbb{Z}}$. (In what follows we will abuse notation and simply say that the group splits over ${\mathbb{Z}}$.) Namely, if a finite simplicial graph $\Gamma$ contains two proper induced subgraphs $\Gamma_1,\Gamma_2 \subset \Gamma$ such that $\Gamma_1 \cup \Gamma_2 = \Gamma$ and $\Gamma_1 \cap \Gamma_2 = v \in \Gamma^0$, then $A(\Gamma)$ splits over ${\mathbb{Z}}$. Indeed, in this case we have $A(\Gamma) = A(\Gamma_1) \ast_{A(v)} A(\Gamma_2)$. If $\Gamma$ has at least three vertices, such subgraphs exist if and only if $\Gamma$ is disconnected or has a cut vertex, i.e., $\Gamma$ is not biconnected. Our first theorem, proved in Section \[sec:split\], states that this condition is necessary as well. \[thmA\] Suppose $\Gamma$ is a finite simplicial graph that has at least three vertices. Then $\Gamma$ is biconnected if and only if $A(\Gamma)$ does not split over ${\mathbb{Z}}$. If $\Gamma$ has one vertex, then $A(\Gamma) \cong {\mathbb{Z}}$, which does not split over ${\mathbb{Z}}$. If $\Gamma$ has two vertices, then $A(\Gamma) \cong F_2$ or $A(\Gamma) \cong {\mathbb{Z}}^2$, both of which do split over ${\mathbb{Z}}$ as HNN-extensions. \[rem:1-ended\] We recall for the reader the characterization of splittings of right-angled Artin groups over the trivial subgroup. Suppose $\Gamma$ is a finite simpicial graph with at least two vertices. Then $\Gamma$ is connected if and only if $A(\Gamma)$ is freely indecomposable, equivalently 1–ended. See for instance [@ar:BM01]. In Section \[sec:jsj\], for 1–ended right-angled Artin groups $A(\Gamma)$ we describe a certain graph of groups decomposition, ${\mathcal{J}}(\Gamma)$, with infinite cyclic edge groups. The base graph for ${\mathcal{J}}(\Gamma)$ is defined by considering the biconnected components of $\Gamma$, taking special care with the $K_2$ components that contain a valence one vertex from the original graph $\Gamma$. Our second theorem shows that this decomposition is a JSJ–decomposition. \[thmB\] Suppose $\Gamma$ is a connected finite simplicial graph that has at least three vertices. Then ${\mathcal{J}}(\Gamma)$ is a JSJ–decomposition for $A(\Gamma)$. Acknowledgements {#acknowledgements .unnumbered} ---------------- Thanks go to Matthew Day for posing the questions that led to this work. Also I thank Vincent Guirardel and Gilbert Levitt for suggesting the use of their formulation of a JSJ–decomposition which led to a simplification of the exposition in Section \[sec:jsj\]. Finally, I thank Denis Ovchinnikov and the anonymous referee for noticing an error in a previous version in the proof of Proposition \[prop:biconnected\] which resulted in a simplification in the proof of Theorem \[thmA\]. Splittings of RAAGs over ${\mathbb{Z}}$ {#sec:split} ======================================= This section contains the proof of Theorem \[thmA\]. The outline is as follows. First, we will exhibit a family of right-angled Artin groups that do not split over ${\mathbb{Z}}$. Then we will show how if $A(\Gamma)$ is sufficiently covered by subgroups that do not split over ${\mathbb{Z}}$, then neither does $A(\Gamma)$. Finally, we will show how to find enough subgroups to sufficiently cover $A(\Gamma)$ when $\Gamma$ has at least three vertices and is biconnected. Property ${\mathbf{F}}({\mathcal{H}})$ {#property-mathbffmathcalh .unnumbered} -------------------------------------- We begin by recalling some basic notions about group actions on trees, see [@bk:Serre03] for proofs. In what follows, all trees are simplicial and all actions are without inversions, that is $ge \neq \bar{e}$ for all $g \in G$ and edges $e$. When a group $G$ acts on a tree $T$, the *length* of an element $g \in G$ is $|g| = \inf \{ d_{T}(x,gx) \mid x \in T \}$ and the *characteristic subtree* is $T_g = \{ x \in T \mid d_{T}(x,gx) = |g| \}$. The characteristic subtree is always non-empty. If $|g| = 0$, then $g$ is said to be *elliptic* and $T_{g}$ consists of the set of fixed points. Else, $|g| > 0$ and $g$ is said to be *hyperbolic*, in which case $T_{g}$ is a linear subtree, called the *axis* of $g$, and $g$ acts on $T_{g}$ as a translation by $|g|$. The following property puts some control over the subgroups that a given group can split over. \[def:FH\] Suppose ${\mathcal{H}}$ is a collection of groups. We say a group $G$ *has property* ${\mathbf{F}}({\mathcal{H}})$ if whenever $G$ acts on a tree, then either there is a global fixed point or $G$ has a subgroup isomorphic to some group in ${\mathcal{H}}$ that fixes an edge. If ${\mathcal{H}}= \{ H \}$ we will write ${\mathbf{F}}(H)$. \[rem:FH\] Bass–Serre theory [@bk:Serre03] implies that if $G$ has property ${\mathbf{F}}({\mathcal{H}})$ and $G$ splits over a subgroup $Z$, then $Z$ has a subgroup isomorphic to some group in ${\mathcal{H}}$. For the sequel we consider the collection ${\mathcal{H}}= \{ F_2,{\mathbb{Z}}^2\}$, where $F_2$ is the free group of rank 2. We can reformulate the question posed in the title using the following proposition. \[prop:Fz2f2\] Suppose $\Gamma$ is a finite simplicial graph that has at least three vertices. Then $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$ if and only if $A(\Gamma)$ does not split over ${\mathbb{Z}}$. Bass–Serre theory (Remark \[rem:FH\]) implies that if $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$ then $A(\Gamma)$ does not split over ${\mathbb{Z}}$. Conversely, suppose that $A(\Gamma)$ does not split over ${\mathbb{Z}}$ and $A(\Gamma)$ acts on a tree $T$ without a global fixed point. The stabilizer of any edge is non-trivial as freely decomposable right-angled Artin groups whose defining graphs have at least three vertices split over ${\mathbb{Z}}$ (Remarks \[rem:cut vertex\] and \[rem:1-ended\]). We claim the stabilizer of any edge contains two elements that do not generate a cyclic group. As a subgroup generated by two elements in a right-angled Artin group is either abelian or isomorphic to $F_2$ [@ar:Baudisch81], this shows that $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$. To prove the claim, let $Z$ denote the stabilizer of some edge of $T$ and suppose ${\langle}g, h {\rangle}\cong {\mathbb{Z}}$ for all $g,h \in Z$. Thus $Z$ is abelian. Since abelian subgroups of right-angled Artin groups are finitely generated (as the Salvetti complex is a finite $K(A(\Gamma),1)$ [@col:CD95]) we have $Z \cong {\mathbb{Z}}$. But this contradicts our assumption that $A(\Gamma)$ does not split over ${\mathbb{Z}}$. Thus we are reduced to proving that property ${\mathbf{F}}({\mathcal{H}})$ is equivalent to biconnectivity for right-angled Artin groups whose defining graph has at least three vertices. A family of right-angled Artin groups that do not split over ${\mathbb{Z}}$ {#a-family-of-right-angled-artin-groups-that-do-not-split-over-mathbbz .unnumbered} --------------------------------------------------------------------------- The following simple lemma of Culler–Vogtmann relates the characteristic subtrees of commuting elements. As the proof is short, we reproduce it here. \[lem:cv\] Suppose a group $G$ acts on a tree $T$ and let $g$ and $h$ be commuting elements. Then the characteristic subtree of $g$ is invariant under $h$. In particular, if $h$ is hyperbolic, then the characteristic subtree of $g$ contains $T_h$. As $h(T_g) = T_{hgh^{-1}}$ if $g$ and $h$ commute then $h(T_g) = T_g$. If $h$ is hyperbolic, then every $h$–invariant subtree contains $T_h$. \[cor:cv\] If ${\mathbb{Z}}^{2}$ acts on a tree without a global fixed point, then for any basis $\{g,h\}$, one of the elements must act hyperbolically. Suppose that both $g$ and $h$ are elliptic. As $hT_{h} = T_{h}$ and $hT_{g} = T_{g}$ by Lemma \[lem:cv\], the unique segment connecting $T_{g}$ to $T_{h}$ is fixed by $h$ and hence contained in $T_{h}$. In other words $T_{g} \cap T_{h} \neq \emptyset$ and therefore there is a global fixed point. Recall that a *Hamiltonian cycle* in a graph is an embedded cycle that visits each vertex exactly once. \[lem:Hamiltonian\] If $\Gamma$ is a finite simplicial graph with at least three vertices that contains a Hamiltonian cycle, then $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$. Enumerate the vertices of $\Gamma$ cyclically along the Hamiltonian cycle by $v_1,\ldots,v_{n}$. Notice that $G_i = {\langle}v_i,v_{i+1} {\rangle}\cong {\mathbb{Z}}^2$ for all $1 \leq i \leq n$ where the indices are taken modulo $n$. Suppose that $A(\Gamma)$ acts on a tree $T$ without a global fixed point. Further suppose that $G_i$ does not fix an edge, for all $1 \leq i \leq n$. There are now two cases. [*Case I Each $G_i$ fixes a point.*]{} The point fixed by $G_i$ is unique as $G_i$ does not fix an edge, denote it $p_i$. If the points $p_i$ are all the same, then there is a global fixed point, contrary to the hypothesis. Consider the subtree $S \subset T$ spanned by the $p_i$. Let $p$ be an extremal vertex of $S$. There is a non-empty proper subset $P \subset \{1,\ldots,n\}$ such that $p = p_i$ if and only if $i \in P$. Let $i_1,j_0 \in P$ be such that the indices $i_0 = i_1 - 1 \mod n$ and $j_1 = j_0 + 1 \mod n$ do not lie in $P$. See Figure \[fig:Cn-I\]. It is possible that $i_1 = j_0$ or $i_0 = j_1$. (0.7,0.7) circle \[radius=0.02\]; (-1,0) circle \[radius=0.02\]; (0,-1) circle \[radius=0.02\]; (0.75,0.7) node\[right\] [$p = p_{i_1} = p_{j_0}$]{}; (-1,0.15) node [$p_{i_0}$]{}; (0.2,-1) node [$p_{j_1}$]{}; (0.7,0.7) – (0,0); (-1,0) – (0,0); (0,-1) – (0,0); The element $v_{i_1} \in G_{i_0} \cap G_{i_1}$ stabilizes the non-degenerate segment $[p,p_{i_0}]$ and the element $v_{j_1} \in G_{j_0} \cap G_{j_1}$ stabilizes the non-degenerate segment $[p,p_{j_1}]$. As $p$ is extremal, these segments overlap and thus ${\langle}v_{i_1},v_{j_1} {\rangle}$ fixes an edge in $T$. This subgroup is isomorphic to either $F_2$ or ${\mathbb{Z}}^2$. [*Case II Some $G_i$ does not fix a point.*]{} Without loss of generality, we can assume that $G_1$ does not fix a point and by Corollary \[cor:cv\] that $v_2$ acts hyperbolically. By Lemma \[lem:cv\], $v_1$ leaves $T_{v_2}$ invariant and so there are integers $k_1,k_2$, where $k_1 \neq 0$, such that $v_1^{k_1}v_2^{k_2}$ fixes $T_{v_2}$. Likewise there are integers $\ell_2,\ell_3$, where $\ell_3 \neq 0$ such that $v_2^{\ell_2}v_3^{\ell_3}$ fixes $T_{v_2}$. Hence ${\langle}v_1^{k_1}v_2^{k_2},v_2^{\ell_2}v_3^{\ell_3} {\rangle}$ fixes $T_{v_2}$, in particular, this subgroup fixes an edge. This subgroup is isomorphic to either $F_{2}$ or ${\mathbb{Z}}^{2}$. In either case, we have found a subgroup isomorphic to either $F_2$ or ${\mathbb{Z}}^2$ that fixes an edge. Hence $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$. Promoting property ${\mathbf{F}}({\mathcal{H}})$ {#promoting-property-mathbffmathcalh .unnumbered} ------------------------------------------------ We now show how to promote property ${\mathbf{F}}({\mathcal{H}})$ to $A(\Gamma)$ if enough subgroups have property ${\mathbf{F}}({\mathcal{H}})$. \[prop:decompose\] Suppose $\Gamma$ is a connected finite simplicial graph with at least three vertices and suppose that there is a collection ${\mathcal{G}}$ of induced subgraphs $\Delta \subset \Gamma$ such that: 1. for each $\Delta \in {\mathcal{G}}$, $A(\Delta)$ has property ${\mathbf{F}}({\mathcal{H}})$, and\[enum:FH\] 2. each two edge segment of $\Gamma$ is contained in some $\Delta \in {\mathcal{G}}$.\[enum:two edge\] Then $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$. Suppose $A(\Gamma)$ acts on a tree $T$ without a global fixed point. If for some $\Delta \in {\mathcal{G}}$, the subgroup $A(\Delta)$ does not have a fixed point, then by , $A(\Delta)$, and hence $A(\Gamma)$, contains a subgroup isomorphic to either $F_2$ or ${\mathbb{Z}}^2$ that fixes an edge. Therefore, we assume that each $A(\Delta)$ has a fixed point. In particular, each vertex of $\Gamma$ acts elliptically in $T$. Also, given three vertices $u, v, w \in \Gamma^{0}$, such that $u$ and $v$ span an edge as do $v$ and $w$, the subgroup ${\langle}u, v, w {\rangle}$ by is contained in some $A(\Delta)$ and hence has a fixed point. We further may assume the fixed point of such a subgroup ${\langle}u,v,w {\rangle}$ to be unique for else ${\langle}u, v {\rangle}\cong {\mathbb{Z}}^{2}$ fixes an edge. As there is no global fixed point, there are vertices $v, v' \in \Gamma^{0}$ that do not share a fixed point. Consider a path from $v$ to $v'$ and enumerate the vertices along this path $v = v_{1},\ldots,v_{n} = v'$. If for some $1 < i < n-1$, the fixed point of ${\langle}v_{i-1},v_{i},v_{i+1} {\rangle}$ is different from that of ${\langle}v_{i}, v_{i+1}, v_{i+2} {\rangle}$, then ${\langle}v_{i},v_{i+1} {\rangle}\cong {\mathbb{Z}}^{2}$ fixes an edge as this subgroup stabilizes the non-degenerate segment between the fixed points. If the fixed points are all the same then $v$ and $v'$ have a common fixed point, contrary to our assumptions. Proof of Theorem \[thmA\] {#proof-of-theoremthma .unnumbered} ------------------------- Theorem \[thmA\] follows from Proposition \[prop:Fz2f2\] and the following proposition. \[prop:biconnected\] Suppose $\Gamma$ is a finite simplicial graph that has at least three vertices. Then $\Gamma$ is biconnected if and only if $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$. Suppose $\Gamma$ is biconnected. Consider the collection ${\mathcal{G}}$ of induced subgraphs $\Delta \subseteq \Gamma$ with at least three vertices that contain a Hamiltonian cycle. By Lemma \[lem:Hamiltonian\], each $\Delta \in {\mathcal{G}}$ has property ${\mathbf{F}}({\mathcal{H}})$. Consider vertices $u,v,w \in \Gamma^0$ such that $u$ and $v$ span an edge $e$ and $v$ and $w$ span an edge $e'$. As $\Gamma$ is biconnected, there is an edge path from $u$ to $w$ that avoids $v$. Let $\rho$ be the shortest such path and let $\Delta$ be the induced subgraph of $\Gamma$ spanned by $v$ and vertices of $\rho$. The cycle $e \cup e' \cup \rho$ is a Hamiltonian cycle in $\Delta$ and hence $\Delta \in {\mathcal{G}}$. The two edge segment $e \cup e'$ is contained in $\Delta$ by construction. Hence using the collection ${\mathcal{G}}$, Proposition \[prop:decompose\] implies that $A(\Gamma)$ has property ${\mathbf{F}}({\mathcal{H}})$. Conversely, If $\Gamma$ is not biconnected, then $A(\Gamma)$ splits over ${\mathbb{Z}}$ and hence does not have property ${\mathbf{F}}({\mathcal{H}})$ (Remark \[rem:cut vertex\] and Proposition \[prop:Fz2f2\]). JSJ–decompositions of 1–ended RAAGs {#sec:jsj} =================================== We now turn our attention towards understanding all ${\mathbb{Z}}$–splittings of a 1–ended right-angled Artin group. These are exactly the groups $A(\Gamma)$ with $\Gamma$ connected and having at least two vertices (Remark \[rem:1-ended\]). The technical tool used for understanding splittings over some class of subgroups are *JSJ–decompositions*. There are several loosely equivalent formulations of the notion of a JSJ–decomposition of a finitely presented group, originally defined in this setting and whose existence was shown by Rips–Sela [@ar:RS97]. Alternative accounts and extensions were provided by Dunwoody–Sageev [@ar:DS99], Fujiwara–Papasogalu [@ar:FP06] and Guirardel–Levitt [@un:GL]. We have chosen to use Guirardel and Levitt’s formulation of a JSJ–decomposition as it avoids many of the technical definitions necessary for the other formulations—most of which have no real significance in the current setting—and as it is particularly easy to verify in the current setting. In this section we describe a JSJ–decomposition for a 1–ended right-angled Artin group (Theorem \[thmB\]). It is straightforward to verify, given the arguments that follow, that the described graph of groups decomposition is a JSJ–decomposition in the other formulations as well. JSJ–decompositions \` a la Guirardel and Levitt {#jsjdecompositions-a-la-guirardel-and-levitt .unnumbered} ----------------------------------------------- The defining property of a JSJ–decomposition is that it gives a parametrization of all splittings of a finitely presented group $G$ over some special class of subgroups, here the subgroups considered are infinite cyclic. The precise definition is as follows. Suppose ${\mathcal{A}}$ is a class of subgroups of $G$ that is closed under taking subgroups and that is invariant under conjugation. An *${\mathcal{A}}$–tree* is a tree with an action of $G$ such that every edge stabilizer is in ${\mathcal{A}}$. An ${\mathcal{A}}$–tree is *universally elliptic* if its edge stabilizers are elliptic, i.e., have a fixed point, in every ${\mathcal{A}}$–tree. \[def:jsj\] A *JSJ–tree* of $G$ over ${\mathcal{A}}$ is a universally elliptic ${\mathcal{A}}$–tree $T$ such that if $T'$ is a universally elliptic ${\mathcal{A}}$–tree then there is a $G$–equivariant map $T \to T'$, equivalently, every vertex stabilizer of $T$ is elliptic in every universally elliptic ${\mathcal{A}}$–tree. The associated graph of group decomposition is called a *JSJ–decomposition*. We will now describe what will be shown to be the JSJ–decomposition of a 1–ended right-angled Artin group. Suppose $\Gamma$ is a connected finite simplicial graph with at least three vertices. By $B_\Gamma$ we denote the *block tree*, that is, the bipartite tree with vertices either corresponding to cut vertices of $\Gamma$ (black) or bicomponents of $\Gamma$, i.e., maximal biconnected induced subgraphs of $\Gamma$, (white) with an edge between a black and a white vertex if the corresponding cut vertex belongs to the bicomponent. See Figure \[fig:jsj\] for some examples. For a black vertex $x \in B_\Gamma^0$, denote by $v_x$ the corresponding cut vertex of $\Gamma$. For a white vertex $x \in B_\Gamma^0$, denote by $\Gamma_x$ the corresponding bicomponent of $\Gamma$. A white vertex $x \in B_\Gamma^0$ is call *toral* if $\Gamma_x \cong K_2$, the complete graph on two vertices. A toral vertex $x \in B_\Gamma$ that has valence one in $B_\Gamma$ is called *hanging*. Associated to $\Gamma$ and $B_\Gamma$ is a graph of groups decomposition of $A(\Gamma)$, denoted ${\mathcal{J}}_0(\Gamma)$. The base graph of ${\mathcal{J}}_0(\Gamma)$ is obtained from $B_\Gamma$ by attaching a one-edge loop to each hanging vertex. The vertex group of a black vertex $x \in B_\Gamma^0$ is $G_x = A(v_x) \cong {\mathbb{Z}}$. The vertex group of a non-hanging white vertex $x \in B_\Gamma$ is $G_x = A(\Gamma_x)$. The vertex group of a hanging vertex $x \in B_\Gamma$ is $G_x = A(v)$ where $v \in \Gamma_x^0$ is the vertex that has valence more than one in $\Gamma$. Notice, in this latter case $v$ is a cut vertex of $\Gamma$. For an edge $e = [x,y] \subseteq B_\Gamma$ with $x$ black we set $G_e = A(v_x) \cong {\mathbb{Z}}$ with inclusion maps given by subgraph inclusion. If $e$ is a one-edge loop adjacent to a hanging vertex $x$, we set $G_e = G_x$ where the two inclusion maps are isomorphisms and the stable letter corresponding to the loop is $w$ where $w \in \Gamma_x^0$ is the vertex that has valence one in $\Gamma$. By collapsing an edge adjacent to each valence two black vertex we obtain a graph of groups decomposition of $A(\Gamma)$, which we denote ${\mathcal{J}}(\Gamma)$. It is not necessary for what follows, but we remark that the graph is groups ${\mathcal{J}}(\Gamma)$ is *reduced in the sense of Bestvina–Feighn [@ar:BF91]*, that is, for each vertex of valence less than three the edge groups are proper subgroups of the vertex group. This property is required for a JSJ–decomposition as defined by Rips–Sela. Observe that all edge groups of ${\mathcal{J}}(\Gamma)$ are of the form $A(v)$ for some vertex $v \in \Gamma^0$ and in particular maximal infinite cyclic subgroups. By $T_{{\mathcal{J}}(\Gamma)}$ we denote the associated Bass–Serre tree. \[ex:jsj\] Examples of $B_\Gamma$, ${\mathcal{J}}_0(\Gamma)$ and ${\mathcal{J}}(\Gamma)$ for two different graphs are shown in Figure \[fig:jsj\]. We have $A(\Gamma_1) \cong F_3 \times {\mathbb{Z}}$. The graph of groups decomposition ${\mathcal{J}}_0(\Gamma_1)$ is already reduced so ${\mathcal{J}}(\Gamma_1) = {\mathcal{J}}_0(\Gamma_1)$. In ${\mathcal{J}}(\Gamma_1)$ all of the vertex and edge groups are infinite cyclic and all inclusion maps are isomorphisms. Considering the other example, ${\mathcal{J}}(\Gamma_2)$ corresponds to the graph of groups decomposition $A(\Gamma_2) = {\mathbb{Z}}^3 \ast_{\mathbb{Z}}{\mathbb{Z}}^2 \ast_{\mathbb{Z}}{\mathbb{Z}}^3$ where the inclusion maps have image a primitive vector and the images in ${\mathbb{Z}}^2$ constitute a basis of ${\mathbb{Z}}^2$. (-2,0) – (2,0); (0,-1) – (0,1); (-2,-2.5) – (2,-2.5); (0,-1.5) – (0,-3.5); (-1,0.5) – (-0.5,0.5); (-1,0.5) – (-1.25,0.933); (-1,0.5) – (-1.25,0.067); (-1,0.5) circle \[radius=0.02\]; (-0.5,0.5) circle \[radius=0.02\]; (-1.25,0.933) circle \[radius=0.02\]; (-1.25,0.067) circle \[radius=0.02\]; (-1.75,0.5) node [$\Gamma_1$]{}; (1,0.5) – (1.5,0.5); (1,0.5) – (0.75,0.933); (1,0.5) – (0.75,0.067); (1,0.5) circle \[radius=0.02\]; (1.5,0.5) circle \[radius=0.02\]; (0.75,0.933) circle \[radius=0.02\]; (0.75,0.067) circle \[radius=0.02\]; (0.3,0.5) node [$B_{\Gamma_1}$]{}; (-1,-0.5) – (-0.75,-0.5); (-0.65,-0.5) circle \[radius=0.1\]; (-1,-0.5) – (-1.125,-0.7165); (-1.175,-0.8031) circle \[radius=0.1\]; (-1,-0.5) – (-1.125,-0.2835); (-1.175,-0.1969) circle \[radius=0.1\]; (-1,-0.5) circle \[radius=0.02\]; (-0.75,-0.5) circle \[radius=0.02\]; (-1.125,-0.7165) circle \[radius=0.02\]; (-1.125,-0.2835) circle \[radius=0.02\]; (-1.75, -0.5) node [${\mathcal{J}}_0(\Gamma_1)$]{}; (0.3, -0.5) node [${\mathcal{J}}(\Gamma_1)$]{}; (1,-0.5) – (1.25,-0.5); (-0.65+2,-0.5) circle \[radius=0.1\]; (1,-0.5) – (-1.125+2,-0.7165); (-1.175+2,-0.8031) circle \[radius=0.1\]; (1,-0.5) – (-1.125+2,-0.2835); (-1.175+2,-0.1969) circle \[radius=0.1\]; (1,-0.5) circle \[radius=0.02\]; (1.25,-0.5) circle \[radius=0.02\]; (-1.125+2,-0.7165) circle \[radius=0.02\]; (-1.125+2,-0.2835) circle \[radius=0.02\]; (-1.1,-2) – (-0.7,-2); (-1.1,-2) – (-1.4,-1.8268) – (-1.4,-2.1732) – (-1.1,-2); (-0.7,-2) – (-0.4,-1.8268) – (-0.4,-2.1732) – (-0.7,-2); (-1.4,-1.8268) circle \[radius=0.02\]; (-1.4,-2.1732) circle \[radius=0.02\]; (-0.4,-1.8268) circle \[radius=0.02\]; (-0.4,-2.1732) circle \[radius=0.02\]; (-1.1,-2) circle \[radius=0.02\]; (-0.7,-2) circle \[radius=0.02\]; (-1.75,-2) node [$\Gamma_2$]{}; (0.6,-2) – (1.8,-2); (0.6,-2) circle \[radius=0.02\]; (0.9,-2) circle \[radius=0.02\]; (1.2,-2) circle \[radius=0.02\]; (1.5,-2) circle \[radius=0.02\]; (1.8,-2) circle \[radius=0.02\]; (0.3,-2) node [$B_{\Gamma_2}$]{}; (-1.4,-3) – (-0.2,-3); (-1.4,-3) circle \[radius=0.02\]; (-1.1,-3) circle \[radius=0.02\]; (-0.8,-3) circle \[radius=0.02\]; (-0.5,-3) circle \[radius=0.02\]; (-0.2,-3) circle \[radius=0.02\]; (-1.75,-3) node [${\mathcal{J}}_0(\Gamma_2)$]{}; (0.8,-3) – (1.6,-3); (0.8,-3) circle \[radius=0.02\]; (1.2,-3) circle \[radius=0.02\]; (1.6,-3) circle \[radius=0.02\]; (0.3,-3) node [${\mathcal{J}}(\Gamma_{2})$]{}; Proof of Theorem \[thmB\] {#proof-of-theoremthmb .unnumbered} ------------------------- Theorem \[thmB\] follows immediately from the following lemma. \[lem:Jvertex-elliptic\] Suppose $\Gamma$ is a connected finite simplicial graph that has at least three vertices and let ${\mathcal{A}}$ be the collection of all cyclic subgroups of $A(\Gamma)$. Every vertex stabilizer of $T_{{\mathcal{J}}(\Gamma)}$ is elliptic in every ${\mathcal{A}}$–tree. In particular, every edge stabilizer of $T_{{\mathcal{J}}(\Gamma)}$ is elliptic in every ${\mathcal{A}}$–tree and so $T_{{\mathcal{J}}(\Gamma)}$ is universally elliptic and every vertex stabilizer of $T_{{\mathcal{J}}(\Gamma)}$ is elliptic in every universally elliptic ${\mathcal{A}}$–tree. Let $T$ be an ${\mathcal{A}}$–tree. As $A(\Gamma)$ is 1–ended, every edge stabilizer of $T$ is infinite cyclic. As the vertex groups of a black vertex is a subgroup of the vertex group of some white vertex, we only need to consider white vertices. The vertex group of every non-toral vertex of ${\mathcal{J}}(\Gamma)$ is elliptic by Proposition \[prop:biconnected\]. Let $x \in B_\Gamma$ be a non-hanging toral vertex. Denote the vertices of $\Gamma_x \cong K_2$ by $v_1$ and $v_2$. Then there are vertices $w_1, w_2 \in \Gamma^0$ such that $[v_i, w_j] = 1$ if and only if $i = j$. In other words, the vertices $w_1,v_1,v_2,w_2$ span an induced subgraph of $\Gamma$ that is isomorphic to the path graph with three edges. If $v_1 \in G_x = A(\Gamma_x) \cong {\mathbb{Z}}^2$ acts hyperbolically, then by Lemma \[lem:cv\] the characteristic subtree of both $w_1$ and $v_2$ contains $T_{v_1}$, the axis of $v_1$. As in the proof of Lemma \[lem:Hamiltonian\], we find integers $k_0,k_1,\ell_0,\ell_1$ with $k_1, \ell_1 \neq 0$ such that ${\langle}v_1^{k_0}w_1^{k_1}, v_1^{\ell_0}v_2^{\ell_1} {\rangle}\cong F_2$ fixes $T_{v_1}$ and hence fixes an edge. As every edge stabilizer of $T$ is infinite cyclic, this shows that $v_1$ must have a fixed point. By symmetry $v_2$ must also have a fixed point. Since $A(\Gamma_x) = {\langle}v_1,v_2 {\rangle}\cong {\mathbb{Z}}^2$, by Corollary \[cor:cv\] this implies that $A(\Gamma_x)$ acts elliptically. Finally, let $x \in B_\Gamma$ be a hanging vertex. Either $G_x$ is a subgroup of some non-hanging white vertex subgroup and so $G_x$ acts elliptically by the above argument, or $A(\Gamma) \cong F_n \times {\mathbb{Z}}$ for $n \geq 2$ where $G_x$ is the ${\mathbb{Z}}$ factor as is the case for $\Gamma_1$ in Example \[ex:jsj\]. In the latter case, as $G_{x}$ is central, by Lemma \[lem:cv\] if $G_{x}$ acts hyperbolically, then $F_{n} \times {\mathbb{Z}}$ acts on its axis. Therefore there is a homomorphism $F_n \times {\mathbb{Z}}\to {\mathbb{Z}}$ whose kernel fixes an edge. As every edge stabilizer of $T$ is infinite cyclic, $G_x$ must act elliptically. We record the following corollary of Lemma \[lem:Jvertex-elliptic\]. \[cor:thmB\] Suppose $\Gamma$ is a connected finite simplicial graph that has at least three vertices. If $A(\Gamma)$ acts on a tree $T$ such that the stabilizer of every edge is infinite cyclic, then every $v \in \Gamma^{0}$ that has valence greater than one acts elliptically in $T$. This follows from Lemma \[lem:Jvertex-elliptic\] as each such vertex is contained in some bicomponent $\Gamma_{x}$ for some non-hanging $x \in B_{\Gamma}$ and hence acts elliptically in $T_{{\mathcal{J}}(\Gamma)}$.

--- abstract: 'We present a non-LTE model atmosphere analysis of [[*Chandra*]{}]{}  HRC-S/LETG and XMM-[[*Newton*]{}]{}  RGS spectroscopy of the prototypical supersoft source CAL 83 in the Large Magellanic Cloud. Taken with a 16-month interval, the [[*Chandra*]{}]{}  and XMM-[[*Newton*]{}]{}  spectra are very similar. They reveal a very rich absorption line spectrum from the hot white dwarf photosphere, but no spectral signatures of a wind. We also report a third X-ray off-state during a later [[*Chandra*]{}]{}  observation, demonstrating the recurrent nature of CAL 83. Moreover, we found evidence of short-timescale variability in the soft X-ray spectrum. We completed the analysis of the LETG and RGS spectra of CAL 83 with new NLTE line-blanketed model atmospheres that explicitly include 74 ions of the 11 most abundant species. We successfully matched the [[*Chandra*]{}]{} and XMM-[[*Newton*]{}]{}  spectra assuming a model composition with LMC metallicity. We derived the basic stellar parameters of the hot white dwarf, but the current state of atomic data in the soft X-ray domain precludes a detailed chemical analysis. We have obtained the first direct spectroscopic evidence that the white dwarf is massive ($M_{\rm WD} \ga 1 M_\odot$). The short timescale of the X-ray off-states is consistent with a high white dwarf mass. Our analysis thus provides direct support for supersoft sources as likely progenitors of SN Ia.' author: - | Thierry Lanz, Gisela A. Telis, Marc Audard, Frits Paerels,\ Andrew P. Rasmussen, and Ivan Hubeny title: 'NLTE Model Atmosphere Analysis of the LMC Supersoft X-ray Source CAL 83' --- Introduction {#Intro} ============ Because of the overall importance of Supernovae type Ia (SNe Ia) in astrophysics and in cosmology, the identification of their progenitors is a pressing issue. Theoretical models suggest that SNe Ia arise from the thermonuclear explosion of a carbon-oxygen (CO) white dwarf (WD) that has grown to the Chandrasekhar mass, [$M_{\rm CH}$]{}  [@hoyle60; @arnett69]. To date, the most promising formation channel resulting in SNe Ia involves accreting WD’s that sustain steady nuclear burning close to their surface. When they form, CO WD’s have masses between 0.7 and 1.2$M_\sun$ depending on the star initial mass [@weidemann87]. SN Ia progenitors, therefore, must be in close binaries where the WD can gain several tenths of a solar mass donated by a companion to reach [$M_{\rm CH}$]{}. Because we lack a direct determination of the progenitor properties, we have to rely on indirect arguments to determine their nature. Two main scenarios have been proposed, involving either the merger of two WD’s [the double-degenerate (DD) scenario; @iben84; @webbink84], or a single WD accreting from a normal companion [the single-degenerate (SD) scenario; @whelan73]. The DD scenario provides a natural way to explain the absence of hydrogen lines in SN Ia spectra, but hardly explains why SNe Ia could be “standard candles” since the explosion should depend on the respective properties of the two WD’s. Moreover, while the DD scenario provides an obvious way to form a more massive object, possibly reaching [$M_{\rm CH}$]{}, the few double WD systems identified to date that could merge in a Hubble time have a total mass smaller than [$M_{\rm CH}$]{} [@saffer98]. New large-scale searches for DD systems, however, have recently yielded promising results [@napiwotzki02; @napiwotzki03], and good candidates might be discovered soon. Finally, the merger scenario has been suggested to lead to an accretion-induced collapse rather than to a SN Ia event [@nomoto85; @saio85]. The SD scenario is thus generally favored today. The principal issue of the SD scenario is, however, to ascertain if the WD can at all accrete sufficient mass to lead to a SN Ia event. This topic has been hotly debated in the last 10 years to sort out the competing effects of mass ejection and accretion. [@nomoto79] first pointed out that an accreting WD would quickly ignite a small amount of accreted material in a hydrogen shell-burning and undergo a weak shell flash. Most of the accreted mass then accumulates in an extended, supergiant-like envelope that might be lost during a subsequent common-envelope phase. [@hachisu96; @hachisu99] argued that starting at a certain critical accretion rate a strong wind develops, regulating the mass transfer between the mass-losing star and the accreting WD. The WD accretes at most at the critical rate, $\dot{M}_{\rm cr} = 9.0\,\times\,10^{-7}\,(M_{\rm WD}/M_\odot - 0.50)\,M_\odot$yr$^{-1}$, and the rest is blown off in the wind. At a later stage the accretion rate drops, and the strong wind stops; the final outcome depends on the mass accreted before this point, because stronger flashes will develop at lower accretion rates. [@cassisi98] made the opposite argument. The basic physical processes are the same, but they argued that dynamical helium-burning flashes probably would hamper accretion and, therefore, the WD most likely would not reach [$M_{\rm CH}$]{}. The most recent theoretical calculations support the idea that the accreting WD may eventually reach [$M_{\rm CH}$]{}. [@yoon03] have built the first detailed binary star evolution model where the WD gains enough mass from a $1.6\,M_\odot$ helium star. This model involves higher accretion rates (few $10^{-6}\,M_\odot$yr$^{-1}$). Subsequently, @yoon04 have shown that rotation tends to stabilize the helium-burning shell, thus increasing the likelihood that accreting WD would reach the stage of central carbon ignition. Owing to the difficulties met by theoretical works, an observational approach is essential to characterize SN Ia progenitors. The most promising candidates are Close Binary Supersoft X-ray Sources (CBSS) that were revealed by ROSAT as a new class of close binaries with ultrasoft X-ray spectra showing no emission above $0.5 - 1$keV [@trumper91]. @vdH92 [vdH92] argued that such a soft emission is not consistent with accreting neutron stars or black holes, which show emission peaking at $1 - 10$keV. They proposed the now classical model of an accreting WD, where a relatively high-mass WD $(0.7 - 1.2\,M_\odot)$ sustains steady burning of the hydrogen-rich accreted material. They argued that accretion has to occur at a finely-tuned rate, $1.0\,\times\,10^{-7}\la\dot{M}\la 4.0\,\times\,10^{-7}\,M_\odot$yr$^{-1}$. At lower rates, hydrogen burning is unstable and occurs in flashes, while an extended envelope forms at higher rates. The stellar luminosity is then dominated by hydrogen burning which liberates an order of magnitude or more of energy than accretion itself. Typical temperatures, $kT\approx 30 - 80$eV, are derived from blackbody simulations and WD model atmospheres after correction for interstellar extinction [@kahabka97]. Based on the vdH92 model, [@rappaport94] discussed the formation and evolution of CBSS, reproducing their typical luminosities, effective temperatures and orbital periods. They estimated that the rate of Galactic SNe Ia associated with the evolution of CBSS might reach 0.006 yr$^{-1}$. Most recently, [@ivanova04] extended the vdH92 work, developing a semianalytical model to investigate the evolution of binaries consisting of main-sequence stars with WD companions in the thermal mass-transfer phase. They accounted for the stabilizing effect of the WD wind, and characterized the different conditions leading to different outcomes, double WD’s, (sub-)[$M_{\rm CH}$]{}  SNe Ia, or accretion-induced collapse. To evolve towards a SN Ia explosion, [@ivanova04] argued that the WD and the donor star should be initially relatively massive, $M_{\rm WD}\ga 0.8\,M_\odot$ and $M_{\rm d}\ga 2\,M_\odot$, with a mass ratio $M_{\rm d}/M_{\rm WD}$ smaller than about 3. The SD scenario recently gained further observational support. [@hamuy03] reported the detection of a narrow H$\alpha$ emission line in SN2002ic, providing the first evidence of hydrogen-rich circumstellar material associated with a SN Ia. This detection might be the missing smoking gun for the SD scenario, though [@livio03] argued that this conclusion might be premature based on the absence of a detection in all other SN Ia spectra. The exact nature of SN Ia progenitors thus remains an open problem. Determining the properties of CBSS may, therefore, provide the strongest empirical case supporting the SD scenario. [[*Chandra*]{}]{}  and XMM-[[*Newton*]{}]{}’s capabilities in obtaining high-resolution spectra in the soft X-ray domain now open the possibility to determine spectroscopically the stellar parameters of the WD’s in CBSS, in particular the surface gravity and hence the WD mass. We present in this paper a non-LTE (NLTE) analysis of the [[*Chandra*]{}]{}  and XMM-[[*Newton*]{}]{}  spectra of the prototypical CBSS CAL 83. We start by summarizing previous results on CAL 83 in §2, and the new observations are discussed in §3 and §4. The NLTE model atmospheres are detailed in §5. Sect. 6 presents the spectrum analysis, leading to a conclusion that the WD in CAL 83 is massive (§7). The CBSS CAL 83 {#CAL83} =============== The [*Einstein*]{} observatory survey of the Large Magellanic Cloud (LMC) revealed two sources with an ultrasoft spectrum, CAL 83 and CAL 87 [@long81]. CAL 83 (= RX J0543.5-6823) was identified with a variable, blue, $V\approx 17$, point-like source with an orbital period of 1.04 days [@cowley84; @smale88]. Soft X-ray spectra of CAL 83 have been subsequently obtained with ROSAT PSPC [@greiner91] and [*Beppo*]{}SAX LECS [@parmar98]. Because of the limited spectral resolution of these observations, no spectral features were visible. The spectral energy distribution was modeled using first blackbodies and then WD model atmospheres. A blackbody analysis of the ROSAT observations implies the puzzling result that CBSS have radii typical of WD’s but radiate at or above the Eddington limit [@greiner91]. [@heise94] constructed the first LTE model atmospheres of CBSS. They showed that these models predict a higher flux in the soft X-ray range, and hence do not require super-Eddington luminosities to fit the ROSAT data. [@hartmann97] extended Heise et al.’s work, investigating the importance of departures from LTE. NLTE effects are expected in high gravity objects if they are sufficiently hot [@dreizler93; @NLTE2]. Indeed, Hartmann & Heise found significant differences between LTE and NLTE model spectra in the temperature range of WD’s in CBSS. The main effect consists in the overionization of heavy species. NLTE model atmospheres and NLTE effects are further discussed in §5 and §6. Significant advances are now expected from [[*Chandra*]{}]{}  and XMM-[[*Newton*]{}]{} spectrometers because of their higher spectral resolution. XMM-[[*Newton*]{}]{}  RGS spectroscopy of CAL 83 was first obtained aiming at deriving the fundamental stellar and binary parameters [@paerels01]. The spectrum shows a very rich line structure. The application of NLTE model atmospheres to analyze the RGS spectrum was, however, only marginally successful and no satisfactory, detailed match to the RGS data was achieved. This paper takes over from the Paerels et al. paper and includes new [[*Chandra*]{}]{}  LETG spectroscopy extending beyond the RGS cutoff at 40Å  as well. This extension to lower energies provides crucial new data to normalize the NLTE model spectra and, hence, to determine the parameters of CAL 83. Observations {#Observ} ============ CAL 83 was observed in April 2000 by XMM-[[*Newton*]{}]{}  and at 3 epochs (November 1999, August 2001, October 2001) by [[*Chandra*]{}]{}, see Table \[LogTbl\] for a log of the observations. In November 1999 and in October 2001, CAL 83 was found to be in a X-ray off-state (see §\[XVar\] and Greiner & Di Stefano 2002). The XMM-[[*Newton*]{}]{}  RGS observation was described by [@paerels01]. We detail here only the August 2001 [[*Chandra*]{}]{}  observation. On 2001 August 15, we observed CAL 83 for 35.4 ksec with [[*Chandra*]{}]{}, using the High Resolution Camera (HRC-S) and the Low Energy Transmission Grating (LETG). This setup affords a spectral coverage between 1 and 175Å  at a nominal resolution of 0.05Å  and, therefore, provides full coverage from the Wien tail of the energy distribution to the cut-off due to interstellar extinction. The data were processed with CIAO, version 3.0.2[^1]. We applied the observation-specific status and Good Time Interval filters, as well as an additional, non-standard filter on the pulse-height value as a function of dispersed-photon wavelength. Wargelin & Ratzlaff[^2] suggested to use a light pulse-height filter which can reduce the background rate in the HRC-S/LETG configuration by 50-70% with minimal X-ray losses and virtually no possibility of introducing spurious spectral features. To test the effects of such a filter on our data, we simultaneously completed the data reduction with and without the filter. After a comparison of the output event lists and spectra confirmed that no significant changes had been made to the valid photon events, we retained the pulse-height filter and used the filtered data for the remainder of the analysis. We used the canned, on-axis first-order grating redistribution matrix file (gRMF) for the LETG spectrum, but we ran the CIAO task “fullgarf” to generate the grating ancillary response files (gARFs) for the positive and negative first spectral order. Furthermore, we multiplied the gARFs by the provided encircled energy auxiliary response file in order to correct for the efficiency of the LETG “bow-tie” extraction region. The positive and negative order spectra were co-added to increase the signal-to-noise ratio. We merged the corresponding gARFs. Note that we did not correct for the effect of higher spectral orders since they are negligible in the $20-70$ Å  faint spectrum of CAL 83. Fig. \[LETGFig\] compares the RGS spectrum (we only use RGS1 data) and the LETG spectrum, calibrated in flux with the instrumental response file. We chose to rebin the spectra by a factor of 8, as the best compromise between signal-to-noise ratio and resolution, thus resulting in a resolution slightly lower than the nominal resolution. The two spectra are very similar, both in term of the spectral energy distribution and of the line features. The LETG spectrum is very noisy between 40 and 44Å, because of the very low effective area due to instrumental (HRC-S) absorption by the carbon K edge at 43.6Å. This instrumental absorption is clearly seen in Figs. \[MFitFig\]-\[ChemFig\] and is well represented in the response function applied to the NLTE model spectra. X-ray Variability {#XVar} ================= Although the main scope of the [[*Chandra*]{}]{}  and XMM-[[*Newton*]{}]{}  observations were to obtain the first high-resolution X-ray spectra of CAL 83, we have extended our analysis to look for X-ray flux variations during the observations. A New X-Ray Off-State {#sect:offstate} --------------------- X-ray off-states have been rarely caught, with two exceptions in April 1996 by ROSAT [@kahabka96] and in November 1999 by [[*Chandra*]{}]{}  [@greiner02]. We report here another X-ray off-state. Because of solar flare activity, our [[*Chandra*]{}]{}  HRC-S/LETG observation in August 2001 was interrupted after 35 ksec and rescheduled for completion in October 2001. This latter exposure, however, did not detect X-rays from CAL 83 in 62 ksec, indicating a X-ray off-state. We extracted counts at the expected position (obtained from the August [[*Chandra*]{}]{} LETG observation which detected CAL 83) using a circle of 14 radius (95% encircled energy for the mirror point-spread function, PSF), and extracted background counts from a concentric annulus having an area 50 times larger. We obtained 34 and 1223 counts, respectively, for a 61.55 ksec exposure. We then followed the approach of @kraft91 to determine the upper confidence limit using a Bayesian confidence level of 95%. This upper limit is 21.045 counts. Our adopted unabsorbed model spectrum (see §\[Resu\]) corresponds to $L_\mathrm{X} = 2.7 \times 10^{37}$ ergs s$^{-1}$ ($0.1 - 10$ keV), i.e., a zeroth order count rate of 0.10 ct s$^{-1}$ (i.e. an absorbed flux at Earth of $8.3 \times 10^{-12}$ ergs s$^{-1}$ cm$^{-2}$). Consequently, the X-ray off-state corresponds to an upper limit of $L_\mathrm{X} < 9.2 \times 10^{34}$ ergs s$^{-1}$ (the upper limit of the absorbed flux at Earth is $F_\mathrm{X} < 2.8 \times 10^{-14}$ ergs s$^{-1}$ cm$^{-2}$). The upper limit on $L_\mathrm{X}$ is similar to that reported by @greiner02 during the 1999 X-ray off-state. Assuming a constant bolometric luminosity, they have then infered a temperature limit, $kT\la 15$eV, that agrees well with our own estimate based on NLTE model atmospheres. Two important caveats need to be mentioned, however. First, the extraction radius probably underestimates the encircled energy since the zeroth order PSF with LETG is not exactly similar to the mirror PSF. Thus, in principle, a larger extraction region (e.g., $3\arcsec$) should contain an encircled energy fraction closer to 100%. Since the zeroth order effective area is provided for a fraction of 100%, such a larger radius could, therefore, provide a more accurate estimate of the upper limit. Nevertheless, the second issue, that is the assumption of a similar model during the on-state and the off-state for estimating the count-to-energy conversion factor, affects the determination of the upper limit much more severely. [@greiner02] discussed several mechanisms to explain the relation between X-ray off-states and the optical variability. Since they were not able to reach a definitive conclusion, we do not know for instance if the spectrum shifts to lower energies, which thus leaves a serious uncertainty on the way of establishing an upper limit to the X-ray luminosity during the off-states. The calculated upper limit, therefore, remains indicative only. The original 1996 off-state was interpreted in two contrasted ways. On one hand, [@alcock97] considered a model of cessation of the steady nuclear burning related to a drop in the accretion rate. The WD would need to cool quickly to explain the observed off-state, and the timescale of the off-state implies a massive WD. This model was based on the combination of optical and X-ray variability, and was later criticized by @greiner02 who found that optical low states were delayed by about 50 days relative to the X-ray off-states. On the other hand, [@kahabka98] argued that the off-state is caused by [*increased*]{} accretion resulting in the swelling and cooling of the WD atmosphere. Assuming adiabatic expansion at constant luminosity and a realistic accretion rate ($\dot{M}\approx 10^{-6}~M_\odot\,{\rm yr}^{-1}$), he argued from the characteristic timescale of the off-state that the WD is massive, $M_{\rm WD}\ga 1.2 M_\odot$. @greiner02 discussed extensively the X-ray off-states and optical variability of CAL 83. They believe that a cessation of nuclear burning was highly unlikely because the short turnoff time requires the WD to have a mass close to [$M_{\rm CH}$]{}, while earlier spectrum analyses (e.g., Parmar et al. 1998) suggested that the WD mass was not so high. Assuming then that the luminosity stays roughly the same, Greiner & Di Stefano investigated two ways to explain the off-states, namely an increase of the photospheric radius or absorption by circumstellar material that would in both cases shift the emission to another spectral domain. From a careful study of the variation patterns, they raised a number of issues for the two models indicating that the optical variability could not be understood with a simple model but requires some complex interaction between the photospheric expansion and the disk. CAL 83 was considered as the prototypical CBSS undergoing steady nuclear burning. However, the observation of a third off-state now demonstrates the recurrent nature of the phenomenon in CAL 83. We defer further discussion on the origin of the off-states after the spectrum analysis. Long-Term and Short-Term Time Variability {#sect:timevar} ----------------------------------------- Figure \[VarFig\] shows the light curves of CAL 83 with XMM-[[*Newton*]{}]{}  (EPIC pn: top panel, RGS: middle panel) and [[*Chandra*]{}]{} (LETG1: bottom panel) in units of flux observed at Earth. We use our final model spectrum in combination with the respective response matrices to obtain conversion factors. Note that, in the case of EPIC pn, the observation was cut into three pieces of similar length but using different filters. To avoid pile-up and optical contamination, we extracted EPIC events in an annulus ($0.2-0.8$ keV range), and used a nearby region for the background. In the case of RGS and LETG, the background was obtained from events “above” and “below” the dispersed spectrum. We used events in the range $20-37$ Å  for RGS, and $20-65$ Å  for LETG, since no signal was present outside these ranges. Although the absolute observed flux still contains some uncertainty (e.g., cross-calibration between instruments, imperfect model), similar flux levels were obtained during both the XMM-[[*Newton*]{}]{}  and [[*Chandra*]{}]{}  observations, despite their time separation of $1.25$ yr. [@greiner02] also noted that the flux observed by XMM-[[*Newton*]{}]{}  was comparable to earlier observations with ROSAT and [*Beppo*]{}SAX. Not only the flux levels were the same, but we found that the RGS and the LETG spectra were remarkably similar (Fig. \[LETGFig\]). On the other hand, we have detected relative, short-timescale variations in the X-ray light curves. Their typical timescale is much shorter than the 1.04-day binary orbital period. These flux variations can reach up to 50% of the average flux. Hardness ratios light curves do not show significant variations, suggesting that the flux variations are not due to temperature fluctuations. However, because of the limited signal-to-noise ratio due to the small effective areas, we stress that we cannot exclude that temperature effects played some role in the flux variations. Simultaneous UV and optical photometry with a similar time sampling would be necessary to study stochastic effects in the accretion process. NLTE Model Atmospheres {#NLTE} ====================== We have constructed a series of NLTE line-blanketed model atmospheres of hot WD’s with our model atmosphere program, TLUSTY, version 201. Detailed emergent spectra are then calculated with our spectrum synthesis code, SYNSPEC, version 48, using the atmospheric structure and the NLTE populations previously obtained with TLUSTY. TLUSTY computes stellar model photospheres in a plane-parallel geometry, assuming radiative and hydrostatic equilibria. Departures from LTE are explicitly allowed for a large set of chemical species and arbitrarily complex model atoms, using our hybrid Complete Linearization/Accelerated Lambda Iteration method [@NLTE1]. This enables us to account extensively for the line opacity from heavy elements, an essential feature as the observed spectrum is suggestive of strong line opacity (see Fig. \[LETGFig\]). We have implemented in our two codes several specific upgrades for computing very hot model atmospheres as well as detailed X-ray spectra. Starting with version 200, TLUSTY is a universal code designed to calculate the vertical structure of stellar atmospheres and accretion disks. With this unification, we directly benefit from the upgrades implemented to calculate very hot accretion disk annuli. This concerns in particular a treatment of opacities from highly ionized metals and Compton scattering [@AGN4]. Originally, @AGN4 implemented a simple description of metal opacities to explore the basic effect of these opacity sources on accretion disk spectra. Adopting one-level model atoms, they treated all ions, from neutrals to fully stripped atoms, of the most abundant chemical species: H, He, C, N, O, Ne, Mg, Si, S, Ar, Ca, and Fe. Therefore, only photoionization from the ground state of each ion is considered. Photoionization cross-sections, including Auger inner-shell photoionization, were extracted from the X-ray photoionization code XSTAR [@kallman00]. We made the simplifying assumption that, if an Auger transition is energetically possible, then it occurs and the photoionization results in a jump by two stages of ionization to a ground state configuration. Fluorescence and multiple Auger electron ejection arising from inner shell photoionization are neglected. To handle dielectronic recombination, we followed the description of @AGN4, based on data from @aldrovandi73, @nussbaumer83, and @arnaud92, which are used in XSTAR. These models only include bound-free opacities from the ground states. Therefore, we constructed multi-level model atoms in order to account for the effect of metal lines in our NLTE model atmospheres. We did so only for the most populated ions, and ions with very low populations are excluded. We retained the ionization and recombination data discussed above for ground state levels, and expanded the model atoms using data calculated by the Opacity Project [@IOP95; @IOP97]. It is essential to explicitly incorporate highly-excited levels in the model atoms, because X-ray lines (where the model atmosphere flux is maximal) are transitions between low-excitation (or ground state) and these highly-excited levels. To deal with the large number of excited levels, we merged levels with close energies into superlevels assuming that they follow Boltzmann statistics relative to each other (that is, all individual levels in a superlevel share the same NLTE departure coefficient). Proper summing of individual transitions is applied for transitions between superlevels. Refer to @NLTE1 and @OS02 for details on NLTE superlevels and on the treatment of transitions in TLUSTY. Table \[IonTbl\] lists the ions included in the model atmospheres, the number of explicit NLTE levels and superlevels, and the corresponding numbers of superlevels and lines. We recall here that OP neglected the atomic fine structure, and these figures refer to OP data (hence, the actual number of levels and lines accounted for is higher by a factor of a few). For each ion, we list the original publications unless the calculations were only published as a part of the OP work. Data for one-level model atoms are from [@kallman00]. The second upgrade in TLUSTY deals with electron scattering, and we have considered the effect of Compton [*vs.*]{} Thompson scattering. At high energies, Compton scattering provides a better physical description. Compton scattering is incorporated in the radiative transfer equation in the nonrelativistic diffusion approximation through a Kompaneets-like term. Details of the implementation are described in @AGN4. Through a coupling in frequencies, this increases significantly the computational cost. We thus decided to explore the differences resulting from using Compton or Thompson scattering. We performed a small number of tests comparing model atmospheres and predicted spectra which were computed using these two approaches. At the considered temperatures ($T\approx 500,000$K), the differences are very small and become visible only in the high-energy tail, above 2keV. At these energies, the predicted flux is very low and, indeed, no flux has been observed in CBSS at these energies. Moreover, the changes result in little feedback on the calculated atmospheric structure. We may therefore safely use Thompson scattering in modeling CBSS atmospheres. Detailed spectra are produced in a subsequent step with our spectrum synthesis program SYNSPEC, assuming the atmospheric structure and NLTE populations calculated with TLUSTY. Upgrades made in the TLUSTY program have been transported in SYNSPEC when necessary. A detailed line list for the soft X-ray domain was built, combining essentially two sources. The initial list was extracted from Peter van Hoof Atomic Line List[^3]. This list contains transitions between levels with measured energies[^4], and all lines thus have accurate wavelengths. These lines represent, however, a small fraction of all lines in the soft X-ray range because the energy of most highly-excited levels has not been measured from laboratory spectra. We thus complemented this initial list with a complete list of transitions from OP. The OP lines do not account for fine structure, and wavelengths are derived from theoretical energies. The expected accuracy of the theoretical wavelengths is expected to be of the order of 0.5Å  or better. This is, however, not as good as the spectral resolution achieved with [*Chandra*]{} or XMM-[*Newton*]{} spectrometers, and potential difficulties may be expected when comparing model spectra to observations. Yet the global opacity effect of these lines is important. The line list only contains lines from the explicit NLTE ions. The spectra are calculated using the NLTE populations from the TLUSTY models, and SYNSPEC does not require partition functions in this case. Finally, we examined the issue of line broadening. As discussed by @SA78, natural broadening becomes the main effect for lines of highly-charged ions, and already dominates the linear Stark broadening for lines. Therefore, we included natural broadening data when available, and we neglected Stark broadening. Departures from LTE are illustrated in Fig. \[LTEFig\] for a model atmosphere with [$T_{\rm eff}$]{}  = 550,000K, $\log g = 8.5$, and a LMC metallicity. The ionization fractions show that the NLTE model is overionized compared to the LTE model, which is a typical behavior of NLTE model stellar atmospheres where the radiation temperature is higher than the local electronic temperature. As expected, the NLTE ionization fractions return to their LTE values at depth ($\tau_{\rm Ross}\ga 2$). The overionization remains limited because the local temperatures are higher in the LTE model compared to the NLTE model as a result of stronger continuum and line absorption at depths around $10^{-3}\la\tau_{\rm Ross}\la 1$. This stronger absorption is indeed visible in the predicted emergent spectrum: the LTE predicted flux is significantly lower below 45Å. Deriving a lower surface gravity, hence a lower WD mass, would therefore be a consequence of a LTE analysis compared to the NLTE analysis (see also §\[Resu\] and Fig. \[MFitFig\]). Spectrum Analysis {#Analysis} ================= Methodology {#Method} ----------- We display in Fig. \[ResolFig\] a typical spectral region of the [[*Chandra*]{}]{}  LETG data together with the predicted spectrum of the stellar surface, that is without any instrumental convolution. This clearly illustrates that all observed features are actually blends of many metal lines, mostly from , , , , , , , and . It is thus impractical to identify these features individually. Moreover, the fact that we had to use theoretical wavelengths for many predicted lines compounds the difficulties. Therefore, the spectrum analysis cannot be based on fitting a few key features to derive the properties of CAL 83, but we rather need to achieve an overall match to the observed spectrum. To this end, we do not use any statistical criterion but we simply match the model to the data by eye. A $\chi^2$ statistics would primarily measure the line opacity that is missing in the model (a systematic effect) rather than efficiently discriminate between different values of the stellar parameters (see §\[Resu\]). Errors are analogously estimated by eye, that is, models outside the error box clearly exhibit a poor match to the observations. We feel that this approach provides conservative error estimates. We started from the stellar parameters derived by [@paerels01], and built a small grid of NLTE model atmospheres around these initial estimates covering a range in effective temperature and surface gravity, $450,000\la T_{\rm eff}\la 600,00$K and $8\la \log g\la 9$. For each set of parameters, we constructed the final model in a series of models of increasing sophistication, adding more explicit NLTE species, ions, levels, and lines. Table \[IonTbl\] lists the explicit species included in the final models. This process was helpful numerically to converge the models, but this was primarily used to investigate the resulting effect on the predicted spectrum of incorporating additional ions in the model atmospheres. We have assumed that the surface composition of CAL 83 reflects the composition of the accreted material, i.e. typical of the LMC composition. We have adopted a LMC metallicity of half the solar value [@rolleston02]. Additional models with a composition reflecting an evolved star donor with CNO-cycle processed material and enhanced $\alpha$-element abundances have also been calculated to explore the effect of changing the surface composition on the derived stellar parameters. To compare the model spectra to the [[*Chandra*]{}]{}  LETG and XMM-[[*Newton*]{}]{}  RGS data, we applied to the model a correction for the interstellar (IS) extinction, the appropriate instrumental response matrix (see §3), and a normalization factor $(R_{\rm WD}/d)^2$. Our IS model is based on absorption cross-sections in the X-ray domain compiled by [@BCMC92] for 17 astrophysically important species. Specifically, the effective extinction curve was calculated with Balucinska-Church & McCammon’s code[^5], assuming a solar abundance mix [@grevesse98]. The total extinction is then proportional to the hydrogen column density. We used the value, $N_{\rm H} = 6.5\pm 1.0~10^{20}$ cm$^{-2}$, measured by [@gansicke98] from Ly$\alpha$ [*HST*]{} GHRS observations of CAL 83. After the IS extinction and instrumental corrections, the model spectra were scaled to the [[*Chandra*]{}]{}  LETG spectrum to match the observed flux level longward of 45Å, yielding the normalization factor $(R_{\rm WD}/d)^2$. The WD radius immediately follows from the adopted distance to the LMC, $d = 50\pm 3$ kpc, based on RR Lyrae and eclipsing binaries [@alcock04; @clausen03]. Results and Discussion {#Resu} ---------------------- We present in Fig. \[MFitFig\] and \[TFitFig\] our best model fit to the RGS and LETG spectra along with the model sensitivity to [$T_{\rm eff}$]{}  and $\log g$. The best model spectrum has been normalized to match the observed LETG flux between 45 and 50Å, yielding a WD radius of $R_{\rm WD} = 7.3~10^8$cm $\approx 0.01\,R_\odot$. The same normalization factor is then applied for comparing the model spectrum to the RGS spectrum. The benefit of having a broader spectral coverage with LETG is instantly apparent for deriving an accurate model normalization. The top two panels of Fig. \[MFitFig\] generally show a satisfactory agreement between the model and the two observed spectra. Above 34Å, the match to the RGS spectrum is not as good as for the LETG spectrum, probably resulting from the difficulty of correcting the RGS data for fixed-pattern noise in this range. The general energy distribution and most spectral features are well reproduced, for instance the features at 24, 29, 30, 32, and 36Å. On the other hand, the observed absorption at $\approx$ 27Å  is not reproduced by our model. A cursory look might suggest that the model predicts an emission feature there but, overplotting the predicted continuum flux, we see that the model essentially misses line opacity around 27-28Å. In this respect, our analysis definitively demonstrates that we observe a photospheric absorption spectrum, with no obvious evidence of emission lines. Fig. \[MFitFig\] presents two models with a different surface gravity, clearly demonstrating that our data allow us to determine $\log g$. The higher flux in the low gravity model is the result of a higher ionization in the low gravity model atmosphere, decreasing the total opacity in the range between 20 and 30Å. This effect is large enough to determine the surface gravity with a good accuracy, typically $\pm~0.1$ dex on $\log g$. Fig. \[TFitFig\] shows three models with different effective temperatures. The surface gravity was adjusted to provide the closest match to the observed spectrum, and the appropriate model normalizations and WD radii were determined. The differences between the models are not as large as in the case of gravity, but we may exclude models as cool as 500,000K or as hot as 600,000K. We thus adopt [$T_{\rm eff}$]{} = 550,000 $\pm$ 25,000K ($k$[$T_{\rm eff}$]{} = 46 $\pm$ 2 eV). The WD mass and luminosity straightforwardly follow, and our results are summarized in Table \[ResuTbl\]. Uncertainties are propagated as dependent errors, formally yielding a WD mass of $M_{\rm WD} = 1.3\pm 0.3 M_\odot$. The mass-radius relation for cold WD’s predicts a WD radius of about 0.004 $R_\odot$ for such a high WD mass [@hamada61], implying that CAL 83 has a swollen atmosphere with a measured radius 2.5 times the expected value. @kato97 calculations of WD envelopes predict an even larger radius ($\approx 0.025 R_\odot$) for the derived mass and temperature, suggesting that the observed emission may emerge only from a hot cap on the WD surface. The derived bolometric luminosity, $L_{\rm bol} = 3.5\pm1.2~10^{37}$ ergs s$^{-1}$, is higher than the luminosity range given by [@gansicke98] but consistent with [@kahabka98] estimate, and corresponds to about 0.3 $L_{\rm Edd}$. The surface composition of WD’s in CBSS is yet unknown. It might reflect either the composition of the accreted material or the nucleosynthetic yields of nuclear burning close to the WD surface. Characterizing this composition would be of major interest for understanding the physical processes occurring on the accreting WD. However, our first concern is to verify how sensitive the determination of stellar parameters is with respect to the assumed composition. So far, we have assumed a composition typical of the LMC metallicity. If the donor star has already evolved off the main sequence, its surface composition might already be altered and might show enhanced abundances of $\alpha$ elements and/or CNO abundances resulting from the mixing with CNO-cycle processed material. We have therefore constructed a NLTE model atmosphere, assuming the same effective temperature and surface gravity, but all $\alpha$ elements are overabundant by a factor of 2, carbon (1/50 the LMC abundance) and oxygen (1/5) are depleted, and nitrogen (8 times) is enhanced. Fig. \[ChemFig\] compares the match of the two models to the LETG spectrum. A careful examination shows that indeed the model with the altered composition reveals some stronger absorption lines, but the changes remain small. We conclude first that the determination of the stellar parameters is little affected by the assumed surface composition (within some reasonable range; exotic compositions will not match the observations), and second that a detailed chemical composition analysis would require improved atomic data. Velocities are a further factor affecting the strength of the absorption line spectrum. In particular, we adopted a microturbulent velocity of 50 kms$^{-1}$. The ratio of this value to the sound speed is rather typical of values derived in other hot stars. Nevertheless, we varied the microturbulent velocity from low values up to values close to the sound velocity to investigate the effect of different microturbulences on the predicted spectrum. The line absorption spectrum indeed strengthened at larger microturbulent velocities, but the overall match to the LETG data is not much changed. The derived parameters remain, therefore, insensitive to the assumed microturbulent velocity. We also explored the role of an outflow in strengthening and broadening the spectral lines because of the presence of a strong wind from the WD has been proposed to regulate the accretion [@hachisu96]. We carefully examined the LETG spectrum, using different binning factors. We did not find evidence of emission or line asymmetries that could reveal an outflow. Note, however, that the spectral resolution would only allow us to detect fast outflows ($>$2000 kms$^{-1}$). For the same reason, we can only exclude very fast [*apparent*]{} rotation velocities, e.g., values close to the critical velocity ($\approx$7000 kms$^{-1}$). It is likely, however, that the inclination angle of the WD rotation axis is small because the absence of eclipses indicates that the accretion disk is almost seen face-on. Therefore, our observations cannot rule out a fast WD rotation which is advocated by @yoon04 for the WD to gain mass efficiently. The uncertainty on the IS transmission might be the most severe issue. Our IS model (see §\[Method\]) predicts a transmission as low as 6% at 50Å increasing up to about 60% at 20Å. A change in the assumed hydrogen column density toward CAL 83 might thus significantly affect the results. Fortunately, this column was measured accurately from Ly$\alpha$ observations by [@gansicke98], $N_{\rm H} = 6.5\pm 1.0~10^{20}$ cm$^{-2}$. We use their error range to explore the resulting changes. Thanks to the relatively tight range, the spectral shape between 25 and 40Å  is little changed, and stays within changes predicted from the adopted errors on [$T_{\rm eff}$]{}  and $\log g$. Therefore, the error on the column density primarily translates into different normalization factors, i.e. a range of WD radii. A higher column density yields a larger WD radius, hence a larger WD mass (which should not exceed [$M_{\rm CH}$]{}). Conversely, a smaller column density yields a lower WD radius, hence a lower luminosity. Since the measured luminosity is close the boundary of stable steady-state nuclear burning [@iben82; @vdH92], it is unlikely that the WD luminosity is much smaller. Therefore, the actual value of the IS density column likely is within an even tighter range. The uncertainty on the WD radius listed in Table \[ResuTbl\] remains relatively small, because it only combines the uncertainty on the normalization factor and on the LMC distance which are well determined. Fully accounting for [@gansicke98] error bar on $N_{\rm H}$ would translate to unrealistically large errors on the WD radius and on the WD mass. We revisit now the issue of the X-ray off-states in the light of our result that the WD has a high mass. @kato97 calculations of WD envelopes showed that the X-ray turnoff time after the cessation of nuclear burning could be very short for massive WDs. An observed turnoff time as short as 20 days [@kahabka98] implies $M_{\rm WD} \ga 1.35 M_\odot$. [@gansicke98] noted that CAL 83 has a low luminosity for a CBSS and is found close to the stability limit of steady burning, which is thus consistent with the idea of unstable burning and might also be related to our discovery of short term variability of the X-ray flux. Although the actual process responsible of the off-states cannot be definitively established, the characteristic timescale supports the idea of a massive WD in CAL 83. Conclusions {#Conclu} =========== Table \[ResuTbl\] summarizes our results. Our spectroscopic analysis indicates that CAL 83 contains a massive WD, $M_{\rm WD} = 1.3\pm 0.3 M_\odot$. Obviously, the upper limit is well over [$M_{\rm CH}$]{}, but a more interesting issue to tie CBSS and SN Ia progenitors is the lower limit for the mass. Because low-mass models do not provide a match to the observed spectrum that is as good as the fits achieved with high-mass models, we have concluded that $M_{\rm WD} > 1.0~M_\odot$ is a robust lower limit for the WD mass. Nevertheless, we need to emphasize here that this conclusion depends on our current description of the soft X-ray opacities, particularly between 20 and 40Å. We believe that the OP data already provide a good description and, although progress in this regard would be very helpful, we do not expect changes drastic enough to modify our conclusion. The issue of IS extinction is potentially more serious, but we have argued that we have an accurate estimate of the hydrogen column density toward CAL 83. In addition, our model is consistent with the WD surface having a hydrogen-rich composition with LMC metallicity. Finally, our analysis does not reveal any evidence of an outflow from the WD. We have reported a third X-ray off-state, showing that CAL 83 is a source undergoing unstable nuclear burning. This is consistent with its low luminosity. The short timescale of the off-states (about 50 days) provides a supporting evidence that CAL 83 WD is massive ($M_{\rm WD}\ga 1.35 M_\odot$). A better characterization of the off-states is required to definitively establish the mechanism(s) responsible of the off-states. Moreover, we confirm that the X-ray luminosity is fairly constant on a long-term basis during on-states. However, on the other hand, we found variations up to 50% of the soft X-ray luminosity in very short times (less than 0.1 orbital period). Simultaneous X-ray and optical photometry covering continuously a duration longer than a full orbital period would be very valuable to study stochastic accretion processes. Within the model of SN Ia progenitors proposed by [@hachisu96], our results would place CAL 83 in a late stage, after the strong accretion and wind phase when the accretion rate drops below the critical rate for sustaining steady nuclear burning. Our results therefore make CAL 83 a very likely candidate for a future SN Ia event. We plan to conduct a similar NLTE analysis of the few other CBSS already observed with [[*Chandra*]{}]{}  LETG spectrometer to further support our results and quantitatively study differences between these objects. 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J., Bhattacharya, D., Nomoto, K., & Rappaport, S. A. 1992, , 262, 97 Webbink, R. F. 1984, , 277, 355 Weidemann, V. 1987, , 188, 74 Whelan, J., & Iben, I. Jr. 1973, , 186, 1007 Yoon, S-C., & Langer, N. 2003, , 412, L53 Yoon, S-C., Langer, N., & Scheithauer, S. 2004, , 425, 217 [llrlll]{} [[*Chandra*]{}]{}  & HRC-S/LETG & 52.3 ksec & 1999-11-29 06:33 & 1999-11-29 21:27 & Off-state\ [[*Chandra*]{}]{}  & ACIS-S & 2.1 ksec & 1999-11-30 14:50 & 1999-11-30 15:47 & Off-state\ XMM-[[*Newton*]{}]{}  & RGS & EPIC& 45.1 ksec & 2000-04-23 07:34 & 2000-04-23 20:04 &\ [[*Chandra*]{}]{}  & HRC-S/LETG & 35.4 ksec & 2001-08-15 16:03 & 2001-08-16 02:10 &\ [[*Chandra*]{}]{}  & HRC-S/LETG & 61.6 ksec & 2001-10-03 11:35 & 2001-10-04 05:12 & Off-state\ [lrrrl]{} & 9 & 80 & 172 &\ & 1 & 1 & &\ \[2mm\] & 15 & 15 & 105 &\ & 1 & 1 & &\ \[2mm\] & 9 & 19 & 27 & @ADOC7\ & 15 & 15 & 105 &\ & 1 & 1 & &\ \[2mm\] & 5 & 14 & 28 & @ADOC9\ & 9 & 19 & 27 & @ADOC7\ & 15 & 15 & 105 &\ & 1 & 1 & &\ \[2mm\] & 1 & 1 & &\ & 8 & 20 & 60 & @ADOC9\ & 13 & 41 & 107 & @ADOC7\ & 15 & 15 & 105 &\ & 1 & 1 & &\ \[2mm\] & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ & 5 & 14 & 28 & @ADOC9\ & 9 & 19 & 27 & @ADOC7\ & 15 & 15 & 105 &\ & 1 & 1 & &\ \[2mm\] & 1 & 1 & &\ & 12 & 54 & 284 & @ADOC9\ & 15 & 109 & 565 & @ADOC7\ & 1 & 1 & &\ \[2mm\] & 22 & 489 & 11900 & @ADOC11\ & 24 & 313 & 7531 & @IOP95\ & 16 & 184 & 2599 & @ADOC14\ & 12 & 54 & 284 & @ADOC9\ & 9 & 19 & 27 & @ADOC7\ & 1 & 1 & &\ \[2mm\] & 20 & 481 & 13665 & @IOP95\ & 23 & 512 & 17966 & @IOP95\ & 23 & 587 & 16253 & @ADOC11\ & 23 & 301 & 7181 & @IOP95\ & 13 & 184 & 2599 & @ADOC14\ & 12 & 54 & 284 & @ADOC9\ & 9 & 19 & 27 & @ADOC7\ & 1 & 1 & &\ \[2mm\] & 1 & 1 & &\ & 23 & 197 & 2877 & @IOP95\ & 19 & 287 & 6836 & @IOP95\ & 20 & 514 & 15943 & @IOP95\ & 22 & 546 & 21675 & @IOP95\ & 26 & 617 & 19248 & @ADOC11\ & 19 & 355 & 9755 & @IOP95\ & 16 & 184 & 2599 & @ADOC14\ & 1 & 1 & &\ \[2mm\] & 1 & 1 & &\ & 1 & 1 & &\ & 24 & 203 & 3098 & @IOP95\ & 15 & 308 & 8043 & @IOP95\ & 26 & 544 & 17703 & @IOP95\ & 22 & 629 & 27835 & @IOP95\ & 24 & 730 & 24897 & @ADOC11\ & 21 & 381 & 11511 & @IOP95\ & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ \[2mm\] & 1 & 1 & &\ & 1 & 1 & &\ & 26 & 253 & 4797 & @ADOC19\ & 17 & 52 & 285 & @IOP95\ & 30 & 211 & 3409 & @IOP95\ & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ & 1 & 1 & &\ \[2mm\] [ll]{} Effective Temperature & $T_{\rm eff} = 5.5\pm 0.25~10^5$ K\ Surface Gravity & $\log g = 8.5\pm 0.1$ (cgs)\ WD Radius & $R_{\rm WD} = 0.01\pm 0.001~R_\odot$\ WD Luminosity & $L_{\rm WD} = 9\pm 3~10^3~L_\odot$\ WD Mass & $M_{\rm WD} = 1.3\pm 0.3~M_\odot$\ [^1]: http://cxc.harvard.edu/ciao/ [^2]: http://cxc.harvard.edu/cal/Letg/Hrc\_bg/ [^3]: http://www.pa.uky.edu/$\sim$peter/atomic/ [^4]: from the NIST Atomic Spectra Database at http://physics.nist.gov/cgi-bin/AtData/main\_asd [^5]: Available at http://cdsweb.u-strasbg.fr/viz-bin/Cat?VI/62

--- abstract: 'We demonstrate radiation-pressure cavity-cooling of a mechanical mode of a micromirror starting from cryogenic temperatures. To achieve that, a high-finesse Fabry-Pérot cavity ($F\approx 2200$) was actively stabilized inside a continuous-flow $^{4}$He cryostat. We observed optical cooling of the fundamental mode of a $50~\mu$m$\times50~\mu$m$\times5.4~\mu$m singly-clamped micromirror at $\omega_m=3.5$ MHz from 35 K to approx. 290 mK. This corresponds to a thermal occupation factor of $\langle n\rangle\approx 1\times10^4$. The cooling performance is only limited by the mechanical quality and by the optical finesse of the system. Heating effects, *e.g.* due to absorption of photons in the micromirror, could not be observed. These results represent a next step towards cavity-cooling a mechanical oscillator into its quantum ground state [@Groeblacher2008a].' author: - Simon Gröblacher - Sylvain Gigan - 'Hannes R. Böhm' - Anton Zeilinger - Markus Aspelmeyer title: 'Radiation-pressure self-cooling of a micromirror in a cryogenic environment\' --- Optomechanical interactions in high-finesse cavities offer a new promising route for the ongoing experimental efforts to achieve the quantum regime of massive mechanical systems [@LaHaye2004; @Schwab2005]. They allow to cool mechanical degrees of freedom of movable mirrors via radiation-pressure backaction [@Braginsky2002], in principle even into their quantum ground state [@Marquardt2007; @Wilson-Rae2007; @Genes2007]. The working principle of this cooling method has been demonstrated in a series of recent experiments [@Gigan2006; @Arcizet2006b; @Schliesser2006; @Corbitt2007]. Ground-state cooling will eventually require to realize the scheme in a cryogenic environment. Optomechanical feedback cooling [@Mancini1998; @Cohadon1999; @Kleckner2006b; @Arcizet2006a; @Poggio2007], another quantum limited strategy [@Courty2001; @Vitali2002; @Genes2007], has recently taken this step by demonstrating cooling of a 3.8 kHz resonator mode from a starting temperature of 2 K to an effective noise temperature of 2.9 mK (or $\langle n\rangle\approx 2.1\times10^4$) [@Poggio2007]. To achieve and surpass such a performance for radiation-pressure backaction schemes requires stable operation of a high-finesse cavity inside a cryostat [@Tittonen1999] and sufficiently strong optomechanical coupling [@Gigan2006; @Arcizet2006b; @Schliesser2006; @Corbitt2007]. Here we report the combination of these requirements in a single experiment using a high-reflectivity micromechanical resonator. We observe radiation-pressure backaction cooling of the fundamental mode of the micromirror at $\omega_m/2\pi=557$ kHz from 35 K to 290 mK (or $\langle n\rangle\approx 1\times10^4$), limited only by the optical finesse of the cavity and by the mechanical quality of the micromirror.\ How does radiation-pressure cooling work? The basic setup comprises an optical cavity of frequency $\omega_c$, pumped by a laser at frequency $\omega_l$, that is bounded by a mechanical oscillator of resonance frequency $\omega_m$. By reflecting photons off the mechanical resonator, in our case a movable micromirror, the intracavity field exerts a radiation-pressure force on the mechanical system. Detuning of the optical cavity ($\Delta=\omega_c-\omega_l\neq 0$) can result in a net positive ($\Delta<0$) or negative ($\Delta>0$) energy transfer from the radiation field to the mechanical oscillator, corresponding to either heating or cooling of the mechanical mode. There are different views to understand the cooling effect. Considering the full dynamics of the system, radiation-pressure forces in a detuned cavity behave as a viscous force that modifies the mechanical susceptibility [@Braginsky2002; @Metzger2004; @Arcizet2006c]. Cooling occurs as a consequence of the delayed (retarded) force response to thermal fluctuations of the mechanical resonator, which is caused by the finite cavity decay rate $\kappa$. It is worth noting that retardation-based optomechanical cooling is not restricted to radiation pressure and its principle was in fact for the first time demonstrated using photothermal forces [@Metzger2004]. Going beyond (semi-)classical descriptions, a full quantum treatment [@Paternostro2006; @Wilson-Rae2007; @Vitali2007; @Marquardt2007] can provide an interesting interpretation of the cooling effect as quantum state transfer between two oscillators, *i.e.* the cavity field and the mechanical mode [@Zhang2003]. This is related to the thermodynamic analogy, by which an entropy flow occurs from the thermally excited mechanical mode to the low-entropy laser field. Finally, the comparison of the photon-phonon interaction with three-wave mixing leads to the intuitive picture of sideband-cooling [@Marquardt2007; @Wilson-Rae2007], as is well known from laser-cooling of atoms and ions [^1].\ Our mechanical objects are oscillating micromirrors of high reflectivity that consist solely of a dielectric Bragg-mirror coating [@Boehm2006]. Compared to our previous work [@Gigan2006] we have used a different coating material to achieve both higher reflectivity and lower inherent absorption. This allowed us to increase the radiation-pressure coupling and to avoid residual photothermal effects. For the fabrication process we start from a high-reflectivity coating ($R > 0.9999$) made out of 40 alternating layers of $Ta_2O_5$ and $SiO_2$ deposited on silicon. We used reactive ion etching to define the resonator shape and selective dry etching of the substrate to free the structures. All mechanical resonators form singly clamped cantilevers with a thickness of $5.4~\mu$m, a width of $50~\mu$m and a length between $50~\mu$m and $300~\mu$m (fig. \[fig1\]b). We found mechanical quality factors $Q\approx 1000 - 3000$ and reflectivities of $R > 0.9999$.\ ![Experimental Scheme. (a) The pump beam is spatially and spectrally filtered in a ring cavity locked to the laser frequency. After phase modulation using an electro-optic modulator (EOM), for Pound-Drever-Hall (PDH) locking, the pump is injected into the micromirror Fabry-Pérot (FP) cavity, which is mounted inside a $^4$He cryostat. The beam reflected from the FP cavity is detected behind a polarizing beam splitter (PBS). The PDH signal is obtained by demodulating the detected signal by the EOM driving frequency and is used for actively stabilizing the cavity length and for monitoring the dynamics of the mechanical mode. Alignment is done via a CCD camera. (b) SEM picture of a group of micromirrors. (c) Cavity mounting inside the cryostat (see text).[]{data-label="fig1"}](Figure1.jpg){width="80.00000%"} The full experimental setup is sketched in fig. \[fig1\]a. We use the micromirror as an end mirror in a high-finesse Fabry-Pérot (FP) cavity, which is pumped by a an ultrastable Nd:YAG laser operating in continuous-wave mode at a wavelength of 1064 nm. The input coupler of the FP cavity is a concave massive mirror (radius of curvature: $25$ mm; reflectivity at 1064 nm: $0.9993$) that is attached to a ring piezo (PZT) in order to actively modify the cavity length. We chose the length $L$ of the cavity slightly shorter than for the semi-concentric case ($L=25$ mm) in order to have a stable cavity and a small cavity-mode waist $w_0$ on the micromirror ($w_0\approx 10~\mu$m). The cavity is mounted inside a continuous-flow $^{4}$He cryostat (fig. \[fig1\]c). The input coupler is attached to the outer shield of the cryostat and therefore always maintains at room temperature. The silicon wafer that holds the micromirrors is glued on a sample holder that is in thermal contact with the cryostat cold finger. A 3-axis translation stage allows precise positioning of the micromirror on the chip with respect to the footprint of the cavity beam. We monitor both position and size of the cavity mode via an external imaging system. In operation, the cryostat is first evacuated to $10^{-6}$ mbar. Cryogenic cooling is achieved by a continuous flow of helium in direct contact with the cold finger. The additional cryogenic freeze out reduces the pressure to below $3\times 10^{-7}$ mbar. On cooling the cryostat from room temperature to approx. 6 K (measured temperature at the cold finger), the thermal contraction of the cavity ($1-2$ mm in total) can be compensated by the 3-axis translation stage. The temperature of the sample holder is monitored via an additional sensor directly attached to it. For a measured cold-finger temperature of 6 K we observe a sample holder temperature of approx. 20 K and an actual sample temperature of 35 K, which we infer from the calibrated power spectrum of the micromirror motion as mode temperature at zero optical detuning (see below). We attribute the temperature gradient to heating of the sample by blackbody radiation from the input coupler, which is kept at 295 K only a few millimeters away from the sample, in combination with finite thermal conductivity between sample, sample holder and cold finger. Both at room temperature and at cryogenic temperatures we observe stable locking of the cavity for a finesse of up to $8000$. We achieve typical mode matching efficiencies into the cavity of $80\%$.\ ![Modified micromirror dynamics due to cavity detuning. Shown is the micromirror’s effective frequency $\omega_{eff}/2\pi$ and effective damping $\gamma_{eff}$ both at room temperature and at 35 K for various detuning values at a laser power of 1 mW. Maximal cooling is obtained approximately at a detuning of $\omega_{m}$, where the net phonon transfer to the optical field is maximized. The solid lines are fits to the data based on the semi-classical model for radiation-pressure backaction (see text).[]{data-label="fig2"}](Figure2.jpg){width="60.00000%"} ![Radiation-pressure backaction. The data follow the curve from the top-left to the bottom-left. The solid lines are fits to the data based on the semi-classical model for pure radiation-pressure backaction (see text). The dotted lines show the expected behavior for bolometric (photothermal) forces when using the same parameters. Even at low temperature a clear deviation from photo-thermal behavior is observed and the data is well described by radiation-pressure effects.[]{data-label="fig3"}](Figure3.jpg){width="60.00000%"} To observe the desired backaction cooling we monitor the dynamics of the different eigenmodes of the micromirror vibration by measuring its displacement power spectrum $S_x(\omega)$ [@Paternostro2006]. This is done by analyzing the Pound-Drever-Hall (PDH) signal in the light backreflected from the FP cavity [@Tittonen1999; @Arcizet2006a; @Gigan2006], a method which is based on the interference of phase-modulated side bands of the pump laser [@Gigan2006; @Arcizet2006b]. The main idea is that the PDH error signal of a locked cavity is proportional to the cavity length. While we use the low-frequency part of the PDH signal as an error signal to actively stabilize the cavity length to the wanted detuning $\Delta$, the high frequency part is directly proportional to the displacement power spectrum $S_x$ of the micromirror [@Paternostro2006] [^2]. One can evaluate the effective mode temperature via the area of the measured power spectrum as $T_{eff}=\frac{m\omega_0^2}{k_B}\langle x^2\rangle$ ($m$: effective mass at the probing point, $\omega_0$: mode frequency, $k_B$: Boltzmann’s constant, $\langle x^2\rangle=\int_{-\infty}^{+\infty}d\omega S_x(\omega)$).\ Backaction cooling is accompanied by a modified dynamics of the mechanical mode, specifically by a shift both in resonance frequency $\omega_{eff}$ and in damping $\gamma_{eff}$. This can be used to identify the nature of the backaction force: for a known effective mass and optical pump power, radiation-pressure forces are uniquely determined by the time dependence of the cavity decay and can therefore be distinguished from forces of dissipative nature such as photothermal forces [@Marquardt2007]. We obtain these effective values directly via the power spectrum $S_x$, which, for a classical harmonic oscillator, is given by $$S_x(\omega)=\frac{4 k_B T \gamma_0}{\pi m}\frac{1}{(\omega_{eff}^2-\omega^2)^2+4 \gamma_{eff}^2\omega^2}, \label{eq:Sx}$$ where $\gamma_0$ is the mechanical damping of the unperturbed mechanical oscillator, *i.e.* the damping at zero detuning. To minimize radiation pressure effects we used very low input power ($\approx 30~\mu$W) and probed the mode at a point of high effective mass, *i.e.* close to a node of vibration. The values for $\gamma_{eff}$ and $\omega_{eff}$ were obtained from fits to the measured power spectra using eq. (\[eq:Sx\]).\ We first confirmed that our optomechanical system is dominated by radiation-pressure backaction. For that purpose, we monitor the modified dynamics of the mechanical mode of a micromirror and compare it with the theoretical predictions for radiation-pressure effects. The results for various cavity detunings are shown in fig. \[fig2\]. The solid lines are fits to the data using the semi-classical approach described in [@Arcizet2006c]. We obtain a fitted cavity finesse $F=2300$ and a fitted effective mass of $m=125$ ng. These values are consistent with our independent estimate of $F=2800\pm600$ and $m=(110\pm30)$ ng (obtained from $S_{ref}$) [^3]. Note that the finesse is measured by slowly scanning the cavity length. The corresponding measurement uncertainty arises from mechanical vibrations of the setup. We also performed a measurement on the mirror at 35 K (fig. \[fig2\]), however with a reduced detuning range (for technical reasons the full detuning range was not available at low temperature). Again, the fit values of $F=2200$ and $m=40$ ng are consistent with our estimates of $F=2800\pm800$ and $m=(30\pm10)$ ng and therefore confirm the radiation-pressure nature of the interaction. In contrast to radiation-pressure forces, photothermal forces are always subject to an exponential retardation due to the dissipative nature of the force and therefore produce a different dynamics on detuning [@Marquardt2007]. We have used the same parameters to simulate the expected behavior resulting from such a force (fig. \[fig3\]), which can clearly not serve as an explanation for our data.\ ![Radiation-pressure cooling in a cryogenic high-finesse cavity. Shown are effective temperature $T_{eff}$ and the effective damping $\gamma_{eff}$ in a detuned cavity for various laser powers. Different laser powers correspond to different symbols. Values of detuning (in units of $\omega_m$) are encoded in color. Starting from cryogenic temperatures (the given cryostat temperature is the measured sample holder temperature) we observe backaction cooling down to 290 mK (or $\langle n\rangle\approx1\times10^4$). The cooling performance is not limited by heating but by optical finesse and mechanical quality factor of the optomechanical system.[]{data-label="fig4"}](Figure4.jpg){width="60.00000%"} Finally, we demonstrate radiation-pressure backaction cooling in a cryogenic cavity. Figure \[fig4\] shows measurements performed on the fundamental mechanical mode at $\omega_m=2\pi\times 557$ kHz of the micromirror. For each detuning and optical power level we obtained $\langle x^2\rangle$, $\omega_{eff}$ and $\gamma_{eff}$ directly from the fits to the measured displacement spectrum $S_x$. The effective mass $m_{eff}$ is obtained as described in the previous paragraph by fitting the data sets of same optical power (at a given cryostat temperature) using a semi-classical approach to radiation-pressure backaction. The effective temperature is obtained by plotting $m_{eff}\cdot\omega_{eff}^2\cdot\langle x^2\rangle$ normalized to the value obtained at zero detuning at room temperature (295 K). When cooling the cavity down to a sample holder temperature of 20 K we find a measured mode temperature at zero detuning (corresponding to $\gamma_0^{35K}=2\pi\times 269$ Hz) of approximately 35 K. On detuning, the mode temperature decreases as expected for both starting temperatures. For a given laser power the effective mode temperature decreases with increasing detuning until $\Delta\approx \omega_m$, where the cooling is optimal. The effective temperature increases again on further increasing the detuning. When starting from room temperature we observe a minimum temperature of approximately 17 K at an input laser power of 3.7 mW. Starting with a cryogenic cavity we observe a minimum mode temperature of approximately 290 mK for 14 mW laser power. This corresponds to a thermal occupation factor of $\langle n\rangle\approx1\times10^4$.\ The cooling performance is not limited by residual heating effects. In the ideal (semi-)classical case $T_{eff}\approx T_{0}\frac{\gamma_0}{\gamma_{eff}}$ (for $\omega_{eff}\ll\gamma_{eff}$ and $T_0$: environment temperature), as one can see from integrating eq. (\[eq:Sx\]) and by using the equipartition theorem. We observe this behavior as linear dependence on the double-logarithmic scale of fig. \[fig4\]. In case of heating, *e.g.* by absorption of photons, one would expect a dependence of the mode temperature on the laser power even for the same effective damping $\gamma_{eff}$. In other words, data points taken at different laser powers would not fall on the same line. The fact that we observe no deviation from the linear dependence for increasing laser power indicates that no significant heating of the mode occurs. We should also note that our experimental parameters ($F=2200, \omega_m=3.5\times10^6$) fulfill the threshold condition for ground-state cooling, because $\omega_{m}/\kappa=0.2>1/\sqrt{32}$ [@Wilson-Rae2007]. Our present cooling performance is only limited by the initial temperature $T_0$ of the environment, *i.e.* the performance of the cryostat, and by the achieved damping ratio $\frac{\gamma_0}{\gamma_{eff}}$. Future improvements will have to include a further reduction of $T_0$, *e.g.* by including a radiation shield to protect the sample from blackbody radiation, a decrease in $\gamma_0$, *i.e.* a larger mechanical Q, and an increase of optical intracavity power, in particular via an increase of finesse. We have demonstrated radiation-pressure backaction cooling of a micromirror in a high-finesse cavity at cryogenic temperatures. Starting from a sample temperature of approximately 35 K we achieve an effective mode temperature of 290 mK ($\langle n\rangle\approx 1\times 10^4$), limited only by the micromirror’s mechanical quality factor and by its optical reflectivity. We consider this a next step towards exploiting the rich structure promised by optomechanical systems when entering the mechanical quantum regime [@Bose1997; @Pinard2005b; @Pirandola2006; @Vitali2007]. We believe that the combination of cryogenic cooling with (active or passive) feedback techniques [@Naik2006; @Poggio2007; @Thompson2007] will be an essential step to achieve this goal. We are grateful to J. B. Hertzberg and K. Schwab for valuable support in sample preparation, and to K. Gugler, T. Paterek, M. Paternostro and D. Vitali for discussion. We acknowledge financial support by the FWF (Projects P19539-N20 and L426-N20), by the IST funded Integrated Project QAP (Contract 015846) of the European Commission, by the City of Vienna and by the Foundational Questions Institute fqxi.org (Grant RFP1-06-14). S. Gröblacher is recipient of a DOC-fellowship of the Austrian Academy of Sciences. [29]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} ****, (). , , , , ****, (). , (). , ****, (). , , , , ****, (). , , , , ****, (). , , , , , (). , , , , , , , , , , ****, (). , , , , , ****, (). , , , , , ****, (). , , , , , , , , , , , ****, (). , , , ****, (). , , , ****, (). , ****, (). , , , , , , , , , , ****, (). , , , , ****, (). , , , ****, (). , , , , ****, (). , , , , , , , , , ****, (). , ****, (). , , , , ****, (). , , , , , , ****, (). , , , , , , , , , ****, (). , , , ****, (). , , , , , , , , , ****, (). , , , ****, (). , , , , , , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , (). [^1]: Note that in our case radiation pressure originates from the reflection of photons off the mirror surface and not from absorption and re-emission as is the case in conventional laser cooling. Still, the cooling mechanism of both schemes is completely analogous. [^2]: The ratio between PDH power spectrum and displacement power spectrum $S_x$ depends on the cavity detuning $\Delta$. We can eliminate the unwanted detuning dependence by normalizing $S_x$ via a reference signal of a known constant displacement power spectrum $S_{ref}$ that is generated by frequency modulation of the pump laser. In addition, $S_{ref}$ is an absolute calibration of the effective mass of the mechanical oscillator, as is outlined in detail *e.g.* in  [@Gigan2006]. [^3]: The reduction in finesse compared to the value of $8000$ is due to our choice of the optimal working point on the cantilever close to the tip of the micromirror, where edge diffraction increased the losses in the cavity.

--- abstract: 'We write down and apply the linearized fluid and gravitational equations consistent with pseudo-Newtonian simulations, whereby Newtonian hydrodynamics is used with a pseudo-Newtonian monopole and standard Newtonian gravity for higher multipoles. We thereby eliminate the need to use mode function matching to identify the active non-radial modes in pseudo-Newtonian core-collapse supernova simulations, in favor of the less complex and less costly mode frequency matching method. In doing so, we are able to measure appropriate boundary conditions for a mode calculation.' author: - 'John Ryan Westernacher-Schneider' bibliography: - 'fluidbib.bib' title: 'Consistent perturbative modeling of pseudo-Newtonian core-collapse supernova simulations' --- Introduction ============ There is increasing attention to gravitational wave asteroseismology of core-collapse supernovae (CCSNe) from a theoretical perspective (eg. [@murphy2009model; @muller2013new; @cerda2013gravitational; @fuller2015supernova; @torres2017towards; @morozova2018gravitational; @torres2018towards; @westernacher2018turbulence; @torres2019universal; @vartanyan2019temporal; @sotani2019dependence; @westernacher2019multimessenger; @warren2019constraining]). One challenge is identifying which hydrodynamical modes of the system are producing gravitational wave (GW) emission in simulations. This requires modeling in post-process. One strategy is to use simulation snapshots as background solutions for a perturbative mode calculation. Once the perturbative mode spectrum is obtained, a matching procedure is necessary to determine which modes are actually active in the simulation. A mode frequency matching procedure has been used frequently [@torres2017towards; @morozova2018gravitational; @torres2018towards], whereby the evolution of perturbative mode frequencies are overlaid on simulation gravitational wave spectrograms, and then matching is judged by frequency coincidence over time. However, some mode classes (particularly $p$-modes) tend to have frequencies which are roughly constant multiples of each other over time, with neighboring modes having frequencies being $\sim 5\,$-$10\%$ away. Frequency mismatches between simulations and perturbative calculations can arise due to the use of different equations of motion in the simulations versus those used in the perturbative calculation. For example, in [@torres2017towards; @torres2018towards] the general relativistic hydrodynamic equations were used in the perturbative calculation, with either no metric perturbations [@torres2017towards] or a subset of possible metric perturbations [@torres2018towards]. Their simulations correspondingly use general relativistic hydrodynamics and a spatially-conformally flat metric approximation for spacetime. As another example, [@morozova2018gravitational] uses for their perturbative equations general relativistic hydrodynamics with either no metric perturbations or only lapse perturbations, supplemented with a Poisson equation to solve for the lapse perturbation. Their simulations on the other hand use Newtonian hydrodynamics and pseudo-Newtonian gravity. The ensuing frequency mismatches generated by the use of different equations may result in mode misidentification during a mode frequency matching procedure, particularly due the absence of the lapse function in the hydrodynamic fluxes in the simulations. In [@westernacher2018turbulence; @westernacher2019multimessenger] a mode function matching procedure was followed instead. This entails comparing the mode functions computed perturbatively with the velocity data in the simulation. As in [@morozova2018gravitational], the simulations were pseudo-Newtonian, whereas the perturbative calculation used the general relativistic hydrodynamic equations in the Cowling approxmation (no metric perturbations), with the lapse function being the only non-zero metric component. The mode function matching procedure produced convincing mode identification despite the use of perturbative equations that are not consistent with the simulation, because neighboring mode functions have distinct enough morphology that the best-fitting mode function is clearly superior to the next-best-fitting one (provided the mode’s excitation is large enough with respect to stochastic or nonlinear motions). A frequency mismatch between the best-fitting mode functions and the simulation frequencies of order $\sim 15\%$ was observed in [@westernacher2018turbulence; @westernacher2019multimessenger], which is large enough to have caused a mode misidentification via mode frequency matching. During targeted modeling of the next galactic core-collapse supernova, this would have produced incorrect inferences about the source. Furthermore, mode misidentification in simulations can misinform analytic or semi-analytic modeling efforts of these systems. However, mode function matching is considerably more complex and expensive than mode frequency matching. It is more complex because frequency masks have to be determined in order to apply appropriate spectral filtering on the velocity data from the simulation. It is more expensive because the entire fluid data in the system must be saved with sufficient temporal cadence such that the spectral resolution allows a clean Fourier extraction of individual mode activity. In [@westernacher2018turbulence; @westernacher2019multimessenger] axisymmetric simulations were performed, which alleviates the storage issue, but one wishes to identify modes in fully 3D simulations as well. Large searches of the CCSN progenitor parameter space would be hampered by the need to perform mode function matching. It would therefore be desirable to use the perturbative equations that are consistent with simulations, which, removing the need for the expensive mode function matching procedure. In this work, we write down and apply the consistent linearized equations appropriate for pseudo-Newtonian codes such as `PROMETHEUS/VERTEX` [@rampp2002radiation; @muller2010new; @muller2012new; @muller2013new; @muller2014new], `FLASH` [@fryxell2000flash; @dubey2009extensible], `FORNAX` [@skinner2019fornax], `CHIMERA` [@bruenn2018chimera]. As long as one does not solve for radial modes, these equations are simply the standard Newtonian ones. During testing we identify and correct a mistreatment of the boundary conditions [@morozova2018gravitational; @westernacher2018turbulence; @westernacher2019multimessenger] for the gravitational potential perturbation. We are able to reproduce the quadrupolar mode frequencies of an equilibrium star evolved using `FLASH`. When applied to a CCSN simulation, we find the best-fitting mode functions have the correct frequency (i.e. agreeing with the simulation) at the $2\%$ or sub-$1\%$ level, depending on the boundary conditions used. We also perform a residual test with the spherically-symmetric Euler equation, showing that the state of hydrostatic equilibrium (assumed in the perturbative calculation) is satisfied only at the $\sim 5\%$ level, whereas the terms coming from a time-dependent or non-steady ($v\neq 0$) background solution are negligible. This serves as a cautionary note for future applications of this perturbative modeling, but also suggests that including a time-dependent or non-steady background would not affect the calculation significantly. We find that the outer boundary condition on the fluid variables yielding the most precise matching with simulations (sub-$1\%$ level) is that of [@torres2017towards], where the radial displacement is taken to vanish at the shockwave location. The agreement is so striking that we are tempted to conclude that this is the physically correct boundary condition in the early post-bounce regime we are considering. Note that the consistent perturbative modeling of pseudo-Newtonian simulations that we present here does *not* answer the question of whether such simulations yield the correct mode excitation. Previously in [@westernacher2018turbulence; @westernacher2019multimessenger], it was shown that, *if* the perturbative modeling does not use the linearization of the equations being simulated, then mode function matching is necessary to correctly identify the active modes in a simulation. In this work, we simply use the consistent linearization to show that correct identification of active modes in a simulation is possible with mode frequency matching alone, and interesting physics can then be extracted (such as the physically correct boundary conditions for the perturbations). The question of whether the mode excitation itself is correct in pseudo-Newtonian simulations is left for future work. Previous studies indicate that mode frequencies are systematically shifted with respect to general relativity (see e.g. [@mueller2008exploring]), and overestimated in particular [@mueller2008exploring; @westernacher2018turbulence; @westernacher2019multimessenger], but one cannot know for sure without directly identifying the excited modes in each case (e.g. by mode function matching). The pitfalls found in [@westernacher2018turbulence; @westernacher2019multimessenger] in using pseudo-Newtonian simulations to study mode frequencies were anticipated clearly in [@mueller2008exploring]. We give a brief summary of the results of [@westernacher2018turbulence; @westernacher2019multimessenger] in Sec. \[sec:summary\]. We described our methods in Secs. \[sec:methods\] & Appendix \[sec:bcs\], and discuss our results in Sec. \[sec:results\]. Tests are presented in Appendix \[sec:tests\]. We use geometric units $G=c=1$ throughout, unless units appear explicitly. Simulations and background information {#sec:summary} ====================================== We analyze the non-rotating $20\,M_{\odot}$ zero-age main sequence mass CCSN progenitor presented previously in [@westernacher2018turbulence; @westernacher2019multimessenger]. It was simulated in axisymmetry using `FLASH` [@fryxell2000flash; @dubey2009extensible] until $\sim 100$ ms post-bounce. Mild excitation of hydrodynamic modes are excited at bounce, the amplitude of which is expected to be artificially enhanced due to asymmetries introduced during collapse by the cylindrical computational grid. However, the strength of excitation does not concern us here – we simply seek to demonstrate mode identification. We defer to [@westernacher2019multimessenger] for a more detailed description of the simulation details. We also defer details regarding the mode function matching method to [@westernacher2018turbulence], where they are described in the most depth. The method involves using spectrogram filter kernels to extract mode motions from the velocity data in the simulations, followed by vector spherical harmonic decompositions to extract the angular harmonic components. The resulting fields are then normalized before their overlaps with perturbative mode functions are computed. Our main purpose here is to apply a consistent linear perturbative scheme to a snapshot from the simulation at $t\sim 40$ ms post-bounce, which was previously analyzed [@westernacher2018turbulence; @westernacher2019multimessenger], to study multiple quadrupolar modes ($l=2,\, m=0$) of the system which are excited weakly at bounce. The first mode has a peak frequency of $515$ Hz[^1]. This mode was found in [@westernacher2018turbulence; @westernacher2019multimessenger] to have a radial order $n=4$, and we make the same conclusion here. The second quadrupolar mode we study has a less well-defined peak frequency (we estimate $1241$ Hz from the GW spectrum), and was not reported in [@westernacher2018turbulence; @westernacher2019multimessenger]. Note that due to an analysis error, perturbative mode frequencies in [@westernacher2018turbulence] should be corrected by multiplying them by $\sim 1.5$. Perturbative scheme {#sec:methods} =================== We begin with the Newtonian perfect fluid and gravity equations, $$\begin{aligned} \partial_t \rho + \nabla_i \left( \rho v^i \right) &=& 0 \label{eq:restmass}\\ \partial_t \left( \rho v_i \right) + \nabla_j \left( \rho v^j v_i\right) + \partial_i P &=& - \rho \partial_i \Phi \label{eq:euler} \\ \nabla^2 \Phi &=& 4\pi \rho. \label{eq:poisson}\end{aligned}$$ We linearize these equations with respect to a spherically symmetric equilibrium background solution, $\rho=\rho(r)$, $v^i(r) = (v^r(r),0,0)$, $P=P(r)$, $\Phi=\Phi(r)$, $\partial_r P/\rho = -\partial_r \Phi$. Denote Eulerian perturbations with $\delta$ and Lagrangian ones with $\Delta$, and substitute eg. $\rho \rightarrow \rho + \delta \rho$ into Eqs. -. Also use the condition of adiabatic perturbations coming from the energy equation, $$\begin{aligned} \frac{\Delta P}{\Delta \rho} = c_s^2 \label{eq:adiab}\end{aligned}$$ where $c_s^2 = P\Gamma_1/\rho$ is the sound speed squared, $\Gamma_1$ is the adiabatic index for the perturbations, and eg. $\Delta P = \delta P + \xi^i\nabla_i P$ where $\xi^i$ is the perturbative Eulerian fluid element displacement vector. The displacement vector $\xi^i$ is related to the velocity perturbation via $\delta v^i = \partial_t \xi^i + v^j\nabla_j \xi^i - \xi^j\nabla_j v^i$, which simplifies to $\delta v^i = \partial_t \xi^i$ when the background velocity is zero. Linearization of Eqs. - assuming axisymmetric perturbations $\xi^i = (\xi^r,\xi^{\theta},0)$ yields $$\begin{aligned} 0 &=& \delta \rho + \rho \xi^i \partial_i \ln{\sqrt{\gamma}} + \rho \partial_i\xi^i + \xi^r\partial_r\rho \label{eq:restmasspert1}\\ 0 &=& \partial_t^2 \xi^r + \frac{1}{\rho}\partial_r \delta P + \partial_r \delta \Phi - \frac{\delta\rho}{\rho^2}\partial_r P \label{eq:euler_rpert1}\\ 0 &=& r^2 \partial_t^2 \xi^{\theta} + \frac{1}{\rho}\partial_{\theta} \delta P + \partial_{\theta} \delta\Phi \label{eq:euler_thpert1}\\ 0 &=& \nabla^2 \delta\Phi - 4\pi \delta \rho \label{eq:poissonpert1}\end{aligned}$$ where $\sqrt{\gamma} = r^2 \sin\theta$ is the square root of the flat 3-metric determinant in spherical coordinates. In deriving Eq.  we integrated in time, setting the integration constant to zero [@poisson2014gravity]. In Eq.  note the appearance of the factor $r^2$ in front of the time derivative, which comes from raising the index using the metric via $\partial_t^2 \xi_{\theta} = \gamma_{i\theta} \partial_t^2 \xi^i = \gamma_{\theta\theta} \partial_t^2 \xi^{\theta} = r^2 \partial_t^2 \xi^{\theta} $. Using the axisymmetric spherical harmonics $Y_l$ ($m=0$) and harmonic time dependence, we insert a separation of variables ansatz $$\begin{aligned} \delta\rho &=& \delta\hat{\rho}(r) Y_{l} e^{-i \sigma t} \nonumber\\ \delta P &=& \delta\hat{P}(r) Y_{l} e^{-i \sigma t} \nonumber\\ \delta\Phi &=& \delta\hat{\Phi}(r) Y_{l} e^{-i \sigma t} \nonumber\\ \xi^r &=& \eta_r(r) Y_{l} e^{-i\sigma t} \nonumber\\ \xi^{\theta} &=& \frac{\eta_{\perp}(r)}{r^2} \partial_{\theta} Y_{l} e^{-i \sigma t}.\end{aligned}$$ We will assume $l\neq 0$. The angular frequency is $\sigma = 2 \pi f$. Note that we are using the coordinate basis $\{ (\partial_r)^i,(\partial_{\theta})^i,(\partial_{\phi})^i \}$ rather than the normalized coordinate basis $\{ \hat{\boldsymbol{r}}, \hat{\boldsymbol{\theta}}, \hat{\boldsymbol{\phi}} \}$, which explains the last ansatz having $\eta_{\perp}/r^2$ rather than $\eta_{\perp}/r$. Plugging these ansatz into Eq.  gives us a relation to eliminate $\delta\hat{P}$ via $$\begin{aligned} \delta\hat{P} = \rho \left(\sigma^2 \eta_{\perp} - \delta\hat{\Phi}\right).\end{aligned}$$ The adiabatic condition then yields a relation which can be used to eliminate $\delta\hat{\rho}$ via $$\begin{aligned} \delta\hat{\rho} = \rho\left( \frac{\sigma^2}{c_s^2}\eta_{\perp} - \frac{\delta\hat{\Phi}}{c_s^2} - \mathcal{B}\eta_r \right),\end{aligned}$$ where we have defined $\mathcal{B} \equiv \partial_r\ln\rho - (1/\Gamma_1)\partial_r \ln P$ as the Schwarzschild discriminant. In what follows, we also define $\tilde{G} \equiv \partial_r P/\rho = -\partial_r \Phi$, and the Brunt-V[ä]{}is[ä]{}l[ä]{} frequency squared is $N^2 = \tilde{G}\mathcal{B}$. The linearization of the remaining Eqs.  &  &  yields $$\begin{aligned} 0 &=& \partial_r \eta_r + \left[\frac{2}{r} + \frac{\partial_r P}{\Gamma_1 P}\right]\eta_r \nonumber\\ &\phantom{=}& \phantom{\partial_r \eta_r} + \left[\frac{\sigma^2}{c_s^2} - \frac{l(l+1)}{r^2}\right] \eta_{\perp} - \frac{1}{c_s^2}\delta\hat{\Phi} \label{eq:restmasspert2}\\ 0 &=& \partial_r \eta_{\perp} - \left[1-\frac{N^2}{\sigma^2}\right] \eta_r + \mathcal{B}\eta_{\perp} - \frac{\mathcal{B}}{\sigma^2} \delta\hat{\Phi} \\ 0 &=& \partial_r \delta\hat{\Phi} - F \\ 0 &=& \partial_r F + \frac{2}{r}F + 4\pi\rho \mathcal{B} \eta_r - 4\pi\rho\frac{\sigma^2}{c_s^2} \eta_{\perp} \nonumber\\ &\phantom{=}& \phantom{\partial_r F + \frac{2}{r}F} + \left[ \frac{4\pi\rho}{c_s^2} - \frac{l(l+1)}{r^2} \right] \delta\hat{\Phi}, \label{eq:poissonpert2}\end{aligned}$$ where we defined $F\equiv \partial_r \delta\hat{\Phi}$ to reduce the system to first order. In obtaining these equations we used the identity $\partial_{\theta}^2 Y_{l} + \cot\theta \partial_{\theta}Y_{l} = -l(l+1)Y_{l}$. Note these perturbative equations are the same equations as in [@christensen1991solar] Eqs. (31-33), after changing the definitions $\delta\hat{\Phi} \leftrightarrow - \Phi^\prime$, $\xi_h \leftrightarrow \eta_{\perp}/r$. The latter identification comes both from different definitions of $\eta_{\perp}$ vs $\xi_h$ as well as the use of different basis vectors – $\{ (\partial_r)^a, (\partial_\theta)^a, (\partial_{\phi})^a \}$ in our case vs $\{ \hat{\boldsymbol{r}}, \hat{\boldsymbol{\theta}}, \hat{\boldsymbol{\phi}} \}$ in [@christensen1991solar]. To solve these equations, we integrate from a small non-zero radius $r_0$ (typically $dr/5$ where $dr$ is the grid resolution), where we impose regularity conditions (see Appendix \[sec:bcs\]) in the form (assuming $l\neq 0$) $$\begin{aligned} \eta_r = A_0 r^{l-1}, &\;\;& \phantom{\partial_r} \eta_{\perp} = \frac{A_0}{l} r^{l\phantom{-1}} \nonumber\\ \delta\hat{\Phi} = C_0 r^{l\phantom{-1}}, &\;\;& \partial_r \delta\hat{\Phi} = l C_0 r^{l-1},\end{aligned}$$ where $A_0$ is specified as a small number ($10^{-5}$ in our case) which encodes the overall amplitude of the perturbation, and $C_0$ is searched for via a root-finding algorithm such that an outer boundary condition on $\delta\hat{\Phi}$ is satisfied – see Appendix \[sec:bcs\] for a detailed description. This outer boundary condition on $\delta\hat{\Phi}$ was not imposed in [@morozova2018gravitational], where instead $\delta\hat{\Phi}\vert_{r_0} = 0 = \partial_r \delta \hat{\Phi}\vert_{r_0}$ was used. This error was repeated in subsequent work, including [@westernacher2018turbulence; @radice2019characterizing; @westernacher2019multimessenger], but does not affect any of the results obtained in the Cowling approximation. We validate our current Newtonian perturbative scheme on a Newtonian polytropic star in Appendix \[sec:tests\], and demonstrate that the effect of ignoring the outer boundary condition on $\delta\hat{\Phi}$ is large mode frequency errors for modes of low radial order. We also demonstrate in Appendix \[sec:tests\] that our current Newtonian perturbative scheme recovers the non-radial modes of equilibrium stars evolved in a pseudo-Newtonian system using `FLASH`. This system has a phenomenologically modified monopole gravitational potential designed to mimic relativistic stars ([@marek2006exploring] Case A). This demonstrates that we can solve for non-radial modes even though we do not have an equation of motion for the monopole potential. Such an equation never appears in our derivation above, because we assumed $l\neq 0$. Having the consistent perturbative scheme for such pseudo-Newtonian simulations allows us to investigate how well other aspects of the approximation (the assumption of equilibrium background, zero background velocity, and spherical averaging) actually affect the mode identification. The other outer boundary condition concerning the fluid variables is considerably more uncertain. In [@morozova2018gravitational] it was taken to be $\Delta P \vert_{r=R} = 0$ for some outer boundary $R$ representing the proto-neutron star (PNS) surface, and in [@torres2017towards] was taken to be $\eta_r\vert_{\mathrm{shockwave}}=0$. With the consistent perturbative equations, we can instead simply plug in the frequency observed in the simulation and see whether the resulting mode function matches the simulated velocity data well. We can also try to infer an appropriate outer boundary condition on the fluid variables in this way. Thus, we can turn the problem around and attempt to *measure* the appropriate boundary condition. Theoretically, the boundary condition must account for the Rankine-Hugoniot jump conditions across the accretion shock, which in turn depend upon the state of the supersonically accreting material upstream from the shockwave (see e.g. [@foglizzo2007instability; @laming2007analytic]). ![Normalized GW spectrum averaged over $t\in [30,50]$ ms post-bounce, computed using a Bohman window with 35 ms width. Two frequencies of the best-fit mode functions are indicated at 507 Hz and 1238 Hz, corresponding to weakly excited quadrupolar modes. These compare well with the peaks in the GW spectrum at $515$ and $1241$ Hz. The shaded areas indicate the frequency extent of the spectral filter used in [@westernacher2018turbulence; @westernacher2019multimessenger] to extract the velocity data, against which perturbative mode functions are matched.[]{data-label="fig:gwspec"}](gwspec.pdf){width="48.00000%"} Results {#sec:results} ======= We show the GW spectrum in Fig. \[fig:gwspec\], which is computed using a Bohman window with 35 ms width, and averaged over times $t\in [30,50]$ ms. The grey shaded intervals indicate the frequency extent of the spectral filters used to extract the velocity data from the simulation. A snapshot of that data near $t=40$ ms is then matched with perturbative solutions, with the frequency as the free parameter in the perturbative solutions. The perturbative solutions whose modefunction matches the velocity data best have frequencies of $507$ and $1238$ Hz, which compares well with the peaks in Fig. \[fig:gwspec\]. Our first finding is that plugging in the simulation frequency $f\sim 515$ Hz (disregarding any outer boundary condition on the fluid variables) yields a perturbative solution that fits the simulation data well – see Fig. \[fig:515Hz\]. In the top two panels we show the $515$ Hz perturbative solution (weighted by $\rho^{1/4}$) for various boundary conditions on $\delta\hat{\Phi}$, namely the vacuum one (Eq. ) imposed at various radii, as well as the in-matter one (Eq. ) which does not depend on the outer boundary location. Note we plot on an arbitrary linear vertical scale. The result obtained using the vacuum boundary condition approaches the in-matter one rapidly as the outer boundary moves out, because the density perturbation $\delta\hat{\rho}$ becomes negligible for $r\gtrsim 60$ km (see bottom panel). For the rest of our results we use the in-matter boundary condition Eq. . ![Upper two panels: Normalized perturbative solutions plotted on a linear vertical scale, with frequency corresponding to the simulation, $f=515$ Hz, for varying outer boundary condition on the Newtonian potential perturbation. The vacuum boundary condition Eq.  is imposed at $r_{\mathrm{bc}}=\{15,30,100\}\,$km, and is seen to approach the in-matter boundary condition case  as the boundary is placed farther out. The perturbative solutions are a poor representation of the simulation beyond $\sim 10$ km. Bottom panel: The density $\rho$ and density perturbation $\vert \delta \hat{\rho} \vert$ are displayed for reference. The density perturbation becomes negligible beyond $\sim 60$ km. The shockwave is located at $r\sim 125$ km at this time $40$ ms post-bounce.[]{data-label="fig:515Hz"}](515Hz_vacbcs.pdf){width="48.00000%"} Next we do a search over frequency (again disregarding outer boundary conditions for the fluid variables) to find the best-fitting perturbative solution to the simulation data. The fit quality is computed by normalizing the $\sqrt{\rho}$-weighted velocities and computing a Frobenius norm of their difference (see [@westernacher2019multimessenger]). The result is shown in Fig. \[fig:515Hz\_bfm\]. Despite not smoothing the simulated data, the agreement is nonetheless striking. We again weight the velocity by $\rho^{1/4}$ to allow easier visual inspection (compared to a $\sqrt{\rho}$-weighting). We stress that this is an unforgiving way of displaying the agreement. The radial nodes of the best-fit perturbative solution are consistent with those found in [@westernacher2018turbulence; @westernacher2019multimessenger], i.e. $n=4$ when counted within the shockwave (which is located at $r\sim 125$ km at this snapshot). Note that since our background is not actually in equilibrium, we have an ambiguity in how we apply the perturbative scheme. Namely, we can set $\tilde{G} = \partial_r P/\rho$ or $\tilde{G} = -\partial_r \Phi$[^2]. We show both cases in Fig. \[fig:515Hz\_bfm\], which yield best-fit solutions with frequencies of $507$ Hz and $523$ Hz, respectively. Both choices are equally accurate for this mode, but unless otherwise specified we will use $\tilde{G} = \partial_r P/\rho$. In Fig. \[fig:1240Hz\_bfm\] we show the analogous plot for the $1241$ Hz frequency mode, showing a similar level of agreement. The best-fitting perturbative solutions have frequencies of $1238$ and $1245$ Hz for the cases $\tilde{G}=\partial_r P/\rho$ and $\tilde{G}=-\partial_r \Phi$, respectively. This is $0.24\%$ and $0.32\%$ disagreement, respectively. ![The best-fit perturbative solutions for two different choices $\tilde{G}=\partial_r P/\rho$ and $\tilde{G}=-\partial_r \Phi$, which in a true spherically symmetric equilibrium would yield the same result. These choices yield frequencies of $507$ Hz and $523$ Hz, respectively. This is a mistmatch with the simulation frequency $515$ Hz by $\pm$1.6%. These perturbative solutions have radial order $n=4$ if counted up to the shockwave location $r=125$ km.[]{data-label="fig:515Hz_bfm"}](515Hz_bestfit_modefunc.pdf){width="48.00000%"} ![Same as Fig. \[fig:515Hz\_bfm\] but for the $\sim 1240$ Hz peak.[]{data-label="fig:1240Hz_bfm"}](1240Hz_bestfit_modefunc.pdf){width="48.00000%"} We now reinstate outer boundary conditions for the fluid variables. Our purpose is to “measure" the boundary conditions which will yield a mode function spectrum such that the best-fit mode function has a frequency which is (at least similar to) the simulation. If such a boundary condition existed, then one could safely identify modes in pseudo-Newtonian simulations by doing frequency matching alone, removing the need for the complicated and expensive mode function matching procedure described in [@westernacher2018turbulence; @westernacher2019multimessenger]. In Fig. \[fig:515Hz\_DeltaP\] we plot the absolute value of the Lagrangian pressure perturbation corresponding to the best-fitting perturbative solutions for the $515$ Hz mode in Fig. \[fig:515Hz\_bfm\] on an arbitrary logarithmic scale. The analogous plot for the $1241$ Hz mode is displayed in Fig. \[fig:1240Hz\_DeltaP\]. The Lagrangian pressure perturbation is overlaid on the background density profile, which is plotted on a faithful logarithmic scale. We indicate the location of the zero-crossings of $\Delta P$ with dotted lines, and also indicate the corresponding density value there. Zero-crossings for the $515$ Hz case occur near $\{6\times 10^{13},10^{12},10^{11},10^{10}\}$ g$\,$cm$^{-3}$. A common definition for the PNS surface is e.g. $\rho=10^{11}\,$g$\,$cm$^{-3}$, and a zero-crossing at that location also occurs for the $1241$ Hz mode in Fig. \[fig:1240Hz\_DeltaP\]. These zero-crossings are not enforced, and if they are not mere coincidences then they could be physically meaningful if they work for different modes. In Tables \[table515\] & \[table1240\], for various outer boundary conditions on the fluid variables we show the mode properties with nearest and next-nearest frequencies to the simulation (subscripts $_{\mathrm{best}}$ and $_{\mathrm{next}}$, respectively). All choices listed, aside from $\Delta P \vert_{\rho=10^{12}}=0$ which fails to reproduce the $1241$ Hz mode, yield a clear relative distinction between the best-fit and the next-best one, and could therefore be regarded as safe to use during a mode frequency matching procedure. However, the boundary condition of [@torres2017towards], $\eta_r \vert_{\mathrm{shockwave}}=0$, yields remarkable sub-$1\%$ agreement for both modes, suggesting it is the physically correct one in this regime. ![Lagrangian pressure perturbation $\vert \Delta P\vert$ corresponding to the best-fit perturbative solutions in Fig. \[fig:515Hz\_bfm\] displayed on an arbitrary logarithmic scale. The rest mass density of the spherically-averaged background is also displayed on an accurate logarithmic scale. Zeros of the Lagrangian pressure perturbation are indicated, which suggest appropriate values of $\rho$ at which $\Delta P = 0$ should be enforced during a mode search.[]{data-label="fig:515Hz_DeltaP"}](515Hz_bestfit_modefunc_DeltaP.pdf){width="48.00000%"} ![Same as Fig. \[fig:515Hz\_DeltaP\] but for the $\sim 1240$ Hz mode. The zeros of $\Delta P$ occuring near $\rho = 10^{12},10^{11}$ g$\,$cm$^{-3}$ are shown.[]{data-label="fig:1240Hz_DeltaP"}](1240Hz_bestfit_modefunc_DeltaP.pdf){width="48.00000%"} Bdy. condition $f_{\mathrm{best}}$ \[Hz\], diff $n_{\mathrm{best}}$ $f_{\mathrm{next}}$ \[Hz\], diff $n_{\mathrm{next}}$ ---------------------------------------- ---------------------------------- --------------------- ---------------------------------- --------------------- $\Delta P\vert_{\rho\sim 10^{12}}=0$ 504, -2.1% 4 381, -26% 4 $\Delta P\vert_{\rho\sim 10^{11}}=0$ 504, -2.1% 4 436, -15% 5 $\Delta P\vert_{\rho\sim 10^{10}}=0$ 503, -2.3% 4 463, -10% 5 ${\eta_r}\vert_{\mathrm{shockwave}}=0$ 513, -0.4% 4 491, -4.7% 5 : Modes with nearest ($f_{\mathrm{best}}$) and next-nearest ($f_{\mathrm{next}}$) frequencies to the simulation value of $515$ Hz, for varying boundary conditions. We use $\tilde{G}=\partial_r P/\rho$. The subscripts on $\Delta P$ (eg. $\Delta P\vert_{\rho\sim 10^{12}}$) indicate density in units of g$\,$cm$^{-3}$. The nearest modes are highlighted in bold.[]{data-label="table515"} Bdy. condition $f_{\mathrm{best}}$ \[Hz\], diff $n_{\mathrm{best}}$ $f_{\mathrm{next}}$ \[Hz\], diff $n_{\mathrm{next}}$ ---------------------------------------- ---------------------------------- --------------------- ---------------------------------- --------------------- $\Delta P\vert_{\rho\sim 10^{12}}=0$ 1101, -11% 8 1532, +23% 12 $\Delta P\vert_{\rho\sim 10^{11}}=0$ 1239, -0.2% 9 1073, -14% 8 $\Delta P\vert_{\rho\sim 10^{10}}=0$ 1235, -0.5% 9 1357, +9.3% 10 ${\eta_r}\vert_{\mathrm{shockwave}}=0$ 1248, -0.6% 9 1137, -8.4% 8 : Same as Table \[table515\] but for the $\sim 1240$ Hz mode. Nodes are counted up to the shockwave at $r=125$ km.[]{data-label="table1240"} Outlook and conclusions {#ch:CCSNconc} ======================= In this work, we presented and tested perturbative equations which are the consistent linear approximation of pseudo-Newtonian systems whereby one uses Newtonian hydrodynamics, standard Newtonian gravity for non-radial components of the potential, and some non-standard monopole potential such as that of [@marek2006exploring] Case A. This system of equations allows one to solve for non-radial modes, thereby allowing identification of active modes in pseudo-Newtonian simulations (eg. `PROMETHEUS/VERTEX` [@rampp2002radiation; @muller2010new; @muller2012new; @muller2013new; @muller2014new], `FLASH` [@fryxell2000flash; @dubey2009extensible], `FORNAX` [@skinner2019fornax], `CHIMERA` [@bruenn2018chimera]) using mode frequency matching. This alleviates the need to perform the complex and expensive mode function matching procedure of [@westernacher2018turbulence; @westernacher2019multimessenger]. We found that the imposing vanishing radial displacement as an outer boundary condition (as in [@torres2017towards]) yields remarkable sub-$1\%$ agreement between perturbative mode frequencies and the simulation, suggesting that this is the physically correct choice. However, imposing a vanishing Lagrangian pressure perturbation at the radii where $\rho=\{10^{11},10^{10}\}$ g$\,$cm$^{-3}$ (the last value being used in [@morozova2018gravitational]) should also prevent mode misidentification. These conclusions ought to be tested in other regimes, eg. later times $t>100$ ms and different progenitor stars. We thank Evan O’Connor for comments and insight regarding neutrino pressure gradients, and both Evan O’Connor and Sean M. Couch for `FLASH` code development and running the simulations analyzed in this work. We also thank an anonymous referee for providing a deeper context of this work within the existing literature. This research was supported by National Science Foundation Grant No. PHY-1912619 at the University of Arizona. Software: Matplotlib [@Hunter:2007], FLASH [@fryxell2000flash; @dubey2009extensible; @couch13; @o2018two], SciPy [@scipy]. Boundary conditions {#sec:bcs} =================== In this section we give details of how boundary conditions are derived, for the purpose of being pedagogical. We use the strategy of [@hurley1966oscillations], except applied directly to our equations -. We wish to determine the behavior of $\{\eta_r,\eta_{\perp},\delta \hat{\Phi}\}$ in a neighborhood of the origin $r=0$. For this purpose, we make the ansatz $$\begin{aligned} \eta_r &=& r^a \sum_{n=0}^{\infty}{A_n r^n} \nonumber\\\eta_{\perp} &=& r^b \sum_{n=0}^{\infty}{ B_n r^n} \nonumber\\ \delta \hat{\Phi} &=& r^c \sum_{n=0}^{\infty}{ C_n r^n}, \nonumber\end{aligned}$$ where $A_n, B_n, C_n$ are constant coefficients nonzero when $n=0$ (do not confuse $n$ in this context with the radial order of modes), and $a,b,c$ are constant exponents to be determined. We require $ a,b,c\geq 0$ by regularity at the origin. This ansatz is a generalization of the Frobenius method to a system of equations. The derivatives we need are $$\begin{aligned} \partial_r \delta \hat{\Phi} &=& r^c \sum_{n=0}^{\infty}{(n+c)C_n r^{n-1}} \\ \partial_r^2 \delta \hat{\Phi} &=& r^c \sum_{n=0}^{\infty}{(n+c)(n+c-1)C_n r^{n-2}},\end{aligned}$$ and similar expressions for $\partial_r \eta_r,\partial_r \eta_{\perp}$. Plugging these ansatz into our equations - and collecting terms proportional to $r^a,r^b,r^c$, we schematically obtain $$\begin{aligned} 0 &=& Q_a r^a + Q_b r^b + Q_c r^c \nonumber\\ 0 &=& R_a r^a + R_b r^b + R_c r^c \\ \label{eq:bdyschem} 0 &=& S_a r^a + S_b r^b + S_c r^c,\nonumber\end{aligned}$$ where the coefficients are $$\begin{aligned} Q_a &=& \sum_{n=0}^{\infty}{nA_nr^{n-1}} + \left[\frac{2+a}{r} + \frac{\partial_r P}{\Gamma_1 P} \right] \sum_{n=0}^{\infty}{A_n r^n} \nonumber\\ Q_b &=& \left[\frac{\sigma^2}{c_s^2} - \frac{l(l+1)}{r^2}\right] \sum_{n=0}^{\infty} B_n r^n \nonumber\\ Q_c &=& -\frac{1}{c_s^2} \sum_{n=0}^{\infty}{C_n r^n} \nonumber\\ R_a &=& -\left[1-\frac{N^2}{\sigma^2}\right] \sum_{n=0}^{\infty}{A_n r^n} \nonumber\\ R_b &=& \sum_{n=0}^{\infty}{nB_n r^{n-1}} + \left[ \frac{b}{r} + \mathcal{B} \right] \sum_{n=0}^{\infty}{B_n r^n} \nonumber\\ R_c &=& -\frac{\mathcal{B}}{\sigma^2} \sum_{n=0}^{\infty}{C_n r^n} \nonumber\\ S_a &=& 4\pi\rho \mathcal{B} \sum_{n=0}^{\infty}{A_n r^n} \nonumber\\ S_b &=& -4\pi\rho \frac{\sigma^2}{c_s^2} \sum_{n=0}^{\infty}{B_n r^n} \nonumber\\ S_c &=& \sum_{n=0}^{\infty}{n^2 C_n r^{n-2}} + \left[ \frac{2c+1}{r} \right] \sum_{n=0}^{\infty}{nC_n r^{n-1}} \nonumber\\ &+& \left[\frac{c(c+1)-l(l+1)}{r^2} + \frac{4\pi\rho}{c_s^2}\right] \sum_{n=0}^{\infty}{C_n r^n}.\end{aligned}$$ Since Eqs.  hold in a neighborhood of the origin, the full coefficients in front of each power of $r$ (once collected) must vanish independently. We are interested in the vanishing of the lowest order terms. In the Frobenius method, only one equation is being solved. This means only one unknown exponent (eg. $a$ above) appears in the equation once the ansatz is plugged in. This makes identifying orders in $r$ straightforward. In our case, we have a system of equations and multiple unknown exponents $a,b,c$ appear in each equation. This makes identifying orders in $r$ more complicated, but we can proceed by considering all possible cases and systematically eliminating them. This is what we do next. Since we are interested in the lowest nontrivial order, it suffices to truncate every sum after the first nonzero term. We also need to consider the order carried by the background quantities. In particular, since the pressure and density are spherically-symmetric quantities with even parity, we have $P \simeq P\vert_{0} + P^{\prime\prime} r^2/2$ and $\rho \simeq \rho\vert_{0} + \rho^{\prime\prime} r^2/2$, where we use a double prime superscript to denote a second radial derivative evaluated at the origin, to avoid cumbersome notation. This means $\partial_r P = P^{\prime\prime} r = \mathcal{O}(r)$ and $\partial_r \rho = \rho^{\prime\prime} r = \mathcal{O}(r)$. Thus $\mathcal{B} = \partial_r\rho/\rho - \partial_r P/(\Gamma_1 P) \simeq [\rho^{\prime\prime}/\rho - P^{\prime\prime}/(\Gamma_1 P)] r = \mathcal{O}(r)$. Similarly, $\tilde{G} = \partial_r P/\rho \simeq P^{\prime\prime} r/\rho = \mathcal{O}(r)$, and so by extension $N^2 = \tilde{G} \mathcal{B} = \mathcal{O}(r^2)$. Inserting these expansions into Eqs.  and keeping lowest-order terms for each of the $r^a,r^b,r^c$ terms separately, we obtain $$\begin{aligned} 0 &=& (2+a)A_0 r^{a-1} - B_0 l(l+1) r^{b-2} - \frac{C_0}{c_s^2} r^c \label{eq:low1} \\ 0 &=& -A_0 r^a + B_0 b r^{b-1} - \frac{C_0}{c_s^2} \left[ \frac{\rho^{\prime\prime}}{\rho} - \frac{P^{\prime\prime}}{\Gamma_r P} \right] r^{c+1} \label{eq:low2} \\ 0 &=& 4\pi\rho\left[\frac{\rho{\prime\prime}}{\rho} - \frac{P{\prime\prime}}{\Gamma_1 P}\right] A_0 r^{a+1} - 4\pi\rho \frac{\sigma^2}{c_s^2} B_0 r^b \nonumber\\ &+& \left[ c(c+1) - l(l+1) \right] C_0 r^{c-2}. \label{eq:low3}\end{aligned}$$ At this stage we do not know whether we have kept consistent orders in $r$, since we do not know the relationship between the exponents $a,b,c$. However, when considering Eq. , notice that the exponents will not depend upon the background solution if and only if the $r^{c-2}$ term is the lowest order one. Independence from the background solution is a property we desire[^3], thus we demand that the $r^{c-2}$ term must vanish, i.e. $c=l$. This also implies $c-2 < a+1$ and $c-2 < b$. The same consideration applied to Eq.  means that one or both of the $r^{a-1}$ and $r^{b-2}$ terms must be lowest order. If the $r^{b-2}$ term is lowest order by itself, that implies $l=0$. If we are not interested in radial modes (in this work, we are not), then we can discard this possibility. On the other hand, if the $r^{a-1}$ term is lowest order by itself, that implies $a=-2$ which would violate regularity at the origin. Thus we must conclude that both terms are lowest order, i.e. $a=b-1$ and $(2+a)A_0 = B_0 l(l+1)$. Lastly, consider Eq. . If the exponents are to be independent of the background quantities, then one or both of the $r^a$ and $r^{b-1}$ terms must be lowest order. But we already established that $a=b-1$, thus they are both lowest order. This yields $A_0 = b B_0$. Combining this relation with the one obtained previously from Eq.  and using $a=b-1$, we finally find $$\begin{aligned} a=l-1, \;\; b=l, \;\; c=l.\end{aligned}$$ Therefore, in a neighborhood of the origin, $$\begin{aligned} \eta_r = A_0 r^{l-1}, &\;\;& \phantom{\partial_r} \eta_{\perp} = \frac{A_0}{l} r^{l\phantom{-1}} \nonumber\\ \delta\hat{\Phi} = C_0 r^{l\phantom{-1}}, &\;\;& \partial_r \delta\hat{\Phi} = l C_0 r^{l-1}. \label{eq:origin}\end{aligned}$$ Beware that we are not using the normalized coordinate basis. In the normalized basis, one instead has $\eta_{\perp} = (A_0/l) r^{l-1}$. In the numerical integration, we begin a small distance away from the origin (eg. $dr/5$, where $dr$ is the grid resolution) and use Eqs.  as initial conditions. This requires specification of $A_0, C_0$ and the angular frequency $\sigma$. The choice of $A_0$ amounts to an arbitrary amplitude, which we choose to be $A_0 = 10^{-5}$. For each value of angular frequency $\sigma$, we perform a root-finding procedure to converge upon the value of $C_0$ such that at the outer boundary $r=R$ we have [@christensen1991solar] $$\begin{aligned} \left[\partial_r \delta\hat{\Phi} + \frac{l+1}{r} \delta \hat{\Phi}\right]\vert_{r=R} = 0. \label{eq:phiobdy}\end{aligned}$$ This relation can be derived from the solution for the $l$th spherical harmonic moment of the Poisson equation [@poisson2014gravity] $$\begin{aligned} \delta\hat{\Phi} = -\frac{4\pi}{2l+1}\frac{1}{R^{l+1}} \int_0^r{\delta \hat{\rho}(\tilde{r}) \tilde{r}^{l+2}d\tilde{r}}, \label{eq:phiobdy_full}\end{aligned}$$ valid when $\delta \hat{\rho}(r) = 0$ for $r>R$. In the case of our CCSN system, $l$th moment rest mass perturbations $\delta \hat{\rho}$ likely escape out through $r=R$, but to the extent that it is of small amplitude and leaks into different harmonics $l^\prime\neq l$, it can be ignored. If it cannot be ignored, then one should instead integrate the perturbative system beyond $r=R$ and then impose $$\begin{aligned} \!\!\!\!\!\!\!\!\left[\partial_r \delta\hat{\Phi} + \frac{l+1}{r} \delta \hat{\Phi}\right]\vert_{r=R} = -4\pi R^{l-1}\!\!\! \int_R^{\infty}{\frac{\delta\hat{\rho}}{r^{l-1}}dr}, \label{eq:phiobdy2}\end{aligned}$$ where the infinite upper limit of integration is understood to be replaced by an appropriate outermost radius, eg. the grid boundary or the CCSN shockwave. When using Eq. , one must integrate past $R$ in order to obtain $\delta\hat{\rho}$ over the domain of interest. The choice of $R$ is irrelevant. Note that $$\begin{aligned} \delta \hat{\rho} = \rho \left( \frac{\sigma^2}{c_s^2} \eta_{\perp} - \frac{\delta\hat{\Phi}}{c_s^2} - \mathcal{B}\eta_r \right). \label{eq:deltarho}\end{aligned}$$ Also, it is advisable to enforce Eq.  at the outer boundary rather than Eq. , in order to get control of the first derivative $\partial_r \delta\hat{\Phi}$. The root-finding loop for $C_0$ is nested inside a root-finder for the angular frequency $\sigma$, which yields either vanishing Lagrangian pressure perturbation at the outer boundary $$\begin{aligned} \Delta P\vert_{R} = [\rho \sigma^2 \eta_{\perp} - \rho \delta\hat{\Phi} + \eta_r \partial_r P]\vert_{R} = 0, \label{eq:DeltaP0}\end{aligned}$$ corresponding to a free surface, or vanishing radial displacement $$\begin{aligned} \eta_r\vert_{R} = 0, \label{eq:etar0}\end{aligned}$$ depending on one’s choice. Tests of perturbative scheme {#sec:tests} ============================ In this section we demonstrate the accuracy of our mode solver on both a Newtonian polytropic star and a pseudo-Newtonian “TOV” star. ![Comparison between axisymmetric $l=1$ and $l=2$ mode frequencies obtained in this work versus past work [@horedt2004polytropes] pg. 387, for a $\Gamma = 5/3$ polytrope. The frequencies are displayed in dimensionless form $\omega = \sqrt{\sigma^2/4\pi G \rho_c}$ where $\sigma=2\pi f$ is the angular frequency and $\rho_c$ is the central density. The wrong boundary condition $\delta\hat{\Phi}\vert_{r_0}=0=\partial_r \delta\hat{\Phi}\vert_{r_0}$ (green dots) has a large error for the lower overtones. With the correct boundary conditions (Eqs. ), we obtain at worst $\sim 0.4 \%$ residual for the fundamental $n=0$ mode.[]{data-label="fig:newt_test"}](newtonian_test.pdf){width="48.00000%"} ![Comparison between axisymmetric $l=2$ mode frequencies obtained perturbatively in this work versus using full nonlinear `FLASH` evolutions in past work [@westernacher2018turbulence; @westernacher2019multimessenger], for a $\Gamma = 2$ polytropic star with $P=\kappa \rho^\Gamma$, $\rho_c = 1.28\times 10^{-3}$, and $\kappa=100$ in geometrized units.[]{data-label="fig:FLASHTOV_test"}](FLASHTOV_test.pdf){width="48.00000%"} Newtonian polytropic star ------------------------- Fig. \[fig:newt\_test\] displays a comparison between $l=1$ and $l=2$ mode frequencies we obtain for a $\Gamma=5/3$ Newtonian polytropic star. The polytropic constant $\kappa$, where $P=\kappa \rho^\Gamma$ is arbitrary, and we display the frequencies in dimensionless form $$\begin{aligned} \omega \equiv \sqrt{\frac{\sigma^2}{4 \pi G \rho_c}},\end{aligned}$$ where $\sigma = 2\pi f$ is the angular frequency and $\rho_c$ is the central rest mass density. We impose a vanishing Lagrangian pressure perturbation at the surface, Eq. . We terminate the frequency search when the update becomes smaller than $0.5$ Hz (we set the stellar mass to $M=1.4 M_{\odot}$ and radius to $R = 12$ km, yielding mode frequencies $\gtrsim 2$ kHz). The frequencies compare favorably with past work ([@horedt2004polytropes] pg. 387 and references therein), except when the outer boundary condition for the Newtonian potential is disregarded (setting $\delta\hat{\Phi} =\partial_r \delta \hat{\Phi}=0$ at the starting point of outward integration), as done in [@morozova2018gravitational] and repeated in subsequent work, including [@westernacher2018turbulence; @radice2019characterizing; @westernacher2019multimessenger]. `FLASH` Tolman-Oppenheimer-Volkoff star --------------------------------------- Fig. \[fig:FLASHTOV\_test\] displays a comparison between $l=2$ modes computed perturbatively in this work with those extracted in [@westernacher2018turbulence; @westernacher2019multimessenger] from a fully nonlinear `FLASH` simulation of an equilibrium $\Gamma=2$ star with $\kappa = 100$ and $\rho_c = 1.28\times 10^{-3}$ in geometrized units. We impose vanishing Lagrangian pressure perturbation at the surface, Eq. . The frequency search terminates when the update is less than $0.5$ Hz. This test demonstrates that the non-radial modes of pseudo-Newtonian systems, as simulated in eg. `FLASH` [@fryxell2000flash; @dubey2009extensible], `FORNAX` [@skinner2019fornax], `CHIMERA` [@bruenn2018chimera], are determined by a purely Newtonian perturbative calculation. Radial perturbations of the gravitational potential, which would require knowledge of an equation of motion determining the “effectively GR” monopole ([@marek2006exploring] Case A), do not arise anywhere when one solves for non-radial modes. CCSN system ----------- We know based on the previous tests that the perturbative system is the consistent linearization of the equations of motion being simulated. However, when applying it to the CCSN system, we are dealing with a non-spherical system which we subject to a spherical averaging before performing the perturbative calculation, and it is not in hydrostatic equilibrium. In Fig. \[fig:bgresidual\] we compare the magnitude of different terms in the spherically-symmetric Euler equation $$\begin{aligned} 0= \partial_t \left(\rho v^r\right) + \frac{1}{r^2} \partial_r \left( r^2 \rho v^r v^r \right) + \partial_r P + \rho \partial_r \Phi,\end{aligned}$$ as a percentage comparison to $\vert \partial_r P\vert$. The equilibrium condition $\partial_r P + \rho \partial_r \Phi$ is satisfied at the $\sim 5$% level. Note that neutrino pressure gradients should also have a contribution to this balance, but their perturbations would introduce additional equations of motion so we have decided to neglect them. Furthermore, neutrino pressure gradients should gradually decouple from the fluid as one moves away from the PNS center, so introducing them into the background solution requires care. The level of violation of the hydrostatic equilibrium condition should be taken as a cautionary note when applying this perturbative calculation to dynamical systems such as CCSNe. By comparison, the other terms which encode time-dependence of the background solution ($\partial_t(\rho v^r)$) or its non-steadiness ($v^r=\,$constant$\,\neq 0$) are not large enough to account for the degree of non-equilibrium (sub-0.1% for $r<50$ km rising to 1% around $r=100$ km). This suggests that generalizing the perturbative scheme to a time-dependent or unsteady background would not yield significant improvements in the perturbative calculations presented in this work. ![A comparison between the magnitude of different terms in the spherically-symmetric Euler equation, as applied to the spherically-averaged snapshot of the CCSN system at $40$ ms. The equilibrium condition $\partial_r P + \rho\partial_r \Phi = 0$ is only satisfied at the $\sim 5\%$ level, which is commensurate with the frequency mismatch between the simulation and the best-fit perturbative solution. The non-equilibrium terms $\partial_t (\rho v^r)$ and $r^{-2} \partial_r (\rho v^r v^r)$ give a negligible contribution to the balance at $r<50$ km (sub-0.1%), and rises to $\sim 1$% around $r=100$ km.[]{data-label="fig:bgresidual"}](spherical_residual.pdf){width="48.00000%"} [^1]: Note that the mode is described in [@westernacher2019multimessenger] as having a frequency of $483$ Hz, which is the middle value of the spectrogram filter kernel used to extract it. However, $515$ Hz is the location of the peak Fourier amplitude in the GW signal. [^2]: This is not the only ambiguity. Wherever a pressure gradient or gravitational potential gradient appears, one could switch it out with the other using $\partial_r P = -\rho\partial_r \Phi$. [^3]: Although it would be interesting to know whether “special" perturbations of stars with exponents depending upon the background solution are ever relevant in practice.

--- abstract: 'It is considered gravitational interaction within the framework of the Newton theory and the quantum field theory. It is introduced the Planck neutrino $\nu_{Pl}$. Gravitational interaction of the fields $\psi\psi$ includes short-range interaction $\psi\nu_{Pl}$ and long-range interaction $\nu_{Pl}\nu_{Pl}$. Gravitational radiation can be identified with the Planck neutrino. The theory predicts the decay of proton into Planck neutrino. It is assumed that the Planck mass built from three fundamental constants $\hbar$, $c$ and $G$ is fixed in all the inertial frames. This leads to that the lifetime of proton relative to the decay into Planck neutrino decreases with the Lorentz factor as $\sim \gamma^{-5}$. Such a dependence of the lifetime of proton on the Lorentz factor yields a cut-off in the EHECRs spectrum. It is shown that the first “knee” in the EHECRs spectrum $E\sim 3\times 10^{15}\ {\rm eV}$ corresponds to the lifetime of proton equal to the lifetime of the universe, the second “knee” $E\sim 10^{17}-10^{18}\ {\rm eV}$ corresponds to the lifetime of proton equal to the thickness of our galactic disc. The EHECRs with the energies $E>3 \times 10^{18}\ {\rm eV}$ can be identified with the Planck neutrinos.' --- 6.5in = 44=-1.0truein =0truein [**Decay of proton into Planck neutrino in the theory of gravity**]{} [D.L. Khokhlov]{} [*Sumy State University, R.-Korsakov St. 2\ Sumy 40007 Ukraine\ e-mail: [email protected]*]{} Introduction ============ The principle of equivalence of inertial and gravitational masses underlines the theory of gravity. In the Einstein theory of gravity [@M], this leads to that the free gravitational field is nonlocalized. Under the presence of the matter, the gravity is described by the Einstein equations $$G_{ik}=T_{ik} \label{eq:GT}$$ where $G_{ik}$ is the Einstein tenzor, $T_{ik}$ is the tenzor of momentum-energy of the matter. Free gravitational field defined by the absence of the matter $T_{ik}=0$ is described by the equations $$R_{ik}=0 \label{eq:R}$$ where $R_{ik}$ is the Ricci tenzor. The localized field must be described by the tenzor of momentum-energy. Einstein characterized the momentum-energy of the gravitational field by the pseudo-tenzor defined as $$t^{ik}=H^{ilkm}_{\ \ \quad ,lm} -G^{ik} \label{eq:tik}$$ where $H^{ilkm}_{\ \ \quad ,lm}$ is the linearized part of $G_{ik}$. Thus in the Einstein theory gravitational field is nonlocalized. The natural way proposed by Lorentz and Levi-Civita [@Pau] is to take $G_{ik}$ as the momentum-energy of the gravitational field. However in this case $G_{ik}$ is equal to zero for the free gravitational field $G_{ik}=R_{ik}=0$. Such a situation may be interpreted as that the gravitational interaction occurs without gravitational field. Then the problem arises as to how to introduce gravitational radiation. The possible resolution of the problem is to introduce some material field as a radiation. Theory ====== Consider gravitational interaction within the framework of the Newton theory and the quantum field theory. The Lagrangian of the Newton gravity is given by $$L=G\frac{m^2}{r}, \label{eq:L1}$$ with the mass m being the gravitational charge. While expressing the Newton constant $G$ via the charge $g=(\hbar c)^{1/2}$ and via the Planck mass $m_{Pl}=(\hbar c/G)^{1/2}$, the Lagrangian (\[eq:L1\]) can be rewritten in the form $$L=G\frac{m^2}{r}=\frac{g^2}{m_{Pl}^2}\frac{m^2}{r}. \label{eq:L2}$$ The Lagrangian of the Newton gravity in the form (\[eq:L2\]) describes gravitational interaction by means of the charge $g$. In this way gravity may be implemented into the quantum field theory. Rewrite the Lagrangian (\[eq:L2\]) in the form of the effective Lagrangian of interaction of the spinor fields [@B] $$L=\frac{g^2}{m_{Pl}^2}J_{\mu}(x)J^{\mu}(x). \label{eq:L5}$$ The term $1/m_{Pl}^2$ in the Lagrangian (\[eq:L5\]) reads that gravitational interaction takes place at the Planck scale. At the same time gravity is characterized by the infinite radius of interaction. To resolve the problem consider the scheme of gravitational interaction which includes both the short-range interaction and the long-range interaction $$L=L_{short}+L_{long}. \label{eq:L3}$$ Let us introduce the Planck neutrino $\nu_{Pl}$. Let the Planck neutrino is the massless particle of the spin 1/2. Suppose that the Planck neutrino interacts with the other fields at the Planck scale $$\psi\rightarrow \nu_{Pl} \label{eq:psnu}$$ where $\psi$ denotes all the fields of the spin 1/2. This interaction is of short-range and is governed by the Lagrangian (\[eq:L5\]) $$L_{short}=\frac{g^2}{m_{Pl}^2}J_{\mu}(x)J^{\mu}(x) \label{eq:L6}$$ where the current $J_{\mu}$ transforms the field $\psi$ into the field $\nu_{Pl}$. Let the interaction of the Planck neutrinos $\nu_{Pl}\nu_{Pl}$ is of long-range and is governed by the Lagrangian identically equal to zero $$L_{long}\equiv 0. \label{eq:L4}$$ The considered scheme allows one to decribe both the classical gravity and the decay of the field $\psi$ into the Planck neutrino. In this scheme gravitational radiation can be identified with the Planck neutrino. Within the framework of the standard quantum field theory, the above scheme of gravitational interaction should include two intermediate fields $\psi\nu_{Pl}$ and $\nu_{Pl}\nu_{Pl}$. Since the Lagrangian of the interaction $\nu_{Pl}\nu_{Pl}$ is identically equal to zero, the energy of the field $\nu_{Pl}\nu_{Pl}$ is identically equal to zero. The field $\psi\nu_{Pl}$ is defined by the Planck mass. In the theory of gravity there is the limit of ability to measure the length equal to the Planck length [@Tr] $\Delta l\geq 2(\hbar G/c^3)^{1/2}=2l_{Pl}$. From this it follows that there is no possibility to measure the field $\psi\nu_{Pl}$ in the physical experiment. Thus both intermediate fields $\psi\nu_{Pl}$ and $\nu_{Pl}\nu_{Pl}$ cannot be measured. This means that the intermediate fields $\psi\nu_{Pl}$ and $\nu_{Pl}\nu_{Pl}$ do not exist. We arrive at the conclusion that gravitational interaction occurs without intermediate fields. The lifetime of proton relative to the decay into Planck neutrino ================================================================= In view of eq. (\[eq:psnu\]), the decay of proton into Planck neutrino occurs at the Planck scale $$p\rightarrow \nu_{Pl}. \label{eq:pnu}$$ The lifetime of proton relative to the decay into Planck neutrino is defined by the Lagrangian (\[eq:L5\]) $$t_p=t_{Pl}\left(\frac{m_{Pl}}{2m_p}\right)^5 \label{eq:tp1}$$ where the factor 2 takes into account the transition from the massive particle to the massless one. This lifetime corresponds to the rest frame. Consider the lifetime of proton in the moving frame with the Lorentz factor $$\gamma=\left(1-\frac{v^2}{c^2}\right)^{-1/2}. \label{eq:gam}$$ In the moving frame the rest mass and time are multiplied by the Lorentz factor $$m'=\gamma m \label{eq:gm}$$ $$t'=\gamma t. \label{eq:gt}$$ The Planck mass $m_{Pl}=(\hbar c/G)^{1/2}$ and the Planck time $t_{Pl}=(\hbar G/c^5)^{1/2}$ are built from three fundamental constants $\hbar$, $c$ and $G$. According to the special relativity [@Pau], the speed of light is fixed in all the inertial frames. Extend the special relativity principle and suppose that the three constants $\hbar$, $c$ and $G$ are fixed in all the inertial frames $$\hbar'=\hbar \qquad c'=c \qquad G'=G. \label{eq:thr}$$ Hence the Planck mass and time are fixed in all the inertial frames. Then the lifetime of proton in the moving frame is given by $$t_p '=t_{Pl}\left(\frac{m_{Pl}}{2\gamma m_p}\right)^5. \label{eq:tp2}$$ For comparison consider the decay of muon which is governed by the Lagrangian of electroweak interaction [@B] $$L=\frac{g^2}{m_{W}^2}J_{\mu}(x)J^{\mu}(x) \label{eq:Lmu}$$ where $m_{W}$ is the mass of W-boson. In the rest frame the lifetime of muon is given by $$t_{\mu}=t_{W}\left(\frac{m_{W}}{m_{\mu}}\right)^5. \label{eq:tmu1}$$ In the moving frame the lifetime of muon is given by $$t_{\mu}'=\gamma t_{W}\left(\frac{m_{W}}{m_{\mu}}\right)^5. \label{eq:tmu2}$$ Thus unlike the usual situation when the lifetime of the particle, e. g. muon, grows with the Lorentz factor as $\sim \gamma$, the lifetime of proton relative to the decay into Planck neutrino decreases with the Lorentz factor as $\sim \gamma^{-5}$. State once again that such a behaviour is due to that the Planck mass built from three fundamental constants $\hbar$, $c$ and $G$ is fixed in all the inertial frames. Extra high energy cosmic rays spectrum in view of the decay of proton into Planck neutrino ========================================================================================== In view of eq. (\[eq:tp2\]), the lifetime of proton relative to the decay into Planck neutrino decreases with the increase of the kinetic energy of proton. Then the decay of proton can be observed for the extra high energy protons. In particular the decay of proton can be observed as a cut-off in the energy spectrum of extra high energy cosmic rays (EHECRs). The EHECRs spectrum above $10^{10}\ {\rm eV}$ can be divided into three regions: two “knees” and one “ankle” [@Yo]. The first “knee” appears around $3\times 10^{15}\ {\rm eV}$ where the spectral power law index changes from $-2.7$ to $-3.0$. The second “knee” is somewhere between $10^{17}\ {\rm eV}$ and $10^{18}\ {\rm eV}$ where the spectral slope steepens from $-3.0$ to around $-3.3$. The “ankle” is seen in the region of $3 \times 10^{18}\ {\rm eV}$ above which the spectral slope flattens out to about $-2.7$. Consider the EHECRs spectrum in view of the decay of proton into Planck neutrino. Let the earth be the rest frame. For protons arrived at the earth, the travel time meets the condition $$t\leq t_p. \label{eq:t}$$ From this the time required for proton travel from the source to the earth defines the limiting energy of proton $$E_{lim}=\frac{m_{Pl}}{2}\left(\frac{t_{Pl}}{t}\right)^{1/5}. \label{eq:E}$$ Within the time $t$, protons with the energies $E>E_{lim}$ decay and do not give contribution in the EHECRs spectrum. Thus the energy $E_{lim}$ defines a cut-off in the EHE proton spectrum. Planck neutrinos appeared due to the decay of the EHE protons may give a contribution in the EHECRs spectrum. If the contribution of Planck neutrinos in the EHECRs spectrum is less compared with the contribution of protons one can observe the cut-off at the energy $E_{lim}$ in the EHECRs spectrum. Determine the range of the limiting energies of proton depending on the range of distances to the EHECRs sources. Take the maximum and minimum distances to the source as the size of the universe and the thickness of our galactic disc respectively. For the lifetime of the universe $\tau_0=14 \pm 2 \ {\rm Gyr}$ [@age], the limiting energy is equal to $E_1=3.9 \times 10^{15}\ {\rm eV}$. This corresponds to the first “knee” in the EHECRs spectrum. For the thickness of our galactic disc $\simeq 300\ {\rm pc}$, the limiting energy is equal to $E_2=5.5 \times 10^{17}\ {\rm eV}$. This corresponds to the second “knee” in the EHECRs spectrum. Thus the range of the limiting energies of proton due to the decay of proton into Planck neutrino lies between the first “knee” $E\sim 3\times 10^{15}\ {\rm eV}$ and the second “knee” $E\sim 10^{17}-10^{18}\ {\rm eV}$. From the above consideration it follows that the decrease of the spectral power law index from $-2.7$ to $-3.0$ at the first “knee” $E\sim 3\times 10^{15}\ {\rm eV}$ and from $-3.0$ to around $-3.3$ at the second “knee” $E\sim 10^{17}-10^{18}\ {\rm eV}$ can be explained as a result of the decay of proton into Planck neutrino. From this it seems natural that, below the “ankle” $E<3 \times 10^{18}\ {\rm eV}$, the EHECRs events are mainly caused by the protons. Above the “ankle” $E>3 \times 10^{18}\ {\rm eV}$, the EHECRs events are caused by the particles other than protons. If Planck neutrinos take part in the strong interactions, they must give some contribution in the EHECRs events. To explain the observed EHECRs spectrum it is necessary to assume that the contribution of Planck neutrinos in the EHECRs spectrum is less compared with the contribution of protons. Suppose that proton decays into 5 Planck neutrinos. Then the energy of the Planck neutrino is $1/5$ of the energy of the decayed proton. For the spectral power law index equal to $-2.7$, the ratio of the proton flux to the Planck neutrino flux is given by $J_p/J_{\nu}=5^{1.7}=15.4$. From the above consideration it is natural to identify EHE particles with the energies $E>3 \times 10^{18}\ {\rm eV}$ with the Planck neutrinos. Continue the curve with the spectral power law index $-2.7$ from the “ankle” $E\sim 3 \times 10^{18}\ {\rm eV}$ to the first “knee” $E\sim 3\times 10^{15}\ {\rm eV}$ and compare the continued curve with the observational curve. Comparison gives the ratio of the proton flux to the Planck neutrino flux $J_p/J_{\nu}\approx 15$. [99]{} C.W. Misner, K.S. Thorne, J.A. Wheeler, [*Gravitation*]{} (Freeman, San Francisco, 1973) W. Pauli, [*Theory of relativity*]{} (Pergamon, New York, 1958) N.N. Bogoliubov and D.V. Shirkov, [*Quantum fields*]{}, 2nd Ed. (Nauka, Moscow, 1993) H.-J. Treder [*in: Astrofisica e Cosmologia Gravitazione Quanti e Relativita*]{} (Giunti Barbera, Firenze, 1979) S. Yoshida, H. Dai, J.Phys. [**G24**]{} (1998) 905 F. Pont, M. Mayor, C. Turon, and D. A. VandenBerg, A&A [**329**]{} (1998) 87

--- abstract: | We analyze the velocity dispersions of individual [H[i]{}]{} and CO profiles in a number of nearby galaxies from the high-resolution HERACLES CO and THINGS [H[i]{}]{} surveys. Focusing on regions with bright CO emission, we find a CO dispersion value [[$\sigma_{\rm{CO}}$]{}]{} $ = 7.3 \pm 1.7$ [kms$^{-1}$]{}. The corresponding [H[i]{}]{} dispersion [$\sigma_{\rm{HI}}$]{}$ = 11.7 \pm 2.3$ [kms$^{-1}$]{}, yielding a mean dispersion ratio [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{}$= 1.4 \pm 0.2$, independent of radius. We find that the CO velocity dispersion increases towards lower peak fluxes. This is consistent with previous work where we showed that when using spectra averaged (“stacked”) over large areas, larger values for the CO dispersion are found, and a lower ratio [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} $ = 1.0 \pm 0.2$. The stacking method is more sensitive to low-level diffuse emission, whereas individual profiles trace narrow-line, GMC-dominated, bright emission. These results provide further evidence that disk galaxies contain not only a thin, low velocity dispersion, high density CO disk that is dominated by GMCs, but also a fainter, higher dispersion, diffuse disk component. author: - 'K.M. Mogotsi , W.J.G. de Blok , A. Cald[ú]{}-Primo , F. Walter , R. Ianjamasimanana , A.K. Leroy' title: HI and CO Velocity Dispersions in Nearby Galaxies --- Introduction ============ Gas velocity dispersions can be used to estimate the kinetic and thermal gas temperatures; to determine the mass distribution and structure of galaxies (e.g., [@petrup07]), and the stability, scale height and opacity of the gas disk. Velocity dispersions are important in studies of star formation, turbulence, the interstellar medium (ISM), and dynamics of galaxies. This is especially true of the vertical velocity dispersion $\sigma_{z}$. The rotation of the galactic disk has no effect on this component of the observed dispersion, and this makes it a useful parameter for studying the vertical structure of galactic disks. Dispersions are used to determine the stability of galactic disks against gravitational collapse using the Toomre parameter [[@toomre64; @ken89]]{}. Another link to star formation and turbulence studies is that dispersions can be used to determine the energy of the ISM (e.g., [[@agtz09; @tamb09]]{}). They are also important in determining the midplane pressure of the gas disk [[@elm89; @ler08]]{} and in star formation laws that consider a variable disk free-fall time [[@elm89; @krumck05; @ler08]]{}. [@lars81]{} used small-scale internal velocity dispersions to determine that molecular clouds are dominated by turbulent motions. Studies at larger scales can be used to determine the level of turbulence found between giant molecular clouds and in large scale motions of gas in galaxies. Studies of velocity dispersions require high spatial and velocity resolution observations (for vertical velocity dispersion studies, galaxies of low inclination are required so as to minimize the contribution from the radial and azimuthal dispersion components). The effective dispersion ($\sigma_{\rm eff}$) can be thought of as a combination of the thermal broadening ($v_{t}$) and turbulent dispersion ($\sigma_{t}$) : $$\sigma_{\rm eff}^{2} = v_{t}^{2} + \sigma_{t}^{2}$$ (e.g., [[@agtz09]]{}). The turbulent component can be decomposed into a radial ($\sigma_{r}$), angular ($\sigma_{\phi}$), and vertical ($\sigma_{z}$) component, or a planar ($\sigma_{xy}$) and vertical component ($\sigma_{z}$). Theory and simulations show that the velocity dispersion is expected to be anisotropic, with $\sigma_{r} > \sigma_{\phi} > \sigma_{z}$ and $\sigma_{xy} \sim 2\sigma_{z}$ [[@agtz09]]{}. When the beam of a telescope is large compared to the rotational velocity gradient in the observed galaxy (e.g., in high-redshift galaxies and highly inclined galaxies), beam smearing can affect the measured dispersion. For gas components with a clumpy structure (e.g., molecular gas), there are additional complications: the observed dispersion $\sigma_{\rm obs}$ is then a combination of the dispersion between clouds (cloud-cloud dispersions $\sigma_{\rm c-c}$) and the internal velocity dispersion within the clouds ($\sigma_{\rm internal}$): $$\sigma_{obs}^{2} = \sigma_{c-c}^{2} + \sigma_{\rm internal}^{2}.$$ The structure of [[H[i]{}]{}]{} is more filamentary and less clumpy than that of molecular gas. Its velocity dispersion is therefore generally not decomposed into internal and cloud-cloud components. [[H[i]{}]{}]{} velocity dispersions ----------------------------------- Since [[H[i]{}]{}]{} is the dominant gas component of galaxies and is easily observable through the $21$cm emission line, it has been extensively studied. [[H[i]{}]{}]{} velocity dispersions of nearby galaxies have been well studied, most notably by [@petrup07]{} and [@tamb09]{}. Early work showed that [[$\sigma_{\rm{HI}}$]{}]{} $\sim 6-13$[kms$^{-1}$]{} (e.g., [@shovnk84]{}, [@vnksho84]{}, [@kmpsanc93]{}), with the dispersions dropping with increasing radial distance from the center (e.g., [@kmpsanc93]{}). [@hunter01]{} and [@hunter11]{} also studied [[$\sigma_{\rm{HI}}$]{}]{} in dwarf galaxies. [@petrup07]{} performed high-resolution and high-sensitivity [[H[i]{}]{}]{} observations of the nearly face-on galaxy NGC 1058 to study its gas velocity dispersion. They found a vertical velocity dispersion of $4-14$[kms$^{-1}$]{}, which decreased with radius. These studies reached resolutions of $\sim 600$pc. @tamb09 used high-resolution [[H[i]{}]{}]{} data from The [[H[i]{}]{}]{} Nearby Galaxies Survey (THINGS; [@walt08]{}) to study [[H[i]{}]{}]{} velocity dispersions. They also found that the dispersions decreased with radius. They found a mean [[$\sigma_{\rm{HI}}$]{}]{} of $\sim 10$[kms$^{-1}$]{} at $r_{25}$, dropping off to $\sim 5 \pm 2$[kms$^{-1}$]{} at larger radii. Stacking analysis was used by [@ianja12]{} to study the velocity dispersions averaged over [H[i]{}]{} disks of the THINGS galaxies. They found [[$\sigma_{\rm{HI}}$]{}]{} $= 12.5 \pm 3.5$[kms$^{-1}$]{} ([[$\sigma_{\rm{HI}}$]{}]{} $= 10.9 \pm 2.1$[kms$^{-1}$]{} for galaxies with inclinations less than 60$^{\circ}$). This stacking analysis allowed them to study the [[H[i]{}]{}]{} velocity profiles at high signal-to-noise, enabling them to decompose the [[H[i]{}]{}]{} profiles into broad and narrow components. Fitting these components with Gaussians, they found [[$\sigma_{\rm{HI}}$]{}]{} $= 6.5 \pm 1.5$[kms$^{-1}$]{} for the narrow (cold) [[H[i]{}]{}]{} component and [[$\sigma_{\rm{HI}}$]{}]{} $= 16.8 \pm 4.3$[kms$^{-1}$]{} for the broad (warm) [[H[i]{}]{}]{} component. A similar analysis by @stilp13 of partially the same data found velocity dispersions of the bulk of the [[H[i]{}]{}]{} of $\sim 6-10$ [kms$^{-1}$]{}. CO velocity dispersions ----------------------- CO velocity dispersions have been less studied than those of [H[i]{}]{}. Mostly this has been due to technical limitations. Early observations of the lowest three CO rotational transitions found dispersions in the range of $5 - 9$ [kms$^{-1}$]{} [@stark84; @wilsco90; @combec97; @walsh02; @wilson11]. Recent instrumental developments have enabled more extensive studies of the CO distribution in galaxies, such as the HERA CO Line Extragalactic Survey (HERACLES; [@ler09]{}); see also Section [\[sec:datamethod\]]{}. HERACLES is a [CO $J = 2 \rightarrow 1$]{} survey of nearby galaxies, covering their entire star-forming disks. It partially overlaps with the THINGS survey, meaning [H[i]{}]{} and CO data are available at comparable resolutions. @cal13 used data from HERACLES and THINGS to compare CO and [[H[i]{}]{}]{} velocity dispersions as averaged over large areas using the stacking technique. They analyzed the dispersions of these stacked [[H[i]{}]{}]{} and CO velocity profiles, stacking by galactocentric radius, star formation, [[H[i]{}]{}]{}, CO and total gas density. They found that [[$\sigma_{\rm{HI}}$]{}]{}$=11.9 \pm 3.1$[kms$^{-1}$]{}, [[$\sigma_{\rm{CO}}$]{}]{}$=12.0 \pm 3.9$[kms$^{-1}$]{} with [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{}$= 1.0 \pm 0.2$. In other words, the CO dispersions they found are very similar to the [[H[i]{}]{}]{} dispersions. @cal13 suggested that this indicates the presence of an additional, more diffuse, higher dispersion molecular disk component that is similar in thickness to the [[H[i]{}]{}]{} disk (see also [@shet14]{}). This finding is in agreement with independent studies by, e.g., [@gar92]{} who find, in addition to a thin molecular disk, a 2–3 kpc thick molecular “halo” around the edge-on galaxy NGC 891. A similar thick molecular disk is also found by @com12 in M33. [@pet13]{} compared interferometric and single-dish observations of M51 (NGC 5194) and also found evidence of an extended molecular disk. Similar results have been found by [@cal15]{}, again by comparing interferometric and single-dish imaging of the molecular gas disks in nearby galaxies. The results presented in @cal13 were based on stacked profiles, i.e., profiles averaged over large regions. In this paper we use the same THINGS and HERACLES data [ as used by @cal13]{} to determine whether evidence for the diffuse molecular component can also be found in individual profiles. In particular, we investigate whether the velocity dispersion of the CO profiles changes as a function of CO intensity, which is what one would expect if a diffuse, high-velocity dispersion component is indeed present. In Section [\[sec:datamethod\]]{} we describe the data used. Section [\[sec:results\]]{} contains a description of the results of our analysis. Section [\[sec:discussion\]]{} contains a discussion of our results and a comparison to other work. In Section 5 we summarize our conclusions. Data and method {#sec:datamethod} =============== We used Hanning-smoothed CO data cubes from HERACLES [@ler09], which is a molecular gas survey of nearby galaxies using the HERA receiver array on the IRAM 30-m telescope. For the neutral hydrogen, we used residual-scaled natural-weighted [[H[i]{}]{}]{} data cubes from THINGS [@walt08], which is a 21-cm survey of 34 nearby spiral and dwarf galaxies. The observations were done with the NRAO Jansky Very Large Array. The work in this paper is based on the analysis done on 13 galaxies (see Table \[tab:props\]) that are common to both surveys and which have CO detections. The properties of these galaxies can be found in Table 1 of @walt08. For convenience, noise values and velocity resolutions of the [H[i]{}]{} and CO observations are listed in Table \[tab:props\]. [ Note that these are the same data as used in the analysis presented in @cal13.]{} [ccccc]{} Galaxy & [[H[i]{}]{}]{} Noise & CO Noise & [[H[i]{}]{}]{} $\Delta V$ & CO $\Delta V$\ & \[mJybeam$^{-1}$\] & \[mK\] & \[kms$^{-1}$\] & \[kms$^{-1}$\]\ (1) & (2) & (3) & (4) & (5)\ NGC 628 & $0.60$ & $21$ & $2.6$ & $5.2$\ NGC 925 & $0.57$ & $16$ & $2.6$ & $5.2$\ NGC 2403 & $0.38$ & $19$ & $5.2$ & $5.2$\ NGC 2841 & $0.35$ & $16$ & $5.2$ & $5.2$\ NGC 2903 & $0.41$ & $21$ & $5.2$ & $5.2$\ NGC 2976 & $0.36$ & $20$ & $5.2$ & $5.2$\ NGC 3184 & $0.36$ & $17$ & $2.6$ & $5.2$\ NGC 3198 & $0.33$ & $17$ & $5.2$ & $5.2$\ NGC 3351 & $0.35$ & $19$ & $5.2$ & $5.2$\ NGC 4214 & $0.69$ & $19$ & $1.3$ & $5.2$\ NGC 4736 & $0.33$ & $21$ & $5.2$ & $5.2$\ NGC 5055 & $0.36$ & $26$ & $5.2$ & $5.2$\ NGC 6946 & $0.55$ & $25$ & $2.6$ & $5.2$\ {width="80.00000%"} \[fig:noisesim\] The CO data generally have a spatial resolution of $13''$. The natural-weighted [[H[i]{}]{}]{} data cubes mostly have resolutions better than this, but were smoothed to $13''$ to match the resolution of the CO data. A number of recent studies of [[H[i]{}]{}]{} velocity profiles have used Gauss-Hermite profiles to take into account asymmetries in the profiles or used multiple Gaussian components to quantify the presence of different components of the ISM (e.g., @deB08, @ianja12). We did explore these fitting functions for our profiles, but found that the CO profiles are better described by simple, single Gaussians. In order to minimize the number of fit parameters, and as we are only interested in the general width of the profile, we therefore use single Gaussians to fit both the [[H[i]{}]{}]{} and CO profiles. In fitting the profiles we imposed a $4S$ noise cutoff on the fitted peak fluxes of the profiles, where $S$ is the rms noise of the profile. Only positions where both [[H[i]{}]{}]{} and CO profiles had peak fluxes greater than $4S$ were retained. For the [[H[i]{}]{}]{} data, determining the $4S$ values was done using non-residual-scaled cubes, as residual-scaling affects the relation between signal and noise (see @walt08 for a full description of the residual scaling procedure). These results were then applied as a mask to the residual-scaled cubes. The remaining velocity profiles of these masked residual-scaled cubes were then fitted and analyzed. In addition to the peak-flux criterion, we also imposed a velocity resolution cutoff where all profiles with fitted dispersions smaller than the velocity resolution of their data cube were removed. We simulated how the uncertainties in the fitted dispersions behave by producing random Gaussian noise at velocity resolutions relevant to our data, adding pre-determined Gaussian velocity profiles to them and re-fitting the data. Simulations were performed with input Gaussian profiles of different amplitudes and dispersions. A thousand iterations of data simulation and fitting were performed for each input amplitude and dispersion value. This was done for different velocity resolutions and the averages of the fit uncertainties are plotted in Fig. \[fig:noisesim\] (top and middle panel). Input dispersions ranged between $2.6$ [kms$^{-1}$]{} and $20$ [kms$^{-1}$]{}. For velocity profiles with peak fluxes greater than $4S$ and velocity resolutions of $2.6$ [kms$^{-1}$]{}, the mean uncertainties in the fitted dispersion were smaller than $\sim 2$ [kms$^{-1}$]{}. For velocity profiles with peak flux equal to $4S$ and velocity resolutions of $5.2$ [kms$^{-1}$]{}, the mean errors in the fitted dispersion were between $1.4$ [kms$^{-1}$]{} and $\sim 3$ [kms$^{-1}$]{}. A small number of the CO spectra showed some minor baseline ripples resulting in a slightly non-Gaussian noise behaviour. We therefore repeated the same procedure but added noise extracted from these CO cubes rather than random Gaussian noise. This was done for profiles with peak values between $1S$ and $16S$. The results are plotted in Fig. \[fig:noisesim\] (bottom panel). The results for the CO noise simulation are consistent with the results from the Gaussian noise simulation down to $4S$ peak flux levels. Results {#sec:results} ======= Comparing [[H[i]{}]{}]{} and CO velocity dispersions {#subsec:pixpixan} ---------------------------------------------------- Using the results from the Gaussian fits, [H[i]{}]{} and CO dispersion maps were made for each galaxy. In addition, we made dispersion difference ([[$\sigma_{\rm{CO}} - \sigma_{\rm{HI}}$]{}]{}) and dispersion ratio ([[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{}) maps for each galaxy by taking the CO and [[H[i]{}]{}]{} dispersion maps and then doing a pixel-by-pixel subtraction or division. The dispersions were binned into $1$ [kms$^{-1}$]{} bins. Histograms of the $\sigma_{\rm{HI}}$ and $\sigma_{\rm{CO}}$ distributions for those positions in each galaxy where [[H[i]{}]{}]{} and CO were both present are shown in Figs. \[fig:HIDisp\] and \[fig:CODisp\]. The distribution of dispersion values from pixels outside the central 0.2$r_{25}$ and with small fit uncertainties ($\Delta \sigma \leq 1.5$ [kms$^{-1}$]{}) are shown as the shaded histograms in the figures. The 0.2$r_{25}$ selection was used in order to minimize the effect of beam smearing, as discussed later in this section. Dispersions are plotted against the number of resolution elements (defined as the ratio of the number of pixels and the number of pixels per beam), or, equivalently, the number of beams. From the histograms it is clear that in regions where there is both [H[i]{}]{} and CO emission, $\sigma_{\rm{HI}}$ values range from $\sim 5 - 30$ [kms$^{-1}$]{} and $\sigma_{\rm{CO}}$ values range from $\sim 5 - 25$ [kms$^{-1}$]{}. The $\sigma_{\rm{HI}}$ modes range from $9- 22$ [kms$^{-1}$]{} and $\sigma_{\rm{CO}}$ modes range from $6 - 15$ [kms$^{-1}$]{} (see Table \[tab:modes\]). Most of the high dispersions have large fitting errors and/or are from pixels in the central regions of galaxies, as shown in Figs. \[fig:HIDisp\] and \[fig:CODisp\]. Such large dispersions are usually due to multiple gas components in the line of sight and/or beam smearing. These give non-Gaussian profiles resulting in bad fits. [rccccccc]{} Galaxy & [[$\sigma_{\rm{CO}}$]{}]{} & [[$\sigma_{\rm{HI}}$]{}]{} & & &\ & (kms$^{-1}$) & (kms$^{-1}$) & (kms$^{-1}$) & (kms$^{-1}$) & (kms$^{-1}$) & &\ (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)\ NGC 628 & 6 (7.5) & 9 (9.1) & 6 (6.5) & $-$1.70 ($\pm$0.04) & $-$2 ($-$1.6) & 1.2 ($\pm$0.01)& 1.2 (1.3)\ NGC 925 & 6 (8.5) & 11 (13.2) & 10 (10.9) & $-$3.5 ($\pm$0.3) & $-$3 ($-$4.7) & 1.4 ($\pm$0.04)& 1.3 (1.7)\ NGC 2403 & 6 (8.3) & 10 (12.3) & 7 (8.7) & $-$3.8 ($\pm$0.1) & $-$4 ($-$4.0) & 1.5 ($\pm$0.02)& 1.4 (1.6)\ NGC 2841 & 8 (10.9) & 10 (18.0) & 9 (15.5) & $-$3.5 ($\pm$0.9) & $-$3 ($-$7.1) & 1.3 ($\pm$0.06)& 1.3 (2.0)\ NGC 2903 & 15 (21.6) & 22 (25.3) & 8 (12.1) & $-$5.5 ($\pm$0.2) & $-$5 ($-$3.7) & 1.3 ($\pm$0.02)& 1.2 (1.4)\ NGC 2976 & 6 (9.3) & 11 (12.2) & 11 (13.2) & $-$3.1 ($\pm$0.2) & $-$5 ($-$2.9) & 1.3 ($\pm$0.05)& 1.1 (1.4)\ NGC 3184 & 8 (8.6) & 10 (11.1) & 7 (8.3) & $-$2.57 ($\pm$0.03) & $-$2 ($-$2.5) & 1.3 ($\pm$0.02)& 1.2 (1.4)\ NGC 3198 & 11 (15.0) & 17 (20.2) & 11 (12.5) & $-$6.6 ($\pm$0.4) & $-$5 ($-$5.2) & 1.4 ($\pm$0.06)& 1.2 (1.6)\ NGC 3351 & 7 (14.1) & 9 (18.9) & 7 (9.8) & $-$2.8 ($\pm$0.3) & $-$3 ($-$4.7) & 1.4 ($\pm$0.04)& 1.2 (1.5)\ NGC 4214 & 6 (8.0) & 16 (14.0) & 6 (7.4) & $-$6.0 ($\pm$0.2) & $-$6 ($-$5.8) & 1.4 ($\pm$0.04)& 1.6 (1.9)\ NGC 4736 & 10 (17.7) & 15 (23.5) & 7 (10.4) & $-$3.4 ($\pm$0.1) & $-$4 ($-$3.2) & 1.2 ($\pm$0.02)& 1.0 (1.2)\ NGC 5055 & 9 (14.9) & 11 (17.9) & 7 (9.9) & $-$2.9 ($\pm$0.1) & $-$3 ($-$3.0) & 1.2 ($\pm$0.01)& 1.3 (1.3)\ NGC 6946 & 9 (11.6) & 13.0 (13.5) & 7 (8.4) & $-$2.4 ($\pm$0.1) & $-$2 ($-$2.0) & 1.2 ($\pm$0.02)& 1.2 (1.3)\ The $\sigma_{\rm{HI}}$ distributions clearly peak at values much larger than the dispersion cutoffs imposed due to the velocity resolution of the data. However, many of the $\sigma_{\rm{CO}}$ distributions have peaks near the dispersion cutoffs. In a few cases clear $\sigma_{\rm{CO}}$ distribution peaks are not seen (e.g., NGC 2403), and therefore the true mean (and mode) $\sigma_{\rm{CO}}$ values for these galaxies are likely to be smaller than $5.2$ km s$^{-1}$. The incomplete sampling and asymmetry of the velocity dispersion histograms means that, especially for the CO, the mean is not a good statistic to characterize the distribution (it will overestimate the typical dispersion value). We therefore also use the mode to describe the dispersion distributions. The modes were calculated after binning the dispersions using a $1$ [kms$^{-1}$]{} bin size. The values are listed in Table \[tab:modes\]. Most of the [[$\sigma_{\rm{CO}}$]{}]{} modes range from $6$ to $11$ [kms$^{-1}$]{} (12/13 galaxies) while their means range from $7$ to $15$ [kms$^{-1}$]{} (11/13 galaxies); most of the [[$\sigma_{\rm{HI}}$]{}]{} modes range from $9$ to $17$ [kms$^{-1}$]{} (12/13 galaxies) and their means range from $9$ to $21$ [kms$^{-1}$]{} (11/13 galaxies). NGC 2841, NGC 2903, NGC 3198 and NGC 3351 were not included in the determination of the average values due to their high inclinations and very asymmetric dispersion distributions. The average [[$\sigma_{\rm{CO}}$]{}]{} mode is $7.3 \pm 1.7$ [kms$^{-1}$]{} (average of the [[$\sigma_{\rm{CO}}$]{}]{} means is $10.5 \pm 3.6$ [kms$^{-1}$]{}), and the average [[$\sigma_{\rm{HI}}$]{}]{} mode is $11.7 \pm 2.3$ [kms$^{-1}$]{} (average of the [[$\sigma_{\rm{HI}}$]{}]{} means is $14.1 \pm 4.3$[kms$^{-1}$]{}). Characteristic values of [[$\sigma_{\rm{HI}}$]{}]{} and [[$\sigma_{\rm{CO}}$]{}]{} are listed in Table \[tab:dispval\]. We also list the median values there for comparison with results from @cal13 [ which were derived from stacking the same data we use]{} (see Section \[sec:discussion\]). {width="80.00000%"} {width="80.00000%"} \[fig:CODisp\] We constructed histograms of the [[$\sigma_{\rm{CO}} - \sigma_{\rm{HI}}$]{}]{} values using bins of $1$ [kms$^{-1}$]{}. For the [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} data we made histograms using bins of size $0.2$. Figure \[fig:DispDiff\] shows the [[$\sigma_{\rm{CO}} - \sigma_{\rm{HI}}$]{}]{} distributions for each of the galaxies. Figure \[fig:DispRat\] shows the [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} distributions. These all have Gaussian shapes and are symmetric and well-sampled. We therefore fitted Gaussians to the distributions. The fitted mean dispersion difference and ratios are shown in Table \[tab:means\]. The CO distribution in NGC 925 and NGC 4214 only encompasses a few resolution elements and care should be taken when interpreting their fits. The mean [[$\sigma_{\rm{CO}} - \sigma_{\rm{HI}}$]{}]{} value is $-3.3 \pm 1.2$ [kms$^{-1}$]{}; the mean [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} is $1.4 \pm 0.2$ [kms$^{-1}$]{}. A summary of the mean, modes, and medians of the dispersion values is shown in Table \[tab:dispval\]. [@cal13]{} quantified the effect of beam smearing in our galaxies by simulating spiral galaxies with $30^{\circ}$, $60^{\circ}$, and $80^{\circ}$ inclinations. They found that beam smearing is greatest in the central regions of galaxies and in highly inclined galaxies. The observed dispersion can be increased by at most a factor of 1.2 at $0.2$$r_{25}$ for galaxies with $30^{\circ}$ inclination, 1.5 for $60^{\circ}$, and 1.8 for $80^{\circ}$, with these factors decreasing quickly toward unity at larger radii. They therefore used an $0.2$$r_{25}$ radial cutoff for their analysis. Even though Figures [\[fig:HIDisp\]]{} and [\[fig:CODisp\]]{} and Table [\[tab:comp\]]{} show that including the inner pixels does not greatly affect our conclusions, in our radial analysis and distribution width analysis we only use pixels with radii greater than $0.2$$r_{25}$ which are therefore not affected by beam smearing. Our [[$\sigma_{\rm{HI}}$]{}]{} values are in agreement with [@ler08]{} ([[$\sigma_{\rm{HI}}$]{}]{}$ = 11 \pm 3$[kms$^{-1}$]{}), [@ianja12]{} ([[$\sigma_{\rm{HI}}$]{}]{}$ = 12.5 \pm 3.5$[kms$^{-1}$]{}) and [@cal13]{} ([[$\sigma_{\rm{HI}}$]{}]{}$ = 11.9 \pm 3.1$[kms$^{-1}$]{}) who all analyzed the THINGS galaxies. [lrrrrr]{} & & & &\ & & & &\ \ mode & 7.3 $\pm$ 1.7 & 11.7 $\pm$ 2.3 & $-$3.4 $\pm$ 1.4 & 1.3 $\pm$ 0.2\ mean & 10.5 $\pm$ 3.6 & 14.1 $\pm$ 4.3 & $-$3.3 $\pm$ 1.4 & 1.4 $\pm$ 0.2\ median & 9.5 $\pm$ 3.0 & 13.1 $\pm$ 3.0 & $-$3.3 $\pm$ 1.2 & 1.4 $\pm$ 0.2\ fitted mean & & & $-$3.3 $\pm$ 1.2 & 1.4 $\pm$ 0.2\ ![Distributions of the dispersion difference [[$\sigma_{\rm{CO}} - \sigma_{\rm{HI}}$]{}]{} values of the galaxies. The lines are colour-coded by galaxy (see legend). The y-axis is in resolution elements (resolution elements = \[number of pixels\]/\[number of pixels per single resolution element\]). Values for NGC 925, NGC 2841, NGC 2903, NGC 3198, NGC 3351, NGC 4214 and NGC 4736 are multiplied by a factor of 5 for better comparison with the other galaxies.[]{data-label="fig:DispDiff"}](f4){width="45.00000%"} ![Distributions of the dispersion ratios [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} of the galaxies for individual resolution elements. The y-axis is in resolution elements (resolution elements = \[number of pixels\]/\[number of pixels per single resolution element\]). Values for NGC 925, NGC 2841, NGC 2903, NGC 3198, NGC 3351, NGC 4214 and NGC 4736 are multiplied by a factor of 5 for better comparison with the other galaxies.[]{data-label="fig:DispRat"}](f5){width="45.00000%"} We also studied radial trends of the dispersions. The radial [[$\sigma_{\rm{HI}}$]{}]{}, [[$\sigma_{\rm{CO}}$]{}]{} and [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} distributions are shown in Figs. \[fig:COHIrad\] and \[fig:DRrad\]. These plots were made using annuli where the filling factors were higher than $10$% and $25$%, respectively. In our analysis we use the results from the 10% annuli. Comparison with the 25% annuli shows that this choice of filling factor has little effect on our results. The [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} values in this analysis were calculated by azimuthally averaging the [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} maps.\ Figure \[fig:COHIrad\] shows [[$\sigma_{\rm{CO}}$]{}]{} and [[$\sigma_{\rm{HI}}$]{}]{} decreasing with radius for most of the galaxies. These also flatten off at larger radii. This behaviour was already seen in [[H[i]{}]{}]{} by [@tamb09]{}. The [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} values remain roughly constant for most of the radial range covered; this can be seen in Fig. \[fig:DRrad\]. {width="80.00000%"} {width="80.00000%"} [l r r r r]{} & & &\ & & &\ [[$\sigma_{\rm{CO}}$]{}]{} mean & 9.3 $\pm$ 2.1 & 8.9 $\pm$ 2.1 & 12.8 $\pm$ 3.9\ [[$\sigma_{\rm{CO}}$]{}]{} median & 8.6 $\pm$ 1.8 & 8.4 $\pm$ 2.0 & 12.0 $\pm$ 3.9\ [[$\sigma_{\rm{HI}}$]{}]{} mean & 12.7 $\pm$ 2.1 & 12.3 $\pm$ 2.3 & 12.7 $\pm$ 3.1\ [[$\sigma_{\rm{HI}}$]{}]{} median & 12.2 $\pm$ 1.9 & 11.9 $\pm$ 2.1 & 11.9 $\pm$ 3.1\ [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{} mean & 1.5 $\pm$ 0.2 & 1.5 $\pm$ 0.3 & 1.0 $\pm$ 0.2\ Comparison with Stacking Results {#sec:discussion} ================================ We now compare our results with the stacking analysis by @cal13. In that study, [ which used the same data sets as the ones used here,]{} the stacking procedure included all possible CO profiles, i.e., there was no rejection based on CO peak-flux and all positions where an H[[i]{}]{} velocity was available for use in stacking the CO profile were used. The only profiles excluded from their analysis were those at radii less than $0.2 r_{25}$. The mean and median dispersion values they found are shown in Table \[tab:comp\]. These [[$\sigma_{\rm{CO}}$]{}]{} values are higher than our mean and median values. To check whether this difference is not caused by the higher uncertainties associated with low peak-flux profiles, we rederived our values for a number of different noise cutoffs between $4S$ and $8S$. Our mean and median dispersion values for data with the central $0.2 r_{25}$ pixels removed and using $4S$ and $8S$ noise cutoffs are shown in Table \[tab:comp\]. Our pixel-by-pixel [[$\sigma_{\rm{CO}}$]{}]{} values are lower irrespective of which noise cutoff we use. Our [[$\sigma_{\rm{HI}}$]{}]{} values remain similar to the @cal13 values. Due to the noise cutoff used here, our analysis does not probe the low peak-flux regime that the stacking analysis in @cal13 is sensitive to. It is therefore possible that the difference found in dispersion values could be caused by profiles with a peak flux lower than $4S$ having systematically higher velocity dispersions. It is, however, difficult to directly and accurately measure the individual velocity dispersions of these low peak-flux profiles. We therefore evaluated the impact of the low peak-flux spectra by creating histograms of the velocity dispersion values for different noise cutoff values. If lower peak-flux profiles do indeed have higher velocity dispersions, then we would expect the fraction of high-dispersion profiles to decrease with increasing noise cutoff. In other words, the prominence of any high-dispersion tail in the histogram should decrease. For this analysis we used data from pixels with radii greater than $0.2$$r_{25}$. An example is shown in the top left panel of Fig. \[fig:width\]. Here we show the normalized $\sigma_{\rm CO}$ distributions for NGC 2403 derived using various noise cutoffs between $4S$ and $10S$. It is clear that the distribution becomes more narrow with increasing cutoff value. We quantify this with the histogram half-width at 20 percent of the maximum, where the half-width is measured in the direction of higher dispersions, with respect to the value of the histogram maximum (or the mode), as indicated in the left panel of Fig. \[fig:width\]. We have repeated this analysis for all our sample galaxies, except for NGC 925, NGC 4214 and NGC 3198, where CO emission is faint and limited in extent, and NGC 2903, where the dispersion values are dominated by streaming motions along the bar. We also excluded the central part of NGC 3351 which is dominated by a compact bar. {width="80.00000%"} ![Differences between the stacked mean [[$\sigma_{\rm{CO}}$]{}]{} dispersion calculated for profiles with peak flux $> 8S$ (${\sigma}_{8}$) and $>5S$ (${\sigma}_{5}$) plotted against radius.[]{data-label="fig:stack2"}](f9){width="45.00000%"} The top-right panel of Fig. \[fig:width\] shows the values of the measured widths as a function of noise cutoff. For the majority of the galaxies shown there, the width becomes narrower toward higher cutoff values and the high-dispersion tail less prominent. We thus find that the fraction of high-dispersion spectra does indeed decrease with increasing peak flux and it is therefore likely that the low peak-flux profiles included in the stacking analysis (but excluded in ours) have systematically higher velocity dispersions. This implies that repeating the stacking analysis of @cal13 with a *higher* noise cutoff should give *lower* dispersion values than their original analysis. We therefore performed a stacking analysis on our data using various cutoffs. As an example, in Figure \[fig:stack2\] we show, as a function of radius, the difference between the $5S$ and the $8S$ stacked dispersions. The average difference between the dispersions is $\sim 1.5$ [kms$^{-1}$]{}, independent of radius, with the $5S$ values systematically higher than the $8S$ ones. To test whether the trend in velocity dispersion we found is not caused by the increasing importance of the noise toward lower peak fluxes (which might be expected to broaden profiles), we repeat our analysis using simulated profiles. We created Gaussian profiles with peak values between $4S$ and $10S$, and with dispersions chosen with a probability distribution equal to the observed [[$\sigma_{\rm{CO}}$]{}]{} distribution of NGC 2403, as shown in Fig. \[fig:CODisp\] (for pixels with radii greater than $0.2$$r_{25}$). We explicitly assume that the velocity dispersion is independent of the peak flux. We compare the dispersion distributions found at various peak-flux levels in Figure [\[fig:width\]]{} (bottom left). The change in these distributions is very different from that as observed for NGC 2403, as shown in the bottom right panel in Figure [\[fig:width\]]{}. The higher velocity dispersions measured at low peak flux are therefore not due to noise affecting the profiles, but due to an increase of the velocity dispersion toward lower peak fluxes. A similar anti-correlation, but for [[H[i]{}]{}]{} rather than CO, was found for a number of dwarf galaxies by @hunter01 [@hunter11]. @hunter11 suggest that this anti-correlation is roughly consistent with a uniform pressure throughout these galaxies, as also found in simulations of magneto-rotational instabilities by @piontek05. Returning to the observed CO velocity dispersions, @cal13 note that the higher dispersion values that are found in the stacked profiles can be explained with a diffuse, extended molecular gas component that pervades our galaxies in addition to the molecular gas in GMCs in the thin, “cold” CO disk. Our pixel-by-pixel analysis is limited to pixels with bright CO emission, which is dominated by the GMCs. These velocity profiles are narrower than those dominated by emission from the diffuse CO disk. These differences therefore are further evidence that a diffuse, high-dispersion component of molecular gas is present in our galaxies in addition to a thin molecular disk. The diffuse component of molecular disks may thus be a common feature in disk galaxies. Summary ======= We have measured the velocity dispersions in individual [H[i]{}]{} and CO profiles of a number of THINGS disk galaxies. We find an [H[i]{}]{} velocity dispersion of [[$\sigma_{\rm{HI}}$]{}]{}$ = 11.7 \pm 2.3$ [kms$^{-1}$]{}. The corresponding CO value is [[$\sigma_{\rm{CO}}$]{}]{}$ = 7.3 \pm 1.7$ [kms$^{-1}$]{}. The ratio between these two dispersions is [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{}$ = 1.4 \pm 0.2$ and is not correlated with radius. In a previous study using the same data, @cal13, by stacking individual velocity profiles, found a systematically higher CO velocity dispersion and a ratio [[$\sigma_{\rm{HI}} / \sigma_{\rm{CO}}$]{}]{}$ = 1.0 \pm 0.2$. This difference can be explained if low peak-flux CO profiles have a systematically higher velocity dispersion than high-peak flux profiles. Our pixel-by-pixel analysis preferrentially selects the bright, high peak-flux CO profiles, in contrast with the stacking analysis which also includes large numbers of low peak-flux CO profiles. The relation of [$\sigma_{\rm{CO}}$]{} decreasing with increasing profile amplitude is consistent with a picture where the bright CO regions (preferentially selected in studies of individual profiles) are dominated by narrow-line GMCs, with a more diffuse, higher dispersion component (more efficiently detected in stacking analyses) becoming more prominent toward lower intensities. A pixel-by-pixel analysis is therefore a good way to study the thin molecular disk component where GMCs dominate the emission. In turn, stacking analyses are more sensitive to the diffuse, high-dispersion extended molecular disk component. Our results thus provide further evidence for the suggestion presented in [@cal13]{} that many disk galaxies have an extended, diffuse molecular disk component in addition to a thin, GMC-dominated, molecular disk. We gratefully thank the anonymous referee for all the comments and suggestions that improved the content of the paper. K.M.M. gratefully acknowledges support from the Square Kilometre Array South Africa (SKA-SA) and the National Astrophysics and Space Science Program (NASSP). K.M.M. would like to thank S. Schutte for the useful discussions during the preparation of the manuscript. The work of W.J.G.d.B. was supported by the European Commission (grant FP7-PEOPLE-2012-CIG \#333939). R.I. acknowledges funding from the National Research Foundation (NRF grant number MWA1203150687) and the University of South Africa (UNISA) postdoctoral grant. Facilities: , . Comparing [[H[i]{}]{}]{} velocity dispersions and second moments ================================================================ In a previous study, @tamb09 determined the second moments of the [[H[i]{}]{}]{} profiles of the THINGS galaxies as an estimate for the velocity dispersions. These second moments were measured as a function of radius over the full extent of the [H[i]{}]{} disk. To gauge how well the second-moment values match the Gaussian dispersion values [$\sigma_{\rm{HI}}$]{}, we derive both parameters for our [H[i]{}]{} profiles, as also measured over the entire radial range and full area of the [H[i]{}]{} disk, i.e., also including regions without CO emission. {width="80.00000%"} Figure \[fig:HIallrad\] shows that there are some slight differences in the second-moment values and Gaussian fitted dispersions. For most galaxies the largest differences between second-moment values and Gaussian fitted dispersions are found in the inner regions of galaxies, with second-moment values being larger than the Gaussian fitted dispersions. The inner regions of the galaxies have more non-Gaussian profiles than the outer regions. This shows that the second moment is more sensitive to non-Gaussianities than profile fits and in these cases should be interpreted with care. We note that the [[H[i]{}]{}]{} dispersions associated with the CO disk (the inner star-forming disk) are higher than the dispersion as measured over the entire [[H[i]{}]{}]{} disk (which includes the outer parts of the galaxy where there is no detectable CO). This is can be explained by the higher star formation rate in the inner disk compared to the outer disk. A further discussion of this is, however, beyond the scope of this paper. 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--- abstract: 'The mechanisms responsible for contention of activity in systems represented by networks are crucial in various phenomena, as in diseases such as epilepsy that affects the neuronal networks, and for information dissemination in social networks. The first models to account for contained activity included triggering and inhibition processes, but they cannot be applied to social networks where inhibition is clearly absent. A recent model showed that contained activity can be achieved with no need of inhibition processes provided that the network is subdivided in modules (communities). In this paper, we introduce a new concept inspired in the Hebbian theory through which activity contention is reached by incorporating a dynamics based on a decaying activity in a random walk mechanism preferential to the node activity. Upon selecting the decay coefficient within a proper range, we observed sustained activity in all the networks tested, viz. random, Barabasi-Albert and geographical networks. The generality of this finding was confirmed by showing that modularity is no longer needed if the dynamics based on the integrate-and-fire dynamics incorporated the decay factor. Taken together, these results provide a proof of principle that persistent, restrained network activation might occur in the absence of any particular topological structure. This may be the reason why neuronal activity does not outspread to the entire neuronal network, even when no special topological organization exists.' author: - Diego Raphael Amancio - 'Osvaldo Novais Oliveira Jr.' - Luciano da Fontoura Costa date: 'Received: date / Accepted: date' title: A decaying factor accounts for contained activity in neuronal networks with no need of hierarchical or modular organization --- Introduction {#intro} ============ The human brain is an example of complex system [@csystem; @csystem2; @csystem3], where the local dynamics is unable to predict the global functionality because the brain components relate to each other in a non-trivial way. This is why the brain has often been represented as complex networks [@newman; @survey] at various scales [@cit2; @cit3], ranging from neurons linked by synapses to interconnected large brain regions [@watts98; @sporns2004]. With this type of representation, one may learn about brain functionality and operation. For example, the network framework allowed the observation of synchronization patterns and the establishment of conditions for epileptic seizures, in terms of the topological organization [@cit3; @info; @percha]. It has been suggested that the human brain works in a critical region analogous to a phase transition dynamics between ordered and chaotic behaviors [@kaiser2]. This critical operation, characterized by a persistent dynamics where the activation neither quickly dies out nor activates the whole network, is essential for studying functional patterns in complex neural networks [@kaiser2; @kaiser]. Because the critical operation is responsible for the maintenance of brain functions [@kaiser2] (e.g., in the critical operation processing capabilities are enhanced [@enhanced1; @enhanced2] and epileptic seizures are prevented [@kaiser], it is crucial to determine the factors leading to such restrictions of activity. While traditional neural networks models require excitatory and inhibitory connections to restrain activity in the critical operation [@curmod1; @curmod2], recent models have shown it to be possible to restrain activity without inhibitory connections [@kaiser2; @kaiser]. More specifically, with nodes representing cortical columns [@corticalcolumns] rather than individual neurons, a simple dynamics could explain restrained activity in hierarchical modular neural networks [@bizenger; @nature2011; @nature2008] under a wide range of initial parameters [@kaiser]. In this paper, we introduce a dynamics that generalizes the approach reported in Ref. [@kaiser]. Taking as inspiration the Hebbian theory, according to which neuronal links are reinforced whenever they are stimulated, we adopted a random walk dynamics preferential to the node activity. We show that the neuronal activity can be topologically constrained even in non-modular structures such as random, geographical and scale-free networks. Therefore, the existence of hierarchy and communities is no longer required. In addition, differently from the model in Ref. [@kaiser], the number of active nodes in the steady state is weakly dependent on the initial localization of the random walker. Description of the Preferential Random Walk {#descricao} =========================================== Random walks have been employed in a wide variety of contexts [@newmanbook; @wang; @newrw1; @newrw2; @teago], including brain networks as a complementary strategy to the analysis of synchronization properties provided by spectral analysis [@bocateli; @estrada; @spectra]. Traditional random walks were used to analyze the interplay between structure and function in cortical and neural networks [@wang; @corrs], while self-avoiding random walks [@sarw] were employed to show that the human cortical network is most resilient to brain injuries [@csystem]. The proposed model of preferential random walk is based on the Hebbian theory [@hebe], which imposes a rule to determine how the weights connecting neurons in neural networks are updated. Formally, the weight $w_{ij}$ linking neurons $i$ and $j$ is updated according to: $$w_{ij} = p^{-1} \sum_{k=1}^p x_i^k x_j^k,$$ where $x_i^k$ is the $k$-th input for neuron $i$ and $p$ is the number of training patterns in the artificial neural network [@ann]. Thus, neurons that are repeatedly fired at the same time will tend to have their links strengthened. This effect is simulated through the introduction of a preferential random walk where the particle performing the random walk is more likely to leap onto a node that has already been frequently accessed in the last iterations. Thus, analogously to the Hebbian theory for edges weights, nodes frequently accessed will tend to increase its activity due to the reinforcement effect. The proposed random walk is preferential to the activity $\mathcal{T}$ of each node at the time step $t$. At $t = t_0 = 1$, every node $i$ has activity $\mathcal{T}_i^{t_0} = 1$. Assuming that the particle performing the random walk over the network is at node $i$, having $j$ as its neighbor, the next node to be visited at $t + 1$ is chosen according to the probability $P^{t+1}_{i \rightarrow j}$: $$P^{t+1}_{i \rightarrow j} = \frac{ \mathcal{T}_{j}^t }{ \sum_k a_{ik} \mathcal{T}_{k}^t },$$ where $$\label{eq.mavg} a_{ik} = \left\{ \begin{array}{ll} 1 & \textrm{ if $i$ and $k$ are connected}, \\ 0 & \textrm{ otherwise.} \\ \end{array} \right.$$ In other words, the particle tends to propagate toward the most active neighbor. The next node visited will have its activity increased by one unity. At the beginning of every time step $t$, all nodes have their activity decaying according to the following rule: $$\label{decaimentoeq} \mathcal{T}_i^{t+1} = \alpha \mathcal{T}_i^{t},$$ where $\alpha$ is the rate of activity preservation, which is restricted to the interval $0 < \alpha < 1$. In the experiments, a given node $i$ was considered activated when $\mathcal{T}_i \geq \tau_c$. Arbitrarily, we adopted $\tau_c = 2.5$. Results and Discussion {#results-and-discussion .unnumbered} ====================== Sustained activity was observed with the preferential random walk process provided that the preservation coefficient $\alpha$ (equation \[decaimentoeq\]) was $\alpha \leq 0.9999$. Figure \[fig1\] shows that after an abrupt onset of activity in the first few time steps, a sharp decrease occurs for all three networks leading eventually to only a single active connected component[^1]. The sharpest decay occurred for the geographical network, since at $t = 4.0 \times 10^4$ activity was restrained to only one component. The activation in geographic networks probably stabilizes faster because these networks are devoid of long-range connections. As such, the activation does not spread to other regions of the network. {width="100.00000%"} Similar results were obtained for the number of active nodes [(Figure S2 of the SI)]{}, since [after $2.0 \times 10^5$ steps]{} activity decayed to a small number of nodes for the $3$ networks. Here, however, important differences were noted in comparing the networks. The maximum number of activated nodes was ca. 7% for the random and BA networks, but for the geographical network this was ca. 1%. This difference may be ascribed to the small-world property inherent in both BA and random networks. As one should expect, the number of active nodes increases with the preservation factor $\alpha$, as shown in [Figure S3 of the SI]{}. Further analysis can be made by considering which nodes remain active throughout the simulations. Figure \[fig2.aa\] shows the evolution of activated nodes in a small geographical network in a given execution of the model (a similar evolution was observed for the other network models). It is worth noting that even when the random walk starts from the same initial node, the set of active nodes at $t=t_f$ is not always the same. Actually, most of the times the final activation is spread along the nearest neighbors from the starting node, as illustrated in Figure \[fig2.a\] (see also Figure S4 of the SI). The amount of active nodes appears to be independent of the starting node for the random walk, as very little impact was observed on the spreading [(see Figure S5 of the SI)]{}. This is in contrast with the results in Ref. [@kaiser]. The set of active nodes was largely preserved throughout the time evolution for the random and the geographical networks, as shown in Figure \[presRate\]. In contrast, only 83% of the active nodes preserved their activity for the BA network, probably because activity was directed toward the hubs. {width="100.00000%"} An important difference was observed with regard to the results reported by [@kaiser]. Rather than being organized in communities [@nature2011] (i.e., in groups of nodes strongly connected to each other and weakly connected to nodes of other groups), the active nodes were not restrained to communities, as indicated by the degree of modularity $Q$ in Figure \[figmodularity\] (the higher the value of $Q$, the greater the tendency to restrain activity in a community). The BA network again behaved differently. {width="100.00000%"} The importance of introducing a decay factor in the dynamics was verified by considering the widely used integrate-and-fire dynamics [@ifire]. The latter can be described mathematically using: $$T_i^{t+1} = \left\{ \begin{array}{ll} T_i^t + \gamma \sum_{j=1}^N \sum_f a_{ij} \delta(t-t_j^f) & \textrm{if } T_i^t < \tau_f \\ \gamma \sum_{j=1}^N \sum_f a_{ij} & \textrm{if } T_i^t \geq \tau_f \\ \end{array} \right.$$ where $T_i^t$ represents the accumulated potential of node $i$ at time $t$, $t_j^f$ is the time when the $f$-th spike occurs and $\gamma$ is the coupling factor. When the potential $T_i$ reaches a given threshold $\tau_f$, the node sends the accumulated activation throughout the outcome links and its internal state is cleaned. In the simulations we considered $\gamma = 1$ and $\tau_f = 3$. The effect of decay was taken into account simply by multiplying the current potential $T_i^t$ by $\alpha$. Figure \[randd\] confirms that activity is sustained for the random network, provided a proper value for the rate of activity preservation is used. Interestingly, the same conclusion was found for the other two network models [(see Figures S6 and S7 of the SI)]{}. Therefore, the observation of contained activity for modular networks reported by [@kaiser] no longer applies when the proposed dynamics is adopted. {width="100.00000%"} Conclusion ========== The study of the brain with the representation provided by the network framework is one of the most promising issues in neuroscience. In the current paper we have employed complex networks to study the conditions influencing the operation in the critical point. Upon introducing a novel random walk preferential to the activity of nodes (i.e., preferential to the recent frequency of access) inspired in the biological model of Hebb [@hebe], we found as a proof of principle that persistent and contained network activation can occur not only in the absence of inhibitory nodes, but also in the absence of a hierarchical clustered organization. In particular, this result can be seen as a generalization of previous findings with similar dynamics [@kaiser] advocating that small-worlds and random networks could not restrict the activity without the presence of inhibitory connections. The detailed analysis of the proposed preferential random walk revealed different behaviors for the three networks studied. While the number of active nodes at the steady state is similar both in random and scale-free networks, this quantity is much smaller in geographic networks. This result suggests that the spread is facilitated by the long-range connections and the small-world effect present in such networks. At the steady state, the size of the active set was found to be similar for all three networks. The dependence of the size of this active core on the activity decay was also verified. In all cases, the number of active nodes increases with the preservation factor. We also found that during the first time steps of the spreading, the active nodes move toward the formation of an anti-community structure [@autovetornewman], in opposition to the behavior observed in other similar dynamics. Finally, we have investigated the influence of the choice of the initial node on some dynamical aspects. More specifically, we found that there is no apparent relationship between the degree of the node locating the random walker at $t=t_0$ and the size of the active core at the steady state. On the other hand, we verified that the choice of the initial node interferes with the localization of the active nodes at $t = t_f$, for the final set of actives nodes spreads along the nearest neighbors from the initial activation. As future work, the approach may be extended by considering other phenomena along the dynamics. 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Supplementary Information {#supplementary-information .unnumbered} ========================= ![\[fig4\] Degree distribution for the (a) random network; (b) geographical network; and (c) scale-free network.](histogramas){width="100.00000%"} ![\[fig2mod\] Number of activated nodes when $t > t_f$ for the (a) random network; (b) geographical network; and (c) scale-free network. In all networks, the dynamics converges to a steady set of activated nodes.](er_ws_ba){width="100.00000%"} ![\[fig3\] Relationship between the rate of activity decay and the number of activated nodes for (a) the random network; (b) the geographical networks; and for (c) scale-free network. The sole parameter present in the dynamics of the preferential random walk is $\alpha$, which models the rate of activity preservation. More specifically, while large values of $\alpha$ leads to a slow activity decrease, low values of $\alpha$ yields a more pronounced activity decrease. The size of the set of active nodes was computed varying $\alpha$ within the range $ 0.99990 \leq \alpha \leq 0.99995$. In all networks, an increasing trend with $\alpha$ could be observed. Specially, a strong correlation occurred for both scale-free and geographical networks. Therefore, the fraction of active nodes can be straightforwardly adjusted by setting the value of the parameter $\alpha$.](er_ws_ba_alpha_tam_comps){width="100.00000%"} ![\[fig2.alf\] Distance between the node starting the preferential random walk and the set of active active nodes at $t=t_f$, for (a) the random network; (b) the geographical network; and (c) the scale-free network. In all cases, the activation spreads along the nearest neighbors.](distancias){width="100.00000%"} ![\[fig5\] Size of the set of activated nodes and its relationship with the degree of nodes starting the preferential random walk for (a) the random network; (b) the geographical networks; and for (c) scale-free network. In all cases, localization had weak impact on spreading. The influence of initial position of the particle was verified by starting the random walk in nodes with distinct degree. Then, the number of activated nodes at $t > t_f$ was computed for each class of starting nodes. Given the degree distributions in Figure \[fig4\], the set of degrees selected to start the walk was: $D_{r} = \{1,2,3,4,5,6,7,8,9,10,11,12 \}$, $D_{g} = \{2,3,4,5,6,7,8,9,10 \}$ and $D_{s} = \{3,5,7,9,11,13,15,17,19,21\}$ respectively for the random, geographical and scale-free networks. For the three network models, weak impact on spreading in the was observed. Interestingly, this result differs from the results reported in Ref. [@kaiser], which showed that a similar dynamics of activated nodes depends on where the initial activation was localized, especially for hierarchical modular networks.](initial_nodes_er_ws_ba){width="100.00000%"} ![\[\] Dynamics of the set of activated nodes (i.e., nodes with $T_i > \tau_c$) submitted to the Integrate-and-Fire dynamics for the geographical network when (a) $\alpha = 0.8050$, (b) $\alpha = 0.8100$ and $\alpha = 0.81125$. While in (a) in (b) the activity is restrained, in (c) it spreads all over the network. ](fire_ws){width="100.00000%"} ![\[\] Dynamics of the set of activated nodes (i.e., nodes with $T_i > \tau_c$) submitted to the Integrate-and-Fire dynamics for the BA network when (a) $\alpha = 0.7440$, (b) $\alpha = 0.7480$ and $\alpha = 0.7452$. While in (a) in (b) the activity is restrained, in (c) it spreads all over the network.](fire_ba){width="100.00000%"} [^1]: A component is a subnetwok in which there is a path between all nodes belonging to the component.

--- abstract: 'Dwarf galaxies (DGs) serve as extremely challenging objects in extragalactic astrophysics. Their origin is expected to be set as the first units in CDM cosmology. Nevertheless they are the galaxy type most sensitive to environmental influences and their division into multiple types with various properties have invoked the picture of their variant morphological transformations. Detailed observations reveal characteristics which allow to deduce the evolutionary paths and to witness how the environment has affected the evolution. Here we refer to general morphological DG types and review some general processes, most of which deplete gas-rich irregular DGs. Moreover, the variety of pecularities is briefly refered, but cannot be comprehensively analyzed because of limited paper space.' address: ' University of Vienna, Institute of Astronomy, Tuerkenschanzst. 17, 1180 Vienna, Austria;\' author: - Gerhard Hensler title: The Morphological Origin of Dwarf Galaxies --- 2[/pc\^2]{} Introduction ============ After a former conference talk on evolutionary models of dwarf galaxies (DGs), in particular, on dwarf irregular galaxies (dIrrs), the blue compact subclass (BCDs) and on starburst (SB)DGs, a senior colleague well-known as one of the pioneers of DG studies asked me with a serious surprise, why DG evolution is so complex and not a straight-forward scenario although their structural appearance looks pretty simple. Nowadays this first impression of DGs has changed totally with the advent of large telescopes, new techniques, and of the accessible panchromatic view. Within the last two decades detailed analyses of DGs has extended but also stirred our view of the classical morphological DG types, dIrrs, SBDGs, and dEs including satellite galaxies, so-called dwarf spheroidals (dSphs), as their extension to lower brightness of about -5$^m$. In their studies of Virgo cluster DGs already [@sb84] found that elliptical DGs (dEs) dominate the cluster galaxy population by far. This stands in contrast to their number fraction in the field where dIrrs are the most frequent DGs but with a smaller fraction than cluster dEs ([@bin88]). For the interpretation of this issue and the occurrence of enhanced SF in dIrrs [@sb84] emphasized already the necessity of various links between the DG types by morphological transitions. From the $\Lambda$CDM cosmology the baryonic matter should settle within Dark Matter (DM) halos which are originally prefered to form low-mass units, so-called subhalos, and hierarchically accumulating to massive galaxies. If the baryonic matter would follow this bottom-up structure formation, the subhalos should also assemble their gas at first and by this also evolve with star fomation (SF) to become the oldest galactic objects in the universe. That this picture seems to be too naive is simply understandable by three major physical principles: the first one is, that the gas assembly works on the free-fall timescale $\tff$, namely, dependent on the gas density as $\rg^{-1/2}$, because gas is accreted through gravitation. At second, the SF timescale $\tSF$ is per definition proportional to M$_g/\Psi$ with $\Psi$ as the SF rate that, on the other hand, in the self-regulated SF mode depends on $\rg^2$ ([@koe95]). Since lower galaxy masses lead to less dense gas, SF is stretched over time for low-mass galaxies. And at least, as SF couples to stellar energy release and since the counteracting cooling process depends on $\rg^2$, the gas expands due to pressure support and reduces the SF rate so that the effect of SF self-regulation is non-linearly amplified. Another important effect that seems to affect the whole network of galaxy formation and evolution is ionizing radiation from the first cosmic objects to the re-ionization of the gas in the universe. This changed not only its thermodynamical state so that its accretion onto low-mass objects was reduced ([@dij04]) but also evaporated gas that was already caught in minihalos ([@bl99]). Since massive objects remained almost unaffected by the re-ionization phase, while DGs should have experienced delayed SF ([@noe07]), this evolutionary dichotomy is observed as downsizing ([@cow96]). Nevertheless, the assumption that all DGs were affected in the re-ionization era and in the same way would request overlapping Stroemgren bubbles in an almost uniformly ionized universe. This assumption, however, is questioned and contrasted by the existence and amplification of density structures ([@par10]). Dwarf irregular and dwarf starburst galaxies {#dIrrs} ============================================ dIrrs are characterized by large gas fractions, ongoing SF, and low metallicities $Z$. That dIrrs consist of the same or a higher gas fraction than giant spiral galaxies and mostly suffer the same SF efficiency, but appear with a wide range, but lower $Z$ than spirals, cannot be explained by simple evolutionary models. When gas is consumed by astration but replenished partly by metal-enriched stellar mass loss, the general analytical derivation relates the element enhancement with the logarithm of the decreasing remaining gas content where the slope is determined by the stellar yields (see e.g. textbooks like by [@pag10] or reviews as e.g. by [@hen10]). The effective yields of gas-rich galaxies decrease, however, with smaller galaxy masses ([@gar02; @zee01]). This means that their element abundances, particularly O measured in regions, are much smaller than those released by a stellar population and confined to a “closed box”. Two processes can reduce the metal abundances in the presence of old stellar populations: loss of metal-enriched gas by galactic outflows or infall of metal-poor to even pristine intergalactic gas (IGM). It is widely believed, that a fundamental role in the chemical evolution of dIrrs is played by galactic winds, because freshly produced metals in energetic events are carried out from a shallow potential well of DGs through a wind (which will be therefore metal-enhanced). Some SBDGs are in fact characterized by galactic winds ([@mar95]) or by large expanding supernova type II (SNeII)-driven X-ray plumes (e.g. [@hen98; @mar02]). Studies have raised doubts to whether the expanding $\Ha$ loops, arcs, and shells mostly engulfing the X-ray plumes, lead really to gas expulsion from the galaxies because their velocities are mostly close to escape, but adiabatic expansion against external gas tends to hamper this. As an extreme, [@bab92] speculated that galactic winds are able to empty DGs from its fuel for subsequent SF events and, by this, transform a gas-rich dIrr to a fading gas-poor system. In order to manifest this scenario and to study mass and element abundance losses through galactic winds numerous numerical models are performed under various, but mostly uncertain conditions and with several simplifications (e.g. [@mlf99; @str04]). The frequently cited set of models by MacLow & Ferrara (self-gravitating, rotationally supported, isothermal disks of dIrrs with fixed structural relations for four different gas masses between M$_{\rm g} = 10^6 - 10^9 \Msun$ and three different SNII luminosities in the center corresponding to SN rates of one per $3\times 10^4$ yrs to 3 Myrs) is mostly misinterpreted: The hot gas is extremely collimated from the center along the polar axis, but cannot sweep-up sufficient surrounding ISM to produce significant galactic mass loss. On the other hand, the loss of freshly released elements from massive stars is extremely high. Moreover, these models lack of realistic physical conditions, as e.g. the existence of an external pressure, self-consistent SF rates, a multi-phase ISM, etc. -- -- -- -- Also more detailed numerical simulations ([@db99]; [@rec06a]), show that galactic winds are not very effective in removing gas from a galaxy. Although galactic winds develop vertically, while the horizontal transport along the disk is very limited, their efficiency depends very sensibly on the galaxy structure and ISM properties, as e.g. on the disk shape ([@rec09]). Fig. \[fig:wind\] reveals clearly that the more eccentric the disk is, the more pronounced does the superbubble expand. On the one hand, the hot SN gas has to act against the galactic ISM, exciting turbulence and mixing between the metal-rich hot with the surrounding gas. Not taken into account in present-day models is the porosity of the ISM, consisting of clouds and diffuse less dense gas. In particular, the presence of clouds can hamper the development of galactic winds through their evaporation. This so-called mass loading reduces the wind momentum and internal energy. Since the metallicity in those clouds are presumably lower than the hot SNII gas, also the abundances in the outflow are diminished as e.g. observed in the galactic X-ray outflow of NGC 1569 ([@mar02]) for which a mass-loading factor of 10 is derived to reduce the metallicity to 1-2 times solar. In recent simulations [@sca10] demonstrate that turbulent mixing can effectively drive a galactic wind. Although they stated that their models lead to a complex, chaotic distribution of bubbles, loops and filaments as observed in NGC 1569, other observational facts have not been compared. Detailed numerical simulations of the chemical evolution of these SBDG by [@rec06b], e.g., could simultaneously reproduce both, the oxygen abundance in the warm gas as well as the metallicity in the hot outflow. [@rec07] show that the leakage of metals from a SBDG is surprisingly not prevented by the presence of clouds, because they pierce holes into the wind shells. This leads to a final metallicity of only a few tenths of dex lower than in models without clouds. Consequently, the basic question must be answered which physical processes trigger such enormous SF rates as observed in SBDGs and would consume all the gas content within much less than the Hubble time. One possibility which has been favoured until almost two decades ago was that at least some of these objects are forming stars nowadays for the very first time. Today it is evident that even the most metal-poor ones (like I Zw 18) contain stars of at least 1 Gyr old ([@mom05]), but most SBDGs have several Gyrs old stellar populations. This means that SF in the past should have proceeded in dIrrs, albeit at a low intensity what can at best explain their chemical characteristics, like for instance the low \[$\alpha$/Fe\] ratio. [ll]{} & \[fig:KS\] In most SBDGs large reservoirs, however, enveloping the luminous galactic body have been detected (NGC 1705: [@meu98], NGC 4449: [@hun98], NGC 4569: [@sti02], NGC 5253: [@kob08], I Zw 18: [@zee98c], II Zw 40: [@zee98b]) with clearly disturbed gas kinematics and disjunct from the luminous galactic body. Nevertheless, in not more than two objects, NGC 4569 ([@mue05]) and NGC 5253 ([@kob08]) gas infall is proven, while for the other cases the gas kinematics obtrudes that the gas reservoir feeds the engulfed DGs. In another object, He 2-10, the direct collision of an intergalactic gas cloud with a DG ([@kob95]) is obviously triggering a huge SB. Yet it is not clear, what happens to dIrrs if they experience on an increasing external pressure what would happen e.g. when they fall into galaxy clusters. In sect.\[dEs\] we will discuss the effect of ram pressure on the structure of the ISM for which numerical models exist for spiral galaxies (e.g. [@roe05]) as well as for dIrrs (e.g. [@mor00]), but only hints from observations. The effect on the SF rate due to compression of the ISM is observed, but not yet fully understood. [@cor06] e.g. observed a coherent enhancement of SF in group galaxies falling into a cluster, therefore, denoted as blue infalling group. The \[$\alpha$/Fe\] vs. \[Fe/H\] behaviour is representative of the different production phases, $\alpha$-elements from the short-living massive stars and iron to 2/3 from type Ia SNe of longer-living binary systems. If the SF duration in a galaxy is very short, type Ia SNe do not have sufficient time to enhance the ISM with Fe and most of the stars will be overabundant in \[$\alpha$/Fe\]. The low average \[$\alpha$/Fe\] ratios in dIrrs, however, compared to large galaxies serve as a hint of a long-lasting mild SF in these galaxies ([@lm04]). ![ The abundance ratio N/O as a function of oxygen abundance observed in spiral and irregular galaxies (shaded area; after: [@zee98a]) overlayed with evolutionary loops due to infall of primordial intergalactic gas clouds. These have different mass fractions $M_{cl}/M_{SF}$ with respect to the mass involved in the SF region The crosses represent evolutionary timesteps of models, the arrows depict the direction of the evolutionary paths. The dashed straight line represents a simple model relation for purely secondary nitrogen production. For discussion see: [@koe05]). []{data-label="fig:NO"}](NO_OH_loop.pdf){width="12cm"} Although the mass-metallicity relation also holds for dIrrs and even steepens its slope ([@tri04]) what can be interpreted by galactic mass loss and the corresponding lower effective yield ([@gar02]), the abundance ratios are unusual. As mentioned above O/Fe reaches already solar values for subsolar oxygen abundances. While this can be explained by means of a long SF timescale, another characteristic and unusual signature is that with O abundances below 1/10 solar the N/O ratio remains at about -1.6, i.e. clearly smaller than in gSs and with a large scatter but no significant correlation with oxygen (see Fig.\[fig:NO\]). Their regime of N/O–O/H values overlaps with those of regions in the outermost disk parts of gSs at around = 8.0 ... 8.5 ([@zee98a]). In the 90th several authors have tried to model these observations by SF variations with gas loss through galactic winds under the assumptions that these dIrrs and BCDGs are young and experience their first epochs of SF (for a detailed review see [@hen99]). Since stellar population studies contradict to the youth hypothesis, another process must be invoked. Since these objects are embedded into envelops and are suggested to suffer gas infall as manifested e.g. for NGC 1569 (see above, [@sti02; @mue05]), the influence of metal-poor gas infall into an old galaxy with continuous SF on particular abundance patterns should be exploited. With the reasonable approach that the fraction of infalling gas increases with decreasing galaxy mass, their results could match not only the observational regime of BCDs in the \[\]-space but also explain the shark-fin shape of observational data distribution ([@koe05]). Dwarf elliptical galaxies {#dEs} ========================= dEs are the most numerous type in clusters and are frequently denoted as examples of “stellar fossil” systems in which the bulk of their SF occurred in the past. They are preferentially located in morphologically evolved environments ([@tre02]), i.e. in regions with high galaxy densities and dominate the morphological types of galaxy in galaxy clusters, as e.g. Virgo, Coma, Fornax, and Perseus. Furthermore, [@tul08] observed that dEs strongly cluster around luminous elliptical/S0 galaxies. The evolution of this galaxy type should be mainly caused by gas and tidal effects on SF and structure and indicates that it is strongly affected by environment. Already [@bot85] found that cluster dEs are usually almost free of interstellar gas and contain few young stars. In trying to understand the dE population, structural regularities and correlations must be studied, as it is known since the 80th between optical surface brightness and luminosity ([@bin84; @kor85]) and between luminosity and stellar velocity dispersion which also correlate with metallicity (e.g. [@pet93]). Furthermore, dEs often have flattened profiles but are mostly kinematically supported by their stellar velocity dispersions rather than by rotation ([@ben91]). The combination of low gas-mass fractions and moderate-to-low stellar metallicities in dE (about 0.1 of solar or less) is a key feature of this class. The lower abundances of stars in dEs ([@haa97]) suggest that extensive gas loss occurred during their evolution and SF ceased due to a lack of raw material rather than exhaustion of the gas supply through SF. Galactic winds are therefore a hallmark of modern models for dE galaxies, starting from the basic consideration by [@lar74] and continued with the study by [@ds86]. They are commonly assumed to have cleaned out dE galaxies soon after their formation. As mentioned in sect. \[dIrrs\] gas expulsion by means of galactic winds even in low-mass systems requires a dark-to-baryonic matter ratio ([@mlf99]) much smaller than assigned to DGs in the classical formation picture (e.g. [@mat98]). There are two competing scenarios for the formation of dEs. On the one hand, those low-mass galaxies are believed to constitute the building blocks in $\Lambda$CDM cosmology and should therefore have evolved congruently with the mass accumulation to the much more massive entities, galaxies and galaxy clusters, by this leading to SF with the downsizing effect through the delay by the re-ionization epoch. Their stellar component is expected to be heated continuously by harrasment of more massive cluster galaxies and thus to be pressure supported. A variety of observations are available which also support diverse scenarios of dEs evolution (see e.g. review by Lisker, conference). Recent HI studies of Virgo cluster dEs ([@con03]) and also those of the Fornax cluster (e.g. [@mic04]) have unveiled that a small but significant fraction of them contains gas, has experienced recent SF, and can be argued from internal kinematics and cluster distribution data to represent an infalling class of different types of gas-rich galaxies in the state of morphological transformation. Recent findings of a significant fraction of rotationally supported dEs in the Virgo cluster ([@zee04]) also supports the possibility of morphological transformation from dIrrs to dEs thru gas exhaustion ([@bos08]). This separation should therefore also be visible in an intermediate age stellar population, flatter figure shape, and rotation. Indeed, [@lis07] found that dEs in the Virgo cluster can be divided in different subclasses which differ significantly in their morphology and clustering properties, however do not show any central clustering, but are distributed more like the late-type galaxies. These subclasses show different disk signatures, such as bars and spiral structures. Also these types of dEs are not spheroidal, but rather thick disk-like galaxies. Similar shapes were also found for the brighter, non-nucleated dEs. There is only a small fraction of nucleated dEs whithout any disk features or cores, which keep the image of spheroidal objects consisting of old stars. [ll]{} & \[fig:VCdE\] A figure analysis of Virgo dEs correlates with the averaged orbit velocity in the sense that flatter (infalling and transformed) dEs show on average a larger orbital velocity (700 km/s) than those originating within the cluster (300 km/s) ([@lis09]). This kinematical dichotomy is expected because galaxies formed in virial equilibrium with the cluster retain their initial kinetic energy while the cluster mass grows. Galaxies falling into the present cluster potential must therefore obtain larger velocities. To obtain information about their both evolutionary stages, the young infalling vs. the late cluster members, [@got09] studied SDSS data. The basic model is that dIrrs which are formed outside the Virgo Cluster and becoming stripped on their infall, by this being transformed into dEs, should reveal properties recognizably different from dEs which have already aged in the cluster, as e.g. colors, effective radius, radial stellar distribution, and abundances. One result by [@got09] is that for the two dE populations, with and without cores, distinguished by their Sersic parameter, there is only a slight indication that non-nucleated dEs are more concentrated towards the inner cluster regions, whereas the fraction of nuclated dE is randomly distributed, while [@lis07] found it to increase with distance. An analysis of the relation between the central surface brightness and the Sersic parameter shows the expected tendency to higher values for brighter galaxies. Furthermore, there were no relations found for the Sersic parameter, the effective radius, or the distance from M87. For deeper insights spectra are urgent. And really, [@tolo09] derived for Coma cluster dEs to be weaker in carbon than dEs in low-density environments, while they are similar in nitrogen. Most recently, [@tol10] found that pressure supported Virgo dEs show higher dynamical mass-to-light ratios than rotationally supported dEs of similar luminosity and further that dEs in the outer parts of the cluster are mostly rotationally supported with disky shapes. Rotationally supported dEs even follow the Tully-Fisher relation. One of [@tol10] fundamental and most spectacular results is, however, that dEs are not DM-dominated galaxies, at least out to the half-light radius. Concerning any metallicity gradient, the picture is not yet clear, but also not what one should expect. While [@spo09] show a tight positive correlation between the total metallicity $Z$ and the mass, [@kol09] do not find any trend involving \[Fe/H\] for Fornax-cluster and nearby-group dEs. However, they found metallicity gradients to exist and argue that this is a lack of sufficient mixing of old stellar populations but compare it also with simulations which found this as a result of galactic winds. Moreover, from the deconvolution of the SF history of their sample dEs with respect to the central 1 arcmin and within the effective radius [@tolo09] allow the conclusion on the existence of SF episodes in the very center even within the last 1 Gyr for a few objects. Correlations of both signatures, SF history and metallicity gradients, for cluster-member dEs vs. infall dEs should be derived for more clusters, but observations are unfortunately very time-expensive if possible at all. Dwarf spheroidal galaxies {#dSphs} ========================= At the faint end of dEs another type of almost gas-free spheroidal DGs exists which is located around massive galaxies like our MWG and M31 and thus orbiting them as satellites. These objects have attracted increasing attention over the last years because at their low-mass end they overlap with Globular Clusters, so that the understanding of their formation and evolution is of substantial relevance for our astrophysical picture of cosmology and galaxy evolution. Four main questions are addressed: 1. How and when did they form? They all harbour a very old stellar population ([@tols09]) and seem to have evolved unaffected of the re-ionization era ([@gre04]). 2. Is their existence as satellite system typical for all massive galaxies? Their origin and DM content is still questioned by some authors ([@kro10]). The large discrepancy of the number of objects really observed and the one expected from $\Lambda$CDM cosmology and because of their orbit concentration to the so-called disk of satellites, also observed around M31, invoked the preferrence of their tidal-tail origin ([@met09]). The large velocity disperion is caused by the tidal effects. 3. How is their evolution determined by the vicinity of the massive mature galaxy? Not only the tidal field must have a disruptive effect, but also a gaseous halo of the central galaxy will interact with the ISM of the dSphs ([@may07]). The tendency of an increasing gas fraction bound to the dSphs with distance from the MWG ([@har01]), points into that direction. 4. Vice versa the question arises, how the satellites influence the structure and evolution of the mature galaxy, here the MWG. The first three questions also concern the morphological origin of dSphs e.g. as tidal-tail DGs formed in the same era as the MWG but in a gas-rich galaxy collision or their transition from gas-rich $\Lambda$CDM satellites to dSphs. The fascinating wealth of data and their precision on stellar ages and kinematics, on their chemical abundances, abundance gradients, and tidal tails of dSphs (for most recent reviews see e.g. [@koc09] and [@tols09]) have triggered numerous numerical models which, however, still lack of the inclusion of the above-mentioned environmental effects. Not before recently [@ph11]; first insights also in this conference proceedings and in [@hen11]) have started to simulate the full sample of a large number of DM subhalos containing baryonic matter by models. Since these models cannot be presented here because of paper space, we wish to highlight only the main issues as follows and refer the interested reader to a forth-coming paper by [@ph11]). In addition to the tidal and gaseous effects by means of the MWG, due to the mutual interactions also between the dSphs disruption, gas and star exhaustion, and merging occur. The metallicity and abundance ratios are traced and comparable with observation. The SF histoies vary dependent on mass and gas re-accretion. Acknowledgement {#acknowledgement .unnumbered} =============== The author is grateful to Joachim Koeppen, Thorsten Lisker, Mykola Petrov, Simone Recchi, Werner Zeilinger, and the SMAKCED collaborations for enlightening and continuous discussions on DGs. 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--- abstract: 'Given an integer $t\ge 1$, a rational number $g$ and a prime $p\equiv 1({\rm mod~}t)$ we say that $g$ is a near-primitive root of index $t$ if $\nu_p(g)=0$, and $g$ is of order $(p-1)/t$ modulo $p$. In the case $g$ is not minus a square we compute the density, under the Generalized Riemann Hypothesis (GRH), of such primes explicitly in the form $\rho(g)A$, with $\rho(g)$ a rational number and $A$ the Artin constant. We follow in this the approach of Wagstaff, who had dealt earlier with the case where $g$ is not minus a square. The outcome is in complete agreement with the recent determination of the density using a very different, much more algebraic, approach due to Hendrik Lenstra, the author and Peter Stevenhagen.' address: 'Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany' author: - Pieter Moree title: 'Near-primitive roots' --- Introduction ============ Let $g\in \mathbb Q\backslash \{-1,0,1\}$. Let $p$ be a prime. Let $\nu_p(g)$ denote the exponent of $p$ in the canonical factorization of $g$. If $\nu_p(g)=0$, then we define $$r_g(p)=[(\mathbb Z/p\mathbb Z)^*:\langle g~{\rm mod~}p\rangle],$$ that is $r_g(p)$ is the residual index modulo $p$ of $g$. Note that $r_g(p)=1$ iff $g$ is a primitive root modulo $p$. For any natural number $t$, let $N_{g,t}$ denote the set of primes $p$ with $\nu_p(g)=0$ and $r_g(p)=t$ (that is $N_{g,t}$ is the set of near-primitive roots of index $t$). Let $\delta(g,t)$ be the natural density of this set of primes (if it exists). For arbitrary real $x>0$, we let $N_{g,t}(x)$ denote the number of primes $p$ in $N_{g,t}$ with $p\le x$.\ In 1927 Emil Artin conjectured that for $g$ not equal to $-1$ or a square, the set $N_{g,1}$ is infinite and that $N_{g,1}(x)\sim c_gA\pi(x)$, with $c_g$ an explicit rational number, $$A=\prod_p\Big(1-{1\over p(p-1)}\Big)\approx 0.3739558,$$ and $\pi(x)$ the number of primes $p\le x$. The constant $A$ is now called Artin’s constant. On the basis of computer experiments by the Lehmers in 1957 Artin had to admit that ‘The machine caught up with me’ and provided a modified version of $c_g$. See e.g. Stevenhagen [@PeterS] for some of the historical details. On GRH this modified version was shown to be correct by Hooley [@Hooley].\ Thus $\delta(g,1)$ is explicitly known (under GRH). Determining similarly $\delta(g,t)$ turns out to be rather more difficult and for ease of exposition we first consider the case where $g>1$ is square free. In this case work of Lenstra [@Lenstra] and Murata [@Murata] suggests the following conjecture (with as usual $\mu$ the Möbius function and $\zeta_k=e^{2\pi i/k}$). \[leo\] Let $g>1$ be a square free integer and $t\ge 1$ an integer. The set $N_{g,t}$ has a natural density $\delta(g,t)$ which is given in Table 1. We have $$N_{g,t}{\it ~is~finite~iff~}\delta(g,t)=0 {\rm ~iff~}g\equiv 1({\rm mod~}4),~2\nmid t,~g|t.$$ We note that if $g\equiv 1({\rm mod~}4)$, $2\nmid t$ and $g|t$, then $N_{g,t}$ is finite. To see this note that in this case we have $({g\over p})=1$ for the primes $p\equiv 1({\rm mod~}t)$ by the law of quadratic reciprocity and thus $r_g(p)$ must be even, contradicting the assumption $2\nmid t$.\ Note that if a set of primes is finite, then its natural density is zero. The converse is often false, but for a wide class of Artin type problems (including the one under consideration in this note) is true (on GRH) as first pointed out by Lenstra [@Lenstra].\ Given an integer $a$ and a prime $q$, we write $a_q$ to denote the $q$-part of $a$ (that is $a_q=q^{\beta}$ with $q^{\beta}|a$ and $q^{\beta+1}\nmid a$). We put $$\label{eetee} B(g,t)=\prod_{p|g,~p\nmid t}{-1\over p^2-p-1},~~ E(t)={A\over t^2}\prod_{p|t}{p^2-1\over p^2-p-1}.$$ Note that if $g|t$, then in the definition of $B(g,t)$ we have the empty product and hence $B(g,t)=1$. It follows that if further $t$ is odd and $g\equiv 1({\rm mod~}4)$, then $\delta(g,t)=0$. The maximal value of $\delta(g,t)$ that occurs is $2E(t)$. Table 1 we took from a paper by Murata [@Murata]. We will show that the densities in Table 1 can be compressed into one equation, namely (\[oneline\]). .8mm $g$ $t_2$ $\delta(g,t)$ -------------------------- ------------ -------------------- $g\equiv 1({\rm mod~}4)$ $t_2=1$ $(1-B(g,t))E(t)$ $2|t_2$ $(1+B(g,t))E(t)$ $g\equiv 2({\rm mod~}4)$ $t_2<4$ $E(t)$ $t_2=4$ $(1-B(g,t)/3)E(t)$ $t_2>4$ $(1+B(g,t))E(t)$ $g\equiv 3({\rm mod~}4)$ $t_2=1$ $E(t)$ $t_2=2$ $(1-B(g,t)/3)E(t)$ $t_2\ge 4$ $(1+B(g,t))E(t)$ \[uno\] Conjecture [\[leo\]]{} holds true on GRH. The proof is postponed untill Section \[sectietwee\]. Generalization to rational $g$ {#sectietwee} ============================== A natural next problem is to study what happens if one relaxes the condition that $g$ should be square free. Our starting point here will be a result due to Wagstaff [@W]. We need some notation. We put $$S(h,t,m)=\sum_{n=1\atop m|nt}^{\infty}{\mu(n)(nt,h)\over nt\varphi(nt)},$$ with $\varphi$ Euler’s totient function. [[@W]]{}. [(GRH)]{}. \[waggel\] \[leno\] Let $g\in \mathbb Q\backslash \{-1,0,1\}$ and $t\ge 1$ be an arbitrary integer. Write $g=\pm g_0^h$, where $g_0\in \mathbb Q$ is positive and not an exact power of a rational and $h\ge 1$ an integer. Let $d(g_0)$ denote the discriminant of $\mathbb Q(\sqrt{g_0})$. The natural density of the set $N_{g,t}$, $\delta(g,t)$, exists and is given by $$\label{musum} \sum_{n=1}^{\infty}{\mu(n)\over [\mathbb Q(\zeta_{nt},g^{1/nt}):\mathbb Q]},$$ which equals a rational number times the Artin constant $A$. Write $g_0=g_1g_2^2$, where $g_1$ is a square free integer and $g_2$ is a rational. If $g>0$, set $m={\rm lcm}(2h_2, d(g_0))$. For $g<0$, define $m=2g_1$ if $2\nmid h$ and $g_1\equiv 3({\rm mod~}4)$, or $h_2=2$ and $g_1\equiv 2({\rm mod~}4)$; let $m={\rm lcm}(4h_2,d(g_0))$ otherwise. If $g>0$, we have $\delta(g,t)=S(h,t,1)+S(h,t,m)$. If $g<0$ we have $$\label{vierergruppe} \delta(g,t)=S(h,t,1)-{1\over 2}S(h,t,2)+{1\over 2}S(h,t,2h_2)+S(h,t,m).$$ In case $g>0$ or $2\nmid h$, Wagstaff expressed $\delta(g,t)$ as an Euler product. By the work of Lenstra [@Lenstra] we know this is also possible in general. The next theorem achieves this. Partial inspiration for it came from recent joint work with Lenstra and Stevenhagen, see Section \[prospect\]. [(GRH)]{}. \[mainz\] Let $g\in \mathbb Q\backslash \{-1,0,1\}$ and $t\ge 1$ be an arbitrary integer. Write $g=\pm g_0^h$, where $g_0\in \mathbb Q$ is positive and not an exact power of a rational and $h\ge 1$ an integer. Let $d(g_0)$ denote the discriminant of $\mathbb Q(\sqrt{g_0})$. Put $F_p=\mathbb Q(\zeta_p,g^{1/p})$. Put $$A(g,t)={(t,h)\over t^2}\prod_{p|t,~h_p|t_p}(1+{1\over p})\prod_{p\nmid t}(1-{1\over [F_p:\mathbb Q]}).$$ Put $$\Pi_1=\prod_{p|d(g_0),~p\nmid 2t}{-1\over [F_p:\mathbb Q]-1}.$$ Put $$\label{e2m2} E_2(m_2)=\begin{cases} 1 & {\rm ~if~}m_2| t_2;\cr -1/3 & {\rm ~if~}m_2 = 2t_2 \ne 2;\cr -1 & {\rm ~if~}m_2 = 2t_2 = 2;\cr 0 & {\rm ~if~}m_2 \nmid 2t_2,\cr \end{cases}$$ We have $$\label{adoora} {A(g,t)\over A}={(t,h)\over t^2}\prod_{p|th}{1\over p^2-p-1}\prod_{p|t\atop pt_p|h_p}p(p-1)\prod_{p|t\atop h_p|t_p}(p^2-1)\prod_{p|h,~p\nmid t_1}p(p-2),$$ where $$t_1=\begin{cases} 2t & {\rm ~if~}g<0,~2|h,~2\nmid t;\cr t & {\rm ~otherwise}. \end{cases}$$ Note that $A(g,t)=0$ iff $g>0$, $2|h$ and $2\nmid t$.\ The natural density of the set $N_{g,t}$ exists, denote it by $\delta(g,t)$.\ Put $v_0={\rm lcm}(2h_2,d(g_0)_2)$ and $v={\rm lcm}(2h_2,d(g)_2)$.\ If $g>0$, then $\delta(g,t)=A(g,t)(1+E_2(v_0)\Pi_1)$.\ If $h$ is odd, then $\delta(g,t)=A(g,t)(1+E_2(v)\Pi_1)$.\ If $g<0$, $2|h$ and $2\nmid t$, we have $\delta(g,t)=A(g,t)$.\ Next assume $g<0$, $2|(h,t)$.\ If $h_2=2$ and $8|d(g_0)$, then $$\label{dubbel} \delta(g,t)=\begin{cases} {1\over 3}A(g,t)(1-\Pi_1) & {\rm ~if~}t_2=2;\cr A(g,t)(1+\Pi_1) & {\rm ~if~}4|t_2.\cr \end{cases}$$ In the remaining cases we have $$\delta(g,t)=\begin{cases} A(g,t)/2 & {\rm ~if~}2t_2|h_2;\cr A(g,t)/3 & {\rm ~if~}t_2=h_2;\cr A(g,t)(1-{1\over 3}\Pi_1) & {\rm ~if~}t_2=2h_2;\cr A(g,t)(1+\Pi_1) & {\rm ~if~}4h_2|t_2.\cr \end{cases}$$ [(GRH)]{}. \[blubbo\] Let $g>1$ be a square free integer. Then $$\label{oneline} \delta(g,t)=(1+E_2({\rm lcm}(2,d(g)_2))B(g,t))E(t).$$ [*Proof*]{}. We have $A(g,t)=S(1,t,1)=E(t)$ (see the remark following Lemma \[t1t2\]). Furthermore, if $2|g$ and $2\nmid t$, then $\Pi_1=-B(g,t)$ and $\Pi_1=B(g,t)$ otherwise. Since $E_2({\rm lcm}(2,d(g)_2))=0$ if $g|2$ and $2\nmid t$, we infer that $E_2({\rm lcm}(2,d(g)_2))\Pi_1=E_2({\rm lcm}(2,d(g)_2))B(g,t)$. Now invoke the theorem. [(GRH)]{}. If $t$ is odd, then $$\delta(g,t)=A(g,t)(1-{1\over 2}(1-(-1)^{h|d(g)|})\Pi_1).$$ [Remark]{}. On putting $t=1$ one obtains the classical result of Hooley [@Hooley].\ [*Proof of Theorem*]{} \[uno\]. On distinguishing cases according to the value of $d(g)_2$, Corollary \[blubbo\] yields Table 1. From Table 1 one easily reads off that if $\delta(g,t)=0$, then $2\nmid t$, $g\equiv 1({\rm mod~}4)$ and $g|t$. In this case we have $(g/p)=1$ for the primes $p\nmid g$ with $p\equiv 1({\rm mod~}t)$ by the law of quadratic reciprocity and hence $N_{g,t}$ is finite and so $\delta(g,t)=0$.\ The proof of Theorem \[mainz\] will be given in Section \[vier\]. It will make use of properties of Wagstaff sums that will be established in the next section. Bringing the Wagstaff sums in Euler product form ================================================ Recall the definition of the Wagstaff sum $$S(h,t,m)=\sum_{n=1\atop m|nt}^{\infty}{\mu(n)(nt,h)\over nt\varphi(nt)}.$$ A trivial observation is that if the divisibility condition forces $n$ to be non-square free, then $\mu(n)=0$ and hence $S(h,t,m)=0$. This happens for example if $m_2\nmid 2t_2$ (cf. Lemma \[shtminprod\]). In case $m=1$ it is easily written as an Euler product (here we use that $\mu$ and $\varphi$ are multiplicative functions). \[t1t2\] [1)]{}. We have $$S(h,t,1)={(t,h)\over t^2}\prod_{p|t,~h_p|t_p}(1+{1\over p})\prod_{p\nmid t}\Big(1-{(p,h)\over p(p-1)}\Big).$$ In particular, $S(h,t,1)=0$ iff $2|h$ and $2\nmid t$.\ [2)]{}. If $2|h$ and $2\nmid t$, then $$S(h,t,2)=-{(t,h)\over t^2}\prod_{p|t,~h_p|t_p}(1+{1\over p})\prod_{p\nmid 2t}\Big(1-{(p,h)\over p(p-1)}\Big).$$ [*Proof*]{}. 1) We have $$S(h,t,1)={(t,h)\over t\varphi(t)}\sum_n {\mu(n)(nt,h)\varphi(t)\over n\varphi(nt)(t,h)} ={(t,h)\over t\varphi(t)}\prod_p\Big(1-{(pt,h)\varphi(t)\over p\varphi(pt)(t,h)}\Big),$$ where we used that the sum $S(h,t,1)$ is absolutely convergent and the fact that the argument in the second sum is a multiplicative function in $n$. The contribution of the primes dividing $t$ to this product is $${(t,h)\over t\varphi(t)}\prod_{p|t,~pt_p|h_p}(1-{1\over p})\prod_{p|t,~h_p|t_p}(1-{1\over p^2}) ={(t,h)\over t^2}\prod_{p|t,~h_p|t_p}(1+{1\over p}),$$ where we used that $\varphi(t)/t=\prod_{p|t}(1-1/p)$. If $p\nmid t$, then $$1-{(pt,h)\varphi(t)\over p\varphi(pt)(t,h)}=1-{(p,h)\over p(p-1)},$$ and part 1 follows.\ 2) We have $$S(h,t,2)=\sum_{2|n}{\mu(n)(nt,h)\over nt\varphi(nt)}=-\sum_{2\nmid n}{\mu(n)(nt,h)\over nt\varphi(nt)}.$$ The latter sum has the same Euler product as $S(h,t,1)$, but with the factor for $p=2$ omitted.\ [Remark]{}. The above lemma and the definition of the Artin constant shows that $E(t)=S(1,t,1)$ and $A=S(1,1,1)$.\ Write $M=m/(m,t)$ and $H=h/(Mt,h)$. Then we have [@W Lemma 2.1] $$S(h,t,m)=\mu(M)(Mt,h)E(t)\prod_{q|(M,t)}{1\over q^2-1}\prod_{q|M\atop q\nmid t}{1\over q^2-q-1} \prod_{q|(t,H)\atop q\nmid M}{q\over q+1}\prod_{q|H\atop q\nmid Mt}{q(q-2)\over q^2-q-1}.$$ The parameter $H$ can be avoided as the formula can be rewritten as $$\label{langlang} {\mu(M)(Mt,h)A\over t^2}\prod_{q|mth}{1\over q^2-q-1}\prod_{q|t,~qt_q|h_q\atop m_q|t_q}q(q-1) \prod_{q|t,~h_q|t_q\atop m_q|t_q}(q^2-1)\prod_{q|h\atop q\nmid mt}q(q-2).$$ (In order to see this it is helpful to consider the cases $m_q|t_q$, that is $M_q=1$, and $qt_q|m_q$, that is $q|M$, separately.) These formulae relate $S(h,t,m)$ to $S(1,t,1)$ ($=E(t)$), respectively to $S(1,1,1)$ ($=A$), however, as we will show, expressions simplify considerably if we relate $S(h,t,m)$ to $S(h,t,1)$. We start by showing how to remove odd prime factors from $m$. \[graadoverkop\] Suppose that $p\nmid 2m$. Then $$S(h,t,mp)=\begin{cases}-S(h,t,m)/({p(p-1)\over (p,h)}-1) & {\rm ~if~}p\nmid t;\cr S(h,t,m) & {\rm ~if~}p|t.\cr \end{cases}$$ [*Proof*]{}. If $p|t$ the summation condition $mp|nt$ in the definition of $S(h,t,mp)$ is equivalent with $m|nt$, that is we have $S(h,t,mp)=S(h,t,m)$.\ Next assume that $p\nmid t$. We have $$S(h,t,mp)=\sum_{m|nt\atop p|n}{\mu(n)(nt,h)\over nt\varphi(nt)}=\sum_{m|nt}{\mu(pn)(pnt,h)\over pnt\varphi(pnt)}=-{(p,h)\over p(p-1)}\sum_{m|nt\atop p\nmid n}{\mu(n)(nt,h)\over nt\varphi(nt)}.$$ On noting that the latter sum can be written as $S(h,t,m)-S(h,t,mp)$, the proof is then completed. \[twooh\] Suppose that we are not in the case where $h$ is even and $t$ is odd. We have $$S(h,t,2t_2)=\begin{cases} -S(h,t,1)/3 & {\rm ~if~lcm}(2,h_2)|t_2;\cr -S(h,t,1) & {\rm ~if~lcm}(2,h_2)\nmid t_2. \end{cases}$$ [*Proof*]{}. We can write $$S(h,t,2t_2)=\sum_{2|n}{\mu(n)(nt,h)\over nt\varphi(nt)}=-{1\over 2}\sum_{2\nmid n}{\mu(n)(2nt,h)\over nt\varphi(2nt)}=\epsilon \sum_{2\nmid n}{\mu(n)(nt,h)\over nt\varphi(nt)},$$ where $\epsilon$ is easily determined (and $\epsilon\ne -1$). Since the latter sum is equal to $S(h,t,1)-S(h,t,2t_2)$, we then infer that $S(h,t,2t_2)={\epsilon\over 1+\epsilon}S(h,t,1)$. Working out the remaining details is left to the reader. \[shtminprod\] Let $m$ be an integer, having square free odd part. Let $h$ and $t$ be integers, with the requirement that $t$ be even in case $h$ is even. Then $$S(h,t,m)=S(h,t,1)E_1(m_2)\prod_{p|m,p\nmid 2t}{-1\over {p(p-1)\over (p,h)}-1},$$ where $$E_1(m_2)=\begin{cases} 1 & {\rm ~if~}m_2| t_2\cr -1/3 & {\rm ~if~}m_2 = 2t_2{\rm ~and~lcm}(2,h_2)|t_2\cr -1 & {\rm ~if~}m_2 = 2t_2{\rm ~and~lcm}(2,h_2)\nmid t_2\cr 0 & {\rm ~if~}m_2\nmid 2t_2,\cr \end{cases}$$ In case $2h_2|m_2$, we have $E_1(m_2)=E_2(m_2)$, where $E_2(m_2)$ is given by [(\[e2m2\])]{}. [*Proof*]{}. By Lemma \[t1t2\] the conditions imposed on $h$ and $t$ imply that $S(h,t,1)\ne 0$. By Lemma \[graadoverkop\] it suffices to show that $S(h,t,m_2)=S(h,t,1)E_1(m_2)$. If $m_2|t_2$, then no divisibility condition on $n$ is imposed in the definition of $S(h,t,m_2)$ and so we obtain $S(h,t,m_2)=S(h,t,1)$ and hence $E_1(m_2)=1$. In case $m_2=2t_2$ we invoke Lemma \[twooh\]. If $m_2\nmid 2t_2$, then the summation condition $m|nt$ implies $4|n$ and hence $\mu(n)=0$ and so $S(h,t,m_2)=0$ and hence $E_1(m_2)=0$.\ The final claim follows on noting that if $2h_2|m_2$ and $m_2=2t_2$, then $h_2|t_2$ and hence lcm$(2,h_2)\nmid t_2$ iff $2\nmid t_2$. Proof of Theorem \[mainz\] {#vier} ========================== The idea of the proof is to express $\delta(g,t)$ in terms of $S(h,t,1)$, except in case $g<0$, $2|h$ and $2\nmid t$, when $S(h,t,1)=0$, in which case we express $\delta(g,t)$ in terms of $S(h,t,2)$. These two Wagstaff sums are then related to $A(g,t)$ using the following lemma. Note that it shows that the dependence of $A(g,t)$ on $g$ is weak, as only $h$ and the sign of $g$ matter. \[stoa\] We have $$A(g,t)=\begin{cases} -S(h,t,2)/2 & {\rm ~if~}g<0,~2|h,~2\nmid t;\cr S(h,t,1) & {\rm ~otherwise}.\cr \end{cases}$$ [*Proof*]{}. Note that if $g<0$ and $2|h$, then $F_2=\mathbb Q(i)$ and $[F_2:\mathbb Q]=2$. In the remaining cases we have $[F_p:\mathbb Q]=p(p-1)/(p,h)$. On invoking Lemma \[t1t2\] the proof is then completed. [*Proof of Theorem*]{} \[mainz\]. Equation (\[adoora\]) follows by Lemma \[stoa\] and (\[langlang\]). We will use a few times, cf. the proof of Lemma \[stoa\], that $$\Pi_1=\prod_{p|d(g_0),~p\nmid 2t}{-1\over [F_p:\mathbb Q]-1}=\prod_{p|d(g_0),~p\nmid 2t}{-1\over {p(p-1)\over (p,h)}-1}.$$ Assume GRH.\ [The case $g>0$]{}.\ By Theorem \[waggel\] we have $\delta(g,t)=S(h,t,1)+S(h,t,m)$, with $m={\rm lcm}(2h_2,d(g_0))$. First assume that $2|h$ and $2\nmid t$. Then, by Lemmas \[t1t2\] and \[stoa\], we have $S(h,t,1)=A(g,t)=0$ and we need to show that $\delta(g,t)=0$. Since $S(h,t,1)=0$ it remains to show that $S(h,t,m)=0$. Since for the $n$ in the summation we have $4|2h_2|n$, this is clear. Next assume we are in the remaining case, that is either $h$ is odd, or $2|(h,t)$. Then $S(h,t,1)=A(g,t)$ by Lemma \[stoa\]. Note that $m_2=v_0$. By Lemma \[shtminprod\] we then find that $\delta(g,t)=S(h,t,1)(1+E_1(v_0)\Pi_1)=A(g,t)(1+E_2(v_0)\Pi_1)$, where we have used that $2h_2|v_0$.\ [The case $h$ is odd]{}.\ If $g>0$ then $v=v_0$ and we are done, so assume that $g<0$. The formula for $m$ in Theorem \[waggel\] can be rewritten as lcm$(2,|d(g)|)$, and one finds that $\delta(g,t)=S(h,t,1)+S(h,t,{\rm lcm}(2,|d(g)|))$. This is the same formula as in case $g>0$ and $2\nmid h$, but with $d(g_0)$ replaced by $|d(g)|$. On noting that the odd part of $d(g_0)$ equals the odd part of $d(g)$, the result then follows.\ [The case $g<0$, $2\nmid t$ and $2|h$]{}.\ We have $S(h,t,1)=S(h,t,m)=S(h,t,2h_2)=0$ and hence $\delta(g,t)=-S(h,t,2)/2$ by (\[vierergruppe\]). Now invoke Lemma \[stoa\] to obtain $\delta(g,t)=A(g,t)$.\ [The case $g<0$ and $2|(h,t)$]{}.\ Note that $2|m$ and $S(h,t,1)=A(g,t)$. By Lemma \[shtminprod\] we infer that $S(h,t,2)=S(h,t,1)$ and $S(h,t,2h_2)=S(h,t,1)E_2(2h_2)$, where $$E_2(2h_2)=\begin{cases} 1 & {\rm ~if~}2h_2|t_2;\cr -1/3 & {\rm ~if~}h_2=t_2;\cr 0 & {\rm ~if~}h_2\nmid t_2.\cr \end{cases}$$ Note that $$E_2(4)=\begin{cases} 1 & {\rm ~if~}4|t_2;\cr -1/3 & {\rm ~if~}t_2=2.\cr \end{cases}$$ If $h_2=2$ and $8|d(g_0)$, then by Theorem \[waggel\] we have $m=2g_1$, which can be rewritten as $m=d(g_0)/2$ (thus $m_2=4$) and so $$\begin{aligned} \delta(g,t) & = S(h,t,1)-{S(h,t,2)\over 2}+{S(h,t,4)\over 2}+S(h,t,{d(g_0)\over 2})\nonumber\cr & = S(h,t,1)\Big({1\over 2}+{E_2(4)\over 2}+E_2(4)\Pi_1\Big).\nonumber\end{aligned}$$ where we used that, by Lemma \[shtminprod\], $S(h,t,d(g_0)/2)=S(h,t,1)E_2(4)\Pi_1$. Using that $S(h,t,1)=A(g,t)$ and the formula for $E_2(4)$, we then arrive at (\[dubbel\]).\ In the remaining case, $m={\rm lcm}(4h_2,d(g_0))$. Note that $m_2=4h_2$ and $$E_2(4h_2)=\begin{cases} 1 & {\rm ~if~}4h_2|t_2;\cr -1/3 & {\rm ~if~}2h_2=t_2;\cr 0 & {\rm ~if~}2h_2\nmid t_2. \end{cases}$$ We find that $$\begin{aligned} \delta(g,t) & =S(h,t,1)-{S(h,t,2)\over 2}+{S(h,t,2h_2)\over 2}+S(h,t,m)\nonumber\cr & =S(h,t,1)\Big({1\over 2}+{E_2(2h_2)\over 2}+ E_2(4h_2) \Pi_1\Big).\nonumber\cr \end{aligned}$$ Using that $S(h,t,1)=A(g,t)$ and the formulae for $E_2(2h_2)$ and $E_2(4h_2)$ given above, the proof is then completed. Vanishing of $\delta(g,t)$ ========================== The aim of this section is to give a new proof of Theorem \[sixcondi\] (due to Lenstra [@Lenstra], who stated it without proof). The first published proof was given by Moree in [@near1]. He introduced a function $w_{g,t}(p)\in \{0,1,2\}$ for which he proved (see [@near1], for a rather easier reproof see [@near2]) under GRH that $$N_{g,t}(x)=(h,t)\sum_{p\le x,~p\equiv 1({\rm mod~}t)}w_{g,t}(p){\varphi((p-1)/t)\over p-1} +O\Big({x\log \log x\over \log^2 x}\Big).$$ This function $w_{g,t}(p)$ has the property that, under GRH, $w_{g,t}(p)=0$ for all primes $p$ sufficiently large iff $N_{g,t}$ is finite. Since the definition of $w_{g,t}(p)$ involves nothing more than the Legendre symbol, it is then not difficult to arrive at the cases 1-6. E.g. in case 1 $g$ is a square modulo $p$, and thus $2|t$, contradicting $2\nmid t$. Likewise for the other 5 cases the obstructions can be written down (it turns out $r_g(p)_2\ne t_2$ in each case). For the complete list of obstructions we refer to Moree [@near1 pp. 170-171].\ Regarding the six vanishing cases Wagstaff [@W p. 143] wrote: ‘It is easy to verify directly that our expression for $\delta(g,t)$ vanishes in each of Lenstra’s cases, but it is tedious to check that these are the only cases in which it vanishes’. We will show that once Wagstaff’s result is brought into Euler product form, as done in Theorem \[mainz\], it is straightforward to establish Theorem \[sixcondi\]. A more conceptual, shorter and elegant (but less elementary) proof of Theorem \[sixcondi\] will appear in [@LMS]. [(GRH)]{}. \[sixcondi\] The set $N_{g,t}$ is finite iff $\delta(g,t)=0$ iff we are in one of the following six (mutually exclusive) cases:\ [1)]{} $2\nmid t$, $d(g)|t$.\ [2)]{} $g>0$, $2h_2|t_2$, $3\nmid t$, $3|h$, $d(-3g_0)|t$.\ [3)]{} $g<0$, $h_2=1$, $t_2=2$, $3\nmid t$, $3|h$, $d(3g_0)|t$.\ [4)]{} $g<0$, $h_2=2$, $t_2=2$, $d(2g_0)|2t$.\ [5)]{} $g<0$, $h_2=2$, $t_2=4$, $3\nmid t$, $3|h$, $d(-6g_0)|t$.\ [6)]{} $g<0$, $4h_2|t_2$, $3\nmid t$, $3|h$, $d(-3g_0)|t$. [Example]{}. (GRH). If $g>1$ is square free, then case 1 is the only one to take into account and we find $\delta(g,t)=0$ iff $2\nmid t$, $d(g)|t$, that is iff $2\nmid t$, $g|t$, $g\equiv 1({\rm mod~}4)$. 1 2 3 4 5 6 --------- --------- ----------- -------------- ------------ ------------ ------------ $(g,t)$ $(5,5)$ $(3^3,4)$ $(-15^3,10)$ $(-6^2,6)$ $(-6^6,4)$ $(-3^3,4)$ [*Proof of Theorem*]{} \[sixcondi\]. If one of 1-6 is satisfied, then $N_{g,t}$ is finite. This can be shown by elementary arguments only involving quadratic reciprocity (see Moree [@near1 pp. 170-171]). It is thus enough to show that $\delta(g,t)=0$ iff one of the six cases is satisfied. For the proof we will split up case 6 into two subcases:\ 6a) $g<0$, $2|h_2$, $4h_2|t_2$, $3\nmid t$, $3|h$, $d(3g_0)|t$.\ 6b) $g<0$, $h_2=1$, $4|t_2$, $3\nmid t$, $3|h$, $d(3g_0)|t$.\ (For our proof it is more natural to require $d(3g_0)|t$, which, since $4|t$, is equivalent with $d(-3g_0)|t$.) Let us denote by $d^{*}(g_0)$ the odd part of the discriminant of $g_0$, that is $d^{*}(g_0)=d(g_0)/d(g_0)_2$. Note that $$\label{pie1} \Pi_1=\begin{cases} 1 & {\rm ~if~}d^{*}(g_0)|t;\cr -1 & {\rm ~if~}3|d(g_0),~d^{*}(g_0)|3t,~3\nmid t,~3|h;\cr \in (-1,1) & {\rm ~otherwise}.\cr \end{cases}$$ [The case $2\nmid t$]{}.\ If $2|h$ one has $\delta(g,t)=0$ iff $g>0$, that is iff $d(g)|t$.\ If $2\nmid h$, then $A(g,t)\ne 0$ and we have $\delta(g,t)=0$ iff $E_2({\rm lcm}(2,d(g)_2))=-1$ and $\Pi_1=1$, that is iff lcm$(2,d(g)_2)=2$ and $d^{*}(g)|t$, that is iff $d(g)|t$.\ Thus from now on we may assume that $2|t$. This ensures that $A(g,t)\ne 0$.\ [The case $g>0$ and $2|t$]{}.\ Now the possibility $E_2(m_2)=-1$ cannot occur and thus $\delta(g,t)=0$ iff $E_2(m_2)=1$ and $\Pi_1=-1$. The latter two conditions are both satisfied iff lcm$(2h_2,d(g_0)_2)|t_2$, $3|d(g_0)$, $d^{*}(g_0)|3t$, $3\nmid t$, $3|h$. These conditions can be reformulated as $2h_2|t_2$, $3|d(g_0)$, $d(g_0)|3t$, $3\nmid t$ and $3|h$. Since $3\nmid t$, $3|d(g_0)$, $d(g_0)|3t$ iff $d(-3g_0)|t$, $3\nmid t$, we are done.\ Thus if $g>0$ or $2\nmid t$, then $\delta(g,t)=0$ iff we are in case 1 or in case 2. It remains to consider the case where $g<0$ and $2|t$.\ [The case $g<0$, $2|t$, $2\nmid h$]{}.\ Here we have $\delta(g,t)=0$ iff $E_2(v)=1$ and $\Pi_1=-1$. Note that $E_2(v)=1$ means that we require lcm$(2,d(g)_2)|t_2$.\ If $t_2=2$, then lcm$(2,d(g)_2)|t_2$ and $\Pi_1=-1$ iff we are in case 3.\ If $4|t_2$, then lcm$(2,d(g)_2)|t_2$ and $\Pi_1=-1$ iff we are in case 6b.\ [The case $g<0$, $2|(h,t)$]{}.\ We have $\delta(g,t)=0$ iff we are in one of the following three cases:\ A) $h_2=2,~t_2=2,~8|d(g_0),~\Pi_1=1$;\ B) $h_2=2,~t_2=4,~8|d(g_0),~\Pi_1=-1$;\ C) $2|h_2,~4h_2|t_2,~\Pi_1=-1$.\ It is easily checked that these are merely cases 4, 5 and 6a in different guises.\ To sum up, we have shown that $\delta(g,t)=0$ iff we are in one of the cases 1,2,3,4,5,6a or 6b. Note that the six cases are mutually exclusive.\ We now propose a conjecture on $\delta(g,t)$ for arbitrary rational $g$. It generalizes Conjecture 1. \[netvorige\] The set $N_{g,t}$ has a natural density $\delta(g,t)$ that is given as in Theorem \[mainz\] and is a rational multiple of the Artin constant $A$. The set $N_{g,t}$ is finite iff $\delta(g,t)=0$ iff we are in one of the six cases of Theorem \[sixcondi\]. On combining Theorem \[mainz\] and Theorem \[sixcondi\] we deduce that Conjecture \[netvorige\] holds true on GRH. Conjecture \[netvorige\] is true under GRH. Near-primitive roots density through character sum averages {#prospect} =========================================================== Lenstra, Moree and Stevenhagen [@LMS] show that for a large class of Artin-type problems the set of primes has a natural density $\delta$ that is given by $$\label{pete} \delta=(1+\prod_p E_p)\prod_p A_p,$$ where $\prod_p A_p$ is the ‘generic answer’ to the density problem (e.g. $A$ in the original Artin problem) and $1+\prod_p E_p$ a correction factor. For finitely many primes $p$ one has $E_p\ne 1$ and further $-1\le E_p\le 1$ as $E_p$ is a (real) character sum average over a finite set (and hence the correction factor is a rational number). In particular, it is rather easy in this set-up to determine when $\delta=0$. The character sum method makes use of the theory of radical entanglement as developped by Lenstra [@SanAnton]\ For the near-primitive root problem the method leads rather immediately to the formula $\delta(g,t)=A(g,t)(1+E_2'\Pi_1)$ in case $g>0$. The only harder part is the determination of $E_2'$. For the details the reader is referred to [@LMS].\ Indeed, the great advance of the newer method is that it very directly leads to a formula for the density in Euler product form. The classical method leads to infinite sums involving the Möbius function and nearly multiplicative functions (in our case Wagstaff’s result (Theorem \[waggel\]). It then requires rather cumbersome manipulations to arrive at a density in Euler product form. Indeed, inspired by the predicted result (\[pete\]) the author attempted (and managed) to bring Wagstaff’s result in Euler product form.\ The analogue of Theorem \[mainz\] obtained in this approach, Theorem 6.4 of [@LMS], looks slightly different from Theorem \[mainz\]. However, on noting that $s_2$ as defined in Theorem 6.4 is merely the 2-part of $m$ as defined in Wagstaff’s result Theorem \[waggel\], it is not difficult to show that both methods give rise to the same Euler products for the density. By allowing $g_0$ to be negative in case $h$ is odd and $g<0$, the above 6 cases where vanishing occurs can be reduced to 5 cases (see Corollary 6.5 of [@LMS]). An application ============== Let $\Phi_n(x)$ denote the $n$-th cyclotomic polynomial. Let $S$ be the set of primes $p$ such that if $f(x)$ is any irreducible factor of $\Phi_p(x)$ over $\mathbb F_2$, then $f(x)$ does not divide any trinomial. Over $\mathbb F_2$, $\Phi_p(x)$ factors into $r_2(p)$ irreducible polynomials. Let $$S_1=(\{p>2:2\nmid r_2(p)\}\}\cup \{p>2:2\le r_2(p)\le 16\})\backslash \{3,7,31,73\}.$$ We have $S_1\subseteq S$. The set $S_1$ contains the primes $p>3$ such that $p\equiv \pm 3({\rm mod~}8)$. On GRH the set $S_!$ has density $$\label{densiedens} \delta(S_1)={1\over 2}+A{1323100229\over 1099324800}\approx 0.950077195\cdots$$ [*Proof*]{}. The set $\{p>2:2\nmid r_2(p)\}\}$ equals the set of primes $p$ such that $({2\over p})=-1$, that is the set of primes $p$ such that $p\equiv \pm 3({\rm mod~}8)$. This set has density $1/2$. We thus find, on consulting Table 1, that $$\begin{aligned} \delta(S_1) & = &{1\over 2}+\sum_{2\le j\le 16\atop 2|j}A(2,j)\cr & = &{1\over 2}+E(2)(1+{2\over 3\cdot 4}+{2\over 16}+{2\over 64})+E(6)(1+{2\over 3\cdot 4})+E(10)+E(14),\nonumber\end{aligned}$$ which yields (\[densiedens\]) on invoking the definition (\[eetee\]) of $E(t)$. That $S_1\subseteq S$ is a consequence of the work of Golomb and Lee [@GL].\ [Acknowledgment]{}. Given expressions like (\[langlang\]), my intuition was that expressing $\delta(g,t)$ in Euler product form would lead to very unpleasant formulae and thus I never attempted this. Discussions with Peter Stevenhagen, considering the near-primitive root problem by a much more algebraic method, strongly suggested easier expressions for $\delta(g,t)$ than expected. This led me to try to bring Wagstaff’s result in Euler product form, also with the aim of verifying the results found by the character sum method (alluded to in Section \[prospect\]).\ The memorable discussions with Stevenhagen are gratefully acknowledged. Finally, I like to thank Carl Pomerance for helpful e-mail correspondence. [99]{} S.W. Golomb and P.F. Lee, Irreducible polynomials which divide trinomials over ${\rm GF}(2)$, [*IEEE Trans. Inform. Theory*]{} [**53**]{} (2007), 768–774. C. Hooley, Artin’s conjecture for primitive roots, [*J. Reine Angew. Math.*]{} [**225**]{} (1967), 209–220. 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--- abstract: 'We have performed microscopic calculations of the Rayleigh scattering cross section for arbitrary metallic single-walled carbon nanotubes. The focus of our investigations lies on excitonic effects and their influence on the characteristic features in a Rayleigh scattering spectrum. Our approach is based on density matrix theory including tight-binding energies, the carrier-light coupling as well as the carrier-carrier interaction. Due to the refractive index contribution to the scattering cross section, we observe characteristic features in Rayleigh spectra, such as a strong deviation from the Lorentz peak shape and the larger oscillator strength of the lower-lying transition $M_{ii}^-$ in the double-peaked structure, independently of the chiral angle and the diameter of the investigated nanotubes. We observe excitonic binding energies in the range of $\unit[60-80]{meV}$ for metallic nanotubes with diameters of $\unit[1.5-2.5]{nm}$. The overlap of the excitonic transition with the close-by continuum has a significant influence on the peak shape and a minor influence on the peak intensity ratios. The presented results are in good agreement with recent experimental data.' author: - 'Ermin Malic$^{1}$' - Janina Maultzsch$^2$ - Stephanie Reich$^3$ - Andreas Knorr$^1$ title: 'Excitonic Rayleigh scattering spectra of metallic single-walled carbon nanotubes' --- Introduction ============ Absorption, photoluminescence, and Raman scattering are standard spectroscopy methods to reveal optical properties of nanoscale objects.[@reichbuch; @jorio08] In particular, they have been applied to characterize carbon nanotubes (CNTs) of different chiral angle, diameter, and family.[@bachilo02; @miyauchi04; @telg04] In 2004, Sfeir at al.[@sfeir04] introduced Rayleigh scattering as an additional powerful technique for identifying the optical finger print of individual single-walled CNTs. It allows the investigation of optical properties of *individual* CNTs, since the weak scattering signal is much easier to measure than e.g. the change in intensity due to the absorption. In combination with electron diffraction data, Rayleigh scattering spectroscopy has successfully been applied to determine the electronic structure of individual CNTs, in particular the predicted peak splitting in metallic tubes due to the trigonal warping effect[@saito00; @reich00c] was proven experimentally.[@sfeir06] Recently, excitonic effects in metallic nanotubes have been experimentally proved by measuring their Rayleigh spectra.[@heinz07; @berciaud10] Despite the large screening, metallic nanotubes were shown to exhibit binding energies around $\unit[50]{meV}$, which is small comparing to semiconducting nanotubes,[@wang05b; @maultzsch05c] but still larger than the thermal energy at room temperature. The experimental data on excitonic Rayleigh scattering spectra has not yet been complemented by theoretical studies. In Refs. we studied free-particle Rayleigh scattering spectra of metallic and semiconducting CNTs showing characteristic features in Rayleigh scattering spectra, which distinguish them from corresponding absorption spectra. In this work, we address the question on how the formation of Coulomb-bound electron-hole pairs, i.e. excitons, influences these features. We perform investigations on (i) the excitonic transition and excitonic binding energy, (ii) the trigonal warping splitting as a function of the diameter and the chiral angle, (iii) the relative oscillator strength in the double-peaked spectra of metallic nanotubes, and (iv) the peak shape in Rayleigh spectra of metallic and semiconducting nanotubes. Finally, we compare our results with recent experimental data.[@wu07; @berciaud10] Rayleigh scattering cross section ================================= In our calculations, the Rayleigh scattering cross section is considered for incident light polarized along the nanotube axis accounting for the depolarization effect that strongly suppresses light polarized perpendicular to the nanotube axis.[@ajiki93] Here, we briefly summarize the derivation of the Rayleigh scattering cross section: Nanotubes are regarded as long cylinders with diameters small compared to the wavelength of light. Starting from Maxwell equations, the expression for $\sigma(\omega)$ can be derived by solving the scalar wave equation in cylindrical coordinates,[@bohren] and exploiting the limit of small nanotube radii. The scattering cross section is given by the ratio of the rate $$W_s=\int_A\bm{S}_s\cdot \bm{e}_r \,dA\,,$$ at which energy passes through the scattering surface $A$ and the incident irradiance. The rate $W_s$ is determined by the radial component of the Poynting vector of the scattered field $\bm{S}_s=\frac{1}{2}\mathrm{Re}[\bm{E}_s \times \bm{H}^*_s]$. By introducing vector cylindrical harmonics[@bohren] $ \bm{M}=\nabla \times (\bm{e}_z\psi)$ and $\bm{N}=k^{-1}\nabla \times \bm{M} $ with a scalar function $\psi$, the wave number $k$, and the unit vector $\bm e_z$ parallel to the cylinder axis the problem can be simplified, since these functions satisfy both the vectorial and the scalar wave equation. Once they are calculated, the electric and magnetic field $\bm E_s$ and $\bm H_s$ can be expanded in $\bm M$ and $\bm N$. The scalar function $\psi$ is called a generating function for the vector harmonics $\bm M$ and $\bm N$. Its choice depends on the investigated system, its symmetries and boundary conditions. For Rayleigh scattering from a cylinder, the scalar function has to satisfy the wave equation in cylindrical polar coordinates. An ansatz for the solution is $ \psi(\rho,\phi,z)=Z_n(r)e^{in\phi}e^{ihz} $ with $Z_n(r)$ as Bessel functions of first and second kind of integral order $n$ and with $r=\sqrt{k^2-h^2}$. The quantum number $h$ satisfies the boundary conditions between the cylinder and the surrounding medium. Within the limit of small particles, i.e. for cylinders with a diameter much smaller than the wavelength of light ($\bm{k}\cdot\bm{r}\ll 1$ with $\bm k$ as the wave vector of light and $\bm r$ as the radius of the cylinder), the scattering cross section $\sigma(\omega)$ per unit length can be expressed via the linear susceptibility $\chi(\omega)$ as $$\label{sigma} \sigma(\omega) =\dfrac{\pi^2}{4c^3}r^4 \omega^3 |\chi(\omega)|^2\,,$$ where $\omega$ is the angular frequency of the light and $c$ the speed of light. In contrast to the scattering from a sphere, the cross section is proportional to $\omega^3$ instead of $\omega^4$. This can be traced back to the scattered field $E_s$, which is given asymptotically by $E_s \propto \frac{1}{\sqrt{k\,r}}$ far away from a cylinder, while for a sphere it is $E_s \propto \frac{1}{k\,r}$. The strength of the Rayleigh scattering is determined by the square of the absolute value of the optical susceptibility $\chi(\omega)$. In contrast to the absorption coefficient[@malic06b] $\alpha\propto \mathrm{Im} \chi(\omega)$, Rayleigh scattering has also a contribution from the real part of $\chi(\omega)$ and, hence, includes the influence of the resonant refractive index $n(\omega)\propto \mathrm{Re}\chi(\omega)$ of optical transitions. This leads to important differences in the characteristics of Rayleigh and absorption spectra, which are discussed below. Density matrix approach ======================= To obtain the Rayleigh scattering cross section, we need the optical susceptibility $\chi(\omega)$, which is the linear response function of the perturbed system. Within the $\bm p \cdot \bm A$ approach for the light-matter interaction, it reads[@scully] $$\label{chi2} \chi(\omega) = \dfrac{j(\omega)}{\varepsilon_0 \omega^2 A(\omega)}$$ with the externally driven current density $j(\omega)$ and the vector potential $A(\omega)$. The current density depends on the Fourier transform of the microscopic polarization $p_{\bm k}(t)$ and the optical matrix element $M_{vc}^z(\bm{k})$ along the nanotube axis (here, z-axis)[@malic06b; @hirtschulz08] $$\label{j} j(t) =-i\dfrac{2e_0\hbar}{m_0}\sum_{\bm{k}}\mathrm{Re}\left(M_{vc}^z(\bm{k})p_{\bm{k}} (t)\right).$$ The microscopic polarization $p_{\bm k}(t)=\langle a^+_{\lambda \bm k}a^{\phantom{+}}_{ \lambda' \bm k}\rangle (t)$ is a measure for the transition probability between the two states $|\lambda \bm k\rangle$ and $|\lambda' \bm k\rangle$, where $\lambda, \lambda'$ stand for the band index and $\bm k$ for the wave vector. Our approach is formulated within the formalism of second quantization with $a_{\lambda \bm k}$ and $a^+_{\lambda \bm k}$ as annihilation and creation operators.[@hirtschulz08] As a result, the knowledge of $p_{\bm k}(t)$ allows the calculation of the current density $j(\omega)$, which is required to obtain the optical susceptibility $\chi(\omega)$. Finally, $\chi(\omega)$ determines the Rayleigh scattering cross section $\sigma(\omega)$, cp. [Eq.]{}[sigma]{}. The temporal dynamics of $p_{\bm k}(t)$ is determined within the Heisenberg equation of motion $ i\hbar \frac{d}{dt} p_{\bm k}(t)=[p_{\bm k}(t), H] $ with the Hamilton operator $$\label{H} H=H_\text{0,c}+H_\text{c-f}+H_\text{c-c}\,,$$ which determines the dynamics of a physical system. The first two terms describe the non-interacting carrier system in the presence of the external electromagnetic field. In this work, a semiclassical approach is applied, i.e. the charge carriers are treated quantum mechanically, while the field is considered to be classical. The carrier-field interaction reads $ H_{\text{c-f}}=i\frac{e_0\hbar}{m_0}\sum_{\bm{l,l'}} \bm{M}_{\bm{l,l'}}\cdot \bm{A(t)}\, a^+_{\bm{l}}a^{\phantom{+}}_{\bm{l'}} $ with the optical matrix elements $ \bm{M}_{\bm{l,l'}} $, the vector potential $A(t)$, the electron mass $m_0$, and the elementary charge $e_0$. The carrier-carrier interaction is given by $ H_{\text{c-c}}=\frac{1}{2}\sum_{\bm{l}_1,\bm{l}_2,\bm{l}_3,\bm{l}_4} W^{\bm{l}_1,\bm{l}_2}_{\bm{l}_3,\bm{l}_4}a^+_{\bm{l}_1}a^+_{\bm{l}_2}a^{\phantom{+}}_{\bm{l}_4}a^{\phantom{+}}_{\bm{l}_3} $ with the Coulomb matrix elements $ W^{\bm{l}_1,\bm{l}_2}_{\bm{l}_3,\bm{l}_4}$. The single-particle energy $\varepsilon_{\bm{k}}$ required in the free carrier contribution $H_{0,c}=\sum_{\bm{\bm k}} \varepsilon_{\bm{\bm k}} a^+_{\bm{\bm k}}a^{\phantom{+}}_{\bm{\bm k}}$ is determined within the nearest-neighbor tight-binding (TB) approach.[@reich02b] The periodic boundary conditions around the nanotube circumference are considered by restricting the allowed wave vectors $\bm k$ to lines in the graphene Brillouin zone (zone-folding approximation).[@reichbuch] The two-dimensional wave vector $\bm{k}$ decouples in a continuous component $k_z$ along the nanotube axis and a perpendicular quantized component $k_\perp=\frac{2}{d}m$ with the diameter $d$ and the subband index $m$. Since nanotubes are described by line groups containing a screw axis, two different sets of quantum numbers are possible: linear $(k_z, m)$ and helical $(\tilde{k_z},\tilde{m})$ quantum numbers.[@damnjanovic03] The linear $k_z\in(-\frac{\pi}{a},\frac{\pi}{a}\,]$ corresponds to the pure translational subgroup of the line group. Here, $a$ stands for the translational period along the tube axis. The linear momentum along the tube axis is a conserved quantity. However, the quasi-angular momentum $m\in(-\frac{q}{2},\frac{q}{2}]$ (with $q$ as the number of hexagons in the nanotube unit cells) contains both pure rotations and screw axis operations. As a result, $m$ is not fully conserved and Umklapp rules need to be taken into account, when the Brillouin zone or the $\Gamma$ point are crossed.[@reichbuch] In contrast, the helical angular momentum $\tilde m\in(-\frac{n}{2},\frac{n}{2}\,]$ is a conserved quantity, since it corresponds to pure rotations of the nanotube. The number of helical subbands is considerably smaller compared to linear indices, see Fig. \[energy\]. The Brillouin zone, however, is larger with $\tilde{k}_z\in(-\frac{q}{n}\frac{\pi}{a},\frac{q}{n}\frac{\pi}{a}\,]$, where $n$ is the greatest common divisor of the chiral indices $n_1$ and $n_2$. Figure \[energy\] illustrates the two different sets of quantum numbers by plotting the band structure of the metallic $(22, 13)$ nanotube. In this work, we have applied helical indices taking all subbands and the full Brillouin zone into account. Note, that for nanotubes with small diameters hybridization effects might play an important role.[@blase94] Here, the zone-folded tight-binding wave functions can be inappropriate. In particular, these curvature effects have been shown to significantly contribute to the wide family spread in Kataura plots.[@jiang07] Furthermore, the nearest-neighbor tight-binding approach is known to be a good approximation for transitions close to the $K$ point, whereas it is often insufficient to model peak positions at higher energies. However, in our work we focus on characteristic peak shapes and relative peak intensities in Rayleigh scattering spectra for metallic and semiconducting nanotubes, where we expect hybridization effects to play a minor role. The optical matrix element $M_{vc}(\bm k)=\langle \psi_v(\bm k)|\nabla|\psi_c(\bm k)\rangle$, cp. Fig.\[matrix\], and the Coulomb matrix element $W^{12}_{~34}=\langle \psi_1 \psi_2|W_{Coul}|\psi_3\psi_4\rangle$ with the screened Coulomb potential $W^{12}_{~34}$ enter into the carrier-light Hamiltonian $H_{\text{c-f}}$ and the Coulomb Hamiltonian $H_{\text{c-c}}$ in [Eq.]{}[H]{}, respectively. They are calculated analytically by applying the zone-folded tight-binding wave functions $\psi(k)$.[@reichbuch] Then, all necessary ingredients are available to determine the temporal evolution of the microscopic polarization $p_{\bm k}$ yielding[@hirtschulz08; @malic08b] $$\begin{aligned} \label{bloch1} \dot{p}_{\bm{k}}(t)=-i\tilde{\omega}_{\bm{k}} p_{\bm{k}}(t) +i \tilde{\Omega}_{\bm k}(t)-\gamma p_{\bm{k}}(t).\end{aligned}$$ This Bloch equation is valid in the limiting case of linear optics, where the driving field is considered to be small resulting in negligible change in occupation in valence and conduction band.[@haug] The dynamics of a system is fully determined by the microscopic polarization $p_{\bm k}$. The Coulomb interaction is considered within the Hartree-Fock level.[@hirtschulz08; @malic08b] The Rabi frequency $$\tilde{\Omega}_{\bm k}(t)=\frac{e_0}{m_0}M^{cv}_z(\bm{k})A(t)-\frac{i}{\hbar}\sum_{\bm{k'}}W_{e-h}(\bm{k}, \bm{k'}) p_{\bm{k'}}$$ in [Eq.]{}[bloch1]{} describes the Coulomb renormalized strength of the electron-light interaction. The term includes the renormalization due to the attractive electron-hole interaction[@hirtschulz08; @malic08b] $W_{e-h}(\bm{k}, \bm{k'})$. This term describes the formation of excitons. The strong Coulomb interaction in carbon nanotubes mixes the degenerate states at the $K$ and $K'$ point resulting in a partial lifting of the degeneracy and the formation of bright and dark excitonic states.[@jiang07] In the following, our investigations focus on the optically active (bright) states. The band gap energy $$\tilde{\omega}_{\bm{k}}=\left(\omega_c(\bm{k})-\omega_v(\bm{k})\right)-\frac{i}{\hbar}\sum_{\bm{k'}}W_{e-e}(\bm{k},\bm{k'})$$ contains the renormalization due to the electron-electron coupling $W_{e-e}(\bm{k},\bm{k'})$ corresponding to the self-energy correction, see the red curve in Fig. \[energy\]. The Coulomb interaction is screened within the static limit of the Lindhard equation.[@haug] The Coulomb matrix elements are calculated within the tight-binding approximation by introducing a regularized Coulomb potential, which is parametrized by the Ohno potential.[@zhao04c; @jiang07] More details can be found in Ref. . A phenomenological parameter $\unit[\gamma=(0.0125/\hbar)]{eV}$ is included into [Eq.]{}[bloch1]{}, which determines the linewidth in the calculated spectra.[@spataru04] The parameter describes dephasing processes resulting e.g. from electron-phonon interaction. The influence of phonons and in particular the investigation of phonon sidebands due to the exciton-phonon coupling is beyond the scope of this work and will be in focus of future studies. However, there are simulations on intersubband transitions predicting weak phonon satellites. [@butscher04] The presented approach is similar to the Bethe Salpeter method[@jiang07] within the Hartree Fock level. The advantage of the density matrix theory lies in particular in the description of the ultrafast relaxation dynamics of non-equilibrium charge carriers allowing a microscopic access to their time and momentum-resolved scattering dynamics.[@malic09] Rayleigh scattering spectra =========================== Figure \[spectra\] shows exemplary the excitonic Rayleigh scattering spectrum of the $(22,13)$ metallic nanotube. It is characterized by a series of well-pronounced peaks stemming from optical transitions between conduction and valence bands, cp. the arrows in Fig. \[energy\]. For light polarized along the nanotube axis, transitions are allowed between electronic states with $\Delta m=0$ due to symmetry-imposed selection rules. As a result, the absorption probability is particularly large for transitions between the first valence band $v_1$ to the first conduction band $c_1$ at a minimal energy $E_{11}$, followed by the transition $v_2 \rightarrow c_2$ at $E_{22}$, etc. The corresponding peaks in the spectrum of metallic tubes are denoted with $M_{11}, M_{22}$, etc. Figure \[spectra\] illustrates several characteristic features of Rayleigh scattering spectra of metallic carbon nanotubes: (i) a pronounced double-peaked structure of the optical transitions due to the trigonal warping effect, (ii) stronger intensity of the lower-lying transitions, i.e. the oscillator strength of $M_{ii}^-$ is larger than of $M_{ii}^+$, and (iii) an asymmetry towards lower energies corresponding to an enhanced cross section $\sigma(\omega)$ at the lower-energy wing. Figure \[exc-el\]a) shows a comparison between the Rayleigh scattering spectrum and the absorption spectrum of the exemplary metallic (22,13) nanotube. The largest difference is obtained with respect to the peak shape. The absorption peaks are Lorentzians reflecting the dependence of the absorption coefficient on $\mathrm{Im} \chi(\omega)$. In contrast, the shape of Rayleigh peaks is more complicated showing deviations from the Lorentzian shape on both lower and higher energy side. This can be explained by the interference with the real part of the optical susceptibility, since the Rayleigh scattering cross section is given by $|\chi(\omega)|^2$, as discussed below in detail. Furthermore, the peaks are slightly red-shifted and the intensity ratio is reversed compared to the absorption spectrum. Figure \[exc-el\]b) shows the difference between the excitonic and the corresponding free-particle Rayleigh spectrum of the $(22,13)$ nanotube. The figure illustrates the excitonic effects on Rayleigh scattering spectra: (i) a considerable blue-shift of the free-particle transition energies, as already shown for absorption spectra[@malic09; @hirtschulz08; @malic08b], (ii) the asymmetry towards lower energies remains unchanged, when excitonic effects are included, (iii) the intensity ratio of the double-peaked structure is slightly increased, and (iv) the cross section at the higher-energy side of transitions is reduced. In the following paragraphs, the observed characteristic features of excitonic Rayleigh scattering spectra are discussed in detail. Excitonic binding energies -------------------------- Excitonic effects significantly influence optical properties of carbon nanotubes, as shown for absorption spectra in many previous reports.[@spataru04; @perebeinos04; @mol05; @capaz06; @jiang07; @deslippe07; @hirtschulz08; @malic09] Excitonic binding energies in the range of have been observed for semiconducting[@wang05b; @maultzsch05c] and in the range of for metallic nanotubes.[@heinz07] Our investigation on Rayleigh scattering spectra are in good agreement with these findings. We observe strong shifts due to the formation of bound electron-hole pairs. The binding energies are around for investigated metallic nanotubes with $d \approx \unit[1.5-2.5]{nm}$. For semiconducting nanotubes, we observe excitonic binding energies in the range of . Our approach allows the investigation of a large variety of different carbon nanotubes. The calculation of the Kataura plot reveals the diameter and the chirality dependence of the excitonic transition and binding energy. As already reported in literature,[@jiang07] it exhibits main $1/d$ lines for each transition $M_{ii}$ and a characteristic V-shaped structure reflecting the chirality dependence of the trigonal-warping splitting. Trigonal warping peaks splitting -------------------------------- Trigonal warping describes the deviation of the equi-energy contours from circles in the Brillouin zone of graphene around the $K$ point.[@saito00; @reich00c] Due to its trigonal shape, an energy splitting of Van-Hove singularities stemming from different sides with respect to the $K$ point appears. This strongly depends on the orientation of the triangle: The splitting is maximal for nanotubes with a small chiral angle and it vanishes for armchair tubes. Furthermore, the higher the transition energy, the larger is the trigonal warping effect, since the deviation from circles is larger. Figure \[trig\] shows the peak splitting $\Delta(d,\theta)$ as a function of the chiral angle and the diameter for metallic nanotubes for both the excitonic and the free-particle picture. First, we observe that excitons do not influence the trigonal warping induced splitting. Second, we find that the splitting scales with diameter $d$ as $\Delta(d,\theta_0) \propto A/d^2$ at a constant chiral angle $\theta_0$. The coefficient $A$ only depends on the order of the transition. The larger the diameter, the smaller is the transition energy and the weaker is the trigonal warping effect. For the dependence on the chiral angle $\theta$, the scaling law is $\Delta(d_0,\theta) \propto A_1-A_2 \theta^2$ at a nearly constant diameter $d_0$ confirming that the splitting is maximal for zigzag nanotubes with $\theta=0^o$ and zero for armchair tubes with $\theta=30^o$. Peak intensity ratio -------------------- The lower-lying transitions within the double-peaked structure of Rayleigh scattering spectra show a higher oscillator strength independently of the chiral angle and diameter, i.e. $M_{ii}^-$ is always higher in intensity than $M_{ii}^+$. The intensity ratio $R_{ii}=I(M_{ii}^-)/I(M_{ii}^+)$ increases with decreasing chiral angle. In the limiting case of armchair nanotubes, the ratio is exactly 1 due to vanishing splitting. The described behavior of peak intensity ratios is not significantly influenced by excitons. The relative intensities can be explained by the different behavior of the optical matrix element $M_{vc}(k)$ entering in $H_{\text{c-f}}$ along the two high-symmetry lines $K\varGamma$ and $KM$ in the graphene BZ.[@malic08] The carrier-field interaction turns out to be higher on the $KM$ side, cp. Fig. \[matrix\]. As a result, the lower-lying transitions in the double-peaked structure stemming from the $KM$ side[@thomsen07] are amplified. Following this argumentation, the intensity ratio should increase with the order of transition. However, the dependence of the scattering cross section $\sigma(\omega)$ on $\omega^3$ cancels this effect, since the energetically higher transition $M_{ii}^+$ is enhanced with respect to $M_{ii}^-$ resulting in a decrease of the intensity ratio. Another interesting observation is the inverse intensity ratio $R_{ii}<1$ in absorption spectra, cp. Fig. \[exc-el\]a). This can be ascribed to the overlap of the $M_{ii}^+$ peak with the Van-Hove singularity associated with $M_{ii}^-$. The high-energy tail of the Van-Hove singularity enhances the intensity of $M_{ii}^+$ leading to an intensity ratio $R_{ii}$ smaller than 1. In the case of Rayleigh scattering, the overlap with the continuum is smaller, since here the continuum is not characterized by a Van Hove singularity. As a result, the intensity ratio $R_{ii}$ remains larger than 1 - as expected from the family behavior of the optical matrix element. Peak shape ---------- Figures \[ray\] and \[abs\] illustrate the characteristic peak shape of an exemplary metallic and a semiconducting nanotube in excitonic (solid blue lines) and free-particle (solid red lines) Rayleigh scattering and absorption spectra, respectively. For comparison, the figures also show a fit with a Lorentzian in the background (dashed green lines). Rayleigh peaks are shown to be asymmetric towards lower energies reflecting an enhanced cross section $\sigma(\omega)$ at the lower-energy wing, cp. Fig. \[ray\]. This can be traced back to the refractive part of the optical susceptibility. The latter exhibits a long tail on the low energy side of each transition, which adds up with the resonant response leading to the observed asymmetry. Both metallic and semiconducting nanotubes exhibit this characteristic asymmetry leading to a considerable broadening of the ”Lorentzian-like” Rayleigh peaks[@berciaud10] - in contrast to the corresponding peaks in the absorption spectra, cp. Fig. \[abs\]. Furthermore, we observe an interesting feature on the high-energy side of metallic nanotubes. Here, two effects are competing: On the one side, the spectrally decaying refractive index leads to a reduction of the scattering cross section. On the other hand, due to the small binding energies there is an overlap between the excitonic transition and the continuum lifting up the intensity. As a result, the overall reduction is much smaller compared to semiconducting nanotubes, where the excitonic binding energy is large and the overlap with the continuum is negligibly small. For comparison, Fig. \[abs\] shows the peak shape in excitonic and free-particle absorption spectra for the same exemplary metallic and semiconducting nanotube as in Fig. \[ray\]. Since the absorption coefficient is determined only by the imaginary part of the optical susceptibility, the asymmetry to lower energies and the resulting broadening are not present in absorption spectra. The peaks are perfect Lorentzians (dashed line) reflecting the excitonic character of the transition. There is no interference with the refractive part of the response function resulting in narrow peaks with a width of determined by the parameter $\gamma$ entering [Eq.]{}[bloch1]{}. Note that the peak shape of metallic nanotubes exhibits a higher-energy shoulder due to the small excitonic binding energies and the resulting overlap with the continuum.[@deslippe07] Comparison to experiment ------------------------ Figure \[exp-th\] illustrates the good agreement of theoretical and experimental Rayleigh scattering spectra for two exemplary metallic tubes. Including excitonic effects even further improved the comparison with the experiment.[@malic08; @wu07] As predicted in theory, the experimentally observed Rayleigh spectra show a double-peaked structure with a clearly enhanced scattering intensity at the lower energy wing of transitions. Furthermore, the oscillator strength of the first peak in the double-peaked structure is found to be stronger in intensity. This agrees well with the experiment, where the intensity ratio is even more pronounced. The calculated transition energies, however, differ from the experimental results. They are blue-shifted by approximately $\unit[0.3-0.4]{eV}$ compared to the experiment. This deviation can be traced back to the calculation of the band structure within the nearest-neighbor tight-binding approach, which is known to be a good description for transitions close to the $K$ point.[@reichbuch] For higher energies, however, a considerable blue-shift in comparison to third-nearest neighbor TB or first-principle calculations was observed.[@reich02b] Extensions to third-nearest neighbor TB or extended TB calculations[@popov04b; @jiang07] would further improve the comparison with experimental data. Conclusions =========== We have performed microscopic calculations of the Rayleigh scattering cross section including excitonic effects for arbitrary metallic single-walled carbon nanotubes. Our approach is based on the density matrix formalism combined with zone-folded tight-binding wave functions. While the absorption coefficient is given only by the imaginary part of the optical susceptibility, the Rayleigh scattering cross section also contains the influence of the real part corresponding to refractive index contribution. This leads to characteristic features in Rayleigh scattering spectra, such as the strong deviation from the Lorentz peak shape exhibiting an enhanced cross section on the lower energy wing, and the larger oscillator strength of the lower-lying transition $M_{ii}^-$ in the double-peaked structure independently of the chiral angle and the diameter of the investigated tubes. We discuss the influence of excitonic effects on these characteristic features including a study on the trigonal warping splitting. The comparison with recent experimental data yields a good agreement with respect to the characteristic peak shape and the peak intensity ratios.\ We acknowledge the support from Sfb 658 and the ERC under grant number 210642. Furthermore, we thank T. F. Heinz (Columbia university) for fruitful discussions. [37]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ** (, , ). , , , ** (, ). , , , , , , ****, (). , , , , , ****, (). , , , , , ****, (). , , , , , , , , ****, (). , , , ****, (). , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , ****, (). , , , , , , , , , **** (). , , , , ****, (). , , , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , ****, (). , ****, (). , ** (, , ). , , , , , ****, (). , ** (, ). , , , , , , , ****, (). , , , , ****, (). , , , , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , , , , ****, (). , ** (, ). , ****, (). , , , , ****, (). , , , , ****, (). , , , , ****, (). , , , ****, (). , , , , ****, (). , , , , ****, (). , , , , ****, (). , ** (, , ). , ****, ().

--- abstract: | Let $\Omega$ be a bounded smooth domain in $\RR^N$. We consider the problem $u_t= \Delta u + V(x) u^p$ in $\Omega \times [0,T)$, with Dirichlet boundary conditions $u=0$ on $\partial \Omega \times [0,T)$ and initial datum $u(x,0)= M \varphi (x)$ where $M \geq 0$, $\varphi$ is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of $M$. As a consequence of these estimates we find that, for $M$ large, the blow up set concentrates near the points where $\varphi^{p-1}V$ attains its maximum. address: - 'Departamento de Matem[á]{}tica, Universidad Catolica de Chile, Casilla 306, Correo 22, Santiago, Chile. ' - ' Instituto de Matem[á]{}ticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 123, Madrid, Spain,on leave from Departamento de Matem[á]{}tica, FCEyN UBA (1428)Buenos Aires, Argentina.' author: - Carmen Cortazar - Manuel Elgueta - 'Julio D. Rossi' title: 'The blow-up problem for a semilinear parabolic equation with a potential' --- [^1] [**1. INTRODUCTION**]{} {#introduction .unnumbered} ======================= In this paper we study the blow-up phenomena for the following semilinear parabolic problem with a potential $$\label{problema} \begin{array}{ll} u_t = \Delta u + V(x) \, u^p \qquad & \mbox{ in }\Omega \times (0,T),\\[8pt] u(x,t) = 0 \qquad & \mbox{ on } \partial \Omega \times (0,T), \\[8pt] u(x,0) = M\varphi(x) \qquad & \mbox{ in } \Omega . \end{array}$$ First, let us state our basic assumptions. They are: $\Omega$ is a bounded, convex, smooth domain in $\RR^N$ and the exponent $p$ is subcritical, that is, $1 < p < (N+2)/(N-2)$. The potential $V$ is Lipschitz continuous and there exists a constant $c>0$ such that $V(x) \geq c$ for all $x \in \Omega$. As for the initial condition we assume that $M \geq 0$ and that $\varphi$ is a smooth positive function compatible with the boundary condition. Moreover, we impose that $$\begin{aligned} \label{initial.2} M\Delta \varphi + \frac{\min\limits_{x \in \Omega} V(x)}{2} M^p \varphi^p \ge 0.\end{aligned}$$ We note that holds for $M$ large if $\Delta \varphi$ is nonnegative in a neighborhood of the set where $\varphi$ vanishes. It is known that, and we will prove it later for the sake of completeness, once $\varphi$ is fixed the solution to (\[problema\]) blows up in finite time for any $M$ sufficiently large. By this we understand that there exists a time $T=T(M)$ such that $u$ is defined in $\Omega \times [0,T)$ and $$\lim_{t\to T} \| u(\cdot,t)\|_{L^\infty(\Omega)}=+\infty .$$ The study of the blow-up phenomena for parabolic equations and systems has attracted considerable attention in recent years, see for example, [@B], [@BB], [@GK1], [@GK2], [@GV], [@HV1], [@HV2], [@M], [@Z] and the corresponding references. A good review in the topic can be found in [@GaVa]. When a large or small diffusion is considered, see [@IY], [@MY]. Important issues in a blow-up problem are to obtain estimates for the [*blow-up time*]{}, $T(M)$, and determine the spatial structure of the set where the solution becomes unbounded, that is, the [*blow-up set*]{}. More precisely, the blow-up set of a solution $u$ that blows up at time $T$ is defined as $$B(u) = \{ x / \ \mbox{there exist }x_n \to x, \, t_n\nearrow T, \ \mbox{with } u(x_n, t_n) \to \infty \}.$$ The problem of estimating the blow-up time and the description and location of the blow-up set has proved to be a subtle problem and has been addressed by several authors. See for example [@SGKM], [@GaVa] and the corresponding bibliographies. Our interest here is the description of the asymptotic behavior of the blow-up time, $T(M)$, and of the blow-up set, $B(u)$, as $M \to \infty$. It turns out that their asymptotics depend on a combination of the shape of both the initial condition, $\varphi$, and the potential $V$. Roughly speaking one expects that if $\varphi \equiv 1$ then the blow-up set should concentrate near the points where $V$ attains its maximum. On the other hand if $V\equiv 1$ the blow-up set should be near the points where $\varphi$ attains its maximum. Just to see what to expect, if we drop the laplacian, we get the ODE $ u_t = V(x) u^p$ with initial condition $u(x,0) = M \varphi (x)$. Here $x$ plays the role of a parameter. Direct integration gives $ u(x,t) = C (T-t)^{-1/(p-1)}$ with $$T=\frac{ M^{1-p}}{ (p-1) V(x) \varphi^{p-1}(x) }.$$ Hence, blow-up takes place at points $x_0$ that satisfy $ V(x_0) \varphi^{p-1} (x_0) = \max_x V (x) \varphi^{p-1}$. Therefore, we expect that the quantity that plays a major role is $(\max_x V (x) \varphi^{p-1}(x) )$. \[teo.Mej\] There exists $\bar M >0$ such that if $M \geq \bar M$ the solution of blows up in a finite time that we denote by $T(M)$. Moreover, let $$A= A(\varphi, V) := \frac{1}{ (\max_x \varphi^{p-1}(x) V (x)) },$$ then there exist two positive constants $C_1$, $C_2$, such that, for $M$ large enough, $$\label{T.2} \displaystyle - \frac{C_1}{ M^{\frac{p-1}{4} } } \le T(M) M^{p-1} - \frac{A}{p-1} \displaystyle \le \frac{C_2}{ M^{\frac{p-1}{3} } },$$and the blow-up set verifies, $$\label{V(BU).2} \varphi^{p-1} (a) V (a) \ge \frac{1}{A} -\frac{C}{ M^{\gamma} }, \qquad \mbox{ for all } a \in B(u),$$ where $\gamma= min (\frac{p-1}{4},\frac{1}{3})$. Note that this result implies that $$\lim_{M\to \infty} T(M) M^{p-1} = \frac{A}{p-1}.$$ Moreover, it provides precise lower and upper bounds on the difference $T(M) M^{p-1} - \frac{A}{p-1}$. We also observe that shows that the set of blow-up points concentrates for large $M$ near the set where $\varphi^{p-1} V$ attains its maximum. If in addition the potential $V$ and the initial datum $\varphi$ are such that $\varphi^{p-1} V$ has a unique non degenerate maximum at a point $\bar a$, then there exist constants $c >0$ and $d>0$ such that $$\varphi^{p-1} (\bar a) V(\bar a) - \varphi^{p-1} (x) V(x) \ge c |\bar a - x |^2 \quad \mbox{ for all } x \in B(\bar a,d).$$ Therefore, according to our result, if $M$ is large enough one has $$|\bar a -a| \le \frac{C}{ M^{\frac{\gamma}{2}}} \quad \mbox{ for any } a\in B(u),$$ with $\gamma= min (\frac{p-1}{4},\frac{1}{3})$. Throughout the paper we will denote by $C$ a constant that does not depends on the relevant parameters involved but may change at each step. **[2. Proof of Theorem \[teo.Mej\].]{}** {#proof-of-theorem-teo.mej. .unnumbered} ======================================== We begin with a lemma that provides us with an upper estimate of the blow-up time. This upper estimate gives the upper bound for $T(M) M^{p-1}$ in and will be crucial in the rest of the proof of Theorem \[teo.Mej\]. \[est.T\] There exist a constant $C>0$ and $M_0>0$ such that for every $M\ge M_0$, the solution of blows up in a finite time that verifies $$\label{esti.T} T (M) \le \frac{A}{M^{p-1} (p-1)}+\frac{C}{M^{\frac{p-1}{3}}M^{p-1}}.$$ [[**[Proof: ]{}**]{}]{}Let $\bar a \in \Omega$ be such that $$\varphi^{p-1} (\bar a)V(\bar a)=\max_x \varphi^{p-1}(x)V(x),$$ $L$ the constant of Lipschitz continuity of $V$, and $K$ an upper bound for the first derivatives of $\varphi$ and $L$. In order to get the upper estimate let $M$ be fixed and $\ve=\ve(M) >0$ to be defined latter, small enough so all functions involved are well defined. Pick $$\delta =\frac{\ve}{2 K},$$ then $$V(x) \ge V(\bar a) -\frac{\ve}{2} \quad \mbox{ and } \qquad \varphi(x) \ge \varphi(\bar a)- \ve \quad \mbox{ for all } x \in B(\bar a, \delta).$$ Let $w$ be the solution of $$\begin{array}{ll} w_t = \Delta w + \left(V(\bar a) -\displaystyle \frac{\ve}{2}\right) w^p & \mbox{ in } B(\bar a,\delta) \times (0,T_w), \\[8pt] w=0 & \mbox{ on } \partial B(\bar a, \delta) \times (0,T_w), \\[8pt] w(x,0) = M (\varphi (\bar a)- \ve), & \mbox{ in } B(\bar a,\delta) \end{array}$$ and $T_w$ its corresponding blow up time. A comparison argument shows that $u \ge w$ in $B(\bar a, \delta) \times (0,T)$ and hence $$T \le T_w.$$ Our task now is to estimate $T_w$ for large values of $M$. To this end, let $\lambda_1 (\delta)$ be the first eigenvalue of $-\Delta$ in $B(\bar a,\delta)$ and let $\phi_1$ be the corresponding positive eigenfunction normalized so that $$\int_{B(\bar a, \delta) } \phi_1 (x) \, dx =1.$$ That is, $$\left\{\begin{array}{ll} - \Delta \phi_1 = \lambda_1 (\delta) \phi_1 , \qquad & \mbox{ in } B(\bar a,\delta),\\[8pt] \phi_1 =0 \qquad & \mbox{ on } \partial B(\bar a, \delta). \end{array}\right.$$ Now, set $$\Phi (t) = \int_{B(\bar a,\delta)} w(x,t) \phi_1 (x)\, dx.$$ Then $\Phi(t)$ satisfies $\Phi(0) = M(\varphi (\bar a)- \ve)$ and $$\begin{array}{rl} \Phi ' (t) & = \displaystyle \int_{B(\bar a,\delta)} w_t (x,t) \phi_1 (x) \, dx \\[10pt] & = \displaystyle \int_{B(\bar a,\delta)} \left( \Delta w(x,t) \phi_1 (x) + \left( V(x_1) -\frac{\ve}{2} \right) w^p (x,t) \phi_1(x) \right) \, dx \\[10pt] & \ge \displaystyle -\lambda_1 (\delta) \int_{B(\bar a,\delta)} w(x,t) \phi_1 (x) \, dx \\[10pt] & \qquad \displaystyle+ \left( V(\bar a) -\frac{\ve}{2} \right) \left( \int_{B(\bar a,\delta)} w (x,t) \phi_1(x) \, dx \right)^p \\[10pt] & = \displaystyle -\lambda_1 (\delta) \Phi (t) + \left( V(\bar a) - \frac{\ve}{2}\right ) \Phi (t)^p. \end{array}$$ Let us recall that there exists a constant $D$, depending on the dimension only, such that the eigenvalues of the laplacian scale according to the rule $ \lambda_1 (\delta ) = D \delta^{-2}$. Now, we choose $\ve$ such that $$\lambda_1 (\delta) = D \delta^{-2}=D \left(\frac{\ve}{2 K}\right)^{-2}= \frac{\ve}{2}(M(\varphi (\bar a)- \ve))^{p-1}.$$ So, $\ve$ is of order $$\ve \sim \frac {C}{ M^{\frac{p-1}{3}}}.$$ Choose $M_0$ such that for $M\geq M_0$ the resulting $\ve$ is small enough. Then for any $M \geq M_0$ we have that $$\label{ec.phi} \Phi ' (t) \geq (V(\bar a) - \ve) \Phi (t)^p,$$ for all $t \geq 0$ for which $\Phi$ is defined. Since $\Phi (0)=M (\varphi(\bar a) - \ve)$ and $T_w$ is less or equal than the blow up time of $\Phi$ integrating it follows that $$\begin{array}{rl} T_w & \le \displaystyle \frac{1}{M^{p-1} (p-1) ( V(\bar a) -\ve ) (\varphi (\bar a) -\ve)^{p-1} } \\[12pt] & \displaystyle \leq \frac{1}{M^{p-1} (p-1) V(\bar a) \varphi (\bar a) ^{p-1}}+\frac{C}{M^{\frac{p-1}{3}}M^{p-1} }, \end{array}$$ for all $M \geq M_0$. Now we prove a lemma that provides us with an upper bound for the blow up rate. We observe that this is the only place where we use hypothesis . \[cota.por.arriba\] Assume . Then there exists a constant $C$ independent of $M$ such that $$u(x,t) \le C(T-t)^{-\frac{1}{p-1}}.$$ Let $m = \min\limits_{x \in \Omega} V$. Following ideas of [@FMc], set $$v = u_t-\frac{m}{2} u^p.$$ Then $v$ verifies $$\begin{array}{ll} v_t- \Delta v - V(x)p u^{p-1}v = \displaystyle \frac{m}{2} p(p-1) u^{p-2} |\nabla u |^2 \geq 0 & \mbox{ in }\Omega \times (0,T),\\[8pt] v =0 & \mbox{ on } \partial \Omega \times (0,T), \\[8pt] v(x,0) = M\Delta \varphi + \left(V(x)-\displaystyle \frac{m}{2}\right) M^p \varphi^p \geq 0 &\mbox{ in } \Omega. \end{array}$$ Therefore $v \geq 0$ and hence $$u_t \geq\frac{m}{2} u^p.$$ Integrating this inequality from $0$ to $T$ we get $$u(x,t)\leq \frac {2^{\frac{1}{p-1}}}{( m (p-1) (T-t)))^{\frac{1}{p-1}}}\equiv C(T-t)^{-\frac{1}{p-1}},$$ as we wanted to prove. We are now in a position to prove Theorem \[teo.Mej\]. [**Proof of Theorem \[teo.Mej\]:**]{} The idea of the proof is to combine the estimate of the blow-up time proved in Lemma \[est.T\] with local energy estimates near a blow-up point $a$, like the ones considered in [@GK1] and [@GK2], to obtain an inequality that forces $\varphi^{p-1} (a)V(a)$ to be close to $\max_x \varphi^{p-1}V$. Let us now proceed with the proof of the estimates on the blow-up set. We fix for the moment $M$ large enough such that $u$ blows up in finite time $T=T(M)$ and let $a=a(M)$ be a blow up point. As in [@GK2], for this fixed $a$ we define $$w(y, s) = (T-t)^{\frac{1}{p-1}} u(a+y(T-t)^{\frac{1}{2}},t) |_{t= T(1-e^{-s})} .$$ Then $w$ satisfies $$\label{ec.w} w_s = \Delta w - \frac{1}{2} y \cdot \nabla w -\frac{1} { p-1}w + V(a+yTe^{-\frac{s}{2}}) w^{p},$$ in $\cup_{s \in (0,\infty)} \Omega (s) \times \{s\}$ where $\Omega (s) = \Omega_a(s) = \{ y \, : \, a+yTe^{-\frac{s}{2}} \in \Omega \}$ with $w(y,0) = T^{\frac{1}{p-1}} \varphi(a+y T^{\frac{1}{2}})$. The above equation can rewritten as $$w_s = \frac{1} { \rho} \nabla (\rho \nabla w)- \frac{1} { p-1}w + V(a+y T e^{-\frac{s}{2}}) w^{p}$$ where $\rho(y)=\exp(\frac{-|y|^2} { 4})$. Consider the energy associated with the ”frozen” potential $$V \equiv V(a),$$ that is $$E(w)=\int_{\Omega(s)} \left({\frac{1}{2}} |\nabla w|^2 +{\frac{1}{2(p-1)}} w^2 -{\frac{1}{p+1}} V(a) w^{p +1} \right)\rho(y)\, dy .$$ Then, using the fact that $\Omega$ is convex, we get $$\frac{dE}{ds}\le -\int_{\Omega(s)}(w_s)^2 \rho(y)\, dy + \int_{\Omega(s)} (V(a+yTe^{-\frac{s}{2}})-V(a)) w^p w_s \rho(y)\, dy.$$ Since $V(x)$ is Lipschitz and $w$ is bounded due to Lemma \[cota.por.arriba\], then there exists a constant $C$ depending only on $N$, $p$ and $V$, recall that the constant in Lemma \[cota.por.arriba\] does not depend on $M$, such that $$\frac{dE}{ds} \leq -\int (w_s)^2 \rho(y)\, dy +C e^{-\frac{s}{2}}T \left(\int (w_s)^2 \rho(y)\, dy\right)^{1/2}.$$ Maximizing the right hand side of the above expression with respect to $\int (w_s)^2 \rho(y)\, dy$ we obtain $$\frac{dE}{ds} \leq C e^{-s }T^2$$ and integrating is $s$ we get $$\label{E.decrece} E(w)\leq E(w_0)+ C T^2 .$$ Since $w$ is bounded and satisfies , following the arguments given in [@GK1] and [@GK2], one can prove that $w$ converges as $s\to \infty$ to a non trivial bounded stationary solution of the limit equation $$\label{ec.limite} 0= \Delta z - \frac{1} { 2} y \cdot \nabla z -\frac{1} { p-1}z + V(a) z^{p}$$ in the whole $\RR^N$. Again by the results of [@GK1] and [@GK2], since $p$ is subcritical, $1<p <(N+2)/(N-2)$, the only non trivial bounded positive solution of (\[ec.limite\]) with $V(a) =1$ is the constant $(p-1)^{-\frac{1}{p-1}}$. A scaling argument gives that the only non trivial bounded positive solution of (\[ec.limite\]) is the constant $k = k(a)$ given by $$k(a) =\frac {1}{( V(a) (p-1))^{\frac{1}{p-1}}}.$$ Therefore, we conclude that $$\lim_{s \rightarrow \infty}w=k(a)$$ if $a$ is a blow-up point. Also by the results of [@GK1], [@GK2] we have $$\label{manuel} E(w(\cdot ,s)) \rightarrow E(k(a)) \qquad \mbox{ as } s\to \infty,$$ where $$\begin{array}{rl} E(k(a))= & \displaystyle \int \left({\frac{1}{2(p-1)}}(k(a))^2 - {\frac{1}{p+1}}V(a) (k(a))^{p +1} \right)\rho(y) \, dy \\ \\ = & (k(a))^2 \displaystyle \left({\frac{1}{2(p-1)}}-{\frac{1}{(p+1)(p-1)}}\right) \int \rho(y) \, dy. \end{array}$$ By and we obtain that, if $a$ is a blow-up point, then $$E(k(a))\leq E(w_0)+ C T^2 .$$ where $w_0(y) = w(y,0) = T^{\frac{1}{p-1}}M \varphi(a + y T^{\frac{1}{2}}).$ As $\varphi$ is smooth, $y \rho(y)$ integrable, and $T^{\frac{1}{p-1}}M$ is bounded by Lemma \[est.T\], there are constants $C$ independent of $a$ such that for $ M\geq M_0$ $$\begin{array}{rl} E(w(\cdot ,0)) & \displaystyle =\int_{\Omega(0)} \left({\frac{1}{2}} |\nabla w_0 (y)|^2 +{\frac{1}{2(p-1)}} w_0^2 (y) \right)\rho(y)\, dy \\[10pt] & \qquad \displaystyle -\int_{\Omega(0)} \left({\frac{1}{p+1}} V(a) w_0^{p +1} (y) \right)\rho(y)\, dy \\[10pt] & \displaystyle \leq\int_{\Omega(0)} \left({\frac{1}{2}} (T^{\frac{1}{p-1}}M)^2 T|\nabla \varphi(a)|^2\right)\rho(y)\, dy \\[10pt] &\qquad \displaystyle +\int_{\Omega(0)} \left({\frac{1}{2(p-1)}} (T^{\frac{1}{p-1}}M \varphi(a))^2 \right)\rho(y)\, dy \\[10pt] & \qquad \displaystyle - \int_{\Omega(0)} \left({\frac{1}{p+1}} V(a) (T^{\frac{1}{p-1}}M \varphi(a))^{p +1} \right)\rho(y)\, dy \\[10pt] & \qquad + CT^{\frac{3}{2}}+ CT^{\frac{1}{2}}. \end{array}$$ Therefore, since $|\nabla \varphi|$ is bounded, $$\begin{array}{rl} E(w(\cdot ,0)) & \displaystyle \leq \int_{\Omega (0)} \left({\frac{1}{2(p-1)}} (T^{\frac{1}{p-1}}M \varphi(a))^2 \right)\rho(y)\, dy \\[10pt] & \qquad - \displaystyle \int_{\Omega (0)} \left({\frac{1}{p+1}} V(a) (T^{\frac{1}{p-1}}M \varphi(a))^{p +1} \right)\rho(y)\, dy \\[10pt] & \qquad + CT^{\frac{3}{2}}+ CT^{\frac{1}{2}}. \end{array}$$ Or, since $T\le 1$ for $M$ large $$E(w(\cdot ,0)) \le E(T^{\frac{1}{p-1}}M \varphi(a))+ CT^{\frac{1}{2}}.$$ Hence we arrive to the following bound for $E(k(a))$ $$\label{carmen.*} E(k(a))\leq E(w(\cdot ,0))+ C T^2 \leq E(T^{\frac{1}{p-1}}M \varphi(a))+ CT^{\frac{1}{2}}.$$ Observe that if $b$ is a constant then the energy can be written as $$E(b)=\Gamma F(b),$$ where $\Gamma$ is the constant $$\Gamma =\int \rho (y) \, dy$$ and $F$ is the function $$F(z)=\left({\frac{1}{2(p-1)}} z^2 -{\frac{1}{p+1}} V(a) z^{p +1} \right).$$ As $F$ attains a unique maximum at $k(a)$ and $F''(k(a))=-1$ there are $\alpha$ and $\beta$ such that if $|z-k(a)|\leq \alpha $ then $$F''(z)\leq -\frac{1}{2},$$ and if $|F(z)-F(k(a))|\leq \beta$ then $$|z-k(a)|\leq \alpha.$$ From we obtain $$F(k(a))\leq F(T^{\frac{1}{p-1}}M \varphi(a))+ CT^{\frac{1}{2}}.$$ If $M_1$ is such that $C(T(M_1))^{\frac{1}{2}}= \beta$ then for $M\geq max(M_0,M_1)$ $$\beta \ge CT^{\frac{1}{2}}\geq F(k(a))- F(T^{\frac{1}{p-1}}M \varphi(a)).$$ Hence by the properties of $F$, $$|k(a)- T^{\frac{1}{p-1}}M \varphi(a)| \le \alpha.$$ Therefore $$CT^{\frac{1}{2}}\geq F(k(a))- F(T^{\frac{1}{p-1}}M \varphi(a))\geq \frac{1}{4} (T^{\frac{1}{p-1}}M \varphi(a)-k(a))^2.$$ So, using Lemma \[est.T\], $$\label{carmen.**} \begin{array}{rl} k(a)-CT^{\frac{1}{4}} & \displaystyle \leq T^{\frac{1}{p-1}}M \varphi(a) \\[8pt] & \displaystyle \leq \frac{\varphi(a)}{ (p-1)^{\frac{1}{p-1}} V^{\frac{1}{p-1}}(\bar a) \varphi (\bar a) }+\frac{C \varphi(a)}{M^{\frac{1}{3}}} \\[12pt] & \displaystyle= k(a)\theta(a)+\frac{C \varphi(a)}{M^{\frac{1}{3}}}, \end{array}$$ where $$\theta(a)=\left(\frac{\varphi(a)V(a)^{\frac{1}{p-1}}}{\varphi(\bar a )V(\bar a)^{\frac{1}{p-1}}}\right)$$ and $\bar a$ is such that $$\varphi^{p-1} (\bar a)V(\bar a)=\max_x \varphi^{p-1}(x)V(x).$$ Recall that $$T\leq \frac{C}{M^{p-1}}.$$ Therefore, we get $$k(a)(1-\theta(a))\leq \frac{C \varphi (a)}{M^{\frac{1}{3}}} + \frac{C}{M^{\frac{p-1}{4}}} \leq\frac{C}{M^{\gamma}},$$ with $\gamma= min (\frac{p-1}{4},\frac{1}{3})$. As $V$ is bounded we have that $k(a)$ is bounded from below, hence $$(1-\theta(a))\leq \frac{C}{M^{\gamma}},$$ that is, $$\theta(a)\geq 1-\frac{C}{M^{\gamma}}$$ and we finally obtain $$\label{jorge} \varphi(a)V(a)^{\frac{1}{p-1}}\geq \varphi(\bar a )V(\bar a)^{\frac{1}{p-1}} -\frac{C}{ M^{\gamma}}.$$ This proves . To obtain the lower estimate for the blow-up time observe that from and the fact that $V(a)\geq c>0$ we get $$\label{pepe} \begin{array}{rl} \varphi(a) & \displaystyle \geq\frac{\varphi(\bar a )V(\bar a)^{\frac{1}{p-1}}}{V(a)^{\frac{1}{p-1}}} -\frac{C}{V(a)^{\frac{1}{p-1}} M^{\gamma}} \\[10pt] & \displaystyle \geq\frac{\varphi(\bar a )V(\bar a)^{\frac{1}{p-1}}}{V(a)^{\frac{1}{p-1}}} -\frac{C}{ M^{\gamma}}\\[10pt] & \geq C>0. \end{array}$$ Inequality gives us $$\frac {1}{( V(a) (p-1))^{\frac{1}{p-1}}}- C T^{\frac{1}{4}} \leq T^{\frac{1}{p-1}}M \varphi(a).$$ Hence $$\frac {1}{ \varphi (a) ( V(a) (p-1))^{\frac{1}{p-1}}}-\frac{C T^{\frac{1}{4}}}{\varphi (a) } \leq T^{\frac{1}{p-1}}M.$$ By and $\varphi^{p-1} (\bar a)V(\bar a)=\max_x \varphi^{p-1}(x)V(x)$ we get $$\frac {1}{ \varphi (\bar a) ( V(\bar a) (p-1))^{\frac{1}{p-1}}}- C T^{\frac{1}{4}} \leq T^{\frac{1}{p-1}}M$$ and using $$T \le \frac{C}{M^{p-1}}$$ we obtain $$\frac {1}{ \varphi (\bar a) ( V(\bar a) (p-1))^{\frac{1}{p-1}}}- \frac{C}{M^{\frac{p-1}{4}}} \leq T^{\frac{1}{p-1}}M$$ as we wanted to prove. 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--- abstract: | In the context of the MV-algebras there is a class of them endowed with a product. A subclass of this class of MV-algebras has been studied by Dinola in [@DiNola2]; the MV-algebras product or MVP. Thus for example the MV-algebra $[0,1]$ is closed for the usual product between real numbers. It is known that this product respects the usual order that in turn coincides with the natural order associated with this MV-algebra. Similarly, the algebra of continuous functions of $[0,1]^n$ in $[0,1]$. In the present work the class of MV-algebras with products is characterized in a wider context than that presented by Dinola. From the properties of the universal algebra found in the MV-algebras of closed continuous functions for products it will be shown that this general context is more convenient to work properties analogous to commutative algebra.\ As a result of this characterization a new algebraic structure is defined, which is an MV-algebra endowed with a product operation, which we will call MVW-rig (Weak-Rig Multivalued) because of its close relation with the rigs defined in [@Menni]. This structure is defined with axioms of universal algebra, a good number of natural examples are presented in the MV-algebras environment and the first results concerning homomorphisms, ideals, quotients and subdirect products are established. In particular, his prime spectrum is studied which, with the co-zariski topology defined by Dubuc, Poveda in [@Yuri1] is compact. Consequently, a good number of results analogous to the theory of commutative rings and rigs are presented, with which this theory maintains a close relation.\ **Key words:** MVW-rig, MV-algebras, commutative rigs, spectrum, prime ideals, rigs.\ author: - | Yuri A. Poveda [^1]\ [email protected] - | Alejandro Estrada [^2]\ [email protected] title: 'MVW-rigs' --- Introduction ============ A MV-algebra is a set $(A,\oplus,\neg,0)$ with a closed binary operation $\oplus$ and a unary operation $\neg$ such that for every $x,y,z \in A$ the following equations are satisfied: - MV1) $x\oplus (y\oplus z)=(x\oplus y)\oplus z$ - MV2) $x\oplus y=y\oplus x$ - MV3) $ x\oplus 0=x$ - MV4) $ x\oplus \neg 0=\neg 0 $ - MV5) $ \neg \neg x=x $ - MV6) $\neg (\neg x \oplus y) \oplus y = \neg (\neg y \oplus x) \oplus x$ Given an MV-algebra $A$, we define the constant $u$ and two operations $\odot$ and $\ominus$ as follows: & u =\_[def]{} 0 &\ & x y =\_[def]{} ( x y) &\ & x y =\_[def]{} ( x y) & It is known that all MV-algebra $A$ with the order $x \leq y$ if and only if $x \ominus y =0$, is a reticle with maximum and minimum. Next we will recall the definition of homomorphism, ideal and prime ideal in an arbitrary MV-algebra. Given two MV-algebras $A$ and $B$, a function $f: A \to B$ is a homomorphism of MV-algebras if for all $x,y$ in $A$ - $f(0)=0$ - $ f(x \oplus y)=f(x) \oplus f(y)$ - $ f(\neg x) =\neg f(x)$ Given $ A $ a MV-algebra. A subset $ I $ of $ A $ is called MV-ideal of $ A $ if it satisfies the following properties: - $0 \in I$ - If $a \leq b$ and $b \in I$, then $a \in I$ - If $a,b \in I$, then $a \oplus b \in I$ A subset $ P $ of a MV-algebra $ A $ is a MV-prime ideal if it is an ideal of MV-algebra, and given $a,b \in A$, $a \wedge b \in P$ implies $a \in P$ or $b \in P$ \[TeoChang\] Any non-trivial MV-algebra is a subdirect product of MV-chains. The following result is typical of the MV-algebras, but as it is not found in the literature and we will use it in the present work, we present it together with its proof. \[dismenos\] In a MV-algebra $A$ we have that $(x_1 \oplus x_2) \ominus (y_1 \oplus y_2) \leq (x_1 \ominus y_1) \oplus (x_2 \ominus y_2)$. Of the definitions of MV-algebra and join we have that: $1= \neg (x_1 \oplus x_2) \oplus (x_1 \oplus x_2) = \neg (x_1 \oplus x_2) \oplus (x_1 \vee y_1) \oplus (x_2 \vee y_2) = \neg (x_1 \oplus x_2) \oplus (x_1 \ominus y_1) \oplus y_1 \oplus (x_2 \ominus y_2) \oplus y_2 = \neg ((x_1 \oplus x_2) \ominus (y_1 \oplus y_2)) \oplus (x_1 \ominus y_1) \oplus (x_2 \ominus y_2)$ then $(x_1 \oplus x_2) \ominus (y_1 \oplus y_2) \leq (x_1 \ominus y_1) \oplus (x_2 \ominus y_2)$ by the definition of order.\ **Notation:** We represent the addition as: $ \bigoplus_{i=1}^n x_i = x_1 \oplus x_2 \oplus \cdots \oplus x_n$. If we generalize the last property we have that $$\bigoplus ^{n} _{i=1} x_i \ominus \bigoplus ^{n} _{i_1} y_i \leq \bigoplus ^{n} _{i=1} {(x_i \ominus y_i)}.$$ MVW-rigs ======== \[DefMVWrig\] A **MVW-rig** is a structure type $( A, \oplus, \cdot, \neg ,0 )$ with three operations $\neg$, $\oplus$ and $\cdot $ defined in $A$ (by abbreviation $a\cdot b = ab$), such that it fulfills the following axioms for every $a,b,c \in A$: - $(A, \oplus, \neg , 0 )$ is a MV-algebra. - $ (A, \cdot)$ is an associative operation defined in $A$. - $ a 0 = 0 a = 0$. - $a (b \oplus c) \leq a b \oplus a c $ and $(b \oplus c) a \leq b a \oplus c a $. - $a (b \ominus c) \geq a b \ominus a c $ and $(b \ominus c) a \geq b a \ominus c a $. By properties of the MV-algebra, $ 0 \leq a$ for all $a \in A$. We define $u =_{def} \neg 0$ for any MVW-rig. Then $ a \leq u$ for all $a \in A$. The operation $\neg$ is called **negation**, the operation $\oplus$ **sum** and the operation $\cdot $ **product** or **multiplication**. **Notation**: $a \cdot a = a^2$. In general $a \cdot a \cdot (n$ $times) = a ^n$. \[propiedades\] Given $A$ a MVW-rig, and $a,b,c \in A$. Then - $a \leq b$ implies $ac \leq bc$ and $ca \leq cb$ - $a(b \vee c ) \geq ab \vee ac$ and $(b \vee c)a \geq ba \vee ca$, - $a(b \wedge c ) \leq ab \wedge ac$ and $(b \wedge c)a \leq ba \wedge ca$, - $(a \vee b)^n \geq a^n \vee b^n $ with $n \in \mathbb{N}$. - $(a \wedge b)^n \leq a^n \wedge b^n $ with $n \in \mathbb{N}$. $i)$ Given $a \leq b$ then $a \ominus b =0$, hence $ 0= (a\ominus b)c \geq ac \ominus bc $ and $ 0= c (a \ominus b) \geq ca \ominus cb $ which implies that $ac \ominus bc = 0$ and $ca \ominus cb = 0$, and thus $ac \leq bc$ and $ca \leq cb$. $ii)$ It follows directly from the definition of join, from the fact that $b \vee c \geq b,c$; and by property $(i)$ $a(b \vee c) \geq ab, ac$. $iii)$ Similary to $ii)$. $iv)$ $(a \vee b)^n = (a \vee b)\cdots (a \vee b) \geq a^n, b^n$. Similarly we obtain $v)$ If in a MVW-rig $ A $, there is an element $ s $ that has the property that for all $ x $ in $ A $ $ sx = xs = x $ it is said that $ s $ is a **unitary element** of the MVW-rig and $ A $ will be called **unitary MVW-rig**, in addition it is unique, since if there is an unitary element $ w \in A $ then $ s = sw = ws = w $. A unitary element in an MVW-rig will be denoted as 1. An MVW-rig is **commutative** if for all $x,y \in A$, $ xy = yx $. \[Ejemplousual\] The interval $ [0,1] $ of real numbers with the usual sum of MV-algebra $ [0,1] $ and usual multiplication in $\mathbb{R}$ is a commutative MVW-rig with unitary element where $u=1$ and $\neg x = 1-x$. \[Ejemplosemiusual\] The interval $ [0, u] $ in $\mathbb{R}$ where $0 \leq u < 1$ is a non-unitary commutative MVW-rig with the truncated sum. If $u=0$ the structure is called **trivial MVW-rig**. \[EjemploLukasiewicz\] The set $\L_n = \left\{0,\frac{1}{n-1},\frac{2}{n-1},\cdots,\frac{n-2}{n-1}, 1\right\} $, known as MV-algebra of Łukasiewicz is not a MVW-rig because it is not closed for the product. If we close it for products then we get the MVW-rig $\dot{\L_n} = \{ \frac{m}{n^k} \in \mathbb{Q} $ for every $k \in \mathbb{N}$ and all integer $m$ between $0$ and $n \} $. Given $n \in \mathbb{N}$, the set $Z_n = \{ 0, 1, \dots , n \}$ is a MVW-rig with $u = n$ as a strong unit and operations defined as follows: $x \oplus y = $ min$\{n, x + y \} $, $\neg x = n-x$ and $x y = $ min$\{n, x \cdot y \} $ where operations sum $+$ and product $\cdot$ are the usual ones in the natural numbers and the order relation of the MV-algebra is the usual one in the natural ones. This MV-algebra is isomorphic to the MV-algebra $\L_{n+1}$ by applying $ \phi_n : \L_{n+1} \rightarrow Z_n$, $ \phi_n(x) = nx$, but $\L_{n+1} $ is not an MVW-rig. This MVW-rig has some interesting properties: It has unitary element and is different from $ u $ if $ n> 1 $; has no cancellation property, the product between two elements is greater than or equal to them. This MVW-rig is a good source of counterexamples of properties that may be true for other MVW-rigs. \[EjemploFree\] Given the MV-algebra $Free_1$, through the Mundici functor we get the $l_u$-group $Free_1^*$ which is isomorphic to the set of continuous functions of $[0,1]$ in $\mathbb{R}$ that have the property that each of them is constituted by finite linear polynomials with integer coefficients and that is contained in the $l_u$-ring $\mathcal{C}(\mathbb{R}^{[0,1]})$. Thus we can take the $l_u$-ring generated by $Free_1^*$ in $\mathcal{C}(\mathbb{R}^{[0,1]})$ which we will call $\dot{F}[x]$. This $l_u$-ring is isomorphic to $l_u$-ring of functions in $\mathcal{C}(\mathbb{R}^{[0,1]})$ each consisting of finite polynomials of $\mathbb{Z}[x]$. $\dot{F}[x]$ is a $l_u$-commutative ring. Given $u$ strong unit of $\dot{F}[x]$ such that $u^2 \leq u$ we take $\Gamma( \dot{F}[x],u) = \{f \in \dot{F}[x] \mid 0 \leq f \leq u \} $. This MV-algebra with the usual product of functions is a commutative MVW-rig denoted by $F_u[x]$. The MVW-rig $F_u[x]$ has more elements than the MV-algebra $ Free_1 $ since in $F_u[x]$ there are piecewise polynomial functions whose polynomials are of degree greater than 1 and the intervals of definition have by ends algebraic numbers. However $Free_1 \subset F_u[x]$. See [@Steven]. Unlike the previous example, this MVW-rig satisfies $fg \leq f \wedge g$ for all $f,g \in F_u[x]$. \[Matrices\] *MVW-rig of matrices:* Given $ M_n $ the set of square matrices $n \times n$ with entries in $[0,1/n]$. We define the sum of two matrices $A,B \in M_n$ with the componentwise operation in the MV-algebra $[0,1/n]$, that is to say $A \oplus B = C$ with $ c_{ij} = a_{ij} \oplus b_{ij} $ for every $i,j = 1, \dots , n.$ The negation is defined componentwise in the MV-algebra $[0,1/n]$, that is $ \neg A = C$, where $c_{ij} = \neg a_{ij} $ for all $i,j = 1, \dots , n.$. The set $ M_n $ is a MV-algebra with the described operations and the zero matrix. The natural order in $ M_n $ is given by: $A \leq B \Leftrightarrow_{def} (A)_{ij} \leq (B)_{ij} $ for every $i,j = 1, \dots , n.$ This defines a partial order in $M_n$. We now define the product in $M_n$ as $(AB)_{ij} = \bigoplus_{k=1}^n a_{ik} b_{kj} $ where each product and sum is defined in the MVW-rig $[0,1/n]$. Note that the strong unit in this MVW-rig is the matrix $$U= \begin{pmatrix} 1/n & \dots & 1/n\\ \vdots & \ddots & \vdots \\ 1/n & \dots & 1/n \end{pmatrix}$$ Let’s prove each of the MVW-rig axioms: $i)$ We know that $M_n$ is a MV-algebra. $ii)$ Given three matrices $A,B,C \in M_n$ we have $(A(BC))_{ij}$=$ ( A ( \bigoplus_{k=1}^n b_{ik} c_{kj} ))_{ij} = \bigoplus_{r=1}^n a_{ir} (\bigoplus_{k=1}^n b_{ik} c_{kj}) = \bigoplus_{r=1}^n \bigoplus_{k=1}^n a_{ir} b_{ik} c_{kj} = \bigoplus_{k=1}^n \bigoplus_{r=1}^n a_{ir} b_{ik} c_{kj} = \\ \bigoplus_{r=1}^n ( \bigoplus_{k=1}^n a_{ir} b_{ik}) c_{kj} = \bigoplus_{r=1}^n (AB)_{ik} c_{kj} = ((AB)C)_{ij}$ where the equality in the distributive law is true when the sum doesn’t overpass $u$ and in this case its proved. $iii)$ It follows directly from the definitions. $iv)$ $(A(B \oplus C))_{ij} = \bigoplus_{k=1}^n a_{ik}(b_{kj} \oplus c_{kj}) \leq \bigoplus_{k=1}^n (a_{ik}b_{kj} \oplus a_{ik}c_{kj}) = \bigoplus_{k=1}^n a_{ik}b_{kj} \oplus \bigoplus_{k=1}^n a_{ik}c_{kj} = (AB)_{ij} \oplus (AC)_{ij}$. $v)$. Given $(A(B \ominus C))_{ij} = \bigoplus_{k=1}^n a_{ik}(b_{kj} \ominus c_{kj}) \geq \bigoplus_{k=1}^n (a_{ik}b_{kj} \ominus a_{ik}c_{kj})$ and by proposition (\[dismenos\]) $\bigoplus_{k=1}^n (a_{ik}b_{kj} \ominus a_{ik}c_{kj}) \geq \bigoplus_{k=1}^n a_{ik}b_{kj} \ominus \bigoplus_{k=1}^n a_{ik}c_{kj}\\ = (AB)_{ij} \ominus (AC)_{ij}$. A $l_u$-ring $R$ is a lattice ordered ring such that $(R, +, u)$ is a lattice ordered group with strong unit. \[Ejemplogamma\] Given $l_u$-ring $R$ for which $u^2\leq u$, we have $\Gamma(R,u)$ is a MVW-rig, where $$\Gamma(R,u)=\{ x \mid 0 \leq x \leq u \}$$ with the truncated sum and the product of $ R $. $i)$ $\Gamma(R,u)$ is a MV-algebra [@Yuri3]. $ii)$ As $u^2 \leq u$ then $xy \leq u$ for every $x,y \in \Gamma(R,u)$, then $(x \cdot y ) \cdot z = (xy) \cdot z = (xy)z = x(yz) = x\cdot (yz) = x \cdot (y \cdot z) $ where $\cdot$ is the product in $\Gamma(R,u)$. $iii)$ it follows directly of the definition. $iv)$ $x \cdot (y \oplus z) = x(y \oplus z) = x( (y +z) \wedge u ) \leq x(y+z) \wedge xu \leq x(y+z) \wedge u = (xy + xz) \wedge u =xy \oplus xz = x \cdot y \oplus x \cdot z$. The distributive on the left is similar. $v)$ $x(y \ominus z) = x((y -z) \vee 0) = x(y - z) \vee x 0 = x(y-z) \vee 0 =(xy - xz) \vee 0 = xy \ominus xz$. $v)$ $x \cdot 0 = u \wedge x0 = u \wedge 0 = 0 = u \wedge 0 = u \wedge 0x = 0 \cdot x$. Homomorphisms and ideals in the MVW-rigs ======================================== Given $A$ and $B$ MVW-rigs. A function $f: A \rightarrow B$ is a homomorphism of MVW-rigs if the following properties are true: - $f$ is a homomorphism of MV-algebras. - $f(ab)=f(a)f(b)$. Given the MVW-rig $F_u[x]$ and a function $\hat{a}$ such that it evaluates each function of $F_u[x]$ in a point $a\in [0,1]$. That is, $\hat{a}: F_u[x] \to [0,1]$, $\hat{a}(f)=f(a)$. The funtion $\hat{a}$ is a homomorphism. Let’s see: $\hat{a}(0)=0(a)=0$, where $0$ if the zero funtion of $F_u[x]$. We also have that $\hat{a}(\neg f) = (\neg f)(a)= \neg f(a)=\neg \hat{a}(f)$ for every $f \in F_u[x]$. Let’s take $f,g \in F_u[x]$, then $\hat{a}(f \oplus g)= (f\oplus g )(a)= f(a) \oplus g(a) = \hat{a}(f) \oplus \hat{a}(g)$ and this proves $i)$. Given $f,g \in F_u[x]$ we have that $\hat{a}(f g)= (f g )(a)= f(a) g(a) = \hat{a}(f) \hat{a}(g)$ and this proves $ii)$. The evaluation homomorphism is of great importance for relating sets $F_u[x]$ with $A=[0,u]$. An ideal of an MVW-rig $A$ is a subset $I$ of $A$ that fulfills the following properties: - $I$ is an MV-ideal of $A$ as MV-álgebra. - If $a \in I$ and $b \in A$ then $ab \in I$ and $ba \in I$. In $F_u[x]$ we can take ideals analogous to the MV-ideals of MV-algebra $Free_1$ which are ideal in the MVW-rig $F_u[x]$. Let $z$ be a fixed element of $[0,1]$. - $ f \in I_z \iff f(z)=0 $ - $ f \in I_{z^+} \iff \exists \epsilon > 0 \text{ such that } f(x)=0, \forall x \in [z, z + \epsilon ] $. ($z \neq 1$) - $ f \in I_{z^-} \iff \exists \epsilon > 0 \text{ such that } f(x)=0, \forall x \in [z - \epsilon , z ]$. ($z \neq 0$) - $ f \in I_S \iff f(x)=0 \forall x \in S, S \subset [0,1] $. If $S$ is a subset of a commutative MVW-rig $ A $, then the ideal generated by $ S $ in $ A $ is: $$\langle S \rangle = \{ x \in A \mid x \leq \bigoplus_{i=1}^n a_i s_i, s_i \in S, a_i \in A \text{ or } a_i \in \mathbb{N} \text{ for each } i=1, \dots , n \}$$ **Note:** If $a_i = n \in \mathbb{N}$, $ns= s \oplus \cdots \oplus s$ ($n$ times). $ \langle S \rangle $ is ideal because $0 \in \langle S \rangle $. Given $x , y \in \langle S \rangle $ then $x \leq \bigoplus_{i=1}^n a_i s_i$ and $y \leq \bigoplus_{j=1}^m a_j s_j$, then $ x \oplus y \leq \bigoplus_{i=1}^n a_i s_i \oplus \bigoplus_{j=1}^m a_j s_j$ and so $x \oplus y \in \langle S \rangle $. Given $x \leq y \in \langle S \rangle$ then $x \leq y \leq \bigoplus_{i=1}^n a_i s_i$ and therefore $x \in \langle S \rangle $. Lastly, given $x \in \langle S \rangle$ and $z \in A $ then $ zx \leq \bigoplus_{i=1}^n (z a_i) s_i $ this implies that $zx \in \langle S \rangle $ and the same for the case $xz \in \langle S \rangle$.\ To see that $\langle S \rangle$ is the less ideal of $A$ that contains $S$, let’s take an ideal $I$ from $A$ that contains $S$ and let’s see that $\langle S \rangle \subset I$. Given $x \in \langle S \rangle$ then $x \leq \bigoplus_{i=1}^n a_i s_i$. Since $I$ contains $S$ then every $s_i$ belongs to $I$ and therefore $a_i s_i $ belongs to $I$. Then the sum $\bigoplus_{i=1}^n a_i s_i$ belongs to $I$ and this implies that $x \in I$. \[prohomo\] Given $\phi$ a homomorphism of a MVW-rigs $A$ to a MVW-rig $B$, we have the following properties: - If $S$ is a subMVW-rig of $A$ then $\phi (S) $ is a subMVW-rig of $B$. - $\phi(x) \leq \phi(y)$ if and only if $ x \ominus y \in $ Ker$(\phi)$. - If $J$ is ideal of $B$ then $\phi^{-1}(J)$ is ideal of $A$. - If $A$ is unitary and $\phi(1) \neq 0$ then $\phi(1)$ is the unitary element of $\phi(A)$. - $\phi$ is inyective if and only if Ker$(\phi) = \{ 0 \} $. The definition of homomorphism of MVW-rig is directly followed. We will use this property of MV-algebras in the following proposition: $x \leq y \oplus z \iff x \ominus z \leq y$. Given $I$ an ideal in a MVW-rig $A$ the equivalence relation. $x \equiv _I y \Leftrightarrow_{def} (x \ominus y) \oplus (y \ominus x) \in I$ is a congruence in the category of MVW-rigs. It is known that this relation respects addition and negation. It is enough to show that it respects the product. If $a \equiv b \ mod(I)$ and $c \equiv d \ mod(I)$ then $a \ominus b \in I$ and $c \ominus d \in I$. We have that $ac \leq (a \vee b)(c \vee d) = ((a \ominus b) \oplus b)((c \ominus d) \oplus d)$ by definition of join; using distributive law of the axiom \[DefMVWrig\](iv) we have $ac \leq ((a \ominus b) \oplus b)((c \ominus d) \oplus d) \leq (a \ominus b) ((c \ominus d) \oplus d) \oplus b((c \ominus d) \oplus d) \leq (a \ominus b)(c \ominus d) \oplus (a \ominus b)d \oplus b(c \ominus d) \oplus bd$. So, $ac \ominus bd \leq (a \ominus b)(c \ominus d) \oplus (a \ominus b)d \oplus b(c \ominus d) \in I$ by absorbing property of $I$ regarding the product. Similarly $bd \ominus ac \in I$, then $(ac \ominus bd) \oplus (bd \ominus ac) \in I$ which implies $ac \equiv bd \ mod(I)$. There is a bijection between the ideals of an MVW-rig $ A $ and the congruences in $A$. Given $ \equiv $ a congruence in $A$, the set $ I= \{ x \in A \mid x \equiv 0 \} $ is an ideal of MVW-rig. Just see that it has the absorbing property. If $x \equiv 0$ and $z \in A$ then since $\equiv $ preserves the product we have that $ xz \equiv 0z=0$. On the other hand, it was seen that $ \equiv_I $ is a congruence. The above assignment is bijective. Given $I,J$ ideals of $A$, $x \equiv_I y \iff x \equiv_J y \iff I=J.$ On the other hand, given $\equiv $ a congruence in $A$, $a \equiv b \iff a\ominus b \oplus b \ominus a \equiv 0 \iff a \equiv_I b$ with $I= \{ x \in A \mid x \equiv 0 \}$. Quotient MVW-rig ================ We define the quotient $ A / I $ as the set of equivalence classes of $ x $ for each $x \in A$ which are denoted by $ [x] _I $. The set $ A / I $ has the operations: & \_I = \[x \]\_I \[negCoc\] &\ & \[x\]\_I \_I = \[x y\]\_I \[sumCoc\] &\ & \[x\]\_I \[y\]\_I = \[x y\]\_I \[ProdCoc\] & The following proposition follows that $ \equiv_I $ is a congruence. $A/I$ is a MVW-rig. The correspondence $x \mapsto [x]_I$ defines a surjective homomorphism $ h $ from MVW-rig $ A $ to quotient MVW-rig $ A / I $ called the natural homomorphism of $ A $ onto $ A / I $ with $ \ker (h) = I$. Given $\phi$ a homomorphism of a MVW-rig $A$ to a MVW-rig $B$ with $ \ker (\phi)= K$, then there is a canonical isomorphism between $\phi(A)$ and $A/K$. $$\xymatrix{A \ar[rrrr]^{\phi} \ar[rrd]_{h} & & & & \phi(A) \subset B \\ & & A/K \ar@{^(->}[rru]_{\varphi}^{\cong} }$$ By the proposition \[prohomo\]($i$) $\phi(A)$ is a MVW-rig. We define $\varphi : A/K \rightarrow \phi(A)$ by $\varphi([a]_K) = \phi(a)$. The Theorem 1.2.8 of [@Cignoli] shows that $\varphi$ is well defined, is one-to-one and onto in $\phi(A)$ with $\varphi([a]_K \oplus [b ]_K) = \varphi([a]_K) \oplus \varphi([b]_K)$. However, $\varphi([a]_K[b ]_K) = \varphi([ab]_K) = \phi(ab) = \phi(a) \phi(b)= \varphi([a]_K) \varphi([b]_K)$. So, $\varphi$ is an isomorphism of MVW-rigs. \[Corresp\_ideales\] Given $ A $ a MVW-rig, $ I $ an ideal of $ A $, then there exists a bijective correspondence between the ideals of $ A $ containing $ I $ and the ideals of the quotient MVW-rig $ A / I $ which preserves the relation of inclusion and also the direct and inverse image of an ideal is an ideal. Let $f$ be the natural homomorphism of $A$ over $A/I$. Given $J$ ideal of $A$ that contains $I$ then Ker$(f) = I \subset J$, we want to see that $f(J)$ is ideal in $A/I$: $0 \in f(J)$ because Ker$(f) \subset J$, given $x,y \in f(J)$ and $z \in A/I$, then $x = f(a)$, $y = f(b)$, $z = f(k)$ with $a,b \in J$, $k \in A$; from here results $x \oplus y = f(a \oplus b)$, $zx = f(ka)$, $xz = f(ak)$, with which $x \oplus y,$ $xz$, $kx \in f(J)$, and if $z \leq x$ and $x \in f(J)$ then $k \ominus a \in Ker(f)$ by property (ii) of the proposition (\[prohomo\]), then $k \ominus a \in J$ and since $a \in J$ then $(k \ominus a ) \oplus a = k \vee a \in J$ and this implies that $k \in J$, this is, $f(k)= z \in f(J)$. On the other hand, if $\tilde{J}$ is ideal of $A/I$, again by proposition (\[prohomo\]) $f^{-1}(\tilde{J})$ is ideal of $A$ and contains $I$ because for $a \in I$, $[a]_I = [0]_I \in \tilde{J}$.\ Given $\mathcal{I}$ the collection of all the ideals of $A$ that contain $I$ and $\mathcal{I}_0$ the collection of all the ideals of $A/I$.\ The correspondence $\widetilde{f} : \mathcal{I} \rightarrow \mathcal{I}_0$, $\widetilde{f}(J) := f(J) = \{ [a]_I \mid a \in J \} $ is a bijection.\ $\widetilde{f}$ is injective since given $K,J \in \mathcal{I}$ with $\widetilde{f}(J) = \widetilde{f}(K)$, then $a \in J \Leftrightarrow [a]_I \in \widetilde{f}(J) \Leftrightarrow [a]_I \in \widetilde{f}(K) \Leftrightarrow a \in K$. (The implication $[a]_I \in \widetilde{f}(K) \Rightarrow a \in K$ is because if we have $b \in K$ given that $[a]_I = [b]_I$ then $a \ominus b \in K$, then $(a \ominus b) \oplus b \in K$ and therefore $a \vee b \in K$ and so $a \in K$).\ $\widetilde{f}$ is surjective because given $\tilde{J} \in \mathcal{I}_0$ we know that $f^{-1}(\tilde{J})$ is ideal of $A$ that contain $I$; and since $f$ is surjective $f(f^{-1}(\tilde{J}))= \tilde{J}$, that is to say, $\widetilde{f}(f^{-1}(\tilde{J}))= \tilde{J}$.\ $\widetilde{f}$ preserves the inclusion because given $J \supseteq K$ in $\mathcal{I}$, if $[a]_I \in \widetilde{f}(K)$ then $a \in K$ and $a \in J$, then $[a]_I \in J$ which implies that $\widetilde{f}(J) \supseteq \widetilde{f}(K)$. Prime and maximal ideals ------------------------ An ideal $I$ of a MVW-rig $A$ is prime if $ab \in I$ implies $a \in I$ or $b \in I$. \[homoprimo\] Given $A,B$ MVW-rigs. If $f: A \rightarrow B$ is a homomorphism of MVW-rigs and $P$ is a prime ideal of $B$, then $f^{-1} (P) = \{ a \in A \mid f(a) \in P \}$ is a prime ideal of $A$. In general, being an ideal MV-prime for $ A $ does not imply that it is a prime ideal for MVW-rig $ A $, it is not true either otherwise . The implication is obtained only when we can establish a relation of order between the product and the infimum, as in the following proposition: Given a MVW-rig $A$ where $ab \leq a \wedge b$ for every $a,b \in A$. If $P$ is a prime ideal of $A$ then it is a MV-prime ideal. Given $a,b \in A$ such that $a \wedge b \in P$, then $ab \in P$ by the relation $ab \leq a \wedge b$. Since $P$ is prime ideal of the MVW-rig $A$ then $a \in P $ or $b \in P$, and so $P$ is a MV-prime ideal. Given $M$ an ideal of a MVW-rig $A$. $M$ is a **maximal ideal** if for every $a \in A$ with $a \notin M$, $ \langle M, a \rangle = A$. In a commutative MVW-rig $A$ with unitary element, every maximal ideal is a prime ideal. Given $M$ a maximal ideal and $a,b$ elements of $A$ such that $ab \in M$. Let’s suppose that $a \notin M$, then there exists elements $m \in M$ and $x \in \langle a \rangle$ such that $1 \leq m \oplus x$. Then, $b = b1 \leq b(m \oplus x) \leq bm \oplus bx$, but $bm \in M$ and $bx \in \langle ab \rangle \subset M $ and so $b \in M$. An element $x$ of a MVW-rig $A$ is called **nilpotent** if $x^n=0$ for any $n>0$. The set $N$ of all nilpotent elements of $A$ is called the **nilradical of $A$**. The nilradical $N$ of a commutative MVW-rig $A$ is an ideal of $A$ and $A/N$ doesn’t have nilpotent elements different from zero. $0 \in N$. If $x,y \in N$ then $x^n=0$ and $y^m=0$ for any $n,m \in \mathbb{N}$, then $(x \oplus y)^{m+n-1}$ is a sum of products $x^r y^s$ where $r+s= m+n-1$ and $r>n$ or $s>m$ (since $A$ is commutative), after every product is zero and therefore $(x \oplus y)^{m+n-1}=0$, so $x \oplus y \in N$. If $x \leq y \in N$ then exists $n \in \mathbb{N}$ such that $y ^n =0$, since $x \leq y$ then $x^n \leq y^n = 0$ and so $x^n =0$ therefore $x \in N$. Given $x \in N$ and $y \in A$ then there exists $n \in \mathbb{N}$ such that $x^n = 0$, since $A$ is commutative we have that $x^n y^n = (xy)^n = 0$ and therefore $xy \in N$. This shows that $N$ is an ideal.\ To see that $A/N$ doesn’t have nilpotent elements, let’s take a nilpotent element $[x]_N$ in $A/N$, then there exists an integer $m>0$ such that $[x]_N^m = [0]_N$ in $A/N$, then $[x^m]_N = [0]_N$ by equation (\[ProdCoc\]), this implies that $x^m \in N$ and therefore exists an integer $k>0$ such that $(x^m)^k =0$, then $x \in N$ and so $[x]_N = [0]_N$. \[NilradyPrimo\] The nilradical $N$ of a MVW-rig $A$ is contained in each prime ideal of $A$. Given $x \in N$, exists an integer $n>0 $ such that $x^n = 0$, since $0\in P$ for every prime ideal $P$ of $A$ then $x ^n \in P$ and since $P$ is prime, $x \in P$. Every non-trivial MVW-rig $A$ has a maximal ideal. Being $\Sigma $ the set of all of the proper ideals of $A$. $ \Sigma $ is different from empty since the ideal $0 \in \Sigma $, and $\Sigma $ is ordered by inclusion. Being $ (J_\alpha) $ a chain of ideals $J_1 \subset J_2 \subset J_3 \subset \cdots$ in $\Sigma$. We have that $J= \cup_\alpha J_\alpha $ is an ideal that belongs to $\Sigma$ since $u \notin J$ because $u \notin J_\alpha $ for every $\alpha$. Therefore, $J$ is an upper bound of the chain and by Zorn’s lemma, $\Sigma $ has at least one maximal element. If $I$ is an ideal of $A$ then there is a maximal ideal of $A$ that contains $I$. It follows directly from the previous proposition applied to $ A / I $ and the bijection given in (\[Corresp\_ideales\]). Given $I$ an ideal of a commutative MVW-rig $A$, we will call **radical** of $I$ the set $$\sqrt{I} = \{ x \in A \mid x^n \in I \text{ for some } n > 0 \}$$ The radical of an ideal $ I $ of a commutative MVW-rig $ A $ has the following properties: - $I \subset \sqrt{I}$. - If $I \subset J$ then $\sqrt{I} \subset \sqrt{J}$ with $J$ ideal of $A$. - If $I$ is the prime ideal, then $I = \sqrt{I}$. - $\sqrt{I \cap J} = \sqrt{IJ} $ The radical of an ideal I of a commutative MVW-rig is the intersección of the prime ideals that contain I. $$\sqrt{I} = \bigcap_{P \supset I} P$$ Given $x \in \sqrt{I}$ there exists $n \in N$ that $x^n \in I$, then $x^{n} \in P$ for every $P \supset I$, which implies that $x \in P$ for every $P \supset I$ because $P$ is prime ideal and therefore $x \in \bigcap\limits_{P \supset I} P$. On the other hand, given $x \notin \sqrt{I}$. Being $\Sigma$ the set of ideals $J$ that contain $I$ with the property $$n > 0 \Rightarrow x^n \notin J$$ $\Sigma$ isn’t empty because $I \in \Sigma$. We must show that every chain in $\Sigma$ has a upper bound in $\Sigma$. For $J_0 \subset J_1 \subset J_2...$ the join is $ K= U_{i=0}^{\infty }J_i$. $K$ is ideal of $A$ because $0 \in K$; if $b,c \in K$ then $b\in J_{i_1}, c \in J_{i_2}$, let’s suppose that $J_{i_1} \subset J_{i_2}$, then $ b \in J_{i_2}$, then $b \oplus c \in J_{i_2}$ and $b \oplus c \in K$ (the same for $J_{i_2} \supset J_{i_1}$); if $b \in K$ and $a \leq b$ then $b \in J_i$ for any $i$, then $a \in J_i$ and $a \in K$; given $b \in K$ and $a \in A$ we have that $b \in J_i$ for any $i$, then $ba \in J_i$ and so $ba \in K$. Now, by the Zorn’s lemma $\Sigma$ contains at least one maximal element. Given P a maximal element of $ \Sigma $ and we’ll show that it’s pime. Given $z,y \notin P$ then $P \oplus \langle z \rangle $, $P \oplus \langle y \rangle $ contain strictly $P$ and therefore aren’t in $\Sigma$. Then there exists integers $n,m>0$ such that $$x^{n} \in P \oplus \langle z \rangle , \qquad x^{m} \in P \oplus\langle y \rangle$$ Then $x^{n} \leq p_1 \oplus z_1$ and $x^{m} \leq p_2 \oplus y_1$ where $p_1, p_2 \in P$, $z_1 \in \langle z \rangle $ and $y_1 \in \langle y \rangle $. We have that $x^{n+m}=x^{n}x^{m} \leq (p_1 \oplus z_1)(p_2 \oplus y_1) \leq (p_1 \oplus z_1)p_2 \oplus (p_1 \oplus z_1)y_1 \leq (p_1 \oplus z_1)p_2 \oplus p_1y_1 \oplus z_1y_1 = p_3 \oplus z_1y_1$ where $p_3 \in P$ this way we have that $x^{n+m} \in P \oplus \langle zy \rangle $. Then $P \oplus \langle zy \rangle \notin \Sigma$ and therefore $ zy \notin P$. Then $P$ is prime. This way we have a prime ideal $P$ that contains $I$ such that $x \notin P$, then $ x \notin \bigcap\limits_{P \supset I} P$. This concludes the proof. \[Nilradical\] The nilradical $N$ of a commutative MVW-rig $A$ is the intersection of all prime ideals of $A$. By proposition (\[NilradyPrimo\]) all of the prime ideals of $A$ contain the nilradical, then applying the previous proposition we arrive at the result. The prime spectrum of a MVW-rig =============================== Now we will characterize the prime spectrum of a MVW-rig by passing prime-spectrum theorems of unitary commutative rings to said structures. Henceforth, when we speak of MVW-rig it will be understood as a commutative MVW-rig with unitary element. Given a MVW-rig $A$, we call **prime spectrum of $A$** or $Spec(A)$ to the set of prime ideals of $A$ and for every $a \in A $ we define: $$V(a)= \{ P \in Spec(A) \mid a \in P \}$$ The collection $ \{ V(a)\}_{a \in A} $ has the following properties that characterize a base of a topological space: - $V(a) \cap V(b) = V(a \oplus b )$ for every $a,b \in A $ - $V(0) = Spec(A)$ - $V(u) = \emptyset $ $i)$ $P\in V(a) \cap V(b) \Leftrightarrow a \in P$ and $b\in P \Leftrightarrow a\oplus b \in P\Leftrightarrow P\in V(a \oplus b)$. $ii)$ $0 \in P$ for every $P \in Spec(A)$. $iii)$ $u \notin P$ for every $P \in Spec(A)$.\ From the above, we have that the collection $ \{ V(a)\}_{a \in A} $ form a base for a topology, called the **Co-Zariski topology**: Given a MVW-rig $A$, we define the topological space $Spec(A)$ whose points are the prime ideals of $ A $ and whose open ones are generated by the base $ \{ V(a)\}_{a \in A} $. Given $A$ a MVW-rig and $a,b$ elements of $A$, then: - $V(a) \cup V(b) = V (ab)$ - $V (ab) \subset V (a \wedge b)$ - $V(a) \cap V(b) = V(a \vee b)$ - $V (a) = Spec(A)$ if and only if $ a $ is nilpotent. Given $a,b$ elements of a MVW-rig $A$, then $V(a) \subset V(b)$ if and only if $\sqrt{\langle b \rangle } \subset \sqrt{\langle a \rangle }$. Given $x \in \sqrt{\langle b \rangle }$ then $x \in \bigcap\limits_{P \supset \langle b \rangle } P$ with $P$ prime ideal, then $x \in P$ for every $P \supset \langle b \rangle $, in particular $x \in P$ for every $P \supset \langle a \rangle $ because $ V(a) \subset V(b)$, then $ x \in \bigcap\limits_{P \supset \langle a \rangle } P$ and therefore $x \in \sqrt{\langle a \rangle }$. On the other hand, given $\sqrt{\langle b \rangle } \subset \sqrt{ \langle a \rangle }$, $ V (\sqrt{\langle a \rangle }) \subset V (\sqrt{\langle b \rangle })$ then $V(a) = V (\langle a \rangle ) = V (\sqrt{\langle a \rangle }) \subset V(\sqrt{ \langle b \rangle }) = V (\langle b \rangle ) = V(b)$. Given $A$ a MVW-rig, $Spec(A)$ is a topological space $T_0$ Given $P, Q \in Spec(A)$ with $P \neq Q$, then there exists $a \in A , a \notin N$ ($N$ the nilradical of $A$), such that $a \in P$ and $a \notin Q$ or $a \notin P$ and $ a \in Q$, then $ P\in V(a)$ and $ Q \notin V (a)$ or $ P \notin V(a)$ and $ Q \in V (a)$. Let’s remember that given $X$ a set of a topological space, the closure of $X$ noted by $\overline{X}$ is defined as the intersection of all closed sets containing $X$, then a point $x$ belongs to $\overline{X}$ if and only if for every basic opening $B$ that contains $x$, $ B \cap X \neq \emptyset $. \[clausura\] Given $Q,P \in Spec (A)$ for a MVW-rig $A$, then $Q \in \overline{ \{P \} }$ if and only if $Q \subset P$. If $ Q \in \overline{ \{ P \} } $ then for every $b \in Q$ we have that $V(b) \cap \{ P \} \neq \emptyset$, then $P \in V(b)$ for every $b \in Q$ and this implies that $b \in P$ for every $b \in Q$, then $Q \subset P$. On the other hand, if $ Q \subset P$, then for every $b \in Q$, $P \in V(b)$, which implies that $ V(b) \cap \{ P \} \neq \emptyset $ for every $b \in Q$ and therefore $Q \in \overline{ \{ P \} }$. \[clausura1\] Given $Q \in Spec (A)$ for a MVW-rig $A$, and $U$ a subset of $Spec(A)$. If $Q \subset P$ for any $P \in U$ then $Q \in \overline{U} $ \[clausura2\] The opposite of the previous proposition is true if the set $ U $ has a single maximal element. A topological space $X$ is irreducible if $X \neq \emptyset $ and the intersection of two non-empty openings is non-empty. For a MVW-rig $A$, $Spec(A)$ is irreducible if and only if $A$ has a single maximal ideal. Let’s suppose that $A$ has at least two maximal ideals $M_1$ and $M_2$, then given $a \in M_1, a \notin M_2$ there exists $b \in M_2$ such that $ x \oplus b = 1$ with $x \in \langle a \rangle $, and $b \notin M_1$ because $M_1$ is it’s own ideal. It results that $V(x)$ and $V(b)$ are non-empty openings because $M_1 \in V(x)$ and $ M_2 \in V(b)$, then $ \emptyset = V(\langle 1 \rangle ) = V( \langle x \oplus b \rangle ) = V(x \oplus b ) = V(x) \cap V(b)$, that is to say, $V(x) \cap V(b) = \emptyset $ which implies that $Spec(A)$ is not irreducible. On the other hand, if $A$ has exactly a maximal ideal, let’s take $V(a) \neq \emptyset$ and $V(b) \neq \emptyset$, this implies that $a \in M$ and $b \in M$, then $M \in V(a) \cap V(b)$ and therefore $A$ is irreducible. \[funcionSpec\] Given $\phi : A \rightarrow B$ a homomorphism of MVW-rigs, we define $\phi ^{*} : Spec (B) \rightarrow Spec (A)$ such that given $J$ ideal of $B$, $\phi ^{*}(J) = \begin{Bmatrix} x \in A \mid \phi (x) \in J \end{Bmatrix}= \phi^{-1}(J)$. Then: - $\phi ^{*}$ is a continuous function between topological spaces. - If $I$ is an ideal of A then $(\phi^{*}) ^{-1} (V(I)) = V( \phi(I))$ - If $\phi$ is injective, $\phi^* (V(b)) = V ( \phi^{-1} (b))$ - If $\phi$ is bijective, then $\phi^*$ is a homeomorphism between $Spec(B)$ and $V(Ker(\phi))$. - If $\phi$ is injective, then $\phi^*(Spect(B)) = Spec(A)$. <!-- --> - First, let’s see that for $J \in Spec (B), \phi ^{*} (J) \in Spec (A)$. $ 0 \in \phi ^{*} (J)$ because $\phi (0) = 0 \in J$. Given $ x, y \in \phi ^{*} (J)$ then $ \phi (x), \phi (y) \in J$, then $\phi (x) \oplus \phi (y) = \phi ( x\oplus y) \in J$ and so $ x \oplus y \in \phi ^{*} (J)$. Given $x \in \phi ^{*} (J)$ and $y \leq x$, then $ \phi (y) \leq \phi (x)$ and as $\phi (x) \in J$ then $\phi (y) \in J$, so $y \in \phi ^{*}(J)$. Given $x \in \phi ^{*} (J)$ and $a \in A$, then $\phi (x) \in J$, as $J$ is ideal then $\phi (a) \phi (x) = \phi (ax) \in J$, then $ax \in \phi ^{*} (J)$. To prove it’s prime let’s take $ xy \in \phi ^{*} (J)$ then $ \phi (xy) \in J$ and as $J$ is prime then $\phi (x) \in J$ or $ \phi (y) \in J$ then $x \in \phi ^{*} (J)$ or $ y \in \phi ^{*} (J)$.\ Now let’s prove that $\phi^{*}$ is continuous. Given $V(a) \in Spec(A)$ let’s demonstrate that $(\phi^{*}) ^{-1} (V(a))$ is an opnening in $Spec(B)$. $(\phi^{*})^{-1} (V(a)) = \{P \in Spec(B) \mid \phi^{*} (P) \in V(a)\} = \{ P \in Spec(B) \mid a \in \phi^{*} (P) \} = \{ P \in Spec(B) \mid \phi(a) \in P\} = V(\phi(a)) \in Spec(B)$. - Given $P \in Spec(A)$, $P \in (\phi^{*}) ^{-1} (V(I)) \Leftrightarrow \phi^{*} (P) \in V (I) \Leftrightarrow I \subset \phi^{*} (P) \Leftrightarrow \phi (I) \subset P \Leftrightarrow P \in V(\phi (I)) $. - $\phi^*(V(b))= \{ Q \in Spec(A) | Q = \phi^{-1}(P), P \in V(b) \} = \{ Q \in Spec(A) | Q = \phi^{-1}(P), b \in P \} = \{ Q \in Spec(A) | b \in \phi(Q) \} = \{ Q \in Spec(A) | \phi^{-1}(b) \in Q \} = V(\phi^{-1}(b))$ - If $Q \in Spec(B)$ then $Ker(\phi)$ is contained in $\phi^*(Q)$. If $P \in V(Ker(\phi))$ then $P/ Ker(\phi)$ is isomorphic with an ideal $Q$ of $Spec(B)$ under the isomorphism $\bar{\phi}: A/ Ker(\phi) \mapsto B$ given that $\phi $ is surjective. So, $P = \phi^*(Q)$ and $\phi^*$ are surjective over $V(Ker(\phi))$. Now, if $\phi^*(P) = \phi^*(Q)$ then $\phi^{-1}(P) = \phi^{-1}(Q)$ and as $\phi$ is bijective, then $P = Q$, which shows that $\phi^* $ is injective. We had already shown the continuity of $\phi^*$ in ($i$), it only remains to show that $\phi^{-1}$ is continuous, that is to say, that $\phi^*$ is an open function, but this is obtained from ($iii$) by $\phi$ being injective. - Note that $\phi$ is injective, by ($iii$) we have that $\phi^*(Spec(B)) = \phi^*(V(0))= V(\phi^{-1}(0)) = V(Ker(\phi)) = V(0) = Spec(A)$. Compactness of the prime spectrum of a MVW-rig ============================================== In this section we will show that the prime spectrum of a MVW-rig $ A $ is compact, for this the filters will be used. Given $A$ a MVW-rig, a non-empty subset $F$ of $A$ is **filter** of $A$ if fulfills the following conditions for every $a,b \in A$: - If $a \leq b $ and $a \in F$ then $b \in F$, - If $a, b \in F$ then $ab \in F$ We define the **P-filters** as the filters $F$ that fulfill an aditional property: - Given $x \in A$ and $\bigoplus_i b_i x \in F$ a finite addition with $b_i \in A$ for every $i$, then $x \in F$. \[pfilter\] For a MVW-rig $A$ and a set $S \subseteq A$, the P-filter generated by $S$ in $A$ is, $$\langle S \rangle_P = \{ x \in A \mid (\exists s_1, \dots , s_n \in S), (\exists b_1, \dots , b_m \in A): s_1 \cdots s_n \leq \bigoplus_i b_i x \}$$ It is easy to see that $S \subset \langle S \rangle_P$ because $s^2 \leq s^2$ implies that $s \in \langle S \rangle_P$. Let’s see that $\langle S \rangle_P $ is P-filter: $i)$ Given $x \leq y, x\in \langle S \rangle_P$, then there exist $s_1, \dots , s_n \in S$ and $b_1, \dots , b_m \in A$ such that $ s_1 \cdots s_n \leq \bigoplus_i b_i x$, and as the operations conserve the order, then $\bigoplus_i b_i x \leq \bigoplus_i b_i y$ and therefore $ s_1 \cdots s_n \leq \bigoplus_i b_i y$, that is to say, $y \in \langle S \rangle_P$. $ii)$ Given $x,y \in \langle S \rangle_P $, then there exist $s_1, \dots , s_{n_1}, t_1, \dots , t_{n_2} \in S$ and $b_1, \dots , b_{m_1}, c_1, \dots, c_{m_2} \in A$ such that $ s_1 \cdots s_{n_1} \leq \bigoplus_i b_i x$ and $ t_1 \cdots t_{n_2} \leq \bigoplus_j c_j y $, then $s_1 \cdots s_{n_1} \cdot t_1 \cdots t_{n_2}\leq (\bigoplus_i b_i x)(\bigoplus_j c_j y) \leq \bigoplus_k d_k xy $ where $d_k$ are products $b_ic_j$, so $xy \in \langle S \rangle_P$. Now let’s demonstrate that $\langle S \rangle_P$ has the P-filter property: If $\bigoplus_j c_j x \in \langle S \rangle_P$ for some $c_j \in A$ then there exist $s_1, \dots , s_n \in S$ and $b_1, \dots , b_m \in A$ such that $ s_1 \cdots s_n \leq \bigoplus_i b_i (\bigoplus_j c_j x)\leq \bigoplus_{ij} b_i c_j x = \bigoplus_k d_k x$ and therefore $x \in \langle S \rangle_P$.\ Finally, let’s see that it is the smallest P-filter containing $S$. Given $H$ P-filter such that $S \subset H$, we want to see that $\langle S \rangle_P \subset H$. Given $x \in \langle S \rangle_P$, there exist $b_1, \dots , b_m \in A$ such that $ s_1 \cdots s_n \leq \bigoplus_i b_i x$ with $s_1, \dots , s_n \in S \subset H$, then $s_1 \cdots s_n \in H$ by being $H$ a filter, then $\bigoplus_i b_i x \in H$ and as $H$ has the P-filter property, we have that $x \in H$. Then $\langle S \rangle_P $ is the smallest P-filter that contains $S$. In particular, for an element $a$ of $A$ the P-filter generated by $a$ is: $$F_a = \{ x \in A \mid \exists n \in \mathbb{N} \text{ and } b_1, \dots , b_m \in A \text{ such that } a^n \leq \bigoplus_i b_i x \}$$ is a P-filter. Every P-filter F of $A$ satisfies that: $F = \bigcup\limits_{a \in F} F_{a}$ Given $x \in F$, then $x \in F_{x}\subseteq \bigcup\limits_{a \in F} F_{a}$. On the other hand, given $y \in \bigcup\limits_{a \in F} F_{a}$, then $y \in F_a$ for some $a \in F$, there exist $n \in \mathbb{N}$ and $b_1, \dots , b_m \in A$ such that $a^n \leq \bigoplus_i b_i y$, as $a \in F$, then $a^n \in F$ and therefore $\bigoplus_i b_i y \in F$, since $F$ has the P-filter property, $y \in F$. The union of two P-filters is not necessarily a P-filter. The following proposition defines the join and the meet of P-filters, which are P-filters. \[filtroLocal\] Given $A$ a MVW-rig we have that:\ - The P-filter generated by the union of two P-filters $F_a$ and $F_b$ is: $$\langle F_a \cup F_b \rangle_P = \{ x \in A \mid \exists n_1,n_2 \in \mathbb{N} \text{ and } b_1, \dots , b_m \in A \mid a^{n_1} b^{n_2} \leq \bigoplus_i b_i x \}$$ - $ \bigvee_{a \in I} F_a = \langle \bigcup_{a \in I} F_a \rangle_P $ - $F_a \cap F_b = F_{a \vee b}$, - $F_a \cap F_b = F_a \wedge F_b $ - $ \langle F_a \cup F_b \rangle_P = F_a \vee F_b = F_{ab} $, $i)$ and $ii)$ are followed directly from the proposition (\[pfilter\]).\ $iii)$ Given $x \in F_a \cap F_b$ then there exist $n_1,n_2 \in \mathbb{N}$ y $b_1, \dots , b_{m_1}, c_1, \dots , c_{m_2} \in A$ such that $a^{n_1} \leq \bigoplus_i b_i x$ and $b^{n_2} \leq \bigoplus_j c_j x$, then $(a \vee b)^{n_1 + n_2} \leq \bigoplus_k d_k x \oplus \bigoplus_l e_l x$ because $(a \vee b)^{n_1 + n_2} \leq (a \oplus b)^{n_1 + n_2}$ and when expanding we get terms of the form $a^s b^r$ where $s >n_1$ or $r>n_2$, and therefore, without loss of generality, for $s > n_1$, $a^s b^r \leq \bigoplus_i b_i x b^r \leq \bigoplus_k d_k x$, and by expanding the expression arriving at what we wanted to prove. In this way $x \in F_{a \vee b}$. On the other hand, given $x \in F_{a \vee b}$, then there exist $n \in \mathbb{N}$ and $b_1, \dots , b_m \in A$ such that $(a \vee b)^n \leq \bigoplus_i b_i x$ and by property (v) of the proposition (\[propiedades\]) $a^n \vee b^n \leq \bigoplus_i b_i x$, then $a^n \leq \bigoplus_i b_i x$ and $b^n \leq \bigoplus_i b_i x$ and so $x \in F_a \cap F_b$.\ $iv)$ It follows directly from the fact that $F_a \cap F_b $ is a P-filter, like we showed it before.\ $v)$ Given $x \in \langle F_a \cup F_b \rangle_P$ then there exist $n_1,n_2 \in \mathbb{N}$ and $b_1, \dots , b_m \in A$ such that $a^{n_1} b^{n_2} \leq \bigoplus_i b_i x$; if $n_1 \leq n_2$ then $a^{n_2} b^{n_2} \leq \bigoplus_i c_i x$ where $c_i = a^{n_2-n_1} b_i$, then $(ab)^{n_2} \leq \bigoplus_i c_i x$ and therefore $x \in F_{ab}$. On the other hand, given $x \in F_{ab}$ then there exists $n \in \mathbb{N}$ and $b_1, \dots , b_m \in A$ such that $(ab)^n \leq \bigoplus_i b_i x$, this is, $a^n b^n \leq \bigoplus_i b_i x$ and so $x \in \langle F_a \cup F_b \rangle_P$.\ \[L\_A\] Given a MVW-rig $A$, the collection $L_A$ of P-filters of $A$, is a local. We want to see that $ F\wedge \bigvee\limits_{a \in I} F_{a}=\bigvee\limits_{a\in I} ( F \wedge F_{a})$.\ First, note that $F \cap \langle \bigcup\limits_{a \in I}F_{a} \rangle_P = \langle F \cap \bigcup\limits_{a \in I}F_{a} \rangle_P$, in fact, since $F \cap \bigcup\limits_{a \in I}F_{a} \subset F,\bigcup\limits_{a \in I}F_{a}$ then $ \langle F \cap \bigcup\limits_{a \in I}F_{a} \rangle_P \subset \langle F \rangle_P = F$ and $ \langle F \cap \bigcup\limits_{a \in I}F_{a} \rangle_P \subset \langle \bigcup\limits_{a \in I}F_{a} \rangle_P$, then $ \langle F \cap \bigcup\limits_{a \in I}F_{a} \rangle_P \subset F \wedge \langle\bigcup\limits_{a \in I}F_{a} \rangle_P = F \cap \langle \bigcup\limits_{a \in I}F_{a} \rangle_P $. On the other hand, given $x \in F \cap \langle \bigcup\limits_{a \in I}F_{a} \rangle_P$ then $x \in F $ and $x \in \langle \bigcup\limits_{a \in I}F_{a} \rangle_P$, therefore there exist finite $ s_i\in F_{a_i}, b_1 \dots b_m \in A$ such that $ s_1 \cdots s_k \leq \bigoplus_j b_j x$. Note that $(x \oplus s_1) \cdots (x \oplus s_k) \leq x^k \oplus x^{k-1}s_1 \oplus \cdots \oplus s_1 \cdots s_k \leq \bigoplus_l c_l x$. By axiom **F1** of filter, we have that $x \oplus s_i \in F \cap F_{a_i} $ for each $ i= 1, \dots ,k $ therefore $x \in \langle F \cap \bigcup\limits_{a \in I}F_{a} \rangle_P$ Now, if we use subsection (ii) of the proposition (\[filtroLocal\]) we have: $ F\wedge \bigvee\limits_{a \in I} F_{a} = F \wedge \langle \bigcup\limits_{a \in I}F_{a} \rangle_P = F \cap \langle \bigcup\limits_{a \in I}F_{a} \rangle_P = \langle F \cap \bigcup\limits_{a \in I} F_a \rangle_P = \langle \bigcup\limits_{a \in I} ( F \cap F_a) \rangle_P = \bigvee\limits_{a\in I} ( F \wedge F_{a})$. Given $A$ a MVW-rig, then $L_A$ is compact. Given $\bigvee\limits_{a \in I} F_{a}= A$ we want to see that there is a finite subcollection of $\left\{F_{a}\right\}_{a \in J}$ such that $\bigvee\limits_{a \in J} F_{a}= A$, $J$ finite set, $J\subset I$.\ It results that $\bigvee\limits_{a \in I} F_{a} = \langle \bigcup\limits_{a \in I}F_{a} \rangle_P = \{ x\in A \mid \exists n_j \in \mathbb{N},$ with $ 0 \leq j \leq k$ and $b_1, \dots , b_m \in A$ such that $ a_1^{n_1} a_2^{n_2} \cdots a_k^{n_k} \leq \bigoplus_i b_i x \} $.\ Since $0 \in A$, then $0 \geq a_1^{n_1} a_2^{n_2} \cdots a_k^{n_k}$, with $n_j \in \mathbb{N}$, then $0 \in \langle \bigcup\limits_{j=1}^{m}F_{a_{j}} \rangle_P = \bigvee\limits_{j=1}^{m}F_{a_{j}}$, so $\bigvee\limits_{j=1}^{m}F_{a_{j}}=A$. $\mathcal{O}(Spec(A))$ is the set of open subsets of $Spec(A)$ \[local\] For a MVW-rig $A$, the function of locals $\theta$ between $Spec(A)$ and $L_A.$ $\mathcal{O}(Spec(A))\stackrel{\theta}{\rightarrow}L_{A}$ $V(a) \longrightarrow F_{a}$ $\bigcup\limits_{j\in J} V(a_j) \longrightarrow \bigvee\limits_{j\in J}F_{a_{j}} $ Is an isomorphism. First, let’s see that $\theta$ is an isomorphism: - $\theta(V(a) \cup V(b)) = \theta(V(ab)) = F_{ab} = F_a \vee F_b = \theta(V(a)) \vee \theta(V(b))$. - $\theta(V(a) \cap V(b)) = \theta(V(a \vee b)) = F_{a \vee b} = F_a \wedge F_b = \theta(V(a)) \wedge \theta(V(b))$. $\theta$ is surjective: given that $F \in L_A$, we have that $F= \langle \bigcup\limits_{a\in F} F_{a} \rangle_P = \bigvee\limits_{a\in F} F_{a}$ then $\theta\textit{}(\bigcup\limits_{a\in F} V(a))=F$ by definition of $\theta$.\ Now, let’s see that $\theta$ is injective: given $F_{1}, F_{2}\in L_A$, such that $F_{1} \neq F_{2}$ being $x \notin F_{2}$ and $x \in F_{1}$ then the ideal of the MVW-rig $A$ generated by $x$ satisfies that $\langle x \rangle \cap F_2 = \emptyset$, in fact, $\langle x \rangle \cap F_{2} \neq \emptyset$, implies that there exists $z \in (x) \cap F_{2}$, and $z \leq \bigoplus_j b_i x$, since $F_{2}$ is P-filter, $x \in F_{2}$, which is absurd. Let’s consider the set of ideals, $\Sigma =\{I \mid x \in I; I\cap F_{2} = \emptyset \}$ $(x) \in \Sigma$ then $ \Sigma \neq \emptyset$ The set $\Sigma$ is inductively superior: each chain of ideals $ I_{i}\in \Sigma$ has a upper bound $\bigcup I_{i}\in \Sigma$. Then by the Zorn’s lemma, $\Sigma$ contains at least a maximal element. Being $P$ a maximal of $\Sigma$,then $x \in P$ and $P\cap F_2=\emptyset$. We want to see that P is prime ideal: being $y, z \in A$, such that $yz \in P$; we want to see that $y \in P$ or $z \in P$. Let’s suppose that $y \notin P$ and $z \notin P$. By the maximality of $P$ in $\Sigma$, we follow that: $ \langle P \cup \left\{y\right\} \rangle \cap F_{2} \neq \emptyset$ and $ \langle P \cup \left\{z\right\} \rangle \cap F_{2} \neq \emptyset$ Consequently there are $p,q \in P$, $w ,w' \in F_{2}$ and $a_1, \dots, a_{m_1}, b_1, \dots , b_{m_2} \in A $ such that $w \leq p \oplus \bigoplus_i a_i y$ and $w' \leq q \oplus \bigoplus_j b_j z $ Since $F_{2}$ is filter, then $(p \oplus \bigoplus_i a_i y)\in F_{2}$ and $(q \oplus \bigoplus_j b_j z)\in F_{2}$ So, as the product retains order, we have: $ w w' = (p \oplus \bigoplus_i a_i y)(q \oplus \bigoplus_j b_j z) \leq r \oplus \bigoplus c_i yz = r'$ where the axiom (iii) was used and $r$ is an element of $P$ obtained from the sums and products of the elements $p,q \in P$ with other elemets of $A$ that, by absorbent property of $P$ are in $P$. Since $w w' \in F_2$ and $F_2$ is filter $r' \in F_{2}$. We follow that $r' \in P\cap F_{2}$ which contradicts the hypothesis $P\cap F_{2}=\emptyset $; then $y \in P$ or $z \in P$. In consequence $\bigcup\limits_{a\in F_{1}} V(a) \neq \bigcup\limits_{b\in F_{2}} V(b) $. because $P \in \bigcup\limits_{a\in F_{1}} V(a) $, due to $ x \in F_{1}$, $P \in V(x)$, $P \notin \bigcup\limits_{b \in F_{2}}V(b)$ due to $P\cap F_{2}=\emptyset$ Given $A$ a MVW-rig, the $Spec(A)$ with the co-Zariski topology, is a compact topological space. The preceding statements are followed. Conclusions =========== The MV-algebras were founded by Chang [@Chang2] to demostrate a completeness theorem for fuzzy logic. This rich structure has been studied since many years ago. One of the most important difficulties for someone to make commutative fuzzy algebra is the absense of some product operation. The MV-algebra $[0,1]$ has a natural product and this product respects the MV-algebra structure. There are some results about the MV-algebra’s product [@DiNola] and [@DiNola2]. Our focus is related with optaning an adequate theory in order to represent a fuzzy commutative algebra in the best form. There exists a close relationship between the class of the special MVW-rigs and some kind of $l_u-$rings. These categories are equivalent, however we don’t show this result here. [99]{} Atiyah, M.F., Macdonald, I.G. *Introduction to commutative algebra*. Addison-Wesley Series in mathematics, 1969. Castiglioni, J., Menni, M., y Zuluaga, W. *A representation theorem for integral rigs and its applications to residuated lattices*. Journal of Pure and Applied Algebra. Volume 220 issue 10, 2016, p. 3533-3566. Chang, C.C. *Algebraic analysis of many-valued logics*. Transactions of the American Mathematical Society. **88** 1958, p. 467–490. Chang, C.C. *A new proof of the completeness of the Łukasiewicz axioms*. Transactions of the American Mathematical Society. **93** 1959, p. 74–90. Cignoli, R.L, D’ottaviano, I.M.L y Mundici, D. *Algebraic Foundations of Many-valued Reasoning*. Klumer Academic Publishers, Dordrecht. 2000. Di Nola, A., Flondor, P. y Leustean, I. *MV-modules*. Journal of Algebra. Volume 267, Issue 1, 2003, p 21-40. Di Nola A., Dvure$\check{c}$enskij, A. *Product MV-algebras*. Mathematical Institute, Slovak Academy of Sciences Bratislava, 1998. Dubuc, E.J., Poveda, Y.A. *Representation Theory of MV-algebras*, Annals of pure and applied logic, 161, 2010. Dubuc, E.J., Poveda, Y.A. *On the equivalence between MV-Algebras and $l$-groups whit strong unit*, Studia logica/Springer, 2015. Estrada, A. *MVW-rigs*. Master’s Thesis, Universidad Tecnológica de Pereira. 2016. Fuchs, L. *Partially Ordered Algebraic Systems*. Pergamon, Oxford/ London/ New York/ Paris. 1963 López, J.H. *Algunas propiedades del espectro primo en MV-álgebras*. Pereira. Colombia. 2012. Montagna, F. *Subreducts of MV -algebras with product and product residuation*. Algebra Universalis. April 2005, Volume 53, Issue 1, pp 109–137. Mundici, D. *Interpretation of AF C$^*$-algebras in Lukasiewicz setential calculus*, Journal of Functional Analysis 65:15-63, 1986 Zuluaga, S. *Los MVW-rigs provenientes de las MV-álgebras libres*. Master’s Thesis, Universidad Tecnológica de Pereira. 2017. Zuluaga, W.J. *Representación por Haces de riRigs*. PhD Thesis, Universidad Nacional de la Plata. Argentina, 2016. [^1]: Universidad Tecnológica de Pereira. [^2]: Universidad Tecnológica de Pereira.

=10000 epsf.sty Significant progress in describing a strongly interacting 2-dimensional electron system (2DES) in a large magnetic field in terms of composite fermions (CF) in a reduced field has stimulated a large body of theoretical and experimental work [@cfrevs; @rectheory]. One of the most suprising implementations of this idea was put forth in the seminal work of Halperin, Lee, and Read [@HLR], who argued that the ground state of the 2DES at Landau level filling factor $\nu$=$\frac{1}{2}$ is well-described by CF in zero net magnetic field, which therefore exhibit a well-defined Fermi surface. Experiments carried out near $\nu$=$ \frac{1}{2}$ have provided convincing evidence of the existence of the CF Fermi surface [@willett93; @willett95; @kang; @goldman94; @smet]. Despite the overall agreement between theory and experiment to date, several fundamental issues about CF at $\nu$=$\frac{1}{2}$ have yet to be resolved experimentally. For example, do CF form a Fermi gas, a “normal” Fermi liquid, or some kind of “unusual” Fermi liquid? Also, does the picture change when the ground state is only partially Experiments which directly probe the electron spin degree of freedom [*right at*]{} $\nu$=$\frac {1}{2}$, especially near the transition between partially and fully spin-polarized ground states, will help to answer these central questions. In this Letter, we report optically pumped nuclear magnetic resonance (OPNMR) [@opnmrchar] measurements of the Knight Shift $K_S$ and the spin-lattice relaxation rate 1/T$_1$ of $^{71}$Ga nuclei in two different electron-doped multiple quantum well (MQW) samples. The $K_S$ data reveal the spin polarization ${\cal P}(T) $$\equiv$$\frac{\langle S_{z}(T)\rangle} {\text{max}\langle S_{z}\rangle}$, while the 1/T$_1$ data probe the spin dynamics of the 2DES. Taken together, these thermodynamic measurements provide unique insights into the physics of CF at $\nu$=$\frac{1}{2}$. The two samples used in this work were previously studied[@pankaj; @nick] near $\nu$=$\frac{1}{3}$. Sample 40W contains forty $300\,$Åwide GaAs wells separated by $3600\,$Åwide Al$_{0.1}$Ga$_{0.9}$As barriers. Sample 10W contains ten $260\,$Åwide wells separated by $3120\,$Åwide barriers. Silicon delta-doping spikes located in the center of each barrier provide the electrons that are confined in each GaAs well at low temperatures, producing 2DES with very high mobility ($\mu > 1.4\times10^{6}$ cm$^{2}$/Vs). This MQW structure results in a 2D electron density that is unusually insensitive to light, and extremely uniform from well to well[@insensitive]. The low temperature ($0.29\,<$$\,T$$\,<\,$$ 3.5\,$K) OPNMR measurements described below were performed using a sorption-pumped $^{3}$He cryostat. The samples, about in size, were in direct contact with $^{3}$He, mounted on the platform of a rotator assembly in the NMR probe. Data were acquired following the previously described[@pankaj; @nick; @opnmr; @science] OPNMR timing sequence: , modified for use below 1 Kelvin (e.g., $ \tau_{D}\,$$>\,$10s, laser power , low rf voltage levels). A calibrated RuO$_{2}$ sensor, in good thermal contact with the sample, was used to monitor the temperature. The peak in $K_S(\nu)$ at $\nu$=1 was used to determine the electron density $n$ for each sample[@pankaj]. Using the rotator assembly, we could set the angle $\theta$ between the sample’s growth axis and the applied field $B_{\text{tot}}$ so that the filling factor $\nu$=$nhc/eB_{\bot}$ (with $B_{\bot}$$\equiv$$B_{\text{tot}}\!\cos\theta$) equalled $\frac{1}{2}$ for these three cases: Case Sample $B_{\text{tot}}$ \[T\] $n$ \[$\text{cm}^{-2}$\] $w$ \[Å\] ------ -------- ------------------------ ---------------- -------------------------- ----------- I 40W 7.03 38.3$^{\circ}$ 6.69$\times\,$$10^{10}$ 300 II 40W 5.52 0.0$^{\circ}$ 6.69$\times\,$$10^{10}$ 300 III 10W 7.03 24.5$^{\circ}$ 7.75$\times\,$$10^{10}$ 260 Figure \[fig1\] shows OPNMR spectra at $\nu$=$\frac{1}{2}$ and $T$$\approx$$0.5$K, for Cases I–III (a-c, solid lines). Nuclei within the quantum wells are coupled to the spins of the 2DES via the isotropic Fermi contact interaction[@slichter], which shifts the corresponding well resonance (labeled “W" on Fig.\[fig1\](c)) relative to the signal from the barriers (“B")[@opnmr; @science]. We define the Knight shift $K_S$ as the peak-to-peak splitting between “W" and “B". For Case I, all spectra (e.g., Fig.\[fig1\](a)) are well-described by the same two-parameter fit (dotted lines)[@pankaj; @nick] that was used for all spectra at $\nu$=$\frac{1}{3}$. This fit is generated under the assumption that all spins are delocalized, so that $\langle S_{z}(\nu,T)\rangle$, averaged over the NMR time scale ($\sim\,$40$\,\mu$sec), appears spatially homogeneous along the plane of the wells, and thus the resulting lineshape is “motionally-narrowed” [@slichter]. In contrast, for Cases II and III, the well resonance (Fig.\[fig1\](b,c)) is much broader than the same fit (dotted lines). An additional gaussian broadening of just the well resonance leads to a better fit (dashed lines). The full width at half maximum (FWHM) of the additional broadening extracted from these fits is plotted in Fig.\[fig1\](d) for Cases II and III. Earlier measurements at $\nu$$<$$\frac{1}{3}$ and $T$$\approx$0.5K were also poorly described by the “motionally-narrowed” lineshape, but in that case the extra well width was sharply temperature-dependent [@nick], whereas here it is essentially independent of temperature. The extra broadening of the well lineshape for Cases II and III (Fig.\[fig1\](b,c,d)) seems to be homogeneous [@density], if so, the corresponding transverse relaxation time T$_2$ [@slichter] is quite short. The origin of this effect is not understood as it is very hard to explain simultaneously the temperature-independence of the extra broadening and the lack of a similar effect in Case I. =3.0in Figure \[fig2\](a) shows $K_S(T)$ at $\nu$=$\frac{1}{2}$ for Cases I–III. The larger scatter in the $K_S(T)$ data for Cases II and III is a consequence of the large linewidth. Using an empirical relation (all in kHz) $K_{S\text{int}}$= $K_S$$+$$1.1$$\times$$(1$$-$$ \exp(-K_S/2.0))$, we can convert $K_S$ into $K_{S\text{int}}$, which is the intrinsic hyperfine shift for the nuclei in the center of each well. $K_{S\text{int}}\,$=$\,A_c\:{\cal P}\:n/w$ is a direct measure of the electron spin polarization ${\cal P}$, where $w$ is the width of each well and $A_c=(4.5\pm 0.2)\times 10^{-13}\,$cm$^{3}$/s is the hyperfine constant[@pankaj]. For Cases II and III, the same values of $K_{S\text{int}}(T)\,$ are also obtained directly from the dashed line fits (e.g., Fig.\[fig1\](b,c)). $K_{S\text{int}}(T)$ is converted to electron spin polarization using =$K_{S\text{int}}(T)/ K_{S\text{int}}^{{\cal P}=1}$, where the maximum shift for a fully polarized 2DES is known for each sample: $K_{S\text{int}}^{{\cal P}=1}$= $K_{S\text{int}}(\nu$[=]{}$\frac{1}{3},$$ T$$\rightarrow$0) [@pankaj]. Figure \[fig2\](b,c) shows that ${\cal P}(\nu$[=]{}$\frac{1}{2},T)$ does not saturate down to our base temperature of 0.29K, in contrast to earlier measurements at $\nu$=1 and $\frac{1}{3}$ [@pankaj; @opnmr]. Moreover, as the temperature is increased, ${\cal P}(\nu,T)$ falls off much faster at $\nu$=$\frac{1}{2}$ than at $\nu$=$1$ or $\frac{1}{3}$ (e.g., at $T_{Z}$=$|$g$^{\ast}\mu_e B_{\text{tot}}/k_B|$, ${\cal P}(\nu$=1,$T_{Z})$$\approx$$93\%$, while ${\cal P}(\nu$=$\frac{1}{2},$$T_{Z})$$ \approx$40$\%$. Here $\mu_e$ is the Bohr magneton and $k_B$ is the Boltzmann constant). Qualitatively, these results are consistent with a tiny (or vanishing) energy gap for spin-flip excitations at $\nu$=$\frac{1}{2}$ for Cases I–III. However, a quantitative understanding of the ${\cal P}(\nu$=$\frac{1}{2},T)$ data remains a challenge for theory (e.g., we cannot explain the crossing of the Case I and Case II data sets at $T$$\approx$$T_{Z}$ (Fig.\[fig2\](b))). =3.0in Even though saturation is not observed, the Knight shift data for Case II are evidence for a $\nu$=$\frac{1}{2}$ ground state in which the electrons are only [*partially*]{} spin-polarized (i.e., ${\cal P}(\nu$=$\frac{1}{2},T\!\rightarrow0) $$\approx$$70-85\%$). This inference is consistent with data obtained from two other experiments in conditions similar to those of Case II. From their time-resolved photoluminescence measurements, Kukushkin et al.[@kukushkin] estimate ${\cal P}(\nu$=$\frac{1}{2},T\!\rightarrow0) $$\approx$$87\%$ at 5.52T. Surface acoustic wave measurements of Willett et al. obtained a Fermi wave vector which was $\sim90\%$ of the theoretical value for fully polarized CF, consistent with a polarization of at least ${\cal P}(\nu$=$ \frac{1}{2},T\!\rightarrow0)\approx 62\%$.[@willett93] The solid curves in Figure \[fig2\](b,c) are two parameter fits to the $T$$<$$T_Z$ data using expressions for ${\cal P}(\nu$=$\frac{1}{2},T)$ derived within a weakly-interacting composite fermion model (WICFM). In this model, the dispersion relations for spin-up and spin-down states are: $$E_{\uparrow}(k)=\frac{\hbar^2k^2}{2m^{\ast}}, \,\,\,\, E_{\downarrow}(k)=\frac{\hbar^2k^2 }{2m^{\ast}} + E_Z^{\ast}(T)\,, \label{eq1}$$ where an exchange interaction has been included in the model through the effective Zeeman energy: $$E_Z^{\ast}(T)= |$g$^{\ast}\mu_eB_{ \text{tot}}|+ E_{\text{Exch}}= k_B T_Z + k_B J {\cal P}(T)\,. \label{eq2}$$ When $J=0$, this is just the non-interacting composite fermion model[@kukushkin]. When $J>0$, there is a Stoner enhancement of the spin susceptibility. In this model, the chemical potential $\mu$ and the polarization ${\cal P}$ are: $$\mu(T)= k_BT\ln\Bigl(-\gamma + \sqrt{\gamma^2+ \exp(\rho)-1}\Bigr) + \frac{E_Z^{\ast}(T)}{2} \label{eq3}$$ $${\cal P}(T) = \frac{1}{\rho} \ln \Bigl(\frac{1+\exp[\frac{\mu(T)}{k_BT}]} {1+\exp[\frac{\mu(T)}{k_BT}(1-\delta(T))]} \Bigr) \label{eq4}$$ $${\text{where}}\,\,\,\,\,\gamma$=$ \cosh({\frac{E_Z^{\ast}(T)}{2k_BT}})$;\, $\rho$=$\frac{2\pi\hbar^2n}{m^{\ast} k_BT}$;\, and\, $\delta(T)$=$\frac{E_Z^{\ast}(T)}{\mu(T)}.$$ =3.0in Equations 2–4 are solved self-consistently for ${\cal P}(T)$ at each $m^{\ast}$ and $J$. Within this WICFM, the behavior of ${\cal P}(T)$ as $T$$\rightarrow$0 is quite sensitive to the parameter $\delta(0)$. The ground state is only fully polarized (${\cal P}(0)$=1) when $\delta(0)$$\geq$1. We find $\delta(0)$$<$1 for Cases II and III, $\delta(0)$=1 for Case I, and the dashed curve illustrates $\delta(0)$$>$1 (Fig.\[fig2\](b,c)). Thus, within this model, the best-fit curves for Cases II and III yield partially-polarized ground states, while Case I is fully-polarized. As described earlier, Cases II and III also have extra linewidth, while Case I does not (Fig. \[fig1\]). Figure \[fig3\] shows the temperature dependence of the $^{71}$Ga nuclear spin-lattice relaxation rate 1/T$_1$ at $\nu$=$\frac{1}{2}$ for Cases I and II. At each temperature, OPNMR spectra were acquired using a series of dark times 10s $\leq$ $\tau_{\text{D}}$ $\leq$ 2560s (i.e., the longest $\tau_{\text{D}}$$\geq$$4T_1$). The value of 1/T$_1$ was determined by fitting the signal intensity at the “W” peak frequency to the form $S( \tau_{\text{D}}$) = $S_0\exp{ (-\tau_{\text{D}}/T_1)}+S_1$. Note that these $T$$<$$T_Z$ relaxation rates are faster than the rate at $T$$\approx$$T_Z$ for $\nu$=1 [@science]. Qualitatively, this shows that there is a greater overlap of the density of states for electrons with opposite spins at $\nu$=$\frac{1}{2}$ than at $\nu$=1. The isotropic Fermi contact hyperfine coupling between the electron spins and the nuclear spins is responsible for both $K_S$ and 1/T$_1$ [@pankaj; @nick; @opnmr; @science], as is the case for some metals[@slichter; @winter]. Within the WICFM, 1/T$_1$($\nu$=$\frac{1}{2}$,$T$) for $^{71}$Ga nuclei in the center of the quantum well is : $$\frac{1}{\text{T}_1}=\frac{\pi (m^{\ast})^2}{\hbar^3} \Bigl(\frac{K_{S\text{int}}^{{\cal P}=1}}{n}\Bigr)^2 \frac{ k_{\text{B}} T}{1+\exp{[\frac{\mu(T)}{k_BT}(\delta(T)-1)]}}. \label{eq5}$$ This expression is used as a two-parameter fit to the 1/T$_1$($T$) data (Fig.\[fig3\], solid lines), where $\mu(T)$ and $\delta(T)$ are obtained from Eqns.2–4 for each $m^{\ast}$ and $J$. The behavior of 1/T$_1$$(T)$ as $T\!\rightarrow0$ is also quite sensitive to the parameter $\delta(0)$. In Fig.\[fig3\], we illustrate $\delta(0) < 1$ with the dotted curve, $\delta(0)$=1 with the dashed curve, and we find $\delta(0)$$>$1 for Cases I and II (solid curves). In contrast to normal metals, here $\mu(0)$$\sim$$|$g$^{\ast}\mu_eB_{ \text{tot}}|$. =3.23in Figure \[fig4\] shows the best values of $J$ and $m^{\ast}$ obtained for each data set in Figs.\[fig2\](b,c) and \[fig3\]. The correlation between $J$ and $m^{\ast}$ is shown by $\Delta$$\chi$$^2$=1 and $\Delta$$\chi$$^2$=4 contours. These ($J$,$m^{\ast}$) values lie quite close to the curves which mark the transition between fully and partially polarized ground states (i.e., where $\delta(0)$=1). There is negligible overlap between the contours and the line $J$=0, so the non-interacting composite fermion model used by Kukushkin et al. [@kukushkin] does not work here. Moreover, there is no ($J$,$m^{\ast}$) pair which can simultaneously describe the four data sets measured using the same sample and $B_{\perp}$ (Fig. 4 (main)), so we conclude that even the weakly-interacting composite fermion model is a poor description of the $\nu$=$\frac{1}{2}$ state for these Cases. The most glaring inconsistency is that of Case II, where $\delta(0)$$<$1 (i.e., partially-spin polarized at $T$=$0$) is inferred from ${\cal P}(T)$, which is [*incompatible*]{} with the result $\delta(0)$$>$1 (i.e., fully-spin polarized at $T$=0) that is inferred from 1/T$_1$$(T)$. Figure \[fig4\] (inset) shows the ($J$,$m^{\ast}$) values obtained from the ${\cal P}(T)$ data for Cases I and III. These values do not agree, however, sample 10W and 40W also have slightly different electron densities and well widths ($n$,$w$). This would affect our results, since we expect $k_BJ$ $\propto$ $eB_{\perp}/m^{\ast}$ $\propto$ $E_C(\lambda)\equiv e^2/(\epsilon\sqrt{l_0^2+\lambda^2})$, where $l_0$=$\sqrt{\hbar c/eB_{\perp}}$ is the magnetic length, $\epsilon$=13, and the parameter $\lambda\approx \frac{1}{4} w$ modifies the Coulomb energy scale due to the non-zero thickness $w$ of the quantum well [@zhang]. To correct for this, the Case III ($J$,$m^{\ast}$) values are rescaled using: $$\begin{aligned} \frac{J(n_{\text{I}},w_{\text{I}})}{J(n_{\text{III}},w_{\text{III}})} = \sqrt{\frac{n_{\text{I}}}{n_{\text{III}}} \Bigl(\frac{1+\frac{n_{\text{I}}w_{\text{I}}^2} {n_{\text{III}}w_{\text{III}}^2}}{2}\Bigr)}\,,\nonumber\\ \frac{m^{\ast}(n_{\text{I}},w_{\text{I}})}{m^{\ast}(n_{\text{III}},w_{\text{III}})} = \sqrt{\frac{n_{\text{I}}}{n_{\text{III}}}\Bigl(\frac{2}{1+ \frac{n_{\text{I}}w_{\text{I}}^2}{n_{\text{III}}w_{\text{III}}^2}}\Bigr)}\,.\end{aligned}$$ The rescaled contour has a good overlap with the ($J$,$m^{\ast}$) values for Case I (Fig.\[fig4\] (inset)). This rescaling is irrelevant for Fig.\[fig4\] (main), where the results on a single sample are shown. In conclusion, neither a non-interacting nor a weakly-interacting composite fermion model is sufficient to explain our experiments, which probe the electron spin degree of freedom [*right at*]{} $\nu$=$\frac {1}{2}$. Knight shift and 1/T$_1$ data, taken together, provide important new constraints on the theoretical description of the $\nu=\frac {1}{2}$ state. Finally, in addition to fully polarized ground states (Case I, $\frac{k_BT_Z}{E_C(\lambda)}$=0.021), partially spin-polarized ground states (Cases II and III, $\frac{k_BT_Z}{E_C(\lambda)}$=0.017 and 0.019) are experimentally accessible, and exhibit unexpected features (e.g., the extra linewidth). We thank N. Read, R. Shankar, S. Sachdev, and R.L. Willett for many helpful discussions. This work was supported by NSF Grant $\#$DMR-9807184. SEB also acknowledges an Alfred P. Sloan Research Fellowship. For comprehensive reviews, see: [*Composite Fermions*]{}, ed. by O.Heinonen (World Scientific, 1998); [*Perspectives in Quantum Hall Effects*]{}, ed. by S.Das Sarma and A. Pinczuk (Wiley, New York, 1997); J.K. Jain, Adv. Phys. [**41**]{}, 105 (1992); R.L. Willett, [*ibid.*]{} [**46**]{}, 447 (1997). For recent theoretical developments near $\nu$=$\frac{1}{2}$, see: N. Read, Phys. Rev. B [**58**]{}, 16262 (1998); A. Stern, B.I. Halperin, F.v.Oppen, and S.H. Simon, [*ibid.*]{} [**59**]{}, 12547 (1999); R. Shankar, cond-mat/9903064. B.I. Halperin, P.A. Lee, and N. Read, Phys. Rev. B [**47**]{}, 7312 (1993). R.L. Willett, R. R. Ruel, K. W. West, and L. N. Pfeiffer, Phys. Rev. Lett. [**71**]{}, 3846 (1993). R.L. Willett, K.W. West, and L.N. Pfeiffer, Phys. Rev. Lett., 2988 (1995). W. Kang [*et al.*]{}, Phys. Rev. Lett. [**71**]{}, 3850 (1993). V.J. Goldman, B. Su, and J.K. Jain, Phys. Rev. Lett. [**72**]{}, 2065 (1994). J.H. Smet [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 2272 (1996). R.R. Du [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 3926 (1995). I.V. Kukushkin, K.v.Klitzing, and K. Eberl, Phys. Rev. Lett. [**82**]{}, 3665 (1999). K. Park and J.K. Jain, Phys. Rev. Lett. [**80**]{}, 4237 (1998). S.E. Barrett, R. Tycko, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. [**72**]{}, 1368 (1994). P. Khandelwal [*et al.*]{}, Phys. Rev. Lett. [**81**]{}, 673 (1998); also available at cond-mat/9801199. N.N. Kuzma [*et al.*]{}, Science [**281**]{}, 686 (1998); also available at cond-mat/9907279. L.N. Pfeiffer [*et al.*]{}, Appl.Phys.Lett. [**61**]{},1211(1992). S.E. Barrett [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 5112 (1995). R. Tycko [*et al.*]{}, Science [**268**]{}, 1460 (1995). C.P. Slichter, [*Principles of Magnetic Resonance*]{} (Springer, New York, 1990), 3$^{\text{rd}}$ ed. In simulations, we found that an unreasonably large FWHM of 70% or more is required for a gaussian distribution of electron densities along the well to explain the large temperature independent linewidth observed at $\nu$=$\frac{1}{2}$ for Cases II and III. J. Winter, [*Magnetic Resonance in Metals*]{}, (Oxford Univ.Press, London, 1971). F.C. Zhang and S. Das Sarma, Phys. Rev. B [**33**]{}, 2903 (1986).

--- abstract: '[A general diffusive predator-prey model is investigated in this paper. We prove the global attractivity of constant equilibria when the conversion rate is small, and the non-existence of non-constant positive steady states when the conversion rate is large. The results are applied to several predator-prey models and give some ranges of parameters where complex pattern formation cannot occur.]{} [**[Keywords]{}**]{}: Reaction-diffusion; Non-existence; Steady state; Global attractivity' author: - | Shanshan Chen[^1],   Junjie Wei[^2],  Jianhui Zhang\ [Department of Mathematics, Harbin Institute of Technology,]{}\ [Weihai, Shandong, 264209, P.R.China.]{} title: ' **[^3]**' --- Introduction ============ During the past decades predator-prey interaction has been investigated extensively, and there are several reaction-diffusion equations modelling the predator-prey interaction, see [@Du-Hsu; @Du-Lou3; @Du-Lou; @Guo-Wu; @Ko-Ryu; @Leung; @Yi] and references therein. The spatiotemporal patterns induced by diffusion, such as Turing pattern, can be used to explain the complex phenomenon in ecology. A prototypical one is the following diffusive predator-prey system with Holling type-II functional response $$\label{1.1} \begin{cases} {\displaystyle}\frac{\partial u}{\partial t}=d_1\Delta u+au\left(1-{\displaystyle}\frac{u}{k}\right)-{\displaystyle}\frac{buv}{1+mu}, & x\in \Omega,\; t>0,\\ {\displaystyle}\frac{\partial v}{\partial t}=d_2\Delta v-\theta v+{\displaystyle}\frac{euv}{1+mu}, & x\in\Omega,\; t>0, \\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega,\; t>0,\\ u(x,0)=u_0(x)\ge(\not\equiv)0, \;\; v(x,0)=v_0(x)\ge(\not\equiv)0,& x\in\Omega. \end{cases}$$ Yi, Wei and Shi [@Yi] investigated the Hopf and steady state bifurcations near the unique positive equilibrium of system . Peng and Shi [@Peng-Shi] proved that the global bifurcating branches of steady state solutions are bounded loops containing at least two bifurcation points, which improved the result in [@Yi]. Ko and Ryu [@Ko-Ryu] investigated the dynamics of system with a prey refuge. For the case of the homogeneous Dirichlet boundary condition, Zhou and Mu [@muzhou] showed the existence of positive steady states through bifurcation theory and fixed point index theory. Recently, Wang, Wei and Shi [@Wangws] studied a diffusive predator-prey model in the following general form $$\label{1.1w} \begin{cases} {\displaystyle}\frac{\partial u}{\partial t}=d_1\Delta u+g(u)\left(f(u)-v\right), & x\in \Omega,\; t>0,\\ {\displaystyle}\frac{\partial v}{\partial t}=d_2\Delta v+v\left(-\theta +g(u)\right), & x\in\Omega,\; t>0,\\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega,\; t>0,\\ u(x,0)=u_0(x)\ge(\not\equiv)0, \;\; v(x,0)=v_0(x)\ge(\not\equiv)0,& x\in\Omega, \end{cases}$$ where predator functional response $g(u)$ is increasing. They investigated the Hopf and steady state bifurcations near the unique positive equilibrium of system , and the existence and non-existence of non-constant positive steady states were also addressed with respect to the diffusion coefficients $d_1$ and $d_2$. Similar results on the Hopf and steady state bifurcations near the positive equilibrium can be found in [@Jin; @Peng-Yi; @Wangws2]. Moreover, a non-monotonic functional response was proposed to model the prey’s group defense, see [@Freed; @Wolk]. That is, the predator functional response in model is non-monotonic and can be chosen as follows: $$\label{type4} \text{(Holling type-IV)}\;\; g(u)={\displaystyle}{\frac}{bu}{1+nu+m u^2},\;\;\text{where}\;\;b,n,m>0.$$ Related to the work on model with non-monotonic functional response, see [@pangw; @RuanX1; @RuanX2; @ZhuC] and references therein. Another prototypical predator-prey model has the following from $$\label{DL} \begin{cases} {\displaystyle}\frac{\partial u}{\partial t}-d_1\Delta u=u(a-u)-{\displaystyle}\frac{buv}{1+mu}, & x\in \Omega,\; t>0,\\ {\displaystyle}\frac{\partial v}{\partial t}-d_2\Delta v=v(d- v)+{\displaystyle}\frac{euv}{1+mu}, & x\in\Omega,\; t>0,\\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega,\; t>0,\\ u(x,0)=u_0(x)\ge(\not\equiv)0, \;\; v(x,0)=v_0(x)\ge(\not\equiv)0,& x\in\Omega. \end{cases}$$ Here the growth rate of the predator is logistic type in the absence of prey, and it is different from models and that the predator can survive without the specific prey. For $m=0$, Leung [@Leung] proved the global attractivity of constant equilibria, which still holds when saturation $m$ is small [@Brown; @Motoni; @Du-Lou]. Du and Lou [@Du-Lou] investigated the existence and non-existence of non-constant steady states when saturation $m$ is large, and see also [@Du-Lou3] for the case of the homogeneous Dirichlet boundary condition. Peng and Shi [@Peng-Shi] proved the non-existence of non-constant positive steady states. Moreover, Yang, Wu and Nie [@Yang-Wu] considered a diffusive predator-prey model under the homogeneous Dirichlet boundary condition, where the growth rate of the predator is like a Beverton-Holt function, and see [@Cheny] for the case of the homogeneous Neumman boundary condition. Motivated by the above work of [@Du-Lou] and [@Wangws], we analyze a diffusive predator-prey model in the following general form $$\label{1.4} \begin{cases} {\displaystyle}\frac{\partial u}{\partial t}=d_1\Delta u+g(u)\left(f(u)-v\right), & x\in \Omega,\; t>0,\\ {\displaystyle}\frac{\partial v}{\partial t}=d_2\Delta v+v\left(h(v) +cg(u)\right), & x\in\Omega,\; t>0, \\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega,\; t>0,\\ u(x,0)=u_0(x)\ge(\not\equiv)0, \;\; v(x,0)=v_0(x)\ge(\not\equiv)0,& x\in\Omega. \end{cases}$$ Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N\le 3$) with a smooth boundary $\partial \Omega$, $f(u)g(u)$ is the growth rate of prey without predator, $g(u)$ is the predator functional response, $c>0$ is the conversion rate, and $h(v)$ is the growth rate per capita of the predator in the absence of prey. For system , we see that $(u,v)$ is a constant positive equilibrium if and only if $u\in(0,a)$ is a solution of the following equation $$H(u):=h\left(f(u)\right)+cg(u)=0.$$ However, it is hard to determine the exact numbers and expressions of constant positive equilibria of system , not to mention the bifurcations near constant positive equilibria. In this paper, we first assume that $f$ and $g$ satisfy the following assumptions: 1. $f\in C^1(\overline{\mathbb {R}^+})$ and there exists a unique $a>0$ such that $f(u)$ is positive for $u\in[0,a)$ and negative for $u>a$. Moreover, $f'(u)<0$ on $(0,a]$ or there exists ${\lambda}\in(0,a)$ such that $f'(u)>0$ on $(0,{\lambda})$ and $f'(u)<0$ on $({\lambda},a]$, 2. $g\in C^1(\overline{\mathbb {R}^+})$, $g(0)=0$, and $g'(u)>0$ for $u\ge0$, and investigate the effect of small conversion rate on the positive steady states of system . There are several examples of $f$ and $g$ satisfying assumptions $(\mathbf{A_1})$ and $(\mathbf{A_2})$. For example, 1. Richards growth rate for prey and Holling type-II functional response: $$f(u)={\displaystyle}{\frac}{\gamma(1+mu)(a-u^p)}{b}\;\;\text{and}\;\;g(u)={\displaystyle}{\frac}{bu}{1+mu},$$ where $a,b,m,\gamma>0$ and $p\ge1$; 2. weak Allee effect in prey and Holling type-II functional response: $$f(u)={\displaystyle}{\frac}{\gamma(1+mu)(a-u)(u+p)}{b}\;\;\text{and}\;\;g(u)={\displaystyle}{\frac}{bu}{1+mu},$$ where $a,b,p,m,\gamma>0$ and $a>p$; 3. logistic growth rate for prey and Ivlve type functional response: $$f(u)=\begin{cases} {\displaystyle}{\frac}{\gamma u(a-u)}{\alpha\left(1-e^{-\beta u}\right)}, & \text{ for } u\ne 0,\\ {\displaystyle}{\frac}{a\gamma}{\alpha\beta}, & \text{ for }u=0, \end{cases}\;\;\text{and}\;\;g(u)=\alpha\left(1-e^{-\beta u}\right),$$ where $a,\alpha,\beta,\gamma>0$. Then, we consider the case that $f$ and $g$ satisfy the following assumptions: 1. $f\in C^1(\overline{\mathbb {R}^+})$ and there exists a unique $a>0$ such that $f(u)$ is positive for $u\in[0,a)$ and negative for $u>a$, 2. $g\in C^2(\overline{\mathbb {R}^+})$, $g(0)=0$, $g'(0)>0$ and $g(u)>0$ for $u>0$, and investigate the effect of large conversion rate on the positive steady states of system . Here we remark that $(\mathbf{A'_1})$ is more general that $(\mathbf{A_1})$, $g$ may be nonmonotonic for $(\mathbf{A'_2})$, and the assumption that $g\in C^2$ is needed to guarantee the regularity of the positive steady states. There are also several examples of $f$ and $g$ satisfying assumptions $(\mathbf{A'_1})$ and $(\mathbf{A'_2})$. For example, 1. logistic growth rate for prey and Holling type-IV functional response: $$f(u)={\displaystyle}{\frac}{\gamma(1+nu+mu^2)(a-u)}{b}\;\;\text{and}\;\;g(u)={\displaystyle}{\frac}{bu}{1+nu+m u^2},$$ where $a,b,m,n,\gamma>0$. The rest of the paper is organized as follows. In Section 2, we show the global attractivity of constant equilibria of system when the conversion rate is small, which also implies the non-existence of non-constant positive steady states. In Section 3, we prove the non-existence of non-constant positive steady states of system when the conversion rate is large. In Section 4, we apply the obtained theoretical results to some concrete examples. The case of small conversion rate ================================= In this section, we investigate the positive steady states of system when conversion rate $c$ is small. Throughout this section, we assume that $h$ satisfies the following assumption 1. $h\in C^1(\overline{\mathbb {R}^+})$, and there exists a unique $d>0$ such that $h(v)$ is positive for $v\in[0,d)$, and negative for $v>d$. Moreover, $h'(v)< 0$ for $v\ge d$. Then $h(v)$ has a inverse function, denoted by $h^{-1}$, when $v\in[d,\infty)$. We first recall the following well-known result for later application. \[L2.0\] Assume that $H:(0,\infty)\to {{\mathbb R}}$ is a smooth function satisfying $H(w)(w-w_0)<0$ for any $w>0$ and $w\ne w_0$. If $w(x,t)$ satisfies the following problem $$\label{aux} \begin{cases} \displaystyle \frac{\partial w}{\partial t}=d\Delta w+H(w), & \;\; x\in\Omega,\; t>t_0,\\ \displaystyle \frac{\partial w(x,t)}{\partial \nu}=0, & \;\; x\in \partial\Omega,\; t>t_0,\\ \displaystyle w(x,t_0)\ge(\not\equiv)0, & \;\; x\in {\Omega}, \end{cases}$$ where $d>0$, $t_0\in {{\mathbb R}}^+$, then $w(x,t)$ exists for all $t>t_0$, and $w(x,t)\rightarrow w_0$ uniformly for $x\in \overline \Omega$ as $t\rightarrow \infty$. From Lemma \[L2.0\], we give the exact asymptotic bounds of the solutions for system . \[L2.1\] Assume that $f$, $g$ and $h$ satisfy assumptions $(\mathbf{A_1})$, $(\mathbf{A_2})$ and $(\mathbf{A_3})$. If $h(f(0))<-cg(a)$, then there exist $(\underline u,\underline v),\;(\overline u,\overline v)>(0,0)$ satisfy $$\label{re} \begin{split} &f(\overline u)-\underline v\le0 ,\;\;h(\overline v)+cg(\overline u)\le0, \\ &f(\underline u)-\overline v\ge0 ,\;\;h(\underline v)+cg(\underline u)\ge0. \\ \end{split}$$ Moreover, for any initial value $\phi=(u_0(x),v_0(x))$, where $u_0(x)\ge(\not\equiv)0$, $v_0(x)\ge(\not\equiv)0$ for all $x\in {\overline{\Omega}}$, there exists $t_0(\phi)>0$ such that the corresponding solution $(u(x,t),v(x,t))$ of system satisfies $$\label{re2} (\underline u,\underline v)\le(u(x,t),v(x,t))\le(\overline u,\overline v)$$ for any $t>t_0(\phi)$. Since $h(f(0))<-cg(a)$, if follows from assumption $(\mathbf{A_3})$ that there exists $\epsilon>0$ such that $h^{-1}\left(-cg(a+\epsilon)\right)>d$ exists and $$\label{fg1} \epsilon<d,\;\;f(0)-\left[h^{-1}\left(-cg(a+\epsilon)\right)+\epsilon\right]>0,$$ where $h^{-1}$ is the inverse function of $h$. It follows from $${\displaystyle}\frac{\partial u}{\partial t}\le d_1\Delta u+g(u)f(u)$$ and Lemma \[L2.0\] that, for any initial value $\phi$, there exists $t_1(\phi)>0$ such that $u(x,t)\le a+\epsilon$ for $t>t_1(\phi)$, where $a>0$ is the unique zero of $f(u)$. Since $${\displaystyle}\frac{\partial v}{\partial t}\ge d_2\Delta v+vh(v),$$ there exists $t_2(\phi)>t_1(\phi)$ such that $v(x,t)\ge d-\epsilon>0$ for $t>t_2(\phi)$. Consequently, we have $${\displaystyle}\frac{\partial v}{\partial t}\le d_2\Delta v+v\left[h(v)+cg(a+\epsilon)\right]\;\;\text{for}\;\;t>t_2(\phi),$$ and then there exists $t_3(\phi)>t_2(\phi)$ such that $v(x,t)\le h^{-1}\left(-cg(a+\epsilon)\right)+\epsilon$ for $t>t_3(\phi)$. Since $f,g$ satisfy $(\mathbf{A_1})$, $(\mathbf{A_2})$ and Eq. , there exists $\overline a \in(0,a)$, depending on $c$, such that $$f(u)-\left[h^{-1}\left(-cg(a+\epsilon)\right)+\epsilon\right]>0 \text{ for }u\in[0,\overline a],$$ and $$f(u)-\left[h^{-1}\left(-cg(a+\epsilon)\right)+\epsilon\right] <0\text{ for }u\in(\overline a, a].$$ It follows from Lemma \[L2.0\] that there exists $t_4(\phi)>t_3(\phi)$ such that $u(x,t)\ge {\displaystyle}{\frac}{\overline a}{2}>0$ for $t>t_4(\phi)$. Choose $$\label{re3} \begin{split} &\overline u=a+\epsilon ,\;\;\underline u={\displaystyle}{\frac}{\overline a}{2}, \\ &\underline v= d-\epsilon,\;\;\overline v=h^{-1}\left(-cg(a+\epsilon)\right)+\epsilon. \\ \end{split}$$ Then $(\underline u,\underline v)$ and $(\overline u,\overline v)$ satisfy Eq. , and there exists $t_0(\phi)>t_4(\phi)$ such that the corresponding solution $(u(x,t),v(x,t))$ of system satisfies Eq. for any $t>t_0(\phi)$. Then, through the upper and lower solution method [@27; @pao; @4; @5], we have the following results on the global attractivity of the positive equilirium. \[T2.2\] Assume that $f$, $g$ and $h$ satisfy assumptions $(\mathbf{A_1})$, $(\mathbf{A_2})$ and $(\mathbf{A_3})$. If $f(0)>d$, then there exists $c_0>0$, depending on $f$, $g$ and $h$, such that system has a unique constant positive steady state solution $(u_*,v_*)$ for any $c\in (0,c_0)$. Furthermore, for any initial value $\phi=(u_0(x),v_0(x))$, where $u_0(x)\ge(\not\equiv)0$, $v_0(x)\ge(\not\equiv)0$ for all $x\in {\overline{\Omega}}$, the corresponding solution $(u(x,t),v(x,t))$ of system converges uniformly to $(u_*,v_*)$ as $t\rightarrow\infty$. Since $f(0)>d$ (or equivalently, $h(f(0))<0$), there exists $c_1>0$ such that $h(f(0))<-cg(a)$ for $c\in(0, c_1]$. In the following, we always assume that $c<c_1$. From Lemma \[L2.1\], we see that there exist $(\underline u,\underline v),\;(\overline u,\overline v)>(0,0)$, which is a pair of coupled upper and lower solution of system . Since $f$ and $g$ are smooth, there exists $K>0$ such that, for any $(u_1,v_1)$ and $(u_2,v_2)$ satisfying $(\underline u,\underline v)\le(u_1,v_1),(u_2,v_2)\le(\overline u,\overline v)$, we have $$\begin{array}{l} \left|g(u_1)\left(f(u_1)-v_1\right)-g(u_2)\left(f(u_2)-v_2\right)\right|\le K(|u_1-u_2|+|v_1-v_2|),\\ \left|v_1h(v_1)+cg(u_1)v_1-v_2h(v_2)-cg(u_2)v_2\right|\le K(|u_1-u_2|+|v_1-v_2|). \end{array}$$ Define two iteration sequences $(\overline u^{(m)},\overline v^{(m)})$ and $(\underline u^{(m)},\underline v^{(m)})$ as follows: for $m\ge 0$, $$\begin{array}{l} \overline u^{(m+1)}=\overline u^{(m)}+{\displaystyle}\frac{g\left(\overline u^{(m)}\right)}{K}\left[f(\overline u^{(m)})-\underline v^{(m)}\right],\\ \underline u^{(m+1)}=\underline u^{(m)}+{\displaystyle}\frac{g\left(\underline u^{(m)}\right)}{K}\left[f(\underline u^{(m)})-\overline v^{(m)}\right],\\ \overline v^{(m+1)}=\overline v^{(m)}+{\displaystyle}\frac{\overline v^{(m)}}{K}\left[h\left(\overline v^{(m)}\right)+cg\left(\overline u^{(m)}\right)\right],\\ \underline v^{(m+1)}=\underline v^{(m)}+{\displaystyle}\frac{\underline v^{(m)}}{K}\left[h\left(\underline v^{(m)}\right)+cg\left(\underline u^{(m)}\right)\right],\\ \end{array}$$ where $(\overline u^{(0)}, \overline v^{(0)})=(\overline u,\overline v)$ and $(\underline u^{(0)}, \underline v^{(0)})=(\underline u,\underline v)$. Then there exist $(\tilde u,\tilde v)$ and $(\check u, \check v)$ such that $(\underline u,\underline v)\le(\check u, \check v)\le (\tilde u,\tilde v)\le(\overline u,\overline v)$, $\lim_{m \rightarrow \infty}\overline u^{(m)}=\tilde u$, $\lim_{m \rightarrow \infty}\overline v^{(m)}=\tilde v$, $\lim_{m \rightarrow \infty}\underline u^{(m)}=\check u$, $\lim_{m \rightarrow \infty}\underline v^{(m)}=\check v$, and $$\label{2.5} \begin{split} &0=f(\tilde u)-\check v,~0=h(\tilde v)+cg(\tilde u), \\ &0= f(\check u)-\tilde v,~0=h(\check v)+cg(\check u). \end{split}$$ It follows from Eq. that $$\label{fgqu} h\left(f(\tilde u)\right)-c g(\tilde u)=h\left(f(\check u)\right)-c g(\check u).$$ The following proof is given in two cases.\ Case I. If $f'(u)<0$ on $(0,a]$, where $a$ is the unique zero of $f(u)$, then it follows from Eq. and assumptions $(\mathbf{A_1})$-$(\mathbf{A_3})$ that $$\label{inc} f(\check u)=\tilde v=h^{-1}\left(-cg(\tilde u)\right)<h^{-1}\left(-c_1g(a)\right)<f(0)\;\;\text{for any}\;\;c\in(0,c_1).$$ Then there exists $\overline \lambda\in(0,a)$ such that $f(\overline \lambda)=h^{-1}\left(-c_1g(a)\right)$ and $\check u\ge \overline {\lambda}$ for any $c\in (0,c_1)$. Hence $$0=h(f(\tilde u))-c g(\tilde u)-h(f(\check u))+c g(\check u)\ge \left[\min_{u\in[\overline {\lambda},a]}\left[h\left(f(u)\right)\right]'-c\max_{u\in[\overline {\lambda},a]}g'(u)\right](\tilde u-\check u)$$ for any $c\in(0,c_1)$. Since $\min_{u\in[\overline {\lambda},a]}\left[h\left(f(u)\right)\right]'>0$, there exists $c_0\in(0,c_1)$, depending on $f$, $g$ and $h$, such that $\tilde u=\check u$ for any $c\in(0,c_0)$.\ Case II. If there exists ${\lambda}\in(0,a)$ such that $f'(u)>0$ on $(0,{\lambda})$ and $f'(u)<0$ on $({\lambda},a]$, then there exist $\tilde \lambda\in({\lambda},a)$ such that $f(\tilde {\lambda})=f(0)$. It follows from Eq. that $\check u \ge \tilde {\lambda}$ for any $c\in(0,c_1)$. Hence $$0=h(f(\tilde u))-c g(\tilde u)-h(f(\check u))+c g(\check u)\ge \left[\min_{u\in[\tilde {\lambda},a]}\left[h\left(f(u)\right)\right]'-c\max_{u\in[\tilde {\lambda},a]}g'(u)\right](\tilde u-\check u)$$ for any $c\in(0,c_1)$. Since $\min_{u\in[\tilde {\lambda},a]}\left[h\left(f(u)\right)\right]'>0$, there exists $c_0\in(0,c_1)$, depending on $f$, $g$ and $h$, such that $\tilde u=\check u$ for $0<c<c_0$. It follows from the upper and lower solution method [@27; @pao; @4; @5] that, for any $c\in(0,c_0)$, system has a unique constant positive equilibrium $(u_*,v_*)$, which is globally attractive. From Lemma \[L2.0\] and Theorem \[T2.2\], we have the following results on the global attractivity of constant equilibria. \[cc25\] Assume that $f$, $g$ and $h$ satisfy assumptions $(\mathbf{A_1})$, $(\mathbf{A_2})$ and $(\mathbf{A_3})$. Then the following two statements are true. 1. If $f(0)>d$ (or equivalently, $h(f(0))<0$), then there exists $c_0>0$, depending on $f$, $g$ and $h$, such that, for any $c\in(0,c_0)$, system has a unique constant positive steady state, which is globally attractive. Hence system has no non-constant positive steady states for $c\in(0,c_0)$ if $f(0)>d$. 2. If $\max_{u\in[0,a]}f(u)<d$ (or equivalently $h(\max_{u\in[0,a]}f(u))>0$), then the steady state $(0,d)$ of system is globally attractive. Hence system has no positive steady states if $\max_{u\in[0,a]}f(u)<d$. If $f(0)>d$, then it follows from Theorem \[T2.2\] that there exists $c_0>0$ such that, for $c\in(0,c_0)$, system has a unique constant positive steady state, which is globally attractive. Hence system has no non-constant positive steady states for $c\in(0,c_0)$. Then, we consider the case that $\max_{u\in[0,a]}f(u)<d$. Since $${\displaystyle}\frac{\partial v}{\partial t}\ge d_2\Delta v+vh(v),$$ it follows from Lemma \[L2.0\] that for any initial value and $$\epsilon\in\left(0,{\displaystyle}{\frac}{1}{2}\left(d-\max_{u\in[0,a]}f(u)\right)\right),$$ there exists $t_2>0$ such that $v(x,t)\ge d-\epsilon>0$ for $t>t_2$, and consequently, $${\displaystyle}\frac{\partial u}{\partial t}\le d_1\Delta u+g(u)\left(f(u)-v\right)\le d_1\Delta u+g(u)\left(\max_{u\in[0,a]}f(u)-d+\epsilon\right)$$for $t>t_2$, which implies that $u(x,t)$ converges uniformly to $0$ as $t\to\infty$. Therefore, the steady state $(0,d)$ of system is globally attractive. We remark that the above results do not mention the case that $$h\left(\max_{u\in[0,a]}f(u)\right)<0<h(f(0)),$$ that is, $$\max_{u\in[0,a]}f(u)>d>f(0).$$ For this case, the dynamics is complex even when $c=0$. The steady states of Eq. for $c=0$ satisfy $$\label{steady1.4} \begin{cases} -d_1\Delta u=g(u)\left(f(u)-v\right), & x\in \Omega,\\ -d_2\Delta v=vh(v), & x\in\Omega, \\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega,\\ \end{cases}$$ and $(u(x),d)$ is a positive solution of Eq. if and only if $u(x)$ is a positive solution of the following equation $$\label{allee} \begin{cases} -d_1\Delta u=g(u)\left(f(u)-d\right),& x\in\Omega, \\ \partial_\nu u=0,& x\in \partial \Omega.\\ \end{cases}$$ Eq. may exhibit a strong Allee effect, which has a stable constant positive equilibrium and an unstable constant positive equilibrium, see [@dsams; @Wangws2; @wangsjmb] for related work on strong Allee effect. The case of large conversion rate ================================= In this section, we prove the non-existence of non-constant positive steady states of system when conversion rate $c$ is large, and the method used here is motivated by [@Peng-Shi]. Throughout this section, we assume that $f$ and $g$ satisfy $(\mathbf{A'_1})$ and $(\mathbf {A'_2})$. Define $$\label{pu} q(u)=\begin{cases} {\displaystyle}{\frac}{g(u)}{u}, & \text{ if } u>0,\\ g'(0), & \text{ if }u=0, \end{cases}$$ and then $q(u)\in C^1(\overline{\mathbb {R}^+})$. Let $w=cu$, $\rho={\displaystyle}{\frac}{1}{c}$, and then $w$ and $v$ satisfy $$\label{steadywz1.4} \begin{cases} -d_1\Delta w=wq(\rho w)\left[f(\rho w)-v\right], & x\in \Omega,\\ -d_2\Delta v=v\left[h(v) +q(\rho w)w\right], & x\in\Omega, \\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega.\\ \end{cases}$$ Hence the existence/non-existence of positive steady states of system for large $c$ is equivalent to that of solutions of system for small $\rho$. In the following, we first cite the maximum principle for weak solutions from [@Lieberman; @Lou-Ni; @Peng-Shi-Wang2] and the Harnack inequality for weak solutions from [@Lin-Ni; @Peng-Shi; @Peng-Shi-Wang2] for later application. \[maxin\] Assume that $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^N$, and $g\in C(\overline\Omega\times \mathbb R)$. If $z\in H^{1}(\Omega)$ is a weak solution of the inequalities $$\begin{cases} \Delta z+g(x,z)\ge0, & x\in \Omega,\\ \partial_\nu z\le0,& x\in \partial \Omega,\\ \end{cases}$$ and there exists a constant $K$ such that $g(x, z) < 0 $ for $z > K$, then $$z \le K \text{ a.e. in } \Omega.$$ \[harnack\] Assume that $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^N$, and $c(x)\in L^q(\Omega)$ for some $q > N/2$. If $z \in H^{1}(\Omega)$ is a non-negative weak solution of the following problem $$\begin{cases} \Delta u +c(x) u=0, & x\in \Omega,\\ \partial_\nu u=0,& x\in \partial \Omega,\\ \end{cases}$$ then there is a positive constant $C$, which is determined only by $\|c(x)\|_q$, $q$ and $\Omega$, such that $$\sup_{x\in\Omega} u\le C\inf_{x\in\Omega} u.$$ Then, we consider the positive solutions of system when $\rho=0$. \[rho0\] Assume that $f$ and $g$ satisfy assumptions $(\mathbf{A'_1})$ and $(\mathbf{A'_2})$, $h$ satisfies 1. $\left[h(v)-h\left(f(0)\right)\right]\left(v-f(0)\right)<0$ for any $v>0(v\ne f(0))$, and $h\left(f(0)\right)<0$. Then, for $\rho=0$, system has a unique positive steady state $\left(-{\frac}{h(f(0))}{g'(0)},f(0)\right)$. Since $h(f(0))<0$, system has a constant positive steady state $(w_*,v_*)=\left(-{\frac}{h(f(0))}{g'(0)},f(0)\right)$ for $\rho=0$. We construct the following function $$\begin{aligned} G(w,v)&:=&\int_{\Omega}\left\{{\displaystyle}{\frac}{w-w_*}{w}\left[d_1\Delta w+wg'(0)(f(0)- v)\right]\right \}dx\\ &+&\int_{\Omega}\left\{{\displaystyle}{\frac}{v-v_*}{v}\left[d_2\Delta v+v\left(h(v)+ g'(0)w\right)\right]\right\}dx\\ &=&-\int_{\Omega}\left[d_1{\displaystyle}{\frac}{w_*|\nabla w|^2}{w^2}+d_2{\displaystyle}{\frac}{v_*|\nabla v|^2}{v^2}\right]dx\\ &+&\int_{\Omega}(v-v_*)[h(v)-h(v_*)]dx.\end{aligned}$$ Therefore, if $(w(x),v(x))$ is a positive solution of system for $\rho=0$, then $$G(w(x),v(x))=0.$$ Since $h$ satisfies $(\mathbf{A_4})$, it follows that $(w(x),v(x))\equiv(w_*,v_*)$. Based on Lemmas \[maxin\] and \[harnack\], we will give a *priori* estimates for positive solutions of system under the following assumption $(\mathbf{A_5})$. 1. $h\in C^1(\overline{\mathbb {R}^+})$ and there exist $n\in \mathbb{N}^+$, $\{q_i\}_{i=0}^{n}$, $\{k_i\}_{i=0}^n$ and $\{\overline k_i\}_{i=0}^n$ such that $$\sum_{i=0}^n k_i v^{q_i}\le -h(v)\le \sum_{i=0}^n \overline k_i v^{q_i}\;\;\text{for any}\;\;v\ge0,$$ where $0= q_0<q_1<q_2<\dots< q_n$, $q_n>{\displaystyle}{\frac}{1}{2}$ and $k_n,\overline k_n>0$. The above mentioned assumptions $(\mathbf{A_4})$ and $(\mathbf{A_5})$ are not strong, and we will remark that many common used growth rate per capita functions satisfy $(\mathbf{A_4})$ and $(\mathbf{A_5})$ at the end of this section. \[imesti\] Assume that $f$, $g$ and $h$ satisfy assumptions $(\mathbf{A'_1})$, $(\mathbf{A'_2})$ and $(\mathbf{A_5})$, $\Omega$ is a bounded domain in $\mathbb{R}^N(N\le3)$ with a smooth boundary $\partial\Omega$, and $(w_{\rho},v_{\rho})$ is a positive solution of system . Denote $$\label{pu2} v_0=\begin{cases} \max\{v\ge0:h(v)=0\}, & \text{ if } \{v\ge0:h(v)=0\}\ne\emptyset,\\ 0, & \text{ if }\{v\ge0:h(v)=0\}=\emptyset. \end{cases}$$ Then the following two statements are true. 1. There exists $\overline C>0$ such that $\sup_{x\in\Omega} w_{\rho},\sup_{x\in\Omega}v_{\rho}<\overline C$ for all $\rho>0$. 2. If $f(0)>v_0$, then there exists $M>0$ such that $$\inf_{0\le \rho\le M}\inf_{x\in \Omega}w_{\rho}>0\;\;\text{and}\;\; \inf_{0\le \rho\le M}\inf_{x\in\Omega}v_{\rho}>0.$$ Since $h$ satisfies assumption $(\mathbf{A_5})$, we see that $\lim_{v\to\infty}h(v)=-\infty$, which implies that $v_0$ is well defined. From Eq. , we obtain $$\label{eswv} \begin{split} &-\int_{\Omega}v_{\rho}h(v_\rho)dx=\int_{\Omega}w_{\rho}q\left(\rho w_{\rho}\right)f\left(\rho w_{\rho}\right)dx,\\ &\int_{\Omega}\left[h(v_\rho)+w_\rho q(\rho w_\rho)\right]dx=-d_2\int_{\Omega}{\displaystyle}{\frac}{|\nabla v_{\rho}|^2}{v_{\rho}^2}dx\le0. \end{split}$$ Since $\rho w_\rho$ satisfies $$-d_1\rho\Delta w_{\rho}\le\rho w_{\rho} f(\rho w_\rho)g(\rho w_\rho),$$ it follows from Lemma \[maxin\] (see also [@Lieberman; @Lou-Ni; @Peng-Shi-Wang2]) that $\rho w_\rho\le a$, where $a>0$ is the unique zero of $f(u)$. Noticing that $$\max_{u\in[0,a]}f(u),\max_{u\in[0,a]}q(u),\min_{u\in[0,a]}q(u)>0,$$ from Eq. , we have $$\label{hv} -\int_{\Omega}v_{\rho}h(v_\rho)dx\le -{\displaystyle}{\frac}{\max_{u\in[0,a]}f(u)\max_{u\in[0,a]}q(u)}{\min_{u\in[0,a]}q(u)}\int_{\Omega}h(v_\rho)dx.$$ This relation and assumption $(\mathbf{A_5})$ imply $$\label{isum} \begin{split} -\int_{\Omega}h(v_\rho)dx\le& \sum_{i=0}^{n}\overline \sigma_i\|v_\rho\|^{q_i}_{q_n+1},\\ k_n \|v_\rho\|_{q_n+1}^{q_n+1}\le& \sum_{i=0}^{n-1}\sigma_i\|v_\rho\|^{q_i+1}_{q_n+1}+\sum_{i=0}^{n}\overline \sigma_i\|v_\rho\|^{q_i}_{q_n+1}, \end{split}$$ where $\{\sigma_i\}_{i=0}^{n-1}$ and $\{\overline\sigma_i\}_{i=0}^n$ are positive and depend only on $f$, $g$, $\{q_i\}_{i=0}^n$, $\{k_i\}_{i=0}^n$, $\{\overline k_i\}_{i=0}^n$ and $\Omega$. Therefore, there exists a constant $C_1>0$ such that $\|v_\rho\|_{q_n+1}\le C_1$ for all $\rho\ge0$, which implies $$\|q(\rho w)\left[f(\rho w)-v\right]\|_{q_n+1}\le\max_{u\in[0,a]}q(u)\left(\max_{u\in[0,a]}f(u)|\Omega|^{{\frac}{1}{q_n+1}}+C_1\right).$$ Since $N\le 3$ and $q_n+1>{\displaystyle}{\frac}{3}{2}\ge{\displaystyle}{\frac}{N}{2}$, from Lemma \[harnack\] we see that there exists $C_2>0$ such that $$\label{wharnack}\sup_{x\in\Omega} w_\rho \le C_2\inf_{x\in\Omega}w_\rho \;\;\text{for all}\;\;\rho\ge0.$$ It follows from Eqs. and that $$\label{infw} \inf_{x\in\Omega}w_{\rho}<{\displaystyle}{\frac}{\sum_{i=0}^{n}\overline \sigma_iC_1^{q_i}}{\min_{u\in[0,a]}q(u)|\Omega|}\;\;\text{for all}\;\;\rho\ge0.$$ Therefore, from Eq. and , we see that there exists a constant $C_3>0$ such that $$\label{wupper} \sup_{x\in\Omega}w_\rho\le C_3\;\;\text{for all}\;\;\rho\ge0.$$ Consequently, $$\label{vupper} -d_2\Delta v_\rho=v_\rho\left[h(v_\rho) +q(\rho w_\rho)w_\rho\right]\le v_\rho\left[h(v_\rho)+\max_{u\in[0,a]}q(u)C_3\right].$$ It follows from assumption $(\mathbf{A_5})$ that $\lim_{v\to\infty}h(v)=-\infty$, which implies that there exists $C_4>0$ such that $$\label{C4} h(v)+\max_{u\in[0,a]}q(u)C_3<0\;\;\text{for}\;\; v>C_4.$$ Therefore, from Eqs. and and Lemma \[maxin\], we obtain $$\sup_{x\in\Omega} z_\rho\le C_4\;\;\text{for all}\;\;\rho\ge0.$$ Letting $\overline C=\max\{C_3,C_4\}$, we have $$\label{esup}\sup_{x\in\Omega} w_{\rho},\;\sup_{x\in\Omega}v_{\rho}<\overline C\;\;\text{for all}\;\;\rho\ge0.$$ In the following, we find the lower bound of $w_\rho$ and $v_\rho$. We first claim that there exists $M_1>0$ such that $$\label{inzz}\inf_{0\le \rho\le M_1}\inf_{x\in \Omega}w_{\rho}>0.$$ By way of contradiction, there exists a sequence $\{\rho_{j}\}_{j=1}^\infty$ such that $\lim_{j\to\infty}\rho_{j}=0$ and $\lim_{j\to\infty}\inf_{x\in\Omega} w_{\rho_{j}}=0$, which implies that $\lim_{j\to\infty}\sup_{x\in\Omega} w_{\rho_{j}}=0$ from Eq. . Then we only need to consider two cases.\ Case I. $\{v\ge0:h(v)=0\}=\emptyset$. Then $v_0=0<f(0)$. Noticing that $\lim_{v\to\infty}h(v)=-\infty$ from assumption $(A_5)$, we obtain $h(v)<0$ for $v\ge0$ and $\max_{v\ge0}h(v)<0$. Since $\lim_{j\to\infty}\sup_{x\in\Omega} w_{\rho_{j}}=0$, we have $$\int_{\Omega} v_{\rho_{j}}\left[-h(v_{\rho_{j}}) -q(\rho_j w_{\rho_{j}})w_{\rho_{j}}\right]dx\ge\int_\Omega v_{\rho_{j}}\left[-\max_{v\ge0}h(v)-q(\rho_j w_{\rho_{j}})w_{\rho_{j}}\right]>0$$ for sufficiently large $j$, which contradicts with the fact that $$\int_{\Omega} v_{\rho_{j}}\left[-h(v_{\rho_{j}}) -q(\rho_j w_{\rho_{j}})w_{\rho_{j}}\right]dx=0.$$ Case II. $\{v\ge0:h(v)=0\}\ne \emptyset$ and $v_0<f(0)$. Since $\lim_{j\to\infty}\sup_{x\in\Omega} w_{\rho_{j}}=0$, we see that, for any $\epsilon>0$, there exists $j_0(\epsilon)>0$ such that $\sup_{x\in\Omega}|w_{\rho_{j}}|\max_{u\in[0,a]} q(u)<\epsilon$ for any $j>j_0(\epsilon)$, which implies that $$-d\Delta v_{\rho_{j}}\le v_{\rho_{j}}\left[h\left(v_{\rho_{j}}\right)+\epsilon\right]\;\;\text{for}\;\;j>j_0(\epsilon).$$ This relation and Lemma \[maxin\] lead to $$v_{\rho_j}\le v_\epsilon:=\max\{v\ge0:h(v)+\epsilon=0\}\;\;\text{for}\;\;j>j_0(\epsilon).$$ It follows from $\lim_{\epsilon\to0}v_\epsilon=v_0<f(0)$ that $v_\epsilon <f(0)$ for sufficiently small $\epsilon$, and without loss of generality, we assume $v_\epsilon <f(0)$. Since $$\label{ppw} \begin{split} -d_1\rho_{j} \Delta w_{\rho_{j}}=&\rho_{j} w_{\rho_{j}} g\left(\rho_{j} w_{\rho_{j}}\right)\left[f\left( \rho_{j}w_{\rho_{j}}\right)-v_{\rho_{j}}\right]\\ \ge& \rho_{j} w_{\rho_{j}} g\left(\rho_{j} w_{\rho_{j}}\right)\left[f\left( \rho_{j}w_{\rho_{j}}\right)-v_\epsilon\right] \end{split}$$ for $j>j_0(\epsilon)$, it follows from Lemma \[maxin\] and $v_\epsilon <f(0)$ that there exists $w_0>0$ such that $\rho_{j}w_{\rho_{j}}\ge w_0>0$ for $j>j_0(\epsilon)$, which contradicts with the fact that $\lim_{j\to\infty}\rho_{j}=0$ and $\lim_{j\to\infty}\sup_{x\in\Omega} w_{\rho_{j}}=0$.\ Therefore, the claim is proved and Eq. hold. Then, we prove that there exists $M_2>0$ such that $$\label{inzz2}\inf_{0\le \rho\le M_2}\inf_{x\in \Omega}v_{\rho}>0.$$ Assuming the contrary, we see that there exists a sequence $\{\rho_i\}_{i=1}^\infty$ such that $\lim_{i\to\infty}\rho_{i}=0$ and $\lim_{i\to\infty}\inf_{x\in\Omega} v_{\rho_{i}}=0$. Since $$\|h(v_{\rho})+q(\rho w_\rho)w_\rho\|_{\infty}\le \max_{v\in{[0,\overline C]}}|h(v)|+\max_{u\in[0,a]}q(u)\overline C,$$ it follows from Lemma \[harnack\] that $\lim_{i\to\infty}\sup_{x\in\Omega} v_{\rho_{i}}=0$, and consequently, $$\lim_{i\to\infty}\sup_{x\in\Omega} \rho_{i}v_{\rho_{i}}=0.$$ Therefore, $$\int_{x\in\Omega}w_{\rho_{i}}q(\rho_i w_{\rho_{i}})\left[f(\rho_i w_{\rho_{i}})-v_{\rho_{i}}\right]dx>0$$ for sufficiently large $i$, which is a contradiction. Letting $M=\min\{M_1,M_2\}$, we have $$\inf_{0\le \rho\le M}\inf_{x\in \Omega}w_{\rho}>0\;\;\text{and}\;\; \inf_{0\le \rho\le M}\inf_{x\in\Omega}v_{\rho}>0.$$ This completes the proof. From Theorem \[imesti\], we obtain the non-existence of non-constant positive solutions of system for small $\rho$. \[mainr\] Assume that $f$, $g$ and $h$ satisfy assumptions $(\mathbf{A'_1})$, $(\mathbf{A'_2})$ and $(\mathbf{A_5})$, and $\Omega$ is a bounded domain in $\mathbb{R}^N(N\le3)$ with a smooth boundary $\partial\Omega$. Then the following two statements are true. 1. If $f(0)>v_0$ and $h$ satisfies $(\mathbf{A_4})$, where $v_0$ is defined as in Eq. , then there exists $c_0>0$, depending on $f$, $g$, $h$, $d_1$, $d_2$ and $\Omega$, such that system has a unique constant positive steady state and no non-constant positive steady states for any $c>c_0$. 2. If $f(0)<v_0$ and $h(v)>0$ for $v<v_0$, then there exists $c_0>0$, depending on $f$, $g$, $h$, $d_1$, $d_2$ and $\Omega$, such that system has no positive steady states for any $c>c_0$. Since $f(0)>v_0$, it follows from Eq. that $h(f(0))<0$. This relation and Lemma \[rho0\] imply that system has a unique positive solution $(w_*,v_*)=\left(-{\frac}{h(f(0))}{g'(0)},f(0)\right)$ for $\rho=0$. Since $h$ satisfies $(\mathbf{A_4})$, we have $h'(f(0))\le0$, which implies that $(w_*,v_*)$ is non-degenerate in the sense that zero is not the eigenvalue of the linearized problem with respect to $(w_*,v_*)$. Then it follows from the implicit function theorem that there exists $\rho_0>0$ such that system has a constant positive solution $(w_\rho,v_\rho)$ for $0<\rho<\rho_0$. Therefore, the existence is proved, and the uniqueness is proved in the following. By the implicit function theorem, we only need to show that if $(w^\rho,v^\rho)$ is a positive solution of system , then $$(w^\rho,v^\rho)\to (w_*,v_*)\;\;\text{in}\;\;C^{2}( \overline \Omega)\times C^{2}( \overline \Omega)\text{ as }\rho\to0.$$ It follows from Theorem \[imesti\] that $$\label{cstar} \underline C\le w^\rho,v^\rho\le\overline C \;\;\text{for}\;\;x\in\overline \Omega \text{ and } \rho\in[0,M],$$ where $M$ is defined as in Theorem \[imesti\] and $$\underline C=\min\{\inf_{0\le \rho\le M}\inf_{x\in \Omega}w^{\rho}, \inf_{0\le \rho\le M}\inf_{x\in \Omega}v^{\rho}\}>0.$$ This, combined with the $L^p$ theory, implies that $w^\rho$ and $v^\rho$ are bounded in $W^{2,p}(\Omega)$ for any $p>N$. It follows from the embedding theorem that $w^\rho$ and $v^\rho$ are precompact in $C^1(\overline\Omega)$. This implies that, for any sequence $\{\rho_i\}_{i=1}^\infty$ satisfying $\lim_{i\to\infty}\rho_i=0$, there exist a subsequence $\{\rho_{i_k}\}_{k=1}^\infty$ and $(w^*(x),v^*(x))\in C^1(\overline \Omega)\times C^1(\overline \Omega)$ such that $$(w^{\rho_{i_k}},v^{\rho_{i_k}})\to (w^*(x),v^*(x))\;\;\text{in}\;\;C^1(\overline \Omega)\times C^1(\overline \Omega)\text{ as }k\to\infty,$$ where $w^*(x)$ and $v^*(x)$ is positive from Eq. . Note that $$\label{spp} \begin{split} w^{\rho_{i_k}}= &[-d_1\Delta+I]^{-1}\left\{ w^{\rho_{i_k}}+w^{\rho_{i_k}}q(\rho_{i_k}w^{\rho_{i_k}})\left[f(\rho_{i_k}w^{\rho_{i_k}})-v^{\rho_{i_k}}\right]\right\},\\ v^{\rho_{i_k}}=&[-d_2\Delta+I]^{-1}\left\{ v^{\rho_{i_k}}+v^{\rho_{i_k}}\left[f(v^{\rho_{i_k}})+q(\rho_{i_k}w^{\rho_{i_k}})w^{\rho_{i_k}}\right]\right\}, \end{split}$$ and $\lim_{k\to\infty}\rho_{i_k}w^{\rho_{i_k}}=0$ in $C^1(\overline \Omega)$. Then, taking the limit of Eq. as $k\to\infty$ and by the Schauder theorem, we see that $(w^*(x),v^*(x))$ is a positive solution of system for $\rho=0$, and $$(w^{\rho_{i_k}},v^{\rho_{i_k}})\to (w^*(x),v^*(x))\;\;\text{in}\;\;C^{2}(\overline\Omega)\times C^{2}(\overline\Omega)\text{ as }k\to\infty.$$ This completes the proof of part $(1)$. Then, we prove the part $(2)$. It follows from $f(0)<v_0$ and Eq. that $v_0>0$ and $v_0=\max\{v>0:h(v)=0\}$. Then $$\label{inev} h(v)>0\;\;\text{for}\;\; v<v_0,\;\;\text{and}\;\;h(v)<0\;\;\text{for}\;\; v>v_0.$$ Assuming the contrary, we see that there exists a sequence $\{\rho_j\}_{j=1}^\infty$ such that $\lim_{j\to\infty} \rho_j=0$ and system has a positive solution $(w_{\rho_j},v_{\rho_j})$ for $\rho=\rho_j$. Since $$-d_2\Delta v_{\rho_j}=v_{\rho_j}\left[h(v_{\rho_j}) +q({\rho_j} w_{\rho_j})w_{\rho_j}\right]\ge v_{\rho_j} h(v_{\rho_j}),$$ it follows from that $v_{\rho_j}\ge v_0$, which implies that $$-d_1\Delta w_{\rho_j}\le w_{\rho_j}q(\rho_jw_{\rho_j})\left[f(\rho_jw_{\rho_j})-v_0\right].$$ From Eq. , we see that $\lim_{j\to\infty}\rho_{j}w_{\rho_{j}}=0$ uniformly on $\overline\Omega$. This, together with the fact that $f(0)<v_0$, imply that there exists $j_0>0$ such that $f(\rho_j w_{\rho_j})-v_0<0$ for any $j>j_0$. Hence $w_{\rho_j}\le0$ for $j>j_0$. This contradicts with the fact that $w_{\rho_j}$ is positive. \[rema\] We remark that assumptions $(\mathbf{A_4})$ and $(\mathbf{A_5})$ are not strong. Some examples of $h$ satisfying $(\mathbf{A_5})$ are 1. logistic:$$\label{log}h(v)=\beta (d-v),\;\;\beta,d>0,$$ 2. weak Allee effect: $$\label{wall}h(v)=\beta (d-v)(v+p),\;\;d>p>0,\beta>0,$$ 3. strong Allee effect: $$\label{sall} h(v)=\beta (d-v)(v-p),\;\;d,p,\beta>0$$ 4. strong Allee effect: $$\label{sall2} h(v)=\beta {\displaystyle}{\frac}{(d-v)(v-p)}{v+r},\;\;d,p,\beta,r>0.$$ Moreover, Eq. always satisfies $(\mathbf{A_4})$, Eq. satisfies $(\mathbf{A_4})$ for $f(0)>d-p$, Eq. satisfies $(\mathbf{A_4})$ for $f(0)>d+p$, and Eq. satisfies $(\mathbf{A_4})$ for $$f(0)>{\displaystyle}{\frac}{dp+dr+pr}{r}.$$ Applications ============ In this section, we apply the previously obtained results to some concrete predator-prey models. A predator-prey model with Holling-II functional response --------------------------------------------------------- In this subsection, we consider model , which is a predator-prey model with logistic growth rate for predator and Holling type-II functional response. Here $\Omega$ is a bounded domain in $\mathbb{R}^N(N\le3)$ with a smooth boundary $\partial\Omega$, parameters $a$, $b$, $e$, $m$, $d_1$ and $d_2$ are all positive constants, and $d$ may be positive constant, negative constant or zero. Letting $$\label{H1} f(u)={\displaystyle}{\frac}{(1+mu)(a-u)}{b},\;\;g(u)={\displaystyle}{\frac}{bu}{1+mu},\;\;h(v)=d-v,\;\;\text{and}\;\; c={\displaystyle}{\frac}{e}{b},$$ system is transformed to system , and parameter $c$ in system is equivalent to $e$ in system . In this case, $f$, $g$ and $h$ satisfy assumptions $(\mathbf{A'_1})$, $(\mathbf{A'_2})$, $(\mathbf{A_4})$ and $(\mathbf{A_5})$, and $$\label{pu22} v_0=\begin{cases} d, & \text{ if } d\ge0,\\ 0, & d<0, \end{cases}$$ where $v_0$ is defined as in Eq. . Moreover, $h$ satisfies $(\mathbf{A_3})$ if $d>0$. Then, from Theorem \[mainr\], we have the following results on the non-existence of non-constant steady states when conversion rate $e$ is large. This results supplements the result in [@Peng-Shi], which consider the case that $e$ equals to $b$ and is sufficiently large. \[Pen\] 1. If $a>bd$, then there exists $e_0>0$, depending on $a$, $b$, $d$, $m$, $d_1$, $d_2$ and $\Omega$, such that system has a unique constant positive steady state and no non-constant positive steady states for any $e>e_0$. 2. If $a<bd$, then there exists $e_0>0$, depending on $a$, $b$, $d$, $m$, $d_1$, $d_2$ and $\Omega$, such that system has no positive steady states for any $e>e_0$. From Corollary \[cc25\], we have the following results on the global attractivity of constant equilibria for small conversion rate, which also imply the non-existence of non-constant steady states. 1. If $d<0$, then there exists $e_0>0$, depending on $a$, $b$, $m$ and $d$, such that the steady state $(a,0)$ of system is globally attractive for $e\in(0,e_0)$. 2. If $0<d<{\displaystyle}{\frac}{a}{b}$, then there exists $e_0>0$, depending on $a$, $b$, $m$ and $d$, such that, for $e\in(0,e_0)$, system has a unique constant positive steady state, which is globally attractive. 3. If $d>{\frac}{(am+1)^2}{4mb}$, then the steady state $(0,d)$ of system is globally attractive. Similarly, using upper and lower solution method and modifying the arguments in Theorem \[T2.2\], we can prove the global attractivity of the constant positive equilibrium with respect to other parameters, for example, $a$ and $m$. \[p43\] If $d>0$, then there exists $a_0>0$, depending on $b$, $d$, $e$ and $m$, such that, for any $a>a_0$, system has a unique constant positive steady state, which is globally attractive. Letting $a_1:=b\left(d+{\displaystyle}{\frac}{e}{m}\right)$, we have $$h(f(0))=d-{\displaystyle}{\frac}{a}{b}<-cg(a)=-{\displaystyle}{\frac}{ea}{1+ma}$$for any $a>a_1$. This, together with $h(0)=d>0$, imply that the assumption of Lemma \[L2.1\] hold. Then there exist $(\underline u,\underline v),\;(\overline u,\overline v)>(0,0)$, which is a pair of coupled upper and lower solution of system for $a>a_1$. Using the similar arguments as Theorem \[T2.2\], there exist $(\tilde u,\tilde v)$ and $(\check u, \check v)$ such that $(\underline u,\underline v)\le(\check u, \check v)\le (\tilde u,\tilde v)\le(\overline u,\overline v)$, $\lim_{m \rightarrow \infty}\overline u^{(m)}=\tilde u$, $\lim_{m \rightarrow \infty}\overline v^{(m)}=\tilde v$, $\lim_{m \rightarrow \infty}\underline u^{(m)}=\check u$, $\lim_{m \rightarrow \infty}\underline v^{(m)}=\check v$, and Eqs. and hold. If $$a>a_2:=\max\left\{b\left(d+{\displaystyle}{\frac}{e}{m}\right),{\displaystyle}{\frac}{1}{m}\right\},$$ then $$f(\check u)=\tilde v<\left(d+{\displaystyle}{\frac}{e}{m}\right)<{\displaystyle}{\frac}{a}{b}=f(0),$$ which implies that $\tilde u>\check u>\tilde {\lambda}={\displaystyle}{\frac}{am-1}{m}>0$, where $f(\tilde {\lambda})=0$. It follows from Eq. that $$\begin{split} 0=&h(f(\tilde u))-c g(\tilde u)-h(f(\check u))+c g(\check u)\\ =&{\displaystyle}{\frac}{(1+m\check u)(a-\check u)}{b}-{\displaystyle}{\frac}{(1+m\tilde u)(a-\tilde u)}{b}+{\displaystyle}{\frac}{e\check u}{1+m\check u}-{\displaystyle}{\frac}{e\tilde u}{1+m\tilde u}\\ \ge&\left[ {\displaystyle}{\frac}{1}{b}\left(2m\tilde {\lambda}-am+1\right)-e\right](\tilde u-\check u)\\ \ge& \left[ {\displaystyle}{\frac}{1}{b}\left(ma-1\right)-e\right](\tilde u-\check u), \end{split}$$ which implies that $\tilde u=\check u$ for $a>a_3:={\displaystyle}{\frac}{be+1}{m}$. Therefore, for any $a>a_0=\max\{a_1,a_2,a_3 \}$, system has a unique constant positive steady state, which is globally attractive. Similarly, we have the global attractivity of the positive equilibrium with respect to saturation $m$. \[p44\] If If $0<d<{\displaystyle}{\frac}{a}{b}$, then there exists $m_0>0$, depending on $a$, $b$, $d$ and $e$, such that, for any $m>m_0$, system has a unique constant positive steady state, which is globally attractive. Propositions \[p43\] and \[p44\] supplement Theorem 2.3 in [@Du-Lou], which prove the global attractivity of the positive equilibrium for $ma\le1$. A predator-prey model with weak Allee effect in predator and Holling-II functional response ------------------------------------------------------------------------------------------- In this subsection, we consider the following predator-prey model $$\label{4weaka1} \begin{cases} {\displaystyle}\frac{\partial u}{\partial t}-d_1\Delta u= u(a-u)-{\displaystyle}\frac{buv}{1+mu}, & x\in \Omega,\; t>0,\\ {\displaystyle}\frac{\partial v}{\partial t}-d_2\Delta v= \beta v(d-v)(v+p)+{\displaystyle}\frac{euv}{1+mu}, & x\in\Omega,\; t>0,\\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega,\; t>0,\\ u(x,0)=u_0(x)\ge(\not\equiv)0, \;\; v(x,0)=v_0(x)\ge(\not\equiv)0,& x\in\Omega. \end{cases}$$ where $a$, $b$, $d$, $p$, $e$, $m$, $\beta$, $d_1$ and $d_2$ are positive constants, $d>p$, and $\Omega$ is a bounded domain in $\mathbb{R}^N(N\le3)$ with a smooth boundary $\partial\Omega$. Here $d>p$ means that the growth rate of predator is weak Allee type in the absence of prey. Letting $$\label{4H1} f(u)={\displaystyle}{\frac}{(1+mu)(a-u)}{b},\;\;g(u)={\displaystyle}{\frac}{bu}{1+mu},\;\;h(v)=\beta(d-v)(v+p),\;\;\text{and}\;\; c={\displaystyle}{\frac}{e}{b},$$ system is transformed to system , and parameter $c$ in system is equivalent to $e$ in system . In this case, $f$, $g$ and $h$ satisfy $(\mathbf{A_1})$, $(\mathbf{A_2})$, $(\mathbf{A'_2})$, $(\mathbf{A_3})$ and $(\mathbf{A_5})$, $v_0=d$, and $h(v)>0$ for $v<v_0$. It follows from Remak \[rema\] that $h(v)$ satisfies $(\mathbf{A_4})$ if ${\displaystyle}{\frac}{a}{b}=f(0)>d-p$. Then, from Theorem \[mainr\] we have the following results on the non-existence of non-constant steady states when conversion rate $e$ is large. 1. If $a>bd$, then there exists $e_0>0$, depending on $a$, $b$, $d$, $p$, $\beta$, $m$, $d_1$, $d_2$ and $\Omega$, such that system has a unique constant positive steady state and no non-constant positive steady states for $e>e_0$. 2. If $a<bd$, then there exists $e_0>0$, depending on $a$, $b$, $d$, $p$, $\beta$, $m$, $d_1$, $d_2$ and $\Omega$, such that system has no positive steady states for $e>e_0$. From Corollary \[cc25\], we have the following results on the global attractivity of constant equilibria for small conversion rate, which also imply the non-existence of non-constant steady states. \[wpp\] 1. If $d<{\displaystyle}{\frac}{a}{b}$, then there exists $e_0>0$, depending on $a$, $b$, $d$, $p$, $\beta$ and $m$, such that, for $e\in(0,e_0)$, system has a unique constant positive steady state, which is globally attractive. 2. If $d>{\frac}{(am+1)^2}{4mb}$, then the steady state $(0,d)$ of system is globally attractive. A predator-prey model with strong Allee effect in predator and Holling-IV functional response --------------------------------------------------------------------------------------------- In this subsection, we consider the following model, $$\label{4weaka} \begin{cases} {\displaystyle}\frac{\partial u}{\partial t}-d_1\Delta u= u(a-u)-{\displaystyle}\frac{buv}{1+nu+mu^2}, & x\in \Omega,\; t>0,\\ {\displaystyle}\frac{\partial v}{\partial t}-d_2\Delta v=\beta v(d-v)(v-p)+{\displaystyle}\frac{euv}{1+nu+mu^2}, & x\in\Omega,\; t>0,\\ \partial_\nu u=\partial_\nu v=0,& x\in \partial \Omega,\; t>0,\\ \end{cases}$$ which is a diffusive predator-prey model with strong Allee effect in predator. Here $a$, $b$, $d$, $p$, $e$, $m$, $n$, $\beta$, $d_1$ and $d_2$ are positive constants. Letting $$\label{4H2} \begin{split} &f(u)={\displaystyle}{\frac}{(1+nu+mu^2)(a-u)}{b},\;\;c={\displaystyle}{\frac}{e}{b},\\ &g(u)={\displaystyle}{\frac}{bu}{1+nu+mu^2},\;\;h(v)=\beta (d-v)(v-p). \end{split}$$ Then system is transformed to system , and parameter $c$ in system is equivalent to $e$ in system . In this case, $f$, $g$ and $h$ satisfy $(\mathbf{A'_1})$, $(\mathbf{A'_2})$ and $(\mathbf{A_5})$, and $v_0=\max\{d,p\}$. It follows from Remak \[rema\] that $h(v)$ satisfies assumptions $(\mathbf{A_4})$ if ${\displaystyle}{\frac}{a}{b}=f(0)>d+p$. Then from Theorem \[mainr\], we have the following results on the non-existence of non-constant steady states when conversion rate $e$ is large. If $a>b(d+p)$, then there exists $e_0>0$, depending on $a$, $b$, $d$, $p$, $m$, $\beta$, $d_1$, $d_2$ and $\Omega$, such that system has a unique constant positive steady state and no non-constant positive steady states for any $e>e_0$. Conclusions =========== In this paper, we consider a general diffusive predator-prey system. It covers a wide range of predator-prey models which include some well-known ones but also some less studied ones. We find that the conversion rate is a key parameter to affect the dynamics of a general predator-prey model, and there are almost no complex patterns for large and small conversion rate. Hence, this phenomenon can occur commonly for predator-prey models, which was found in [@Cheny] for a special model with a nonlinear growth rate for the predator. For the case of small conversion rate, we show the global attractivity of the unique constant positive steady state even when $h(v)$ is nonmonotonic, and hence it can be applied to model with weak Allee effect in predator. A special case where $h(v)$ is monotonic was analyzed in [@Cheny]. We remark that our result is not a direct conclusion from [@27; @pao; @4; @5], and needs some detailed analysis for the relations between the limits of upper solutions sequence and lower solutions sequence. Moreover, our result supplements some existing ones. For example, Propositions \[p43\] and \[p44\] supplement Theorem 2.3 in [@Du-Lou], which show the global attractivity of the positive equilibrium for $ma\le1$. For the case of large conversion rate, we show the nonexistence of the positive steady states. Our method is motivated by [@Peng-Shi], but we need to modify many of their arguments to derive our result. We find that, even with a nonmonotonic functional response, there exist no nonconstant positive steady states for large conversion rate. Moreover, our result in Theorem \[Pen\] also supplements Theorem 1.2 of [@Peng-Shi], which show the nonexistence in the case that $e$ equals to $b$ and is sufficiently large. 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--- author: - 'Robert P. Kirshner' title: Foundations of supernova cosmology --- Foundations of supernova cosmology ================================== Supernovae and the discovery of the expanding universe ------------------------------------------------------ Supernovae have been firmly woven into the fabric of cosmology from the very beginning of modern understanding of the expanding, and now accelerating universe. Today’s evidence for cosmic acceleration is just the perfection of a long quest that goes right back to the foundations of cosmology. In the legendary Curtis-Shapley debate on the nature of the nebulae, the bright novae that had been observed in nebulae suggested to Shapley (1921) (see Trimble, 1995) that the systems containing them must be nearby. Otherwise, he reasoned, they would have unheard-of luminosities, corresponding to M = -16 or more. Curtis (1921) countered concluding that “the dispersion of the novae in spirals and in our galaxy may reach ten magnitudes...a division into two classes is not impossible.” Curtis missed the opportunity to name the supernovae, but he saw that they must exist if the galaxies are distant. Once the distances to the nearby galaxies were firmly established by the observation of Cepheid variables (Hubble, 1925), the separation of ordinary novae and their extraordinary, and much more luminous super cousins, became clear. A physical explanation for the supernovae was attempted by Baade and Zwicky (1934). Their speculation that supernova energy comes from the collapse to a neutron star is often cited, and it is a prescient suggestion for the fate of massive stars, but not the correct explanation for the supernovae that Zwicky and Baade studied systematically in the 1930s. In fact, the spectra of all the supernovae that they discovered and followed up in those early investigations were of the distinct, but spectroscopically mysterious, hydrogen-free type that today we call SN Ia. They are not powered by core collapse, but by a thermonuclear flame. Baade (1938) showed that the luminosities of the supernovae in their program were more uniform than those of galactic novae, with a dispersion of their peak luminosities near 1.1 mag, making them suitable as extragalactic distance indicators. Right from the beginning, supernovae were thought of as tools for measuring the universe. Nature has more than one way to explode a star. This was revealed clearly by Minkowski (1941) who observed a distinct spectrum for some supernovae, different from those obtained for the objects studied by Baade. SN 1940B had strong hydrogen lines in its spectrum. These are the stars whose energy source we now attribute to core collapse in massive stars. At the time, it seemed sensible to call Baade’s original group Type I (SN I) and the new class Type II (SN II). The small dispersion in luminosity for Baade’s sample resulted from his good luck in having Zwicky discover a string of supernovae that were all of a single type. SN I are generally less luminous than the galaxies in which they occur. (Introductory texts, and introductory remarks in colloquia concerning supernovae usually get this basic fact wrong.) The SN II are, generally speaking, fainter than SN I and have a larger dispersion in their luminosity. Separating the supernovae, on the basis of their spectra, into distinct physical classes is one way they have become more precise as distance indicators. By the late sixties, Kowal (1968) was able to make a Hubble diagram for 19 SN I. The scatter about the Hubble line for this sample, which reached out to the Coma Cluster of galaxies at a redshift of  7000 km/s was about 0.6 magnitudes. These were photographic magnitudes, obtained with the non-linear detectors of the time, and they contained no correction for absorption by dust in the host galaxies, which we now know is an important source of scatter in the observed samples. But this was a promising step forward. {width="85.00000%"} \[\] In 1968, there was plenty of room for improvement in the precision of SN I measurements and in extending the redshift range over which they were studied. As Kowal forecast: “These supernovae could be exceedingly useful indicators of distance. It should be possible to obtain average supernova magnitudes to an accuracy of 5% to 10% in the distances.” He also predicted the future use of supernovae to determine cosmic acceleration: “It may even be possible to determine the second-order term in the redshift-magnitude relation when light curves become available for very distant supernovae.” The “second-order term” would be the one that indicated cosmic acceleration or deceleration. Along with the Hubble constant (which would require reliable distances from Cepheid variables), this deceleration term was expected to provide an account of cosmic kinematics, and, in the context of General Relativity, for the dynamics of the Universe, as sketched for astronomers in the classic paper by Sandage (1961). On the last page of this paper, Sandage worked out the observational consequences of the exponential expansion that would be produced by a cosmological constant. He explicitly shows that you cannot decide between an accelerating universe of this type and the steady-state model (they would both have q$_{0}$ = -1). Yet, in 1968, the measurement of deceleration was presented by Sandage (1968) as a decisive test between the steady-state model, which predicted acceleration, and Friedmann cosmologies where matter would produce deceleration. It is possible that, if cosmic acceleration had been discovered earlier, it might have been taken as evidence in favor of the steady-state model. It was the richer physical context of cosmological information, such as the cosmic microwave background, that led to a much different conclusion in 1998. ### Classifying supernovae In 1968, there was ample room for technical improvement in the measurements themselves, a need for a proper account for the effects of dust, and just as important, well into the 1980s the classification scheme for SN I was still incomplete. Core-collaspe supernovae were mixed in among the thermonuclear explosions that make up most of the Type I supernovae. As described by Zwicky (1965) and later by Oke and Searle (1974) the definition of a SN I was empirical: it meant that the spectrum resembled the bright supernova SN 1937C as extensively studied by Minkowski (1939). The bright supernova SN 1972E, observed with a new generation of spectrophotometric instruments by Kirshner et al. (1973a) in the infrared (Kirshner et al. 1973b) and at late times (Kirshner et al. 1975) provided a rich template for redefining the spectra of Type Ia supernovae. The distinctive feature in Type I supernova spectra is a broad and deep absorption observed at about 6150 Angstroms, attributed by Pskovskii (1968) to absorption by Si II. However, there were a handful of SN I, usually dubbed “peculiar” SN I, whose spectra resembled the other SN I in other respects, but which lacked this distinctive absorption line at maximum light. We now understand that this is not just a minor detail: the SN Ib (and their more extreme cousins, the SN Ic) are completely different physical events, ascribed to core-collapse in massive stars that have lost their hydrogen envelopes in late stages of stellar evolution (Branch and Doggett, 1985; Uomoto and Kirshner, 1985; Wheeler and Levreault, 1985; Wheeler and Harkness, 1990; Filippenko, 1997). The notation SN Ia was introduced to refer to the original class of supernovae, like SN 1937C and SN 1972E, that has no hydrogen or helium lines in the spectrum and the strong Si II feature. Once the SN Ib were distinguished from the SN Ia, the homogeneity of the SN Ia improved, with the scatter about the Hubble line decreasing to 0.65-0.36 mag, depending on which objects were selected and which photometric bands were used (Tammann and Leibundgut, 1990; Branch and Miller, 1990, 1993; Della Valle and Panagia, 1992). This work rested on the assumption that the SN Ia were identical, so that a single underlying template for the light curve (Leibundgut, 1988) could be used to interpolate between the observations of any individual object to determine its apparent brightness at maximum light in the B-band, and put all the objects on a common scale. #### SN II as cosmological distance indicators The idea that supernovae could be used to measure cosmological parameters had more than one component. Another line of work employed Type II supernovae. As pointed out by Kirshner and Kwan (1974), the expanding photospheres of these hydrogen-rich supernovae provide the possibility to measure distances without reference to any other astronomically determined distance. The idea of the Expanding Photosphere Method (EPM) is that the atmosphere was not too far from a blackbody, so the temperature could be determined from the observed energy distribution. If you measure the flux and temperature, that determines the angular size of the photosphere. Since you can measure the temperature and flux many times during the first weeks after the explosion, an observer can establish the angular expansion rate of a supernova. At the same time, absorption lines formed in the expanding atmosphere, from hydrogen and from weaker lines that more closely trace the expansion of the photosphere, give the expansion velocity. If you know the angular rate of expansion from the temperature and flux and the linear rate from the shape of the absorption lines, you can solve for the distance to a Type II supernova. The combination of the supernova’s redshift and distance allows for a measurement of the Hubble constant that does not depend on any other astronomically-determined distance. The departure of the energy distribution for a supernova atmosphere from a blackbody could be computed, as done by Schmidt, Eastman, and Kirshner (1992), and this held out the prospect of making more precise distance measurements to SN II than had been achieved for SN Ia. Wagoner (1977) noted that this approach could be extended to high redshift to measure the effects of cosmic deceleration, and also pointed out that the EPM provided an internal test of its own validity: if the distance determined remained the same, while the temperature and the velocity of the atmosphere changed, this was a powerful sign that the measurement was consistent. This was an important point, since the prospects for using galaxies as the principal tracer of cosmic expansion were dimming, due to evidence that the luminosity of a galaxy could easily change over time due to stellar evolution and galaxy mergers. Even sign of this change was not known for certain. Galaxies might grow brighter over time due to mergers, and they might grow dimmer due to stellar evolution. In either case, unless the effect was carefully calibrated, it could easily swamp the small changes in apparent magnitude with redshift that hold the information on the history of cosmic expansion. Supernovae, though fainter than galaxies, were discrete events that would not have the same set of changes over cosmic time. The use of SN II for cosmology has recently been revived and it promises to provide an independent path to measuring expansion and perhaps even acceleration (Poznanski et al., 2008). For SN II, the expanding photospheres provide a route to distances that can accommodate a range of intrinsic luminosities and still provide accurate distances, because the atmospheres have hydrogen and behave like those of other stars. For SN Ia, the atmospheres are more difficult to analyze, but the hope was simpler: that the physics underlying the explosion of a SN Ia would determine its luminosity. The idea that SN Ia were identical explosions has a theoretical underpinning. In the earliest pictures, the SN Ia were imagined to come from the ignition of a carbon-oxygen white dwarf at the Chandrasekhar mass (Hoyle and Fowler, 1960; Colgate and McKee, 1969). In models of this type, a supersonic shock wave travels through the star, burning it thoroughly into iron-peak isotopes, especially Ni$^{56}$. Such a standard explosion of a uniform mass would lead to a homogenous light curve and uniform luminosity, making SN Ia into perfect standard candles. The exponential light curves that suggest an energy input from radioactivity and the late-time spectra of SN Ia, which are made up of blended iron emission lines were broadly consistent with this picture. Though the simple theoretical idea that SN Ia are white dwarfs that ignite near the Chandrasekhar mass has been repeated many times as evidence that SN Ia must be perfect standard candles, nature disagrees. Observations show that there is a factor of three range in luminosity from the most luminous SN Ia (resembling SN 1991T) to the least luminous (resembling SN 1991-bg). Despite the facts, many popular (and professional!) accounts of SN Ia assert that SN Ia are standard candles because they explode when they reach the Chandrasekhar limit. This is wishful thinking. #### Searching for SN Ia for cosmology Nevertheless, the hope that SN Ia might prove to be good standard candles began to replace the idea that brightest cluster galaxies were the standard candles best suited to measuring the deceleration of the universe. As a coda to his pioneering automated supernova search, Stirling Colgate imagined the way in which a similar search with the Hubble Space Telescope might find distant supernovae (Colgate, 1979). A more sober analysis of the problem by Gustav Tammann estimated the sample size that would be needed to make a significant detection of deceleration using HST (Tammann, 1979). The result was encouraging: depending on the dispersion of the SN Ia, he found between 6 and 25 SN Ia at z$ \sim $0.5 would be needed to give a 3$\sigma$ signal of cosmic deceleration. Tammann got the quantities right– it was only the sign of the effect that was wrong. Unwilling to wait for the advent of the Hubble Space Telescope, a pioneering group from Denmark began a program of supernova observations using the Danish 1.5 meter telescope at ESO (Hansen, Jorgensen, and Norgaard-Nielsen, 1987; Hansen et al., 1989). Their goal was to find distant supernovae, measure their apparent magnitudes and redshifts, and, on the assumption the SN Ia were standard candles, fit for q$_{0}$ from the Hubble diagram. This method is described with precision in the chapter in this book by Pilar Ruiz-Lapuente. The difference between q$_{0}$ = 0.1 and q$_{0}$ of 0.5 is only 0.13 mag at redshift of 0.3. At the time they began their work, there was hope that the intrinsic scatter for SN Ia might be as small as 0.3 mag. To beat the errors down by root-N statistics to make a 3 sigma distinction would take dozens of well-observed supernovae at z $\sim$ 0.3. The Danish group used the search rhythm developed over the decades by Zwicky and his collaborators for finding supernovae. Since the time for a Type Ia supernova to rise to maximum and fall back by a factor of 2 is roughly one month, monthly observations in the dark of the moon are the best way to maximize discoveries. Observations made toward the beginning of each dark run were most useful, since that allowed time to follow up each discovery with spectroscopy and photometry. This is the pattern Zwicky established with the Palomar 18-inch Schmidt and which was used for many years by Sargent and Kowal with the 48-inch Schmidt at Palomar (Kowal, Sargent, and Zwicky, 1970). It is the pattern used by the Danish group, and all the subsequent supernova search teams until the introduction of dedicated searches like that of Kare et al. (1988) and the rolling search led by John Tonry (Barris et al., 2004) that became the model for the recent ESSENCE and SNLS searches. But there was something new in the Danish search. Photographic plates, which are large but non-linear in their response to light, were replaced by a Charge Coupled Device (CCD). The advantages were that the CCD was much more sensitive to light (by a factor of $\sim$ 100!) and that the digital images were both linear and immediately available for manipulation in a computer. Fresh data taken at the telescope could be processed in real time to search for new stars, presumably supernovae, in the images of galaxy clusters. The new image needed to be registered to a reference image taken earlier, the two images appropriately scaled to take account of variations in sky brightness, the better of the two images blurred to match the seeing of the inferior image, and then subtracted. The Danish team implemented these algorithms and demonstrated their success with SN 1998U, a SN Ia in a galaxy at redshift 0.31 (Norgaard-Nielsen et al., 1989). Although this group developed the methods for finding distant supernovae in digital data, the rate at which they were able to find supernovae was disappointingly low. Instead of making steady progress toward a cosmologically-significant sample at a rate of, say, one object per month, they only found one supernova per year. At this rate, it would take 10 years to beat down the measuring uncertainty and to begin to learn about the contents of the universe. And that was in the optimistic case where the intrinsic scatter of SN Ia was assumed to be small. Instead, the observational evidence was pointing in the opposite direction, of larger dispersion among the SN Ia. Another early effort, carried out by the Lawrence Berkeley Laboratory at the 4m Anglo-Australian Telescope had even less luck. Despite building a special-purpose prime focus CCD camera to find supernovae, they reported none (Couch et al., 1991). ### SN Ia as standard candles– not! Starting in 1986, careful observations made with CCD detectors showed ever more clearly that the luminosity and the light curve shapes for SN Ia were not uniform (Phillips et al., 1987). In 1991, two supernovae at opposite extremes of the luminosity scale showed for certain that this variety was real, and needed to be dealt with in order to make SN Ia into effective distance measuring tools. SN 1991bg (Leibundgut et al., 1993; Filippenko et al., 1992) was extremely faint and SN 1991T (Phillips et al., 1992) was extremely bright. Despite hope for a different result, and a theoretical argument why their luminosities should lie in a narrow range, Type Ia supernovae simply are not standard candles: they are known to vary over a factor of three in their intrinsic luminosity. The size of the sample needed to make a cosmological measurement scales as the square of the scatter, so, in 1991, the truly productive thing to harness supernovae for cosmology was not to find more distant supernovae, but to learn better how to reduce the uncertainty in the distance for each object. Using a set of well-sampled SN Ia light curves with precise optical photometry and accurate relative distances, Phillips (1993) demonstrated a correlation between the shape of a SN Ia light curve and the supernova’s luminosity. Supernovae with the steepest declines are the least luminous. More interestingly, even among the supernovae that do not lie at the extremes of the distribution marked by SN 1991T and SN 1991bg, the relation between luminosity and light curve shape provides an effective way to decrease the scatter in the Hubble diagram for SN Ia. Phillips used this correlation to decrease the observed scatter about the Hubble line to about 0.3 mag. This made the path forward a little clearer. What was needed was a well-run supernova search for relatively nearby supernovae that could guarantee accurate follow-up observations. Mark Phillips, Mario Hamuy, Nick Suntzeff and their colleagues at Cerro Tololo Inter-American Observatory and at the University of Chile’s Cerro Calán observatory worked together to conduct such a search, the Calán-Tololo Supernova Search (Hamuy et al., 1993). The technology was a hybrid of the past and the future– photographic plates were used on the venerable Curtis Schmidt telescope (named in honor of Heber D. Curtis, of the debate cited earlier) at Cerro Tololo to search a wide field (25 square degrees) in each exposure. Despite the drawbacks of photographic plates as detectors, this large field of view made this the most effective search for nearby supernovae. The plates were developed on the mountain, shipped by bus to Santiago, and then painstakingly scanned by eye with a blink comparator to find the variable objects. The modern part was the follow-up. Since the search area was large enough to guarantee that there would be objects found each month, CTIO scheduled time in advance on the appropriate telescopes for thorough photometric and spectroscopic follow-up with CCD detectors. The steady weather at Cerro Tololo and the dedicated work at Cerro Calán led to a stream of supernova discoveries and a rich collection of excellent supernova light curves. For example, in 1996, the Calán-Tololo group published light curves of 29 supernovae obtained on 302 nights in 4 colors (Hamuy et al., 1996a). This is what was needed to develop reliable ways to use the supernova light curves to determine the intrinsic luminosity of SN Ia, and to measure the luminosity distance to each object (Hamuy et al., 1996b). The Calán-Tololo Search was restricted to redshifts below 0.1, so it did not, by itself, contain information on the cosmology. However, it provided the data needed to understand how to measure distances with supernovae, and, when used in combination with high-z supernovae, it had the potential to help determine the cosmology. ### Dust or cosmology? However, the accuracy of the distance measurements was compromised by the uncertain amount of dust absorption in the each supernova host galaxy. Two parallel approaches were developed. One, led by Mark Phillips and his colleagues, used the observational coincidence, first noted by Paulina Lira, that the evolution in the color B-V had a very small dispersion at ages from 30 to 90 days after maximum (Phillips et al., 1999). By measuring the observed color at those times, the absorption could be inferred and the true distance measured. The other, based on the same data set, and then later extended through observations at the Whipple Observatory of the Center for Astrophysics, used an empirical method to find that intrinsically faint supernovae are also intrinsically redder. Since the light curve shape, which was the strongest clue to supernova luminosity, was not greatly affected by absorption, it was possible to determine both the distance and the absorption by dust to each supernova. A formal treatment of the extinction using Bayes’ theorem was used to determine the best values and their uncertainty (Riess, Press, and Kirshner, 1996a). This MLCS (Multi-color Light Curve Shape) approach was also used to examine whether the dust in other galaxies was the same as dust in the Milky Way (Riess, Press, and Kirshner, 1996b). While the early indications were that the dust in other galaxies had optical properties that were consistent with those found in the Galaxy, as the samples of supernovae have grown larger and the precision of the measurements has improved, this simple picture is no longer tenable. These early workers recognized that measuring the extinction to individual supernovae was an essential step in deriving reliable information on the cosmology. After all, the dimming due to an accelerating cosmology at redshift 0.5 is only of order 0.2 magnitudes. If instead this dimming were produced by dust like the dust of the Milky Way, the additional reddening would be only 0.07 mag in the B-V color, so good photometry in multiple bands was essential to make reliable inferences on the presence or absence of cosmic acceleration. ### Early results The earliest observations of the Supernova Cosmology Project (SCP) did not take account of these requirements. Their observations of SN 1992bi at z= 0.458 were made in only one filter, making it impossible, even in principle, to determine the reddening (Perlmutter et al., 1995). No spectrum for this object was obtained, but it was completely consistent with being a SN Ia. This was a striking demonstration that the search techniques used by the SCP, which resembled those of the Danish team, could reliably detect transient events in galaxies at the redshifts needed to make a cosmologically interesting measurement. The search was carried out with a 2048 by 2048 pixel CCD camera at the 2.5 m Isaac Newton Telescope, whose increased speed over the Danish system made it plausible that a supernova could be found in each month’s observing. As with the Calán-Tololo search being carried out at low redshift, it was reasonable for the SCP to schedule follow-up observations. The SCP developed the “stretch” method for accounting for the connection between luminosity and light curve shape in the B and V bands. This works very well, but does not, by itself, account for the effects of dust extinction (Goldhaber et al., 2001). The High-Z Supernova Team (HZT) was formed in 1995 by cooperation between members of the Calán-Tololo group and supernova workers at the Harvard-Smithsonian Center for Astrophysics and ESO. The goal was to apply the new methods for determining the intrinsic luminosity and reddening of a supernova, developed from the low-redshift samples, to objects at cosmologically interesting distances. This required mastering the techniques of digital image subtraction. The first object found by the High-Z Team was SN 1995K, at a redshift of 0.479, which, at that time was the highest yet published (Leibundgut et al., 1996). Observations were obtained in two colors, and the supernova’s spectrum showed it was a genuine Type Ia. Leibundgut et al. used the observations to show that the light curve for SN 1995K was stretched in time by a factor of (1 + z), just as expected in an expanding universe. The time-dilation effect had been discussed in 1939 by Olin Wilson (1939), sought in nearby data by Rust (1974), and by Leibundgut (1990). Publications by Goldhaber and his colleagues of the SCP (Goldhaber et al., 1996, 2001) show this effect in their data, though the degeneracy between the light curve shape as analyzed by the “stretch” method and time dilation requires some (quite plausible) constraints on changes in the supernova population with redshift to draw a firm conclusion. Another approach to the same problem uses the evolution of the spectra of SN Ia to show in an independent way that the clocks governing distant supernovae appear to run slower by the factor (1 + z) (Foley et al., 2005; Blondin et al., 2008). In the mid-1990s, important technical developments improved the ability to discover distant supernovae. At the National Optical Astronomy Observatories, new 2K x 2K CCD systems were implemented at the 4-meter telescopes at Kitt Peak and at Cerro Tololo. In 1997, the Big Throughput Camera (Wittman et al., 1998) became available for general use at the 4 meter telescope at Cerro Tololo. This 16 Megapixel camera set the standard for distant supernova searches and was employed by both SCP and HZT as they developed the samples that led to the discovery of cosmic acceleration. But the path to cosmic acceleration was not smooth or straight. In July 1997, based on 7 objects, the SCP published the first cosmological analysis based on supernovae (Perlmutter et al., 1997). Comparing their data from z $\sim$0.4, most of which was obtained through just one filter, to the nearby sample from Calán-Tololo (Hamuy et al., 1996a) they found a best value for $\Omega_M$ of 0.88, and concluded that their results were “inconsistent with Lambda-dominated, low-density, flat cosmologies.” Some theorists had begun to speculate that $\Lambda$ was the missing ingredient to reconcile the observations of a large value for the Hubble Constant (Freedman, Madore, and Kennicutt, 1997), the ages of globular clusters, and a low value for $\Omega_M$ in a flat cosmology (Ostriker and Steinhardt, 1995; Krauss and Turner, 1995). If the universe was flat with a total $\Omega$ of 1, and had $\Omega_M$ of 0.3, then subtraction pointed to a value for $\Lambda$ of 0.7 and you could match the ages of the globular clusters even if the Hubble constant was significantly larger than previously thought. But the initial results of the SCP pointed in the opposite direction, and their evidence for deceleration threw the cold water of data on these artfully-constructed arguments. The situation began to change rapidly late in 1997. Both teams used the Hubble Space Telescope to observe supernovae that had been found from the ground. The precision of the HST photometry was very good, with the supernova well resolved from the host galaxy thanks to the unique angular resolution of HST. Once the difficult task of accurately connecting the HST photometry to the ground-based work was complete, the observations could be combined to provide additional constraints at the beginning of 1998. For the SCP, there was one additional object from HST, at a record redshift of 0.83. When combined with a subset of the data previously published in July, the analysis gave a qualitatively different answer. In their January 1998 Nature paper (submitted on October 7, 1997), the SCP now found that “these new measurements suggest that we may live in a low-mass-density universe.” There was no observational evidence presented in this paper for cosmic acceleration (Perlmutter et al., 1998). For the High-Z Team, the HST-based sample was larger, with 3 objects, including one at the unprecedented redshift of 0.97 (Garnavich et al. 1998). Although the HZT additional sample of ground-based high-redshift observations was meager (just 1995K), using the same MLCS and template-fitting techniques on both the high-z and low-z samples, and augmenting the public low-z sample from Calán-Tololo with data from the CfA improved the precision of the overall result. Taken at face value, the analysis in this paper, submitted on October 14, 1997 and published on January 14, 1998, showed the tame result that matter alone was insufficient to produce a flat universe, and, more provocatively, if you insisted that $\Lambda$ was zero, and the universe was flat, then the best fit to the data had $\Omega_M$ less than 0. This was a very tentative whisper of what, with hindsight, we can now see was the signal of cosmic acceleration. An accelerating Universe ------------------------ ### First results Both teams had larger samples under analysis during the last months of 1997, and it was not long before the first analyses were published. The High- Z Team, after announcing their results at the Dark Matter meeting in February 1998 (Filippenko and Riess, 1998), submitted a long article entitled “Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant” to the Astronomical Journal on March 13, 1998. This appeared in the September 1998 issue (Riess et al., 1998). It used a sample of 16 high-z and 34 nearby objects obtained by High-Z Team members, along with the methods developed at the CfA and by the Calán-Tololo group to determine distances, absorptions, and their uncertainties for each of these objects. The data clearly pointed to cosmic acceleration, with luminosity distances in the high-z sample 10-15% larger than expected in a low-mass density universe without $\Lambda$. The High-Z Team also published a long methods paper (Schmidt et al., 1998) and an analysis of this data set in terms of the dark energy equation of state (Garnavich et al., 1998). The SCP, after showing their data at the January 1998 AAS meeting, cautiously warned that systematic uncertainties, principally the possible role of dust absorption, made it premature to conclude the universe was accelerating. They prepared a long paper for publication that showed the evidence from 42 high redshift objects and 18 low-redshift objects from the Calán-Tololo work. This was submitted to the Astrophysical Journal on September 8, 1998 and appeared in June 1999 (Perlmutter et al., 1999). Although the SCP had no method of their own for determining the reddening and absorption to individual supernovae, they showed that the color distributions of their high-z sample and the objects they selected from the Calán-Tololo sample had similar distributions of restframe color, an indication that the extinction could not be very different in the two samples. They also applied the method of Riess, Press, and Kirshner (1996a) to determine the absorption in the cases where they had the required data. The analysis showed, with about the same statistical power as the High-Z Team paper, that the luminosity distances to supernovae clearly favored a picture in which the universe was accelerating. ### Room for doubt? Two important questions about these results soon surfaced. One was whether the results of the two groups were independent. Some of the machinery for analyzing the data sets, for example, the K-corrections to take account of the way supernova redshifts affect the flux in fixed photometric bands, were based on the same slender database of supernova spectra. Similarly, the low-redshift sample used by the SCP was made up entirely of objects observed by the HZT. The two teams cooperated on observing a few of the high redshift objects and both teams used the data for those objects. A small number of co-authors showed up on both the High-Z Team and the Supernova Cosmology Project publications. But the analysis was done independently, most of the high-redshift samples were disjoint, and the astronomical community generally took the agreement of two competing teams to imply that this result was real. But it was the integrity of the results, not the friction of the personalities, that made this work credible. Another question about the initial results was whether the measured effect– a small, but significant dimming of the distant supernovae relative to nearby ones, was due to cosmology, to some form of dust, or to evolution in the properties of SN Ia with redshift (Aguirre, 1999a,b; Aguirre and Haiman, 2000; Drell, Loredo, and Wasserman, 2000). Aguirre explored the notion that there might be “grey dust” that would cause dimming without reddening. Theoretical difficulties included the limit imposed by using all the available solids, distributing them uniformly, and staying under the limit imposed on the thermal emission from these particles by observations in the far infrared. A direct approach to the possible contribution of dust came from measurements of supernovae over a wider wavelength range– the dust could not be perfectly grey, and a wider range of observations, made with infrared detectors, would reveal its properties more clearly. The earliest application of this was by the HZT (Riess et al., 2000), who observed a supernova at z = 0.46 in the rest-frame I band, with the goal of constraining the properties of Galactic dust or of the hypothetical gray dust. They concluded that the observed dimming of the high-z sample was unlikely to be the result of either type of dust. Much later, this approach was employed by the SCP (Nobili et al., 2005). Dust obscuration, and the relation of absorption to reddening, remains the most difficult problem in using supernova luminosity distances for high-precision cosmology, but the evidence is strong that dust is not responsible for the $\sim $0.25 mag dimming observed at z $\sim $0.5. A second route to excluding grey dust was to extend observations of SN Ia to higher redshift. If the dimming were due to uniformly distributed dust, there would be more of it along the line-of-sight to a more distant supernova. Due to the discovery of a supernova in a repeat observation of the Hubble Deep Field (Gilliland, Nugent, and Phillips, 1999) and unconscious follow-up with the NICMOS program in that field, Adam Riess and his collaborators were able to construct observations of SN 1997ff at the extraordinary redshift of z $\sim $1.7 (Riess et al., 2001). In a flat universe with $\Omega_{\Lambda} \sim 2/3$ and $\Omega_{M} \sim 1/3$, there is a change in the sign of the expected effect on supernova apparent brightness. Since the matter density would have been higher at this early epoch by a factor (1+z)$^{3}$, the universe would have been decelerating at that time, if the acceleration is due to something that acts like the cosmological constant. The simplest cosmological models predict that a supernova at z $\sim$1.7 will appear brighter than you would otherwise expect. Dust cannot reverse the sign of its effect, so these measurements of the light curve of a SN Ia at z $\sim$1.7 provided a powerful qualitative test of that idea. While the data were imperfect, the evidence, even from this single object, was inconsistent with the grey dust that would be needed to mimic the effect of cosmic acceleration at lower redshift. Another way of solidifying the early result was to show that the spectra of the nearby supernova of Type Ia, the supernovae at z $\sim $0.5 that gave the strongest signal for acceleration, and spectra of the most distant objects beyond z of 1 give no sign of evolution. While the absence of systematic changes in the spectra with epoch isn’t proof that the luminosities do not evolve, it is a test which the supernova could have failed. They do not fail this test. The early HZT results by Coil et al. (2000) show that, within the observational uncertainties, the spectra of nearby and the distant supernovae are indistinguishable. This approach was explored much later by the SCP (Hook et al., 2005) with consistent results. ### After the beginning By the year 2000, the context for analyzing the supernova results, which give a strong constraint on the combination ($\Omega_{\Lambda} - \Omega_{M}$) soon included strong evidence for a flat universe with ($\Omega_{\Lambda} + \Omega_{M} = 1$) from the power spectrum of the CMB (de Bernardis et al., 2002) and stronger evidence for the low value of $\Omega_M$ from galaxy clustering surveys (Folkes et al., 1999). The concordance of these results swiftly altered the conventional wisdom in cosmology to a flat $\Lambda$CDM picture. But the concordance of these various methods does not mean that they should lean on each other for support like a trio of drunkards. Instead, practitioners of each approach need to assess its present weaknesses and work to remedy those. For supernovae, the opportunities included building the high-z sample, which was still only a few handfuls, extending its range to higher redshift, augmenting the low-z sample, identifying the systematic errors in the samples, and developing new, less vulnerable methods for measuring distances to supernovae. #### Building the High- z sample The High-Z Team published additional data in 2003 that augmented the High-Z sample and extended its range to z = 1.2 (Tonry et al., 2003). Using the 12K CCD detector at the Canada-France-Hawaii Telescope and the Suprime-Cam at Subaru 8.2m telescope, the HZT then executed a “rolling” search of repeated observations with a suitable sampling interval of 1-3 weeks for 5 months (Barris et al., 2004). This enabled the High-Z Team to double the world’s sample of published objects with z $>$ 0.7, to place stronger constrains on the possibility of grey dust, and improve knowledge of the dark energy equation-of-state. The publication by the SCP of 11 SN Ia with 0.36 $<$ z $<$ 0.86 included high-precision HST observations of the light curves and full extinction corrections for each object (Knop et al., 2003). By this point, in 2003, the phenomenon of cosmic acceleration was well established and the interpretation as the effect of a negative pressure component of the universe fit well into the concordance picture that now included results from WMAP (Spergel et al., 2003). But what was not so clear was the nature of the dark energy. Increasing the sample near z $\sim$0.5 was the best route to improving the constraints on dark energy. One way to describe the dark energy is through the equation of state index $w=p/\rho$. For a cosmological constant, 1 +w = 0. Back-of-the-envelope calculations showed that samples of a few hundred high-z supernovae would be sufficient to constrain w to a precision of $10\%$. As before, two teams undertook parallel investigations. The Supernova Legacy Survey (SNLS), carried out at the Canada-France-Hawaii telescope, included many of the SCP team. The ESSENCE program (Equation of State: SupErNovae trace Cosmic Expansion) carried out at Cerro Tololo included many of the High-Z Team. This phase of constraining dark energy is thoroughly described in the chapter in this book by Michael Wood-Vasey, The SNLS observing program was assigned 474 nights over 5 years at CFHT. They employed the one-degree imager, Megacam, to search for supernovae and to construct their light curves in a rolling search, with a 4 day cadence, starting in August of 2003. In 2006, they presented their first cosmological results, based on 71 SN Ia, that gave a value of 1+w = -0.023 with a statistical error of 0.09, consistent with a cosmological constant (Astier et al., 2006). The ESSENCE program used the MOSAIC II imager at the prime focus of the 4m Blanco telescope. They observed with this 64 Megapixel camera every other night for half the night during the dark of the moon in the months of October, November, and December for 6 years, starting in 2002. The survey is described by Miknaitis et al. (2007) and cosmological results from the first 3 years of data were presented in 2006 (Wood-Vasey et al., 2007). The ESSENCE analysis of 60 SN Ia gave a best value for 1+ w = -0.05, with a statistical error of 0.13, consistent with a cosmological constant and with the SNLS results. Combining the SNLS and ESSENCE results gave a joint constraint of 1 + w = -0.07 with a statistical error of 0.09. We can expect further results from these programs, but the easy part is over. Bigger samples of distant supernovae do not assure improved knowledge of dark energy because systematic errors are now the most important source of uncertainty. These include photometric errors and uncertainties in the light curve fitting methods, but also more subtle matters such as the way dust absorption affects the nearby and distant samples. Collecting large samples is still desirable, especially if the photometric errors are small, but tightening the constraints on the nature of dark energy will also demand improved understanding of supernovae and the dust that dims and reddens them. ![[*Top panels:*]{} Hubble diagram and residuals for MLCS17. The new CfA3 points are shown as rhombs and the OLD and High–z points as crosses. [*Bottom panel:*]{} Hubble diagram of the CfA3 and OLD nearby SN Ia (from Hicken et al., 2009a).](MLCS17_hubble_residuals.ps "fig:"){width="95.00000%"} ![[*Top panels:*]{} Hubble diagram and residuals for MLCS17. The new CfA3 points are shown as rhombs and the OLD and High–z points as crosses. [*Bottom panel:*]{} Hubble diagram of the CfA3 and OLD nearby SN Ia (from Hicken et al., 2009a).](MLCS17_LCpaper_hubblediagram.ps "fig:"){width="100.00000%"} \[\] #### Extending its range While the work of Tonry et al. (2003) and Barris et al. (2004) showed that it was possible, with great effort, to make observations from the ground of supernovae beyond a redshift of 1, the installation of the Advanced Camera for Surveys (ACS) on HST provided a unique opportunity to search for and follow these extremely high-redshift objects (Blakeslee et al., 2003). By enlisting the cooperation of the GOODS survey, and breaking its deep exposures of extragalactic fields into repeated visits that formed a rolling search, the Higher-Z Team, led by Adam Riess, developed effective methods for identifying transients, selecting the SN Ia from their colors, obtaining light curves, determining the reddening from IR observations with NICMOS, and measuring the spectra with the grism disperser that could be inserted into the ACS (Riess et al., 2004a,b, 2007). This program has provided a sample of 21 objects with z $>$ 1, and demonstrated directly the change in acceleration, the “cosmic jerk” , that is the signature of a mixed dark matter and dark energy universe. The demise of the ACS brought this program to a halt. It is possible that the planned servicing mission can restore HST to this rich line of investigation. ![[*Left panels:*]{} Today’s best constraints from the Constitution data set on $\Omega_{M}$ and $\Omega_{\Lambda}$. The lower panel shows the combination of the SN contours with the BAO prior. [*Right panels:*]{} Same for $w$ versus $\Omega_{M}$ in a flat Universe (Hicken et al., 2009b).](SALT.OMOL.ps "fig:"){width="47.50000%"} ![[*Left panels:*]{} Today’s best constraints from the Constitution data set on $\Omega_{M}$ and $\Omega_{\Lambda}$. The lower panel shows the combination of the SN contours with the BAO prior. [*Right panels:*]{} Same for $w$ versus $\Omega_{M}$ in a flat Universe (Hicken et al., 2009b).](SALT.OMw.ps "fig:"){width="47.50000%"} \[\] #### Augmenting the low-z sample Both the High-Z Team (which included members of the Calán-Tololo supernova program) and the SCP depended on low redshift observations of supernovae to establish the reality of cosmic acceleration. The samples at high redshift were assembled, at great effort, and high cost in observing time at the world’s largest telescopes because it was clear that these data could shift our view of the universe. The low-z samples require persistence, careful attention to systematic effects, and promised no shift in world view. They have been slower to develop. Two early steps forward were the publication by Riess of 22 BVRI light curves from his Ph.D. thesis at Harvard (Riess et al., 1999), and the publication of 44 UBVRI light curves from the thesis work of Jha (Jha, Riess, and Kirshner, 2007). The U-band observations in Jha’s work were especially helpful in analyzing the HST observations of the Higher-Z program, since, for the highest redshift objects observed with HST, most of the observations correspond to ultraviolet emission in the supernova’s rest frame. Jha also revised and retrained the MLCS distance estimator that Riess had developed, using this larger data set, and dubbed it MLCS2k2. Recently, Kowalski compiled the “Union” data sample (Kowalski et al., 2008). His work assessed the uncertainties in combining data from diverse sources, and, by applying stringent cuts to the data, provided a set of 57 low-redshift and 250 high-redshift supernovae to derive constraints on dark energy properties. Kowalski noted the imbalance of the low-z and high-z samples and emphasized the opportunity to make a noticeable improvement in the constraints on dark energy by increasing the sample size for the nearby events. A third Ph.D. thesis at Harvard, by Malcolm Hicken, has just been completed that finally brings the low-redshift sample out of the statistical limit created by our slow accumulation of nearby objects and begins to encounter the systematic limit imposed by imperfect distance estimators. Hicken analyzed the data for 185 SN Ia in 11500 observations made at the Center for Astrophysics over the period from 2001 to 2008. This large and homogenous data set improves on the Union data set compiled by Kowalski to form the (more perfect) Constitution data set (Hicken et al., 2009a,b). When Hicken uses the same distance fitter used by Kowalski to derive the expansion history and fits to a constant dark energy, he derives 1 + w = 0.013 with a statistical error of about 0.07 and a systematic error that he estimates at 0.11. As discussed below, one important contribution to the systematic error that was not considered by Kowalski is the range of results that is produced by employing different light curve fitters such as SALT, SALT2, and MLCS2k2 which handle the properties of dust in different ways. This CfA work is a follow-up program that exploits the supernova discovery efforts carried out at the Lick Observatory by Alex Filippenko, Weidong Li, and their many collaborators (Filippenko et al., 2001) as well as a growing pace of supernova discoveries by well-equipped and highly motivated amateur astronomers. Since the selection of the Constitution supernova sample is not homogeneous, information extracted from this sample concerning supernova parent populations and host galaxy properties needs to be handled with caution, but it suggests that even after light curve fitting, the SN Ia in Scd, Sd, or Irregular galaxy hosts are intrinsically fainter than those in Elliptical or S0 hosts, as reported earlier by Sullivan, based on the SCP sample (Sullivan, 2003). The idea of constructing a single fitting procedure for supernovae in all galaxy types has proved effective, but it may be missing a useful clue to distinct populations of SN Ia in galaxies that are and are not currently forming stars. There may be a variety of evolutionary paths to becoming a SN Ia that produce distinct populations of SN Ia in star-forming galaxies that are not exactly the same as the SN Ia in galaxies where star formation ceased long ago (Mannucci et al., 2005; Sullivan et al., 2006; Scannapieco and Bildsten, 2005). Constructing separate samples and deriving distinct light curve fitting methods for these stellar populations may prove useful once the samples are large enough. A step in this direction comes from the work at La Palma, building up the sample at the sparsely-sampled redshift range near z = 0.2. (Altavilla et al., 2009). A comprehensive approach to sampling has been taken by the Sloan Supernova Survey (Frieman et al., 2008). By repeatedly scanning a 300 deg$^2$ region along the celestial equator, the survey identified transient objects for spectroscopic follow-up with excellent reliability and has constructed ugriz light curves for over 300 spectroscopically confirmed SN Ia. With excellent photometric stability, little bias in the supernova selection, and a large sample in the redshift range 0.05 $<$ z $<$ 0.35, this data set will be a powerful tool for testing light curve fitting techniques, provide a low-redshift anchor to the Hubble diagram, and should result in a more certain knowledge of dark energy properties. In the coming years, comprehensive results from the SCP’s SN Factory (Aldering et al., 2002), the Carnegie Supernova Program (Hamuy et al., 2006), and the analysis of the extensive KAIT archive (Filippenko et al., 2001) should change the balance of the world’s sample from one that is just barely sufficient to make statistical errors smaller than systematic errors, to one that provides ample opportunity to explore the ways that sample selection might decrease those systematic errors. Shifting to the infrared ------------------------ Coping with the effects of dust absorption was an important contribution of the early work by Phillips et al. (1999) and by Riess, Press, and Kirshner (1996a), the later work by Knop et al. (2003) and Jha, Riess, and Kirshner (2007) and it continues to be the most difficult and interesting systematic problem in supernova cosmology. The formulations that worked sufficiently well to measure 10% effects will not be adequate for the high precision measurements that are required for future dark energy studies. The analysis of the low-redshift supernova data by Conley et al. (2007) showed that either the ratio of reddening to extinction in the supernova hosts was distinctly different from that of the Milky Way (R$_{V}$ = 1.7 instead of the conventional value of 3.1) or there was a “Hubble Bubble”– a zone in which the local expansion rate departed from the global value. As discussed by Hicken (2009a), today’s larger sample does not show evidence for the Hubble Bubble, but the value of R$_{V}$ that performs best for MLCS2k2 and for SALT is significantly smaller than 3.1. It seems plausible that the sampling for earlier work was inhomogeneous, with highly reddened objects present only in the nearby region. If the correction for reddening in these cases was not carried out accurately, they could contribute to the illusion of a Hubble Bubble. But the evidence for a small effective value of R$_{V}$ has not gone away. It seems logical to separate the contribution due to reddening from the contribution that might result from an intrinsic relation between supernova colors and supernova luminosity, as done in MLCS2k2, but the approach of lumping these together, as done by the fitting techniques dubbed SALT and SALT2, also works well empirically (Guy et al., 2005, 2007). In the ESSENCE analysis, the effects of extinction on the properties of the observed sample were carefully considered, and found to affect the cosmological conclusions. Getting this problem right will be an important part of preparing for higher precision cosmological measurements with future surveys. {width="100.00000%"} \[\] Fortunately, there is a very promising route to learning more about dust, avoiding its pernicious effects on supernova distances, and deriving reliable and precise measures of dark energy properties. That route is to measure the properties of the supernovae in the rest frame infrared. As shown in the pioneering work of Krisciunas, Phillips, and Suntzeff (2004), nearby SN Ia in the Hubble flow behave as very good standard candles when measured in near infrared bands (NIR), typically J, H, and K$_s$. This work has recently been extended by Wood-Vasey et al. who used the PAIRITEL system (a refurbished and automated version of the 2MASS telescope) at Mount Hopkins to obtain near infrared light curves that double the world’s sample (Wood-Vasey et al., 2008) . Even with no correction for light curve shape or dust absorption, the NIR light curves for SN Ia exhibit a scatter about the Hubble line that is typically 0.15 mag. This is comparable to the scatter that is achieved by the output of the elaborate light curve fitters now in use for optical data that correct for the width of the light curve’s peak and use the optical colors to infer dust corrections. This means that the SN Ia actually do behave like standard candles– but in the NIR! What’s more, the effects of dust absorption generally scale as $1/\lambda$, so the effects of extinction on the infrared measurements should be 4 times smaller than at the B band. When combined with optical data, the infrared observations can be used to determine the properties of the dust, and to measure even more accurate luminosity distances. Early steps toward these goals are underway (Friedman et al., 2009). The next ten years ------------------ Goals for the coming decade are to improve the constraints on the nature of dark energy by improving the web of evidence on the expansion history of the universe and on the growth of structure through gravitation (Albrecht et al., 2006, 2009; Frieman, Turner, and Huterer, 2008; Ruiz-Lapuente, 2007). Supernovae have an important role to play because they have been demonstrated to produce results. Precise photometry from homogeneous data, dust absorption determined with near-IR measurements, and constructing useful subsamples in galaxies with differing star formation histories are all areas where we know improvement in the precision of the distance measurements is possible. More speculative, but plausible, would be the use of supernova spectra in a systematic way to improve the distance estimates. Implementation of statistically sound ways to use the light curves (and possibly spectra) to determine distances should make the results more reliable and robust. What is missing is a level of theoretical understanding for the supernova explosions themselves that could help guide the empirical work, and provide confidence that stellar evolution is not subtly undermining the cosmological inferences (Hoeflich, Wheeler, and Thielemann, 1998; Ruiz–Lapuente, 2004). Large samples from Pan-STARRS, the Dark Energy Survey, and, if we live long enough, from JDEM and LSST will eventually be available. The chapter in this book by Alex Kim makes a persuasive case for the effectiveness of a thorough space-based study of supernovae. Our ability to use these heroic efforts effectively depends on improving our understanding of supernovae as astronomical objects in the context of galaxy formation, stellar evolution, and the physics of explosions. 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--- abstract: 'We present the first implementation of the relativistic quantum chemical two- and four-component density matrix renormalization group algorithm that includes a variational description of scalar-relativistic effects and spin–orbit coupling. Numerical results based on the four-component Dirac–Coulomb Hamiltonian are presented for the standard reference molecule for correlated relativistic benchmarks: thallium hydride.' author: - Stefan Knecht - Örs Legeza - Markus Reiher title: 'Four-Component Density Matrix Renormalization Group' --- Owing to remarkable advances in the past decades relativistic quantum chemical methods have become a routinely applicable and indispensable tool for the accurate description of the chemistry and spectroscopy of molecular compounds comprising heavy elements [@perspectives_rel_qc12; @dyall_faegri_qc; @reiher_qc]. Also the complete understanding of the photochemistry and photophysics of first- and second-row molecules requires to encompass relativistic effects — the most important of which are spin–orbit (SO) interactions — needed to calculate intersystem crossing rates [@WCMS:WCMS83]. Major challenges for relativistic quantum chemistry originate from (i) the reduction of non-relativistic (spin and spatial) symmetries caused by magnetic couplings that lead to in general complex wave functions and require the use of double-group symmetry as well as (ii) the large number of (unpaired) valence electrons to be correlated (in particular for heavy elements) and (iii) the occurrence of near-degeneracies of electronic states. Popular quantum chemical methods such as CASSCF/CASPT2/SO-CASPT2 [@caspt2-review-wires2012]  assume an additivity of electron correlation and spin–orbit effects or a weak polarization of orbitals due to spin–orbit interaction, or both. Hence, for heavy-element compounds accuracy is inevitably limited as relativistic effects and static or dynamic electron correlation are often not only large but also counteracting [@pyykkoe88; @perspectives_rel_qc12]. To address the latter issue adequately, a number of genuine relativistic multiconfigurational and multireference approaches have been proposed [@Fleig20122-review-rel-correlation-methods; @neese_socasscf12; @Kim2013]. In this Communication we merge the strengths of the density matrix renormalization group (DMRG) algorithm [@white; @White1999], which has been successfully introduced to the field of non-relativistic quantum chemistry [@ors_springer; @reiher10_dmrg; @chanreview], with a variational description of all relativistic effects in the orbital basis. This new four-component (4c) DMRG *ansatz* goes beyond preceeding scalar-relativistic DMRG approaches [@C3CP53975J; @Moritz2005a] and allows us to efficiently describe first and foremost non-dynamic correlation (or strong correlations) in heavy-element complexes by means of extensive active orbital spaces which would surmount capabilities of any to-date available relativistic multiconfigurational approach. The point of departure for our relativistic DMRG implementation is the time-independent 4c-Dirac–Coulomb(–Breit) Hamiltonian [@saue_primer11] — any suitable (exact) two-component (2c) Hamiltonian approximation is directly amendable, too. The basic four-component electronic eigenvalue equation for a many-particle system is conveniently cast (with positive-energy projectors omitted) in a form which is known from non-relativistic quantum chemistry [@dyall_faegri_qc; @reiher_qc], $$\label{rel-ham} \hspace{-0.30cm} \hat{H} \Psi = \left(\sum\limits_{i}\hat{h}_D(i)+\frac{1}{2} \sum\limits_{i \neq j}\hat{g}(i,j)+V_{NN}\right) \Psi = E_{\rm el} \Psi ,$$ where $\hat{h}_D(i)$ is the one-electron Dirac Hamiltonian for electron $i$, $\hat{g}(i,j)$ is a two-electron operator describing the interaction between electrons $i$ and $j$, $V_{NN}$ is the classical nuclear repulsion energy operator, $E_{el}$ is the energy eigenvalue and $\Psi$ is a four-component wave function. In the absence of any external magnetic field it can be shown that Eq. (\[rel-ham\]) is symmetric under time-reversal [@reiher_qc] from which follows that a fermion four-component spinor functions $\phi_i$ occurs in Kramers pairs $\{\phi_i,\bar{\phi}_i\}$. A spinor $\bar{\phi}_i$ can thus be obtained from the action of the time-reversal operator $\hat{\mathcal{K}} = -i\Sigma_y \hat{\mathcal{K}}_0$ on ${\phi}_i$, that is ${\hat\mathcal{K}}\phi_i = \bar{\phi}_i$. Hence, our 4c- (or 2c-)spinor basis is comprised of Kramers pairs which we will imply in the following derivations. In the no-pair approximation, we can formulate the resulting Hamiltonian in second-quantized and normal ordered form, $$\hat{\widetilde{H}} = \sum\limits_{PQ}^{}\ F_{P}^{Q} \{a_{{P}}^{\dagger}a_Q\} + \frac{1}{4} \sum\limits_{PQRS}^{}\ V_{PR}^{QS} \{a_{{P}}^{\dagger}a_{R}^{\dagger}a_{S}a_{Q}\}\ , \label{rel-h-2nd-quant}$$ where the summation indices $PQRS$ strictly refer to positive-energy orbitals, and $F_{P}^{Q}$ and $V_{PR}^{QS} = (G_{PR}^{QS} - G_{PR}^{SQ} ) $ are Fock-matrix elements and antisymmetrized two-electron integrals $G_{PR}^{QS}$, respectively. The Hamiltonian $\hat{\widetilde{H}}$ in Eq. (\[rel-h-2nd-quant\]) constitutes the starting point for our 4c-DMRG implementation. We benefit from a quaternion symmetry scheme [@saue1999_quaternion-symmetry-dirac] that has been implemented for the binary double groups D$_{2h}^{\ast}$ and subgroups thereof in the `Dirac`  program package [@DIRAC12] to which our DMRG program is interfaced. In this scheme, point group symmetry and quaternion operator algebra are combined advantageously such that the eigenvalue equation, Eq. (\[rel-ham\]), can be solved either using real (double groups D$_{2h}^{\ast}$, D$_{2}^{\ast}$ and C$_{2v}^{\ast}$; resulting number of non-zero real matrices of a quaternion operator matrix representation: NZ = 1), complex (C$_{2h}^{\ast}$, C$_{2}^{\ast}$ and C$_{s}^{\ast}$; NZ = 2) or quaternion algebra (C$_{i}^{\ast}$ and C$_{1}^{\ast}$; NZ = 4). Working in a Kramers-paired spinor basis, one can then show that all operator matrix elements $t_{p\bar{q}}$ of a time-symmetric one-electron operator $\hat{t}$ are zero by symmetry. Furthermore the complete set of two-electron integrals $G_{PR}^{QS}$  of the two-electron (Coulomb) operator $\hat{g}$ in molecular orbital (MO) basis can be cast into a $4 \times 3$ ((NZ,3)) matrix representation (see also Appendix B.3 page 161*ff* of Ref. [@thyssendiss]), $$\begin{aligned} \mathbf{{G}} = \left(\begin{array}{ccc} \mathcal{R}((PQ|RS)) & \mathcal{R}((P\bar{Q}|R\bar{S})) & \mathcal{R}((\bar{P}Q|\bar{R}S)) \\ \mathcal{I}((PQ|RS)) & \mathcal{I}((P\bar{Q}|R\bar{S})) & \mathcal{I}((\bar{P}Q|\bar{R}S)) \\ \mathcal{R}((PQ|R\bar{S})) & \mathcal{R}((P\bar{Q}|R{S})) & \mathcal{R}((\bar{P}Q|{R}S)) \\ \mathcal{I}((PQ|R\bar{S})) & \mathcal{I}((P\bar{Q}|R{S})) & \mathcal{I}((\bar{P}Q|{R}S)) \end{array} \right), \label{dirac:nz-3-format}\end{aligned}$$ where $\mathcal{R}$ and $\mathcal{I}$ denote the real and complex parts of a two-electron integral in MO representation, respectively, and $P,Q,R,S$ label spinor indices. The number of nonzero rows for a given binary double group thus corresponds to the NZ rank as given above. Important symmetry reductions for both the one- and two-electron integrals are therefore being taken into account in a relativistic Kramers-unrestricted DMRG implementation. This scheme not only provides considerable computational savings but also ensures that the DMRG wave function has the correct time-reversal symmetry in case of a closed-shell molecule. In a Kramers-restricted spinor basis all one-electron matrix elements $F_{P}^{Q}$ (see Eq. \[rel-h-2nd-quant\]) among barred and unbarred components will be identical while matrix elements between barred and unbarred are non-zero only in the NZ=4 case. In contrast, a two-electron integral $G_{PR}^{QS}$ may generally be comprised of barred and unbarred spinors. As illustrated by Eq. (\[dirac:nz-3-format\]) for NZ=1 and NZ=2, respectively, only an even number ($n_{\rm barred}=0,2,4$) of barred spinors yields a non-vanishing two-electron integral whereas for NZ=4 all combinations are contributing. Even though integrals can be made real-valued (NZ=1), permutational symmetry is reduced by a factor two compared to the 8-fold permutational symmetry in the non-relativistic case since orbitals are complex in a relativistic framework. In DMRG, electron–electron correlation is taken into account by an iterative procedure that minimizes the Rayleigh quotient corresponding to the electronic Hamiltonian $\hat{\widetilde{H}}$ and eventually converges a full-CI-type wave function within the selected active orbital space. The full configuration Hilbert space of a finite system comprising $N$ MOs, $\Lambda^{(N)}$, is built from tensor product spaces of local orbital (tensor) spaces $\Lambda_i$,[@new_review] which can be written as $\Lambda^{(N)}=\otimes_{i=1}^N \Lambda_i$. Since standard non-relativistic DMRG implementations usually employ a spatial-orbital basis, the dimension of the local Hilbert space of a single molecular orbital, $q={\rm dim}\ \Lambda_i$, becomes 4 while the full dimensionality is ${\rm dim}\ \Lambda^{(N)}=4^N$. In this representation an MO can be either empty, singly occupied with spin up or down, or doubly occupied with paired spins. Our implementation exploits a two-dimensional local Hilbert-space representation, $q=2$, where each spinor can either be empty or singly occupied. The tensor space dimension is then $2^{N}$ with $N$ being the number of spinors. In the two-site DMRG variant [@white], that is the basis for our relativistic DMRG implementation, $\Lambda^{(N)}$ is approximated by a tensor product space of four tensor spaces, i.e., $\Lambda^{(N)}_{\rm DMRG}=\Lambda^{(l)}\otimes\Lambda_{l+1}\otimes\Lambda_{l+2}\otimes\Lambda^{(r)}$. The dimensions of the corresponding local *left* ($l$)  and *right* ($r$) spaces are denoted as $M_l={\rm \dim}\ \Lambda^{(l)}$ and $M_r={\rm \dim}\ \Lambda^{(r)}$, respectively. With $q={\rm \dim}\ \Lambda_{l+1}={\rm \dim}\ \Lambda_{l+2}$ the resulting dimensionality of the DMRG wave function is ${\rm \dim}\ \Lambda^{(N)}_{\rm DMRG}=q^2M_lM_r\ll q^N$. The number of block states, $M_l$ and $M_r$, required to achieve sufficient convergence can be regarded as a function of the level of entanglement among the molecular orbitals. Hence the maximum number of block states $M_{\rm max} = \max{(M_l,M_r)}$ determines the accuracy of a DMRG calculation [@legeza_dbss2]. The success and numerical efficiency of the DMRG algorithm rely on a subsequent application of the singular value decomposition (SVD) theorem [@Schollwock2010; @new_review] while the performance depends on the level of entanglement encoded in the wave function [@legeza_dbss3]. During an SVD step, the finite system is divided into two parts by expressing $\Lambda^{(N)}_{\rm DMRG}=\Lambda^{(L)}\otimes\Lambda^{(R)}$, namely the system and environment blocks, where $\Lambda^{(L)}=\Lambda^{(l)}\otimes\Lambda_{l+1}$, and $\Lambda^{(R)}=\Lambda_{l+2}\otimes\Lambda^{(r)}$. In each DMRG step, the basis states of the system block are then transformed to a new [*truncated basis*]{} set by a unitary transformation based on the preceeding SVD[@schollwoeck]. This transformation depends therefore on how accurately the environment is represented [@Moritz2006] as well as on the level of truncation[@legeza_dbss2]. As a consequence the accuracy of the DMRG method is governed by the truncation error, $\delta \varepsilon_{\rm TR}$, as well as by the environmental error, $\delta \varepsilon_{\rm sweep}$ [@legeza1996]. The latter is minimized in each DMRG macro-iteration by a successive application of the SVD going through the system back and forth (“sweeping"). In order to minimize $\delta \varepsilon_{\rm sweep}$, which is usually largest during the initial sweep of the DMRG approach because of a poor representation of the environment, we take advantage of the Configuration Interaction based Extended Active Space procedure (CI-DEAS) [@legeza_dbss; @legeza2004-leiden] to efficiently construct the environmental basis states by means of an orbital entropy profile [@dmrg-quantum-information-analysis-2011]. The latter is dependent on the orbital ordering along a (fictitious) one-dimensional chain [@legeza_dbss3; @dmrggerrit3] and determines the maximum number of block states $M_{\rm max}=\max(M)$ that is needed to satisfy an *a priori* defined accuracy threshold given by a value $\chi$. The truncation error $\delta \varepsilon_{\rm TR}$ is a function of the total number of block states $M$. Assuming $M_l=M_r=M$ we can exploit a second-order polynomial fit as a function of $1/M$ by taking the limit of zero energy change between two sweeps $E(M, \delta \varepsilon_{\rm sweep}=0)$  for a given $M$ to provide a good estimate for the truncation-free solution [@legeza1996; @barcza2012]. We demonstrate the capabilities of our 4c-DMRG implementation at the example of the thallium hydride molecule since this system has become a standard benchmark molecule for a plethora of relativistic methods [@seijo1995_tlh; @lenthe1996_tlh; @han1999_tlh; @titov2000_tlh; @visscher2001_tlh_ccsdt; @mayer2001_tlh; @choi2001_tlh; @ilias2001_tlh; @choi2003_tlh; @zeng2010-tlh-so] (see also references in Ref. ). Orbitals and MO integrals were computed with a development version of the `Dirac12` program package [@DIRAC12] using the Dirac–Coulomb (DC) Hamiltonian and triple-$\zeta$ basis sets for Tl (cv3z) [@dyall_6p_all; @dyall_6p_core_corr2012] and H (cc-pVTZ) [@dunning89], which include core-correlating functions for Tl. All DMRG calculations were performed with the relativistic development branch of the <span style="font-variant:small-caps;">QC-DMRG-Budapest</span> program [@Legeza]. C$_{2v}^{\ast}$ double group symmetry (NZ=1) was assumed throughout all calculations for TlH. MP2 natural spinors (NSs) [@mp2-no], correlating the Tl $5s5p4f5d6s6p$ and H $1s$ electrons while keeping the remaining core electrons of Tl frozen, served as the orbital basis for all electron-correlation calculations. Since `Dirac12` requires to use uncontracted basis sets in a four-component framework, a virtual orbital threshold was set at 135 hartree, such that the initial virtual correlation space in the MP2 calculation comprised all recommended core-valence and valence-correlation functions. The final active space was then chosen to include all occupied spinors that have MP2-NS occupancies less than $1.98$  as well as all virtuals up to a cutoff of $\approx 0.001$ in the MP2-NS occupation numbers. Given this criterion, an active space of 14 electrons — the occupied Tl $5d6s6p$ plus H $1s$ shells — in 47 Kramers pairs (94 spinors) was used in the CI [@fleig03; @fleig06; @knecht10a] MP2, CC [@mrcceng; @Kallay_2001; @Kallay_2001] and DMRG calculations. The latter are further characterized by the choice of $M_{\rm min}, M_{\rm max}, M_{\rm min}^{\rm DEAS}$ and $\chi$, denoted in the following as DMRG(14,94)\[$M_{\rm min},M_{\rm max},M_{\rm min}^{\rm DEAS},\chi$\]. ![\[fig:entropy\]Left: One-orbital entropy profile, $s_i$, calculated at the experimental internuclear distance $r_e^{\rm exp}$=1.872 Å. The larger the entropy value for a given spinor the larger its contribution to the total correlation energy. Right: Schematic plot of a piecewise orbital entanglement based on the two-orbital mutual information, $I_{ij}$. Entanglement strengths are indicated by different colors.](figure1a "fig:") ![\[fig:entropy\]Left: One-orbital entropy profile, $s_i$, calculated at the experimental internuclear distance $r_e^{\rm exp}$=1.872 Å. The larger the entropy value for a given spinor the larger its contribution to the total correlation energy. Right: Schematic plot of a piecewise orbital entanglement based on the two-orbital mutual information, $I_{ij}$. Entanglement strengths are indicated by different colors.](figure1b "fig:") Figure \[fig:entropy\] depicts the one- $s_i$ and two-orbital $I_{ij}$ entropy profiles [@legeza_dbss3; @dmrg-entanglement12; @boguslawski2013a] at the experimental internuclear distance $r_e^{\rm exp}$=1.872 Å  computed from an initial DMRG\[256,256,256,10$^{-5}$\] calculation. We first note that the one-orbital entropy profile, (left-hand side of Figure \[fig:entropy\]) is nearly perfectly symmetric with respect to the unbarred (\#1–\#47) and barred (\#48–\#94) spinors where any slight deviation is an artefact of the preset low $M_{\rm min},M_{\rm max}$ values. The total quantum information $I_{\rm tot}$ encoded in the wave function, defined as the sum of one-orbital entropies, $I_{\rm tot}=\sum_{i} s_i$, can be taken as a measure of the importance of dynamic (weak) electron correlation. The lower $I_{\rm tot}$ (compared to $I_{\rm tot}^{\rm max}=\sum_i s_i^{\rm max}= N\ln(2)=65.15$), the more important will be an appropriate account of dynamic electron correlation in order to grasp all important correlation effects. In the present case of TlH we have $I_{\rm tot}\simeq 2.23\ll I_{\rm tot}^{\rm max}$ which points to the fact that TlH is a predominantly single-reference close to its equilibrium structure. The two-orbital mutual information, $I_{ij}$, confirms this qualitative picture. $I_{ij}$ values are visualized in the right panel of Figure \[fig:entropy\], where the degree of entanglement between spinors is marked by a color-coded connecting line. While few spinors are weakly entangled (red) the majority is entangled with even smaller strengths (green). Since several spinors are mutually entangled with the same order of magnitude, we expect that large $M_{\rm min}, M_{\rm max}$  values combined with a low quantum information loss threshold $\chi$ are required to reach a fully converged DMRG wave function. To corroborate this hypothesis we compiled in Table \[tlh-results\] total energy differences for various standard wave-function-expansion methods as well as for our 4c-DMRG(14,94)\[4500,1024,2048,10$^{-5}$\] model with respect to a chosen 4c-CCSDTQ reference at $r_e^{\rm exp}$=1.872 Å. The 4c-DMRG wave function was built from an optimized ordering of orbitals based on the entropy profiles given in Figure \[fig:entropy\] and by applying high accuracy settings in the initial CI-DEAS sweep (with ${\rm CI}_{\rm level}=4$ and $\chi_{\rm CI}=10^{-8}$). These initial conditions ensured both a rapid elimination of the environmental error and a fast total convergence towards the global minimum as illustrated by the left-hand side of Figure \[fig:energy\]. The 4c-CISDTQ energy is in fact reached after no more than six sweeps of the 4c-DMRG wave function optimization procedure. method $\Delta$E$_{\rm el}$ ----------------------------------------------------- ---------------------- 4c-CISD(14,94) 41.55 4-CISDT(14,94) 32.80 4c-CISDTQ(14,94) 2.63 4c-MP2(14,94) -13.49 4c-CCSD(14,94) 10.58 4c-CCSD(T)(14,94) -0.32 4c-CCSDT(14,94) 0.33 4c-CCSDT(Q)(14,94) -0.07 4c-DMRG(14,94)\[4500,1024,2048,10$^{-5}$\] 2.57 4c-DMRG(14,94)\[$M\rightarrow\infty$ extrapolated\] 0.7 : Total electronic energy differences $\Delta$E$_{\rm el}$ (in mH) for different correlation approaches with respect to the 4c-CCSDTQ(14,94) reference energy of -20275840.24233 mH for TlH computed at the experimental equilibrium internuclear distance 1.872 Å. \[tlh-results\] Inspection of Table \[tlh-results\] furthermore reveals that the 4c-DMRG energy is, although being below our best variational 4c-CISDTQ energy, still 2.57 mH higher than the reference 4c-CCSDTQ as well as 2.89 mH higher than the single-reference 4c-CCSD(T) energies. From the convergence pattern of our benchmark 4c-DMRG\[4500,1024,2048,10$^{-5}$\] calculation displayed on the left-hand side of Figure \[fig:energy\] the following picture emerges: after having reached the maximum number of block states $M_{\rm max}$ ($\approx$ 3 sweeps) — for the present problem we have a computational limit of $M_{\rm max}=4500$ — the convergence rate slows down significantly and after the sixth sweep the energy is no longer a decreasing function of the iteration steps because the environmental error now starts to fluctuate to a certain extent depending on the actual superblock configuration. As a result the 4c-CCSDTQ reference energy could not be reached (see Table \[tlh-results\]), which is, however, not a fundamental problem of the approach. It must be emphasized again that DMRG is best suited for static-correlation problems while TlH is dominated by dynamic correlation, for which CC approaches are much more suitable. ![Left: Convergence of the ground state energy (shifted by 20275 H) as a function of iteration steps of the 4c-DMRG(14,94)\[4500,1024,2048,10$^{-5}$\] approach at $r_e^{\rm exp}$=1.872 Å. Reference energies calculated by various CI and CC wave function models are also given as horizontal lines. The inset shows that the 4c-DMRG energy drops below the 4c-CI-SDTQ energy. Right: Extrapolation of DMRG energies E($M$)-20275 H for fixed $M$ values towards the limit $E(M\rightarrow\infty)$-20275 H. []{data-label="fig:energy"}](figure2) However, extrapolating the DMRG energy for a given $M$ to the limit $E(M, \delta \varepsilon_{\rm sweep}=0)$ by using an exponential function in $1/\rm{sweep}$ (*vide supra*) provides an effective means to eliminate the truncation error. The [right-hand side]{} of Figure \[fig:energy\] shows the extrapolated energies along with the resulting best-estimate DMRG energy $E(M\rightarrow\infty)=-20275.8395$ H. At $r_e^{\rm exp}$=1.872 Å  $E(M\rightarrow\infty)$ is now as close as +0.7 mH to the CCSDTQ reference energy. Taking further advantage of the extrapolation scheme we exploit a characteristic feature of the DMRG parametrization, namely, that it includes, in contrast to a CI expansion, all excitations required to describe the wave function to a given accuracy. This property implies that the general structure of the DMRG wave function is preserved and can be determined even with smaller $M$ values [@legeza_dbss2]. We found that the resulting 4c-DMRG\[512, $\delta \varepsilon_{\rm sweep}=0$\] potential energy curve does not only effectively reproduce the shape of the 4c-CCSDTQ potential energy curve but also yields accurate spectroscopic constants — compiled in Table \[tlh-spec-results\] — as extracted from a fourth-order polynomial fit. The 4c-CCSDTQ data is in excellent agreement with experiment for the equilibrium internuclear distance $r_e$, harmonic frequency $\omega_e$, and for the anharmonicity constant $\omega_e x_e$ while it turns out to be superior to a selection of other theoretical approaches listed in Table \[tlh-spec-results\]. The current DMRG results in turn show an excellent agreement with experiment for $r_e$ while predicting slightly too high values for $\omega_e$ (+20 cm$^{-1}$) and $\omega_e x_e$ ($+4$ cm$^{-1}$), respectively. We conclude with a note on the computational demands of our 4c-DMRG implementation in comparison to CCSDTQ. The benchmark DMRG\[4500,1024,2048,10$^{-5}$\] calculation required $\approx$ 50 GB of core memory to represent all operators of the left and right blocks while the relativistic MRCC code [@kallay2011_4cCC] had similar memory requirements for the optimization of the various $t$ amplitudes. Our new 4c- and 2c-DMRG approach (no 2c-results are shown here) bears the potential to become a new reference approach comparable to higher-order CC methods, in particular for molecular systems that exhibit rather strong multi-configurational character in their ground- and excited states. To improve on the description of dynamic correlation, a further combination with standard approaches like, for example, multi-reference perturbation theory is possible and will yield a powerful tool for the theoretical chemistry and photophysics of heavy-element molecules like lanthanide and actinide complexes. method r$_e$ \[Å\] $\omega_e$ \[cm$^{-1}$\] $\omega_e x_e$ \[cm$^{-1}$\] ----------------------- ------------- -------------------------- ------------------------------ 4c-DMRG(14,94)\[512\] 1.873 1411 26.64 4c-CISD(14,94) 1.856 1462 23.11 4c-CISDTQ(14,94) 1.871 1405 20.11 4c-MP2(14,94) 1.828 1546 47.27 4c-CCSD(14,94) 1.871 1405 19.36 4c-CCSD(T)(14,94) 1.873 1400 23.52 4c-CCSDT(14,94) 1.873 1398 22.28 4c-CCSDT(Q)(14,94) 1.873 1397 21.01 4c-CCSDTQ(14,94) 1.873 1397 22.24 CCSD(T)$^a$ 1.876 1385 n/a CCSD(T)$^b$ 1.877 1376 n/a MRD-CI$^c$ 1.870 1420 n/a SO-MCQDPT$^d$ 1.876 1391 29.42 experiment$^e$ 1.872 1390.7 22.7 : Spectroscopic constants of $^{205}$TlH obtained from 4c-DMRG\[$512,\delta \varepsilon_{\rm sweep}=0)$\], CI and CC calculations in comparison with other theoretical and experimental work. $^a$ 4c-DC CCSD(T) \[14 electrons\], see Ref. .\ $^b$ 4c-DC-Gaunt CCSD(T) \[36 electrons\], see Ref. .\ $^c$ GRECP spin–orbit MRD-CI, see Ref. .\ $^d$ model-core potential spin–orbit MCQDPT, see Ref. .\ $^e$ experimental data taken from Refs. . \[tlh-spec-results\] This work has been supported financially in part by the Schweizerischer Nationalfonds (project no. 200020\_144458/1) and by the Hungarian Research Fund (OTKA) through Grants No. K100908 and No. NN110360. 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--- abstract: 'Pattern avoiding machines were recently introduced in [@CCF] to gain a better understanding of the classical $2$-stacksort problem. Here we generalize these devices by allowing permutations with repeated elements both as inputs and as forbidden patterns for the first stack. Then we provide a description of those patterns such that the corresponding set of sortable permutations is a class. Finally, we regard a pattern-avoiding stack as an operator and we characterize all the patterns that give rise to a bijective map.' author: - | Giulio Cerbai\ Dipartimento di Matematica e Informatica “U. Dini”\ University of Firenze, Italy\ [email protected] title: | Sorting Cayley Permutations\ with Pattern-avoiding Machines [^1] --- Introduction ============ The problem of sorting a permutation using a stack, together with its many variants, has been widely studied in the literature. The original version was proposed by Knuth in [@Kn]: given an input permutation $\pi$, either *push* the next element of $\pi$ into the stack or *pop* the top element of the stack, placing it into the output. The goal is to describe and enumerate sortable permutations. To sort a permutation means to produce a sorted output, i.e. the identity permutation. An elegant answer can be given in terms of pattern avoidance: a permutation is sortable if and only if it does not contain a subsequence of three elements which is order isomorphic to $231$. A set of permutations that can be characterized in terms of pattern avoidance is called a *class* and the minimal excluded permutations are its *basis*. The notion of pattern avoidance turns out to be a fundamental tool to approach a great variety of problems in combinatorics. We refer the reader to [@B] for a more detailed survey on stack-sorting disciplines, and to [@B2] and [@Ki] for an overview on patterns in permutations and words. It is easy to realize that the optimal algorithm for the classical stacksorting problem has two key properties. First, the elements in the stack are maintained in increasing order, reading from top to bottom. Moreover, the algorithm is *right-greedy*, meaning that it always performs a push operation, unless this violates the previous condition. Note that the expression “right-greedy” refers to the usual (and most natural) representation of this problem, depicted in Figure \[stacksort\_machine\]. Although the classical problem is rather simple, as soon as one allows several stacks connected in series things become much harder. For example, it is known that the permutations that can be sorted using two stacks in series form a class, but in this case the basis is infinite [@M], and still unknown. The enumeration of such permutations is still unknown too. In the attempt of gaining a better understanding of this device, some (simpler) variants have been considered. In his PhD thesis [@W], West considered two passes through a classical stack, which is equivalent to perform a *right-greedy* algorithm on two stacks in series. In [@Sm], Smith considered a decreasing stack followed by an increasing stack. Recently, the authors of [@CCF] considered an even more general device consisting of two stacks in series with a right-greedy procedure, where a restriction on the first stack is given in terms of pattern avoidance. More precisely, the first stack is not allowed to contain an occurrence of a forbidden pattern $\sigma$, for a fixed $\sigma$. West’s device is obtained by choosing $\sigma=21$. The pattern $\sigma=12$ corresponds to the device analyzed in [@Sm], but with a right-greedy (and thus less powerful) algorithm. These devices are called *pattern-avoiding machines*. Other than imposing restrictions on devices and sorting algorithms, one can also allow a larger set of input sequences. Since the notion of pattern itself is inherently more general, it is natural to consider sorting procedures on bigger sets of strings (see [@AAAHH], [@ALW] and [@DK]). Here we pursue this line of research by analyzing the behaviour of pattern-avoiding machines on permutations with repeated letters, which are known as Cayley permutations. A more formal definition of Cayley permutation will be given in Section \[section\_tools\], together with the necessary background and tools. In Section \[section\_pattern\_machine\] we generalize the results of [@CCF] by determining for which patterns $\sigma$ the words that can be sorted by the $\sigma$-machine form a class. In such cases, we also give an explicit description of the basis, which is either a singleton or consists of two patterns. In Section \[section\_operator\], we regard a pattern-avoiding stack as a function that maps an input word into the resulting output, characterizing the patterns $\sigma$ that give rise to a bijective operator. (0,0)(35,20) (-10,15)(7.5,15)(7.5,-5)(17.5,-5)(17.5,15)(35,15) (-10.5,12.5) (35,12.5) (22.5,20)(17,18.5)(14.5,13.5) (10.5,13.5)(8,18.5)(2.5,20) (19,23.5)[$push$]{} (6,23.5)[$pop$]{} (35,17.5)[$\pi_1 \dots \pi_n$]{} (12.5,-10)[$\left\lfloor \begin{array}{c} 2 \\ 1 \end{array} \right\rfloor$]{} (0,0)(35,20) (-15,15)(2.5,15)(2.5,-5)(12.5,-5)(12.5,15)(22.5,15)(22.5,-5)(32.5,-5)(32.5,15)(50,15) (-15.5,12.5) (50,12.5) (27.5,-10)[$\sigma$]{} (7.5,-10)[$\left\lfloor \begin{array}{c} 2 \\ 1 \end{array} \right\rfloor$]{} (-25,17.5)[$$]{} (12.6, 2.5) (22.6, 2.5) (50,17.5)[$\pi_1 \dots \pi_n$]{} (37.5,20)(32,18.5)(29.5,13.5) (5.5,13.5)(3,18.5)(-2.5,20) (25.5,14)(20.5,18.8)(14.5,18.5)(9.5,14) (34,23.5)[$push_{1}$]{} (1,23.5)[$pop$]{} (17.5,23.5)[$push_{2}$]{} Tools and Notations {#section_tools} =================== Let $\mathbb{N}^*$ be the set of strings over the alphabet $\mathbb{N}=\left\lbrace 1,2,\dots \right\rbrace$ of positive integers. Let $x=x_1 \cdots x_n$ and $p=p_1 \cdots p_k$ in $\mathbb{N}^*$, with $k \le n$. The word $x$ *contains* the pattern $p$ if there are indices $i_1<i_2<\cdots <i_k$ such that $x_{i_1} x_{i_2} \cdots x_{i_k}$ is order isomorphic to $p$. This means that, for each pair of indices $u,v$, we have $x_{i_u}<x_{i_v}$ if and only if $p_u<p_v$ and $x_{i_u}=x_{i_v}$ if and only if $p_u=p_v$. In this case, we write $p \le x$ and we say that the subsequence $x_{i_1} x_{i_2} \cdots x_{i_k}$ realizes an *occurrence* of $p$. Otherwise, we say that $x$ *avoids* the pattern $p$. This notion generalizes the usual notion of pattern involvement on permutations. For example, the string $x=142215$ contains the pattern $2113$, since the substring $4225$ is order isomorphic to $2113$. On the other hand, $x$ avoids the pattern $1234$. A *class* is a set of words that is closed downwards with respect to pattern involvement. A class is determined by the minimal set of words it avoids, which is called its *basis*. Now define the set ${\mathcal{C}}$ as the set of strings $x$ on $\mathbb{N}$ where each integer from $1$ to $max(x)$ appears at least once. Following [@MF], we call these strings *Cayley permutations* (they are called *normalized words* in [@DK], and sometimes also surjective words, Fubini words or packed words). Cayley permutations, with respect to their length, are enumerated by sequence $A000670$ in [@Sl]. For example, the only Cayley permutation of length $1$ is the string $1$, there are $3$ Cayley permutations of length $2$, namely $11$, $12$ and $21$, and $13$ Cayley permutations of length $3$, which are $111$, $112$, $121$, $211$, $122$, $212$, $221$, $123$, $132$, $213$, $231$, $312$, $321$. Since only the relative order of the elements is relevant for avoidance and containment, patterns live naturally in the set ${\mathcal{C}}$. More precisely, given $x \in \mathbb{N}^*$, an order-isomorphic string $p \in {\mathcal{C}}$ can be produced by suitably rescaling the elements of $x$, so to remove gaps. For this reason, and because we think that the most natural setting is the one where patterns and words belong to the same set, in the rest of the paper we will work on ${\mathcal{C}}$ rather than on $\mathbb{N}^*$. We denote with ${\mathcal{C}}(p)$ the set of Cayley permutations that avoid the pattern $p$, for $p \in {\mathcal{C}}$; for a set of patterns $B= \{ p_1,\dots,p_k \}$, ${\mathcal{C}}(B)$ will denote the set of Cayley permutations that avoid every pattern $p_1,\cdots,p_k$. $\sigma$-machines on Cayley Permutations {#section_pattern_machine} ======================================== The authors of [@CCF] introduced pattern-avoiding machines on permutations. Here we generalize these devices by allowing Cayley permutations both as inputs and as forbidden patterns. Let $\sigma$ be a Cayley permutation of length at least two. A $\sigma$*-stack* is a stack that is not allowed to contain an occurrence of the pattern $\sigma$ when reading the elements from top to bottom. Before introducing $\sigma$-machines, we recall some useful results. Classical stacksort on $\mathbb{N}^*$ has been discussed in [@DK]. Note that there are two possibilities when defining the analogue of the stacksort algorithm on $\mathbb{N}^*$. One can either allow a letter to sit on a copy of itself in the stack, or force a pop operation if the next element of the input is equal to the top element of the stack. Here we choose the former possibility, leaving the latter for future investigation. This is equivalent to regard a classical stack as a $21$-avoiding stack. The following theorem, proved in [@DK] for $\mathbb{N}^*$, also applies to Cayley permutations. \[hare\_stacksort\] Let $\pi \in {\mathcal{C}}$. Then $\pi$ is sortable using a $21$-stack if and only if $\pi$ avoids $231$. The term $\sigma$*-machine* refers to performing a right-greedy algorithm on two stacks in series: a $\sigma$-stack, followed by a $21$-avoiding stack (see Figure \[stacksort\_machine\]). A Cayley permutation $\pi$ is said to be $\sigma$*-sortable* if the output of the $\sigma$-machine on input $\pi$ is the identity permutation. The set of $\sigma$-sortable permutations is denoted by ${\mathrm{Sort}}(\sigma)$. We use the notation ${s_{\sigma}}(\pi)$ to denote the output of the $\sigma$-stack on input $\pi$. Note that, being ${s_{\sigma}}(\pi)$ the input of the $21$-stack, Theorem \[hare\_stacksort\] guarantees that $\pi \in {\mathrm{Sort}}(\sigma)$ if and only if ${s_{\sigma}}(\pi)$ avoids $231$. This fact will be used repeatedly for the rest of the paper. In [@CCF], the authors provide a characterization of the (permutation) patterns $\sigma$ such that the set of $\sigma$-sortable permutations is a class. The main goal of this section is to prove an analagous result for the set ${\mathcal{C}}$. \[remark\_hat\] For any $\sigma=\sigma_1 \cdots \sigma_k \in {\mathcal{C}}$, if the input Cayley permutation $\pi \in {\mathcal{C}}$ avoids $\sigma^r$, then the restriction of the $\sigma$-stack is never triggered and ${s_{\sigma}}(\pi)=\pi^r$. Otherwise, the leftmost occurrence of $\sigma$ results necessarily in an occurrence of $\hat{\sigma}$ in ${s_{\sigma}}(\pi)$, where $\hat{\sigma}=\sigma_2 \sigma_1 \sigma_3 \sigma_4 \cdots \sigma_k$. From now on, for any $\sigma \in {\mathcal{C}}$, we will use the notation $\hat{\sigma}$ to denote the Cayley permutation obtained from $\sigma$ by interchanging $\sigma_1$ and $\sigma_2$, and $\sigma^r$ to denote the reverse of $\sigma$. \[suff\_class\] Let $\sigma=\sigma_1 \cdots \sigma_k \in {\mathcal{C}}$. If $\hat{\sigma}$ contains $231$, then ${\mathrm{Sort}}(\sigma)={\mathcal{C}}(132,\sigma^r)$. In this case, ${\mathrm{Sort}}(\sigma)$ is a class with basis either $\{ 132,\sigma^r \}$, if $\sigma^r$ avoids $132$, or $\{ 132\}$, otherwise. We start by proving that ${\mathrm{Sort}}(\sigma )\subseteq {\mathcal{C}}(132,\sigma^r )$. Let $\pi \in {\mathrm{Sort}}(\sigma)$. Note that ${s_{\sigma}}(\pi)$ avoids $231$. Suppose by contradiction that $\pi$ contains $\sigma^r$. Then ${s_{\sigma}}(\pi)$ contains $\hat{\sigma}$ due to the Remark \[remark\_hat\] and $\hat{\sigma}$ contains $231$ by hypothesis, which is impossible. Otherwise, if $\pi$ avoids $\sigma^r$, but contains $132$, then ${s_{\sigma}}(\pi)=\pi^r$ due to the same remark. Moreover $\pi^r$ contains $231$ by hypothesis, again a contradiction with $\pi \in {\mathrm{Sort}}(\sigma)$. This proves that ${\mathrm{Sort}}(\sigma )\subseteq {\mathcal{C}}(132,\sigma^r )$. Conversely, suppose that $\pi$ avoids both $132$ and $\sigma^r$. Then what noted above implies that ${s_{\sigma}}(\pi)=\pi^r$, which avoids $132^r=231$ by hypothesis, therefore $\pi$ is $\sigma$-sortable. Thus we also have that ${\mathcal{C}}(132,\sigma^r )\subseteq {\mathrm{Sort}}(\sigma )$, as desired. [$\blacksquare$]{} Next we show that the condition of Theorem \[suff\_class\] is also necessary for ${\mathrm{Sort}}(\sigma)$ in order to be a class. The only exception is given by the pattern $\sigma=12$. \[12\_stack\] ${\mathrm{Sort}}(12)={\mathcal{C}}(213)$. Let $\pi \in {\mathcal{C}}$. Suppose that $\pi$ contains $k$ occurrences of the minimum element $1$ and write $\pi =A_1 1 A_2 1 \cdots A_k 1 A_{k+1}$. It is easy to see that $s_{12}(\pi )=s_{12}(A_1) s_{12}(A_2) \cdots s_{12}(A_k) s_{12} (A_{k+1}) 1 \cdots 1$. Indeed, an occurrence of $1$ can enter the $12$-stack only if the $12$-stack is either empty or it contains only other copies of $1$. Finally, the element $1$ cannot play the role of $2$ in an occurrence of the (forbidden) pattern $12$. Therefore the presence of some copies of $1$ at the bottom of the $12$-stack does not affect the sorting process of the block $A_i$, for each $i$. Now, suppose that $\pi$ contains an occurrence $bac$ of $213$. We prove that $\pi$ is not $12$-sortable by showing that $s_{12}(\pi)$ contains $231$. We proceed by induction on the length of $\pi$. As noted above, we can write $\pi =A_1 1 A_2 1 \cdots A_k 1 A_{k+1}$ and $s_{12}(\pi )=s_{12}(A_1) s_{12}(A_2) \cdots s_{12}(A_k) s_{12} (A_{k+1}) 1 \cdots 1$. Suppose that $b \in A_i$ and $c \in A_j$, for some $i \le j$ (note that $b,c \neq 1$). If $i=j$, then $A_i$ contains an occurrence $bac$ of $213$. Thus $s_{12}(A_i)$ contains $231$ by induction, as wanted. Otherwise, let $i<j$. Then $b \in s_{12}(A_i)$ and $c \in s_{12}(A_j)$ and the elements $b$ and $c$, together with any copy of $1$, realize an occurrence of $231$ in $s_{12}(\pi)$, as desired. Conversely, suppose that $\pi=\pi_1 \cdots \pi_n$ is not sortable, i.e. $s_{12}(\pi )$ contains $231$. We prove that $\pi$ contains $213$. Let $bca$ an occurrence of $231$ in $s_{12}(\pi)$. Note that $b$ has to precede $c$ in $\pi$. This is due to the fact that a non-inversion in the output necessarily comes from a non-inversion in the input, since the stack is $12$-avoiding. However, $b$ is pushed out before $c$ enters. Denote with $x$ the next element of the input when $b$ is extracted. Then we have $x<b$ and also $x \neq c$, since $c>b$. Finally, the triple $bxc$ forms an occurrence of $213$ in $\pi$, as desired. [$\blacksquare$]{} \[necess\_class\] Let $\sigma \in {\mathcal{C}}$ and suppose $\sigma \neq 12$. If $\hat{\sigma}$ avoids $231$, then ${\mathrm{Sort}}(\sigma)$ is not a class. Let $\sigma=\sigma_1 \cdots \sigma_k \in {\mathcal{C}}$, with $k \ge 2$. We show that there are two Cayley permutations $\alpha,\beta$ such that $\alpha \le \beta$, $\beta$ is $\sigma$-sortable and $\alpha$ is not $\sigma$-sortable. This proves that ${\mathrm{Sort}}(\sigma)$ is not closed downwards, as desired. Figure \[table\_alpha\_beta\] shows an example of $\alpha$ and $\beta$ for patterns $\sigma$ of length $2$ and for $\sigma=231$. Now, suppose that $\sigma$ has length at least $3$ and $\sigma \neq 231$. Then the Cayley permutation $\alpha=132$ is not $\sigma$-sortable. Indeed, ${s_{\sigma}}(\alpha)=\alpha^r=231$, since $\alpha$ avoids $\sigma^r$. Next we define the permutation $\beta$ as follows. $\sigma$ $\alpha \notin {\mathrm{Sort}}(\sigma)$ $\beta \ge \alpha, \beta \in {\mathrm{Sort}}(\sigma)$ ---------- ----------------------------------------- ------------------------------------------------------- 11 132 3132 21 132 35241 231 1324 361425 - Suppose that $\sigma_1$ is the strict minimum of $\sigma$, i.e. $\sigma_1=1$ and $\sigma_i \ge 2$ for each $i \ge 2$. Define $\beta=\sigma'_k \cdots \sigma'_3 1 \sigma'_2 \sigma'_1$, where $\sigma'_i=\sigma_i+1$ for each $i$. Note that $\beta \in {\mathcal{C}}$ and $1 \sigma'_2 \sigma'_1$ is an occurrence of $132$ in $\beta$. We prove that $\beta$ is $\sigma$-sortable by showing that ${s_{\sigma}}(\beta)$ avoids $231$. The action of the $\sigma$-stack on input $\beta$ is depicted in Figure \[sorting\_alpha\]. The first $k-1$ elements of $\beta$ are pushed into the $\sigma$-stack, since $\sigma$ has length $k$. Then the $\sigma$-stack contains $1 \sigma'_3 \cdots \sigma'_k$, reading from top to bottom, and the next element of the input is $\sigma'_2$. Note that $\sigma'_2>1$, whereas $\sigma_1<\sigma_2$, therefore $\sigma'_2 1 \sigma'_3 \cdots \sigma'_k$ is not an occurrence of $\sigma$ and $\sigma'_2$ is pushed. The next element of the input is now $\sigma'_1$. Here $\sigma'_1 \sigma'_2 \sigma'_3 \cdots \sigma'_k$ is an occurrence of $\sigma$, thus we have to pop $\sigma'_2$ before pushing $\sigma'_1$. After the pop operation, the $\sigma$-stack contains $1 \sigma'_3 \cdots \sigma'_k$. Again $\sigma'_1>1$, whereas $\sigma_1<\sigma_2$, therefore $\sigma'_1$ is pushed. The resulting string is ${s_{\sigma}}(\beta)=\sigma'_2 \sigma'_1 1 \sigma'_3 \sigma'_4 \cdots \sigma'_k$. We show that ${s_{\sigma}}(\beta)$ avoids $231$. Note that $\sigma'_2 \sigma'_1 \sigma'_3 \sigma'_4 \cdots \sigma'_k \simeq \hat{\sigma}$ avoids $231$ by hypothesis. Moreover, the element $1$ cannot be part of an occurrence of $231$, because $\sigma'_2 > \sigma'_1$ and $1$ is strictly less than the other elements of $\beta$. Therefore ${s_{\sigma}}(\beta)$ avoids $231$, as desired. - Otherwise, suppose that $\sigma_1$ is not the strict minimum of $\sigma$, i.e. either $\sigma_1 \neq 1$ or $\sigma_i=1$ for some $i \ge 2$. Define $\beta=\sigma''_k \cdots \sigma''_2 1 \sigma''_1 2$, where $\sigma''_i=\sigma_i+2$ for each $i$. Note that $\beta \in {\mathcal{C}}$ and $1 \sigma''_2 2$ is an occurrence of $132$ in $\beta$. Consider the action of the $\sigma$-stack on $\beta$. Again the first $k-1$ elements of $\beta$ are pushed into the $\sigma$-stack. Then the $\sigma$-stack contains $\sigma''_2 \cdots \sigma''_k$, reading from top to bottom, and the next element of the input is $1$. Note that $1 \sigma''_2 \cdots \sigma''_k$ is not an occurrence of $\sigma$. Indeed $1<\sigma''_i$ for each $i$, while $\sigma_1$ is not the strict minimum of $\sigma$ by hypothesis. Therefore $1$ enters the $\sigma$-stack. The next element of the input is then $\sigma''_1$, which realizes an occurrence of $\sigma$ together with $\sigma''_2 \cdots \sigma''_k$. Thus $1$ and $\sigma''_2$ are extracted before $\sigma''_1$ is pushed. Finally, the last element of the input is $2$. Again $2$ can be pushed into the $\sigma$-stack because $2$ is strictly smaller than every element in the $\sigma$-stack, whereas $\sigma_1$ is not the strict minimum of $\sigma$ by hypothesis. The resulting string is ${s_{\sigma}}(\beta) = 1 \sigma''_2 2 \sigma''_1 \sigma''_3 \cdots \sigma''_k$. Note that $\sigma''_2 \sigma''_1 \sigma''_3 \cdots \sigma''_k \simeq \hat{\sigma}$ avoids $231$ by hypothesis. Finally, it is easy to realize that the elements $1$ and $2$ cannot be part of an occurrence of $231$, similarly to the previous case. This completes the proof. [$\blacksquare$]{} [|c|c|]{} (-15,27.5)[$$]{} (12.6, 12.5) (40,27.5)[$\sigma'_2 \sigma'_1$]{} (-10,0)[**Step** $\boldsymbol{1}$]{} & (-15,27.5)[$$]{} (12.6, 12.5) (40,27.5)[$\sigma'_1$]{} (-10,0)[**Step** $\boldsymbol{2}$]{} \ (-15,27.5)[${\sigma'_2}$]{} (12.6, 12.5) (40,27.5)[$\sigma'_1$]{} (-10,0)[**Step** $\boldsymbol{3}$]{} & (-15,27.5)[${\sigma'_2}$]{} (12.6, 12.5) (40,27.5)[$$]{} (-10,0)[**Step** $\boldsymbol{4}$]{} \ \[class\_vs\_nonclass\] Let $\sigma \in {\mathcal{C}}$ be a Cayley permutation of length $3$ or more. Then the set of $\sigma$-sortable permutation ${\mathrm{Sort}}(\sigma)$ is not a permutation class if and only if $\hat{\sigma}$ avoids $231$. Otherwise, if $\hat{\sigma}$ contains $231$, ${\mathrm{Sort}}(\sigma)$ is a class with basis either $\{ 132,\sigma^r \}$, if $\sigma^r$ avoids $132$, or $\{ 132\}$, otherwise. Cayley permutations avoiding any permutation pattern of length $3$ are enumerated by sequence $A226316$ of [@Sl]. We end this section by analyzing the $21$-machine. The $11$-machine will be discussed in Section \[section\_operator\], thus completing the analysis of the $\sigma$-machines on Cayley permutations for patterns $\sigma$ of length $2$. The classical permutation analogue of the $21$-machine consists in applying a right-greedy algorithm on two stacks in series, which is exactly the (well known) case of the West $2$-stack sortable permutations (see [@W2]). In this case, although sortable permutations do not form a permutation class, we can describe them efficiently in terms of avoidance of barred patterns. [@W2]\[west\_2stack\] A permutation $\pi$ is not $21$-sortable if and only if $\pi$ contains $2341$ or $\pi$ contains an occurrence of the barred pattern $3 \bar{5}241$, i.e. an occurrence $3241$ which is not part of an occurrence of $35241$. The previous theorem can be reformulated in terms of a more general notion of pattern, which will be useful later when dealing with Cayley permutations. A *mesh pattern* of length $k$ is a pair $(\tau,A)$, where $\tau$ is a permutation of length $k$ and $A \subseteq \left[ 0,k \right] \times \left[ 0,k \right]$ is a set of pairs of integers. The elements of $A$ identify the lower left corners of shaded squares in the plot of $\tau$ (see Figure \[mesh\_west\]). An occurrence of the mesh pattern $(\tau,A)$ in the permutation $\pi$ is then an occurrence of the classical pattern $\tau$ in $\pi$ such that no elements of $\pi$ are placed into a shaded square of $A$. Note that the barred pattern $3 \bar{5} 241$ is equivalent to the mesh pattern $\mathcal{W}= (3241,\{(1,4) \})$ depicted in Figure \[mesh\_west\]. Now, in order to prove an analogous characterization for the $12$-machine on Cayley permutations, we need to adapt the definition of mesh pattern to strings that may contain repeated elements. In other words, we have to allow the shading of squares that correspond to repeated elements. Instead of giving a formal definition, we refer to the example depicted in Figure \[mesh\_west\]. We will use the term *Cayley-mesh pattern* to denote mesh patterns on Cayley permutations. \[increas\_stack\] Let $\pi=\pi_1 \cdots \pi_n \in Cay$. Suppose that $\pi_i < \pi_j$, for some $i < j$. Then $\pi_i$ precedes $\pi_j$ in $s_{21}(\pi)$. It follows from the definition of $21$-stack. [$\blacksquare$]{} A Cayley permutation $\pi$ is not $21$-sortable if and only if $\pi$ contains $2341$ or $\pi$ contains the Cayley-mesh pattern $\mathcal{Z}$ depicted in Figure \[mesh\_west\]. In particular, ${\mathrm{Sort}}(21)$ is not a class. For example, the $21$-sortable Cayley permutation $34241$ contains the non-sortable pattern $3241$. We can basically repeat the argument used by West for classical permutations. We just have to incorporate the new shaded box, which corresponds to an occurrence of $3241$ that is part of an occurrence of $34241$. We sketch the proof anyway for completeness. Let $\pi \in Cay$ and suppose that $\pi$ is $21$-sortable. Suppose by contradiction that $\pi$ contains an occurrence $bcda$ of $2341$ and consider the action of the $21$-stack on $\pi$. By Lemma \[increas\_stack\], $b$ is extracted from the $21$-stack before $c$ enters. Similarly, $c$ is extracted before $d$ enters. Thus $s_{21}(\pi)$ contains the occurrence $bca$ of $231$, against $\pi$ sortable. Otherwise, suppose that $\pi$ contains an occurrence $cbda$ of $3241$. We show that there is an element $x$ between $c$ and $b$ in $\pi$ such that $x \ge d$. If $x<c$ for each $x$ in between $c$ and $b$, then $b$ is pushed into the $21$-stack before $c$ is popped. This results in the occurrence $bca$ of $231$ in $s_{21}(\pi)$, a contradiction with $\pi$ $21$-sortable. Otherwise, suppose there is at least one element $x$ between $c$ and $b$ in $\pi$, with $x \ge c$. If $x=c$, we can repeat the same argument with $xbda$ instead of $cbda$. If $c<x<d$, then $cxda \simeq 2341$, which is impossible due to what said in the above case. Therefore it has to be $x \ge d$, as desired. Conversely, suppose that $\pi$ is not $12$-sortable. Equivalently, let $bca$ be an occurrence of $231$ in $s_{21}(\pi)$. We show that either $\pi$ contains $2341$ or $\pi$ contains an occurrence $cbda$ of $3241$ such that $x<d$ for each $x$ between $c$ and $b$ in $\pi$. Observe that $a$ follows $c$ and $b$ in $\pi$ due to Lemma \[increas\_stack\]. Suppose that $b$ comes before $c$ in $\pi$. Note that $c$ is extracted from the $21$-stack before $a$ enters. Let $d$ the next element of the input when $c$ is extracted. Then $d>c$ and $bcda$ is an occurrence of $2341$, as wanted. Otherwise, suppose that $b$ follows $c$ in $\pi$, and thus $\pi$ contains $cba$. Since $c$ is not extracted before $b$ enters, it has to be $x \le c$ for each $x$ between $c$ and $b$ in $\pi$. Moreover, $c$ is extracted before $a$ enters. When $c$ is extracted, the next element $d$ of the input is such that $d>c$. This results in an occurrence $cbda$ of $3241$ with the desired propriety. [$\blacksquare$]{} Enumerate the $21$-sortable Cayley permutations. The initial terms of the sequence are $1,3,13,73,483,3547,27939,231395$ (not in $\cite{Sl}$). $\sigma$-stacks as Operators {#section_operator} ============================ In this section we regard $\sigma$-stacks as operators. Let $\sigma \in {\mathcal{C}}$ and define the map ${\mathcal{S}^{\sigma}}: {\mathcal{C}}\mapsto {\mathcal{C}}$, where ${\mathcal{S}^{\sigma}}(\pi)={s_{\sigma}}(\pi)$, for each $\pi \in {\mathcal{C}}$. We are interested in the behavior of the map ${\mathcal{S}^{\sigma}}$, for a fixed $\sigma \in {\mathcal{C}}$. This line of inquiry for stacksort operators is not new in the literature. More generally, suppose to perform a deterministic sorting procedure. Then it is natural to consider the map $\mathcal{S}$ that associates an input string $\pi$ to the (uniquely determined) output of the sorting process. Some of the problems that arise are the following. - Determine the *fertility* of a string, which is the number of its pre-images under $\mathcal{S}$. Fertilty under classical stacksort has been recently investigated by Defant (see [@D]). - Determine the image of $\mathcal{S}$, i.e. the strings with positive fertility. These are often called *sorted permutations* (see [@BM]). We start by discussing the case $\sigma=11$. Here we provide a useful decomposition that allows to determine explicitly the image ${\mathcal{S}^{\sigma}}(\pi)$ of any $\pi \in {\mathcal{C}}$. From now on, we denote with $\mathcal{R}$ the *reverse* operator, i.e. $\mathcal{R}(\pi)=\pi^r$, for each $\pi \in {\mathcal{C}}$. \[11\_decom\] Let $\sigma=11$ and let $\pi=\pi_1 \cdots \pi_n$ be a Cayley permutation. Suppose that $\pi$ contains $k+1$ occurrences $\pi_1,\pi_1^{(1)},\dots, \pi_1^{(k)}$ of $\pi_1$, for some $k \ge 0$. Write $\pi=\pi_1 B_1 \pi_1^{(1)} B_2 \cdots \pi_1^{(k)} B_k$. Then $$\mathcal{S}^{11}(\pi)= \mathcal{S}^{11}(B_1) \pi_1 \mathcal{S}^{11}(B_2) \pi_1^{(1)} \cdots \mathcal{S}^{11}(B_k) \pi_1^{(k)}.$$ Consider the action of the $11$-stack on input $\pi$. Since $x \neq \sigma_1$ for each $x \in B_1$, the sorting process of $B_1$ is not affected by the presence of $\sigma_1$ at the bottom of the $11$-stack. Then, when the next element of the input is the second occurrence $\sigma_1^{(1)}$ of $\sigma_1$, the $11$-stack is emptied, since $\sigma_1 \sigma_1^{(1)}$ is an occurrence of the forbidden $11$. The first elements of $\mathcal{S}^{11}(\pi)$ are thus $\mathcal{S}^{11}(B_1) \sigma_1$. Finally, $\sigma_1^{(1)}$ is pushed into the (empty) $11$-stack and the same argument can be repeated. [$\blacksquare$]{} \[11\_bij\] Let $\sigma=11$. Then $(\mathcal{R} \circ \mathcal{S}^{11})$ is an involution on ${\mathcal{C}}$. Moreover, $\mathcal{S}^{11}$ is a length-preserving bijection on ${\mathcal{C}}$. Therefore, the number of $11$-sortable Cayley permutations of length $n$ is equal to the number of $231$-avoiding Cayley permutations of length $n$. We proceed by induction on the length of the input permutation. Let $\pi=\pi_1 \cdots \pi_n$ a Cayley permutation of length $n$. The case $n=1$ is trivial. If $n \ge 2$, write $\pi=\pi_1 B_1 \pi_1^{(1)} B_2 \cdots \pi_1^{(k)} B_k$ as in the previous lemma. Then, using the same lemma and the inductive hypothesis: $$\begin{split} \left[ \mathcal{R} \circ \mathcal{S}^{11} \right]^2(\pi)=& \\ \left[ \mathcal{R} \circ \mathcal{S}^{11} \right]^2 \left( \pi_1 B_1 \pi_1^{(1)} B_2 \cdots \pi_1^{(k)} B_k \right) =&\\ \left[ \mathcal{R} \circ \mathcal{S}^{11} \circ \mathcal{R} \right] \left( \mathcal{S}^{11}(B_1) \pi_1 \mathcal{S}^{11}(B_2) \pi_1^{(1)} \cdots \mathcal{S}^{11}(B_k) \pi_1^{(k)} \right) =& \\ \left[ \mathcal{R} \circ \mathcal{S}^{11} \right] \left( \pi_1^{(k)} \mathcal{R}(\mathcal{S}^{11}(B_k)) \cdots \pi_1^{(1)} \mathcal{R}(\mathcal{S}^{11}(B_2)) \pi_1 \mathcal{R}(\mathcal{S}^{11}(B_1)) \right) =& \\ \mathcal{R} \left( \mathcal{S}^{11}(\mathcal{R}(\mathcal{S}^{11}(B_k))) \pi_1^{(k)} \cdots \mathcal{S}^{11}(\mathcal{R}(\mathcal{S}^{11}(B_2))) \pi_1^{(1)} \mathcal{S}^{11}(\mathcal{R}(\mathcal{S}^{11}(B_1))) \pi_1 \right) =& \\ \pi_1 \left[ \mathcal{R} \circ \mathcal{S}^{11} \right]^2 (B_1) \pi_1^{(1)} \left[ \mathcal{R} \circ\mathcal{S}^{11} \right]^2(B_2) \cdots \pi_1^{(k)} \left[ \mathcal{R} \circ \mathcal{S}^{11} \right]^2(B_k) =& \\ \pi_1 B_1 \pi_1^{(1)} B_2 \cdots \pi_1^{(k)} B_k = \pi & \\ \end{split}$$ Therefore we have $(\mathcal{R} \circ \mathcal{S}^{11})^2(\pi)=\pi$, as desired. Finally, the reverse map $\mathcal{R}$ is bijective, thus $\mathcal{S}^{11}$ is a bijection on ${\mathcal{C}}$ with inverse $\mathcal{R} \circ \mathcal{S}^{11} \circ \mathcal{R}$. [$\blacksquare$]{} Theorem \[11\_bij\] provides a constructive description of the set ${\mathrm{Sort}}(11)$. Indeed, since ${\mathrm{Sort}}(11)=\mathcal{R} \circ \mathcal{S}^{11} \circ \mathcal{R} (\mathcal{C}(231))$, every $11$-sortable permutation $\pi$ is obtained from a $231$-avoiding Cayley permutation by applying $\mathcal{R} \circ \mathcal{S}^{11} \circ \mathcal{R}$. Next we generalize the above result by providing a characterization of all patterns $\sigma$ such that ${\mathcal{S}^{\sigma}}$ is bijective on ${\mathcal{C}}$. The main tool is an encoding of the action of ${\mathcal{S}^{\sigma}}$ as a Dyck path. A *Dyck path* is a path in the discrete plane $\mathbb{Z}\times \mathbb{Z}$ starting at the origin, ending on the $x$-axis, never falling below the $x$-axis and using two kinds of steps (of length $1$), namely up steps ${\mathtt{U}}=(+1,+1)$ and down steps ${\mathtt{D}}=(+1,-1)$. The *height* of a step is its final ordinate. For each up step ${\mathtt{U}}$, there is a unique *matching* step ${\mathtt{D}}$ defined as the first ${\mathtt{D}}$ step after ${\mathtt{U}}$ with height $1$ less than ${\mathtt{U}}$. A *valley* of a Dyck path is an occurrence of two consecutive steps ${\mathtt{D}}{\mathtt{U}}$. A *peak* is an occurrence of two consecutive steps ${\mathtt{U}}{\mathtt{D}}$. The *length* of a Dyck path is the total number of its steps. See Figure \[Dyck\_path\] for an example of Dyck path. It is well known that Dyck paths, according to the semilength, are enumerated by Catalan numbers (sequence $A000108$ in [@Sl]). A *labeled Dyck path* is a Dyck path where each step has a label. In this paper we consider labeled Dyck paths where the label of each up step is the same as the label of its matching down step. Therefore we can represent a labeled Dyck path $\mathcal{P}$ as a pair $\mathcal{P}=(P,\pi)$, where $P$ is the underlying Dyck path and $\pi$ is the string obtained by reading the labels of the up steps from left to right. Given an unlabeled Dyck path $P$ of length $2n$, the *reverse* path $\mathcal{R}(P)$ of $P$ is obtained by taking the symmetric path with respect to the vertical line $x=n$. Now let $\sigma \in {\mathcal{C}}$ and suppose we are applying ${\mathcal{S}^{\sigma}}$ to the input Cayley permutation $\pi$, i.e. we are sorting $\pi$ using a $\sigma$-stack. Then define a labeled Dyck path ${\mathcal{P}_{\sigma}}(\pi)$ as follows. - Insert an up step ${\mathtt{U}}$ labeled $a$ whenever the algorithm pushes an element $a$ into the $\sigma$-stack. - Insert a down step ${\mathtt{D}}$ labeled $a$ whenever the algorithm pops an element $a$ from the $\sigma$-stack. In other words, we define $P_{\sigma}(\pi)$ as the unlabeled Dyck path obtained by recording the push operations of the $\sigma$-stack with ${\mathtt{U}}$ and the pop operations with ${\mathtt{D}}$. Then ${\mathcal{P}_{\sigma}}(\pi)=(P_{\sigma}(\pi),\pi)$. Note that $P_{\sigma}(\pi)$ is a Dyck path. Indeed the number of push and pop operations performed when processing $\pi$ is the same, therefore the number of ${\mathtt{U}}$ steps matches the number of ${\mathtt{D}}$ steps (and thus the path ends on the $x$-axis). Moreover, the path cannot go below the $x$-axis, since this would correspond to performing a pop operation when the $\sigma$-stack is empty, which is not possible. An example of this construction, when $\sigma=11$, is depicted in Figure \[Dyck\_path\]. Some basic properties of ${\mathcal{P}_{\sigma}}(\pi)$ are listed in the following Lemma. \[path\_prop\] Let $\sigma \in {\mathcal{C}}$. Let $\pi=\pi_1 \cdots \pi_n$ be a Cayley permutation of length $n$ and let ${\mathcal{P}_{\sigma}}(\pi)=(P_{\sigma}(\pi),\pi)$. Then: 1. The input $\pi$ is obtained by reading the labels of the up steps of $P_{\sigma}(\pi)$ from left to right. The output ${s_{\sigma}}(\pi)$ is obtained by reading the labels of the down steps from left to right. 2. The height of $P_{\sigma}(\pi)$ after each up (respectively down) step is equal to the number of elements contained in the $\sigma$-stack after having performed the corresponding push (respectively pop) operation. 3. The $\sigma$-stack is empty after a a pop operation if and only if the corresponding ${\mathtt{D}}$ step of $P_{\sigma}(\pi)$ is a return on the $x$-axis. In other words, the decomposition of $\pi$ considered in Lemma \[11\_decom\] corresponds to the decomposition of $P_{\sigma}(\pi)$ obtained by considering the returns on the $x$-axis. 4. The labels of the down steps are uniquely determined by the labels of the up steps. Conversely, the labels of the down steps uniquely determine the labels of the up steps. More precisely, matching steps have the same label. Indeed the element pushed into the $\sigma$-stack by an up step is then popped by the matching down step. 5. Let ${\mathtt{D}}{\mathtt{U}}$ be a valley in $P_{\sigma}(\pi)$. Let $a$ be the label of ${\mathtt{D}}$ and $b$ the label of ${\mathtt{U}}$. Then $b$ plays the role of $\sigma_1$ in an occurrence of $\sigma$ that triggers the restriction of the $\sigma$-stack, whereas $a$ plays the role of $\sigma_2$ in such occurrence. Moreover the number of valleys of $P_{\sigma}(\pi)$ is equal to the number of elements of $\pi$ that trigger the restriction of the $\sigma$-stack. 6. If $\sigma_1=\sigma_2$, then, for each valley ${\mathtt{D}}{\mathtt{U}}$, the labels of ${\mathtt{D}}$ and ${\mathtt{U}}$ are the same. \[dyck\_reverse\] Let $\sigma=\sigma_1 \cdots \sigma_k \in {\mathcal{C}}$. Let $\pi=\pi_1 \cdots \pi_n \in {\mathcal{C}}$ and let $\gamma=\mathcal{R}({\mathcal{S}^{\sigma}}(\pi))$. Consider the two labeled Dyck paths ${\mathcal{P}_{\sigma}}(\pi)=(P_{\sigma}(\pi),\pi)$ and ${\mathcal{P}_{\sigma}}(\gamma)=(P_{\sigma}(\gamma),\gamma)$. 1. If $\sigma_1=\sigma_2$, then $P_{\sigma}(\pi)=\mathcal{R} (P_{\sigma}(\gamma))$. 2. If $P_{\sigma}(\pi)=\mathcal{R} (P_{\sigma}(\gamma))$, then $(\mathcal{R} \circ {\mathcal{S}^{\sigma}})^2 (\pi)=\pi$. <!-- --> 1. Suppose that $\sigma_1=\sigma_2$. We proceed by induction on the number of valleys of $P_{\sigma}(\pi)$. If $P_{\sigma}(\pi)$ has zero valleys, then $\pi$ avoids $\mathcal{R}(\sigma)$ by point $5.$ of Lemma \[path\_prop\]. Therefore ${\mathcal{S}^{\sigma}}(\pi)=\mathcal{R}(\pi)$ and $\gamma=\mathcal{R}^2(\pi) = \pi$. Since $P_{\sigma}(\pi)={\mathtt{U}}^n {\mathtt{D}}^n$ is a pyramid, the thesis follows immediately since each pyramid is equal to its reverse. Now suppose that $P_{\sigma}(\pi)$ has at least one valley. Let $P_{\sigma}(\pi)=p_1 \cdots p_{2n}$ and write $P_{\sigma}(\pi)={\mathtt{U}}^i {\mathtt{U}}^j {\mathtt{D}}^j {\mathtt{U}}^l {\mathtt{D}}Q$, where the steps $p_{i+2j}$ and $p_{i+2j+1}$ form the leftmost valley and $Q=p_{i+2j+l+2} \cdots p_n$ is the remaining suffix of $P_{\sigma}(\pi)$ (see Figure \[figure\_bij\_cases\]). Note that the label of both $p_{i+2j}$ and $p_{i+2j+1}$ is equal to $\pi_{i+1}$ because of points $4.$, $5.$ and $6.$ of Lemma \[path\_prop\]. Point $5.$ also implies that $p_{i+2j+1}$ plays the role of $\sigma_1$ in an occurrence of $\sigma$ that triggers the restriction of the $\sigma$-stack. More precisely, immediately after the push of $\pi_{i+j}$ (i.e. after the up step $p_{i+j}$ in $P_{\sigma}(\pi)$), $\pi_{i+j+1}$ is the next element of the input. Since the next segment of the path is ${\mathtt{D}}^j$, $j$ pop operations are performed before pushing $\pi_{i+j+1}$. This means that the element $\pi_{i+1}$, corresponding to the last down step, plays the role of $\sigma_2$ in an occurrence of $\sigma$, while $\pi_{i+j+1}$ plays the role of $\sigma_1$. Moreover, there are $k-2$ elements in the $\sigma$-stack that play the role of $\sigma_3,\dots,\sigma_k$. Since the elements in the $\sigma$-stack correspond to the labels of the initial prefix ${\mathtt{U}}^i$, $\pi_1 \cdots \pi_i$ contains an occurrence of $\sigma_k \cdots \sigma_3$ (claim I). Then, after $j$ pop operations are performed, the $\sigma$-stack contains $\pi_i \cdots \pi_1$, reading from top to bottom, and the elements $\pi_{i+j+1},\pi_{i+j+2}, \dots, \pi_{i+j+l}$ are pushed (claim II). Now, write $\pi= \underbrace{\pi_1 \cdots \pi_i}_{A} \ \underbrace{\pi_i+1 \cdots \pi_{i+j}}_{B} \ \underbrace{\pi_{i+j+1} \cdots \pi_{i+j+l}}_{C} \ \underbrace{\pi_{i+j+l+1} \cdots \pi_n}_{D}$, where the elements of $A$ correspond to the initial prefix ${\mathtt{U}}^i$ of $P_{\sigma}(\pi)$, $B$ corresponds to ${\mathtt{U}}^j$, $C$ to ${\mathtt{U}}^l$ and $D$ to the remaining up steps. Consider the string $A C D = \pi_1 \cdots \pi_i \pi_{i+j+1} \cdots \pi_n$ obtained by removing the segment $B=\pi_{i+1} \cdots \pi_{i+j}$ from $\pi$. Let $\tilde{\pi}$ the Cayley word that is order isomorphic to $A C D$, i.e. obtained by suitably rescaling the elements of $A C D$, if necessary. Note that ${\mathcal{P}_{\sigma}}(\tilde{\pi})$ is obtained from ${\mathcal{P}_{\sigma}}(\pi)$ by cutting out the pyramid ${\mathtt{U}}^j {\mathtt{D}}^j$, which corresponds to the removed segment $B$. This is because the elements contained in the $\sigma$-stack after having pushed $\pi_i$ are exactly the same as the elements contained in the $\sigma$-stack after having pushed $\pi_{i+j+1}$, thus we can safely cut out the pyramid ${\mathtt{U}}^j {\mathtt{D}}^j$ without affecting the sorting procedure. Therefore ${\mathcal{S}^{\sigma}}(\pi)=\mathcal{R}(B) {\mathcal{S}^{\sigma}}(\tilde{\pi})$ and $\gamma=\mathcal{R}({\mathcal{S}^{\sigma}}(\pi))=\mathcal{R}({\mathcal{S}^{\sigma}}(\tilde{\pi}))B$. Now, since $P_{\sigma}(\tilde{\pi})$ has one valley less than $P_{\sigma}(\pi)$, by inductive hypothesis $P_{\sigma}(\tilde{\pi})=\mathcal{R}(P_{\sigma}( \tilde{\gamma}))$, where $\tilde{\gamma}=\mathcal{R}({\mathcal{S}^{\sigma}}(\tilde{\pi}))$. The only difference bewteen $P_{\sigma}(\pi)$ and $P_{\sigma}(\tilde{\pi})$ is the removed pyramid ${\mathtt{U}}^j {\mathtt{D}}^j$. Therefore, if we show that $P_{\sigma}(\gamma)$ is obtained from $P_{\sigma}(\tilde{\gamma})$ by reinserting the same pyramid ${\mathtt{U}}^j {\mathtt{D}}^j$ in the same place, the thesis follows. We have $\gamma=\mathcal{R}({\mathcal{S}^{\sigma}}(\tilde{\pi}))B$ and $\tilde{\gamma}=\mathcal{R}({\mathcal{S}^{\sigma}}(\tilde{\pi}))$. Consider the last push performed by the $\sigma$-stack when processing $\tilde{\gamma}$, which corresponds to the last up step of ${\mathcal{P}_{\sigma}}(\tilde{\gamma})$. Note that, since $P_{\sigma}(\tilde{\pi})=R(P_{\sigma}( \tilde{\gamma})$, this is also the first down step of $P_{\sigma}(\tilde{\pi})$, and thus the first pop performed when processing $\tilde{\pi}$. Therefore the elements contained in the $\sigma$-stack after the last push performed while processing $\tilde{\gamma}$ are $\pi_{i+j+l} \cdots \pi_{i+j+1} \pi_i \cdots \pi_1$, reading from top top bottom. If we sort $\gamma$ instead of $\tilde{\gamma}$, we have to process the additional segment $B$. Now, the first element of $B$ is $\pi_{i+1}$. Since the same happened when sorting $\pi$ (see claim I), $\pi_{i+1}$ realizes an occurrence of $\sigma$ together with $\pi_{i+j+1}$ (which plays the role of $\sigma_2$) and other $k-2$ elements in $\pi_1 \cdots \pi_i$. The only difference is that, contrary to what happened when sorting $\pi$, the role of $\pi_{i+1}$ and $\pi_{i+j+1}$ are interchanged: here the hypothesis $\sigma_1=\sigma_2$ is relevant. As a result, before pushing the first element $\pi_{i+1}$ of $B$, we have to pop each element of the $\sigma$-stack up to $\pi_{i+j+1}$, $\pi_{i+j+1}$ included. After that, the $\sigma$-stack contains $\pi_i \cdots \pi_1$, reading from top to bottom. Therefore we can push $\pi_{i+1}=\pi_{i+j+1}$ and the remaining elements of $B$ because of claim II. This means that $P_{\sigma}(\gamma)$ is obtained by inserting a pyramid ${\mathtt{U}}^j {\mathtt{D}}^j$ immediately before the last $i$ down steps of $P_{\sigma}(\tilde{\gamma})$, as desired. 2. By hypothesis, $P_{\sigma}(\gamma)=\mathcal{R}(P_{\sigma}(\pi))$, therefore the word $w$ obtained by reading the labels of the down steps of $P_{\sigma}(\gamma)$ (from left to right) is $w=\mathcal{R}(\pi)$. By definition of ${\mathcal{P}_{\sigma}}(\gamma)$, we also have $w={\mathcal{S}^{\sigma}}(\gamma)$. Therefore $\mathcal{R}(\pi)={\mathcal{S}^{\sigma}}(\gamma)={\mathcal{S}^{\sigma}}(\mathcal{R}({\mathcal{S}^{\sigma}}(\pi)))$ and the thesis follows by applying the reverse operator to both sides of the equality. [$\blacksquare$]{} (0,0) – (10,0); (0,0) – (4,4); (4,4) – (7,1); (7,1) – (8,2); (8,2) – (10,0); (0.5,0.5) – (9.5,0.5); (1.5,1.5) – (6.5,1.5); (2.5,2.5) – (5.5,2.5); (3.5,3.5) – (4.5,3.5); (7.5,1.5) – (8.5,1.5); at (0,0) [$\bullet$]{}; at (1,1) [$\bullet$]{}; at (2,2) [$\bullet$]{}; at (3,3) [$\bullet$]{}; at (4,4) [$\bullet$]{}; at (5,3) [$\bullet$]{}; at (6,2) [$\bullet$]{}; at (7,1) [$\bullet$]{}; at (8,2) [$\bullet$]{}; at (9,1) [$\bullet$]{}; at (10,0) [$\bullet$]{}; at (0.75,0.75) [$4$]{}; at (1.75,1.75) [$2$]{}; at (2.75,2.75) [$1$]{}; at (3.75,3.75) [$3$]{}; at (4.75,3.25) [$3$]{}; at (5.75,2.25) [$1$]{}; at (6.75,1.25) [$2$]{}; at (7.75,1.75) [$2$]{}; at (8.75,1.25) [$2$]{}; at (9.75,0.25) [$4$]{}; (0,0) – (6,6); (6,6) – (9,3); (9,3) – (14,8); (14,8) – (15,7); at (0,0) [$\bullet$]{}; at (2,2) [$\bullet$]{}; at (3,3) [$\bullet$]{}; at (4,4) [$\bullet$]{}; at (5,5) [$\bullet$]{}; at (6,6) [$\bullet$]{}; at (7,5) [$\bullet$]{}; at (8,4) [$\bullet$]{}; at (9,3) [$\bullet$]{}; at (10,4) [$\bullet$]{}; at (13,7) [$\bullet$]{}; at (14,8) [$\bullet$]{}; at (15,7) [$\bullet$]{}; at (6,4.5) [${\mathtt{U}}^j {\mathtt{D}}^j$]{}; at (2.5,2.5)[$\pi_{i}$]{}; at (5.5,5.5)[$\pi_{i+j}$]{}; at (13.5,7.5)[$\pi_{i+j+l}$]{}; at (0.5,0.5)[$\pi_{1}$]{}; at (3.5,3.5)[$\pi_{i+1}$]{}; at (8.5,3.5)[$\pi_{i+1}$]{}; at (9.5,3.5)[$\pi_{i+j+1}$]{}; (3.5,3.5) – (8.5,3.5); (1,1) – (2,2); (3,3) – (9,3); \[Dyck\_path\] \[sigma\_bij\] Let $\sigma=\sigma_1 \cdots \sigma_k \in {\mathcal{C}}$. Then ${\mathcal{S}^{\sigma}}$ is bijective if and only if $\sigma_1=\sigma_2$. Suppose that $\sigma_1 \neq \sigma_2$. Then $\hat{\sigma} \neq \sigma$, thus also $\mathcal{R}(\sigma) \neq \mathcal{R}(\hat{\sigma})$. Finally, ${\mathcal{S}^{\sigma}}(\mathcal{R}(\sigma))=\hat{\sigma}={\mathcal{S}^{\sigma}}((\mathcal{R}(\hat{\sigma})))$, therefore ${\mathcal{S}^{\sigma}}$ is not injective. Conversely, suppose that $\sigma_1=\sigma_2$. By Lemma \[dyck\_reverse\], we have that $(\mathcal{R} \circ {\mathcal{S}^{\sigma}})^2$ is the identity on ${\mathcal{C}}$, therefore $\mathcal{R} \circ \mathcal{{\mathcal{S}^{\sigma}}}$ is bijective. Finally, since the reverse map $\mathcal{R}$ is bijective, ${\mathcal{S}^{\sigma}}$ is a bijection too, as desired. [$\blacksquare$]{} Final Remarks ============= In this paper we continued the analysis of pattern-avoiding machines by analyzing their generalization to permutations with repeated letters. We provided a description of $\sigma$-sortable Cayley permutations for patterns $\sigma$ of length $2$, although some enumerative problems remain open. When $\sigma$-sortable permutations form a class, Theorem \[class\_vs\_nonclass\] explicitly describes its basis. All the other cases remain to be solved. In the final section of this work we started the analysis of fertility and sorted permutations under pattern-avoiding machines. The main result is Theorem \[sigma\_bij\], which provides a characterization of the devices where the fertility of any sorted permutation is exactly $1$. ### Acknowledgements\ {#acknowledgements .unnumbered} The author would like to sincerely thank Anders Claesson for the fruitful discussions and the precious data he provided. [20]{} M. Albert, *PermLab: Software for Permutation Patterns*, at http://www.cs.otago.ac.nz/staffpriv/malbert/permlab.php. M. H. Albert, R. E. L. Aldred, M. D. Atkinson, C. Handley, D. Holton, *Permutations of a multiset avoiding permutations of length $3$*, European Journal of Combinatorics, vol. 22, pp. 1021-–1031, 2001. M. D. Atkinson, S. A. Linton, L. A. Walker, *Priority queues and multisets*, The Electronic Journal of Combinatorics, vol. 2, 1995. M. Bona, *A survey of stack sorting disciplines*, The Electronic Journal of Combinatorics, vol. 9, 2002-2003. M. Bona, *Combinatorics of Permutations*, Discrete Mathematics and Its Applications, CRC Press, 2004. M. Bousquet-Melou, *Sorted and/or sortable permutations*, Discrete Mathematics, vol. 225, pp. 25–50, 2000. G. Cerbai, A. Claesson, L. Ferrari, *Stack sorting with restricted stacks*, Journal of Combinatorial Theory, Series A, Vol. 173, 2020. C. Defant, *Postorder preimages*, Discrete Mathematics and Theoretical Computer Science, vol. 19, 2017. C. Defant, N. Kravitz, *Stack-sorting for words*, avalaible at arXiv:1809.09158, 2018. S. Kitaev, *Patterns in permutations and words*, Monographs in Theoretical Computer Science. An EATCS Series, Springer, Heidelberg, 2011. D. E. Knuth, *The art of computer programming, vol. 1, Fundamental Algorithms*, Addison-Wesley, Reading, Massachusetts, 1973. M. M. Murphy, *Restricted permutations, antichains, atomic classes and stack sorting*, PhD thesis, University of St Andrews, 2002. M. Mor, A.S. Fraenkel. *Cayley permutations*, Discrete Mathematics, vol. 48, pp. 101–112, 1984. N. J. A. Sloane, *The On-Line Encyclopedia of Integer Sequences*, at oeis.org. R. Smith, *Two stacks in series: a decreasing stack followed by an increasing stack*, Annals of Combinatorics, vol. 18, pp. 359–-363, 2014. J. West, *Permutations with forbidden subsequences and stack-sortable permutations*, PhD thesis, Massachusetts Institute of Technology, 1990. J. West, *Sorting twice through a stack*, Theoretetical Computer Science, vol. 117, pp. 303–313, 1993. [^1]: The author is member of the INdAM Research group GNCS; he is partially supported by INdAM - GNCS 2019 project “Proprietá combinatorie e rilevamento di pattern in strutture discrete lineari e bidimensionali".

--- author: - 'Tong-Tong Hu,' - 'Shuo Sun,' - 'Hong-Bo Li and' - 'Yong-Qiang Wang' title: Deforming charged black holes with dipolar differential rotation boundary --- Introduction ============ In classical general relativity, due to the uniqueness theorem of black holes [@a1; @1; @2; @3], the asymptotically flat charged black hole solutions with zero angular momentum in four dimensions are named as Reissner-Nordstrom (RN) black holes, which have two spherical event horizons. In four-dimensional anti-de Sitter (AdS) spacetime, it is well-known that except for compact horizons of arbitrary genus, there exist some solutions with noncompact planar or hyperbolic horizons. Because of the development of Anti-de Sitter/conformal field theory (AdS/CFT) correspondence [@Maldacena:1997re; @Maldacena:1998re; @Witten:1998qj; @Aharony:1999ti], it becomes more important to study physical properties of AdS black holes. Because the asymptotically AdS black hole has a boundary metric of conformal structure, we could deform the boundary metric to obtain a family of solutions of black hole with deforming horizon, whose curvature is not a constant value. There are many works in this field with both analytical and numerical methods. For the analytic method, the authors in [@Chen:2015zoa] constructed a family of black hole solutions with deforming horizons in four-dimensional spacetime by using AdS C-metric [@levicivita1917; @weyl1917; @Plebanski:1976gy]. In addition, a class of solutions of four-dimensional AdS black holes with noncompact event horizons of finite area was found in [@Klemm:2014rda; @Gnecchi:2013mja], and black holes with bottle-shaped event horizon were founded analytically in [@Chen:2016rjt]. With numerical methods, the authors in [@Markeviciute:2017jcp] got a family of deforming solutions including soliton and black hole with dipolar differential boundary $\Omega(\theta)=\varepsilon \cos(\theta)$. The constant $\varepsilon$ is the boundary rotation parameter and $\theta$ is polar angle. When $\varepsilon>2$, the norm Killing vector $\partial _t$ becomes spacelike for certain regions which also are called as ergoregions, and deforming AdS black holes with ergoregions may be unstable due to superradiant scattering [@Green:2015kur]. Because of superradiance, both solitons and black holes develop hair at $\varepsilon>2$. Motivated by this work, we also studied deforming solutions with odd multipolar [@Sun:2019qsf] and even multipolar [@Li:2019tsm] differential rotation boundary. Furthermore, in [@Crisford:2018qkz], the authors numerically studied a class of vacuum solutions with a noncompact, differential rotation boundary metric. With AdS C-metric, the effect of changing boundary metric on hyperbolic and compact AdS black holes had been studied in [@Horowitz:2018coe]. Considering the matter fields, the authors in [@a17] constructed the deforming black holes in $D = 5$ minimal gauged supergravity. Until now, the works of deforming AdS black hole with differential boundary [@Markeviciute:2017jcp; @Li:2019tsm; @Sun:2019qsf] are only studied in the situations without charge. It would be interesting to see whether there exist the charged deforming AdS black holes solutions in Einstein-Maxwell-AdS spacetime. In this paper, we would like to numerically solve coupled Einstein-Maxwell equations to obtain a family of solutions of charged deforming black holes. These solutions have the anti-symmetric rotation profile on the equatorial plane, which keeps total angular momentum of black hole being zero. In contract to the situations without charge, there exist some new properties of black holes due to the existence of charge $q$. Firstly, there exists at least one value of horizon for an arbitrary temperature, while there exists no horizon when $T<T_{min}$ for $q=0$. Besides, the extremum of temperature is determined by charge $q$ and divide temperature into several parts according to the charge $q$. In different regions of temperature, the number of values for horizon is different. Specifically, in the region with one value of horizon for a fixed temperature, there exist two families of solutions with same horizon when temperature is lower than the minimal extremum of temperature for RN-AdS black hole $T_{RN}=\frac{\sqrt{6}}{3\pi}$. Furthermore, in the region with three values of horizon for a fixed temperature, it is interesting to find that two small branches have same properties such as horizon geometry, entropy and quasinormal modes, although their horizon radii are not equal. The plan of our work is as follows. In the next section, we introduce our model and numerical method. In Sec. \[Sec3\], we obtain numerical solutions of charged AdS black hole with differential rotation boundary and discuss the effect of the temperature $T$ and the charge $q$ on solutions. We also show more properties of deforming charged AdS black hole including horizon geometry, entropy and stability. The conclusions and outlook are given in the last section. Model and numerical method {#Sec2} ========================== We start with Einstein-Maxwell action in four-dimensional AdS spacetime, whose action is given by $$\begin{aligned} S=\frac{1}{16\pi G}\int \mathrm{d}^4x&\sqrt{-g}\left(R-2\Lambda-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right), \label{eq:action}\end{aligned}$$ where $G$ is the gravitational constant, $\Lambda$ is cosmological constant represented by AdS radius $L$ as $-3/L^2$ in four-dimensional spacetime, $g$ is determinant of metric and $R$ is Ricci scalar. The equations of motion of the Einstein and the Maxwell fields which can be derived from the Lagrangian density (\[eq:action\]) are as follows \[m\] $$\label{equation1} R_{\mu\nu}+\frac {3}{L^2}g_{\mu\nu}-(\frac {1}{2}F_{\mu\lambda}{F_{\nu}}^{\lambda}-\frac {1}{8}g_{\mu\nu}F_{\lambda\rho}F^{\lambda\rho})=0,$$ $$\label{equation2} \nabla_{\mu}F^{\mu\nu}=0.$$ The spherically symmetric solution of motion equations (\[m\]) is the well-known Reissner-Nordstrom-AdS (RN-AdS) black hole. The metric of RN-AdS black hole solution could be written as follows $$\begin{aligned} \label{matric1} ds^{2} &=& -\left(1-\frac{2M}{r}+\frac{q^2}{r^2}+\frac{r^2}{L^2}\right)dt^2+\left(1-\frac{2M}{r}+\frac{q^2}{r^2}+\frac{r^2}{L^2}\right)^{-1}dr^2 +r^2 d\Omega^2, \label{eq:RN}\end{aligned}$$ and the gauge field is written as $$\label{gauge} F=dA, \;\;\; A=\frac{q}{r}dt.$$ Here, $d\Omega^2$ represents the standard element on $S^2$, the constant $M$ is the mass of black hole measured from the infinite boundary, and the constant $q$ is the charge of black hole. The black hole mass is related to the charge $q$ and horizon radius $r_+$ by the relation $$\label{root} M=\frac{1}{2}\left(r_++\frac{q^2}{r_+}+\frac{r_+^3}{L^2} \right),$$ where $r_{+}$ is the largest root. The Hawking temperature $T_H$ of RN-AdS black hole is given by $$\begin{aligned} T_H=\frac{1}{4\pi r_+} \left(1+ \frac{3 r_+^2}{L^2}-\frac{q^2}{r_+^2}\right).\end{aligned}$$ At near infinity the metric is asymptotic to the anti-de Sitter spacetime, and boundary metric is given by $$\label{boundary} ds_\partial^2=-dt^2+d\theta^2+\sin^2\theta d\phi^2.$$ In order to obtain the new asymptotic Anti-de Sitter solution, the authors in [@Markeviciute:2017jcp] add differential rotation to the boundary metric, which is given by $$ds_\partial^2=-dt^2+d\theta^2+\sin^2\theta[d\phi+\Omega(\theta)dt]^2, \label{eq:boundary}$$ with a dipolar differential rotation $\Omega(\theta)=\varepsilon\cos\theta$. The constant $\varepsilon$ is the amplitude of the boundary rotation. The norm of Killing vector $\partial_t$ is $$\|{\partial{t}}\|^2=-1+\frac{\varepsilon^2}{4}\sin^2({2\theta}). \label{eq:timelike}$$ From the above equation, we could find that the maximal value of Killing vector appears at $\theta=\frac{\pi}{4}$. We will take the same dipolar differential rotation boundary (\[eq:boundary\]). In order to get a set of charged deforming black hole solutions, we would like to use DeTurk method [@Headrick:2009pv; @Wiseman2012; @Dias:2015nua] to solve equations of motion (\[m\]). By adding a gauge fixing term, we change equations (\[equation1\]) to elliptic equations: $$\label{DTequation} R_{\mu\nu}+\frac {3}{L^2}g_{\mu\nu}-(\frac {1}{2}F_{\mu\lambda}{F_{\nu}}^{\lambda}-\frac {1}{8}g_{\mu\nu}F_{\lambda\rho}F^{\lambda\rho})-\nabla_{(_\mu}\xi_{\nu)}=0,$$ where the Deturk vector $\xi^\mu=g^{\nu\rho}(\Gamma^\mu_{\nu\rho}[g]-\Gamma^\mu_{\nu_\rho}[\tilde{g}])$ is related to reference metric $\tilde{g}$. It is notable that the reference metric $\widetilde{g}$ should be chosen to have the same boundary and horizon structure with $g$. Using this method to solve equations (\[equation2\]) and (\[DTequation\]), we could obtain a family of charged AdS black hole solutions. Black hole solutions {#Sec3} ==================== To obtain solutions of charged deforming AdS black hole, we start with this ans$\ddot{a}$tze of metric, $$\begin{gathered} \mathrm{d}s^2=\frac{L^2}{(1-y^2)^2}\Bigg\{-y^2\tilde{\Delta}(y)U_1\mathrm{d}t^2+\frac{4\,y_+^2 U_2\,\mathrm{d}y^2}{\tilde{\Delta}(y)}+y_+^2 \Bigg[\frac{4\,U_3}{2-x^2}\left(\mathrm{d}x+x \sqrt{2-x^2}\,y\,U_4\, \mathrm{d}y\right)^2\\+(1-x^2)^2 U_5\,\left(\mathrm{d}\phi+y^2x\sqrt{2-x^2}\,U_6\,\mathrm{d}t\right)^2 \Bigg]\Bigg\}, \label{eq:ansatzbh}\end{gathered}$$ with $$\Delta(y)=\frac{q^2(1-y^2)^2}{L^2y_+^2}+(1-y^2)^2+y_+^2 (3-3y^2+y^4)\,,\quad\text{and}\quad \tilde{\Delta}(y)= \Delta(y) \delta + y_+^2 (1 - \delta)\,,$$\[key\] where the functions $U_i,i\in(1,...,6)$ depend on $x$ and $y$, the parameter $q$ is the charge of black hole, and $y_{+}$ is horizon radius. Here, $y$ is related to radial coordinate $r$ with $r=Ly_+/(1-y^2)$, and $x$ represents polar angle on $S^2$ with $\sin\theta=1-x^2$. When $U_1=U_2=U_3=U_5=\delta=1$ and $U_4=U_6=0$, the line element (\[key\]) can reduce to RN-AdS black hole. Considering an axial symmetry system, we have polar angle reflection symmetry $\theta\rightarrow\pi-\theta$ on the equatorial plane, and thus it is convenient to consider the coordinate range $\theta \in [0,\pi/2] $, i.e $x \in [0,1] $. We require the functions to satisfy the following boundary conditions on the equatorial plane $x=0$, $$\partial_x U_i(0,y)=0, \;\;\;i=1,2,3,4,5,6,$$ and set axis boundary conditions at $x=1$, where regularity must be imposed Dirichlet boundary conditions on $U_4$ and Neumann boundary conditions on the other functions $$\label{abc} U_4(1,y)=0,$$ and $$\label{abc} \partial_x U_1(1,y)=\partial_x U_2(1,y)=\partial_x U_3(1,y)=\partial_x U_5(1,y)=\partial_x U_6(1,y)=0.$$ At $y=1$, we set $U_4=0$, $U_{6}=\varepsilon$ and $U_{1}=U_{2}=U_{3}=U_{5}=1$. Moreover, expanding the equations of motion near $x=1$ gives $U_3(1,y)=U_5(1,y)$. In order to ensure that the number of unsolved functions is the same as that of equations in Deturk method, we introduce the component $A_\phi$ in gauge potential. We choose the following form of gauge potential $$\label{gauge} A = A_t dt+ A_\phi d\phi ,$$ where $A_t$ and $A_\phi$ are all real functions of $x$ and $y$. As for the boundary conditions of vector field, we set $A_t(x,1)=\mu$ and $A_x(x,1)=0$, where the constant $\mu$ is chemical potential which represents the asymptotic behavior of Maxwell field at infinity. At $x=1$, we choose $A_t(1,y)=0$ and $A_x(1,y)=0$. The Hawking temperature of charged deforming black hole under ans$\ddot{a}$tze (\[key\]) takes the following form: $$T=\frac{1}{4\pi}\sqrt{-g^{tt}g^{MN}\partial_{M}g_{tt}\partial_{N}g_{tt}}\mid_{r=r_{+}}=\frac{y_{+}^4+\delta(-q^2+y_{+}^2(1+2y_{+}^2))}{4\pi y_{+}^3}. \label{temperature}$$ When the charge $q=0$, the formula (\[temperature\]) can reduce to the temperature of Schwarzschild-AdS black hole which was also given in [@Markeviciute:2017jcp]. When we fix $\delta=1$, the extremums of temperature $T$ depend on the value of charge $q$: - $q=0$: There is only a local minimum $T_{min}=T_S=\frac{\sqrt{3}}{2\pi}$, which is the minimal temperature of Schwarzschild-AdS black hole. - $0<q<1/6$: There are two extremums of temperature. $$\left\{ \begin{aligned} &T_{min}=\frac{3\sqrt{\frac{3}{2}}\left(-q^2+\frac{1}{12}\left(\sqrt{1-36q^2}+1\right)^2+\frac{1}{6}\left(\sqrt{1-36q^2}+1\right)\right)}{\pi\left(\sqrt{1-36q^2}+1\right)^{3/2}},\\ &T_{max}=\frac{3\sqrt{\frac{3}{2}}\left(-q^2+\frac{1}{12} \left(1-\sqrt{1-36 q^2}\right)^2+\frac{1}{6}\left(1-\sqrt{1-36q^2}\right)\right)}{\pi\left(1-\sqrt{1-36 q^2}\right)^{3/2}} . \end{aligned} \right.$$ - $q=1/6$: $T_{max}=T_{min}=T_{RN}=\frac{\sqrt{6}}{3\pi}$, which is the minimal extremum of temperature for RN-AdS black hole. - $q>1/6$: There exists no extremum of temperature. ![The temperature $T$ as functions of $y_+$ for $\delta$=1. From top to bottom, the black, blue, red and green lines describe charge $q=0$, $\frac{1}{9}$, $\frac{1}{6}$ and $\frac{1}{4}$, respectively. The red and black horizontal dashed lines represent $T_{S}=\frac{\sqrt{3}}{2\pi}$ and $T_{RN}=\frac{\sqrt{6}}{3\pi}$. The red and black vertical lines represent $y_{+}=\frac{1}{\sqrt{3}}$ and $y_+=\frac{1}{\sqrt{6}}$.[]{data-label="temf"}](hu-temperature1.pdf){width="80.00000%"} Next, we will analyze how the charge $q$ and temperature $T$ determine the number of values of horizon. In Fig.$\ $\[temf\], we plot the temperature $T$ as functions of horizon $y_+$ at $\delta=1$ for several values of charge $q$. The black, blue, orange and green lines represent $q=0,\frac{1}{9}, \frac{1}{6}$ and $\frac{1}{4}$, respectively. For $q=\frac{1}{6}$, the intersection of the black horizontal and the vertical dashed lines indicates horizon $y_+=\frac{1}{\sqrt{6}}$ and $T_{RN}=\frac{\sqrt{6}}{3\pi}$. For $q=0$, the intersection of the red horizontal and the vertical dashed lines indicates horizon $y_+=\frac{1}{\sqrt{3}}$ and $T_{S}=\frac{\sqrt{3}}{2\pi}$. The number of values of horizon depends on different ranges of temperature $T$ and charge $q$: 1. $q=0$: 1. $T<T_{S}$: There exists no horizon. 2. $T=T_{S}$: There are two equal values of horizon $y_+=\frac{1}{\sqrt{3}}$. 3. $T>T_{S}$: There are two different values of horizon. 2. $0<q<1/6$: 1. $T<T_{min}$ or $T>T_{max}$: There is one value of horizon. 2. $T=T_{min}$ or $T=T_{max}$: There are three values of horizon and two of them are equal. 3. $T_{min}<T<T_{max}$: There are three different values of horizon. 3. $q=1/6$: 1. $T=T_{RN}$: There are three equal values of horizon $y_+=\frac{1}{\sqrt{6}}$. 2. $T\neq T_{RN}$: There is only one value of horizon. 4. $q>1/6$: There is only one value of horizon. By regulating parameter $\delta$, we can also get three values of horizon below the local minimal temperature $T_{min}$. For simplify, we fix chemical potential $\mu=0.5$ and AdS radius $L=1$ in our numeral calculations. In Fig.$\ $\[u\], we give the typical distributions of $U_4$ as functions of $x$ and $y$ for $T=0.42$, $\varepsilon=1.6$ and $\delta=1$. When fixing $q=0.07057<\frac{1}{6}$, we can obtain three values of horizon. The distributions of $U_4$ for two small branches with $y_+=0.0992$ (left) and $y_+=0.1773$ (right) are given in the top of Fig.$\ $\[u\]. The left of bottom is the distributions of $U_4$ for large branch $y_+=1.5436$. To understand how the charge $q$ influences the distributions of $U_4$, we also plot $U_4$ as functions of $y$ at the equatorial plane $x=1$ for several values of $q$. From top to bottom, the distributions of function $U_4$ with charge $q=$ $0$, $1.7068$, $2.2684$, $3.4299$ are represented by black, red, blue, green and pink lines, respectively. Due to the existence of relation (\[temperature\]), the horizon radius $y_+$ increases with the increasing of charge $q$ for a fixed temperature. ![*Top*: The distributions of $U_4$ as functions of $x$ and $y$ of two small branches for $y_+=0.0992$ (left) and $y_+=0.1773$ (right). *Bottom left*: The distributions of $U_4$ as functions of $x$ and $y$ for large branch $y_+=1.5436$. The three solutions of $U_4$ are given with $T=0.42$, $\varepsilon=1.6$, $\delta=1$ and $q=0.07057$. *Bottom right*: The distributions of $U_4$ as functions of $y$ at the equatorial plane $x=1$ for several values of charge $q$ with $T=0.42$ and $\varepsilon=1.6$. From top to bottom, the black, red, blue, green and pink lines describe the charge $q=0$, $1.7068$, $2.2684$, $2.8363$ and $3.4299$, respectively.[]{data-label="u"}](smallsmallQ4.pdf){width="6.5cm"} ![*Top*: The distributions of $U_4$ as functions of $x$ and $y$ of two small branches for $y_+=0.0992$ (left) and $y_+=0.1773$ (right). *Bottom left*: The distributions of $U_4$ as functions of $x$ and $y$ for large branch $y_+=1.5436$. The three solutions of $U_4$ are given with $T=0.42$, $\varepsilon=1.6$, $\delta=1$ and $q=0.07057$. *Bottom right*: The distributions of $U_4$ as functions of $y$ at the equatorial plane $x=1$ for several values of charge $q$ with $T=0.42$ and $\varepsilon=1.6$. From top to bottom, the black, red, blue, green and pink lines describe the charge $q=0$, $1.7068$, $2.2684$, $2.8363$ and $3.4299$, respectively.[]{data-label="u"}](smallQ4.pdf){width="6.5cm"} ![*Top*: The distributions of $U_4$ as functions of $x$ and $y$ of two small branches for $y_+=0.0992$ (left) and $y_+=0.1773$ (right). *Bottom left*: The distributions of $U_4$ as functions of $x$ and $y$ for large branch $y_+=1.5436$. The three solutions of $U_4$ are given with $T=0.42$, $\varepsilon=1.6$, $\delta=1$ and $q=0.07057$. *Bottom right*: The distributions of $U_4$ as functions of $y$ at the equatorial plane $x=1$ for several values of charge $q$ with $T=0.42$ and $\varepsilon=1.6$. From top to bottom, the black, red, blue, green and pink lines describe the charge $q=0$, $1.7068$, $2.2684$, $2.8363$ and $3.4299$, respectively.[]{data-label="u"}](largeQ4.pdf){width="6.5cm"} ![*Top*: The distributions of $U_4$ as functions of $x$ and $y$ of two small branches for $y_+=0.0992$ (left) and $y_+=0.1773$ (right). *Bottom left*: The distributions of $U_4$ as functions of $x$ and $y$ for large branch $y_+=1.5436$. The three solutions of $U_4$ are given with $T=0.42$, $\varepsilon=1.6$, $\delta=1$ and $q=0.07057$. *Bottom right*: The distributions of $U_4$ as functions of $y$ at the equatorial plane $x=1$ for several values of charge $q$ with $T=0.42$ and $\varepsilon=1.6$. From top to bottom, the black, red, blue, green and pink lines describe the charge $q=0$, $1.7068$, $2.2684$, $2.8363$ and $3.4299$, respectively.[]{data-label="u"}](Q4boundary.pdf){width="6.5cm"} Horizon geometry {#31} ---------------- In this subsection, we will study how the black hole horizon geometry behaves with the increase of boundary rotation parameters $\varepsilon$ and charge $q$. We could use an isometric embedding in the three-dimensional space [@c1; @c2; @c3; @c4; @c5] to investigate the horizon geometry of a two-dimensional surface in a curved space [@Markeviciute:2017jcp; @Gibbons:2009qe]. With the method provided by [@Markeviciute:2017jcp], the horizon of black hole is embedded into hyperbolic $H^3$ space in global coordinates: $$ds^2_{H^3}=\frac{dR^2}{1+R^2/l^2}+R^2\left[\frac{dX^2}{1-X^2}+(1-X^2)d\phi^2\right],$$ where $l$ is the radius of the hyperbolic space and we fix $l=0.73$ in our whole calculation. The induce metric of the horizon of black hole is the following form: $$ds^2_{H}=L^2\left[\frac{4y_{+}^2U_{3}(x,0)}{2-x^2}dx^2+y_{+}^2(1-x^2)^2U_{5}(x,0)d\phi^2\right], \label{reduce1}$$ which can be obtained from the ans$\ddot{a}$tze (\[key\]). The embedding is given by a curve with two parameters $\{R(x),X(x)\}$ and written by: $$ds_{pb}^2=\left[\frac{R(x)'}{1-\frac{R(x)^2}{l^2}}+\frac{R(x)'^2X(x)'^2}{1-X(x)^2}\right]dx^2+R(x)^2(1-X(x)^2)d\phi^2. \label{reduce2}$$ Equating this line element with induce metric (\[reduce1\]), we can get the following first order differential equation: $$\begin{aligned} 0&=&4H(x)P(x)(X(x)^2-1)^2[P(x)-l^2(X(x)^2-1)] \\ &&+4l^2P(x)X(x)(X(x)^2-1)P(x)'X(x)'-(X(x)^2-1)^2l^2P(x)^2(l^2+P(x))^2X(x)'^2, \nonumber\end{aligned}$$ where $H(x)=(2-x^2)^{-1}(4y_+^2U_3(X,0))$ and $P(X)=y_+^2(1-x^2)^2U_5(x,0)$. ![Hyperbolic embedding of the cross section of the large black hole horizon for different values of $\varepsilon$ with $T=0.42$ and $q=0.07057$.[]{data-label="largehorizon"}](42largebranch.pdf){width="90.00000%"} In Fig.$\ $\[largehorizon\], we show the hyperbolic embedding of the cross section of the large black hole horizon for different values of $\varepsilon$ with the charge $q=0.07057$ and the temperature $T=0.42\geq T_{min}=0.2325$. From inner to outer, the black, red, green, orange and blue lines describe the boundary rotation parameter $\varepsilon$=$0.6$, $1.2$ ,$1.6$, $1.8$ and $1.9$, respectively. It is clear that the horizon deforms more dramatically with the increase of boundary rotation parameter $\varepsilon$. ![Hyperbolic embedding of the cross section of three black hole horizons for different values of $\varepsilon$ at $T=0.2585$ and $q=0.07057$. *Left*: The horizon geometry of large branch for $y_+=0.9152$. *Right*: The horizon geometry of two small branches for $y_+=0.3110$(lines) and $y_+=0.0859$(dots).[]{data-label="fig4"}](2585large.pdf) ![Hyperbolic embedding of the cross section of three black hole horizons for different values of $\varepsilon$ at $T=0.2585$ and $q=0.07057$. *Left*: The horizon geometry of large branch for $y_+=0.9152$. *Right*: The horizon geometry of two small branches for $y_+=0.3110$(lines) and $y_+=0.0859$(dots).[]{data-label="fig4"}](2585smallhorizon.pdf) Considering there is only one horizon radius when $T<T_{min}$ with $\delta=1$, we adjust $\delta<1$ to get three values of horizon and study the deformation of horizon for a fixed low temperature. In Fig.$\ $\[fig4\], we present hyperbolic embedding of the cross section of three black hole horizons for different values of $\varepsilon$ with $T=0.2585$ and $q=0.07057$. In the left panel, we show large black hole solutions for $y_+=0.9152$ and find that the size of the deformation of horizon cross section increases as $\varepsilon$ increases, which is similar to the situation in Fig.$\ $\[largehorizon\]. In the right panel, we show the result of two small branches for $y_+=0.3110$(lines) and $y_+=0.0859$(dots). What is different from the left panel is that the size of the deformation of horizon is a decreasing function of boundary rotation parameters $\varepsilon$. For the two small branches with different horizon radii, the horizon radius of the bigger one is nearly four times as that of the smaller one, but it is interesting to find that the two small branches have same embedding graphs of horizon geometry. ![Hyperbolic embedding of the cross section of the large black hole horizon for different values of charge $q$ with $y_+=1.5$ and $\varepsilon=1.6$. []{data-label="fig5"}](horizonQ.pdf){width="80.00000%"} To show the effect of charge $q$ on the deformation of horizon, we give hyperbolic embedding of the cross section of the large black hole horizon for different values of charge $q$ with $y_+=1.5$ and $\varepsilon=1.6$ in Fig.$\ $\[fig5\]. Due to the existence of relation (\[temperature\]), the temperature decreases with the increasing of charge $q$ for a fixed horizon radius. From outer to inner, the red, black, orange and blue lines represent $q=0$, $1.6104$, $2.1712$ and $2.6143$ respectively. The deformation of horizon becomes smaller as the charge $q$ increases. Entropy ------- In this subsection, we will discuss the entropy of deforming charged black hole. The formula of entropy of black hole is written as $$S=\frac{A}{4G_N}=\frac{2\pi y_+^2L^2}{G_N}\int^1_0dx\frac{1-x^2}{\sqrt{2-x^2}}\sqrt{U_3(x,0)U_5(x,0)}.$$ In Fig.$\ $\[42entropy\], we show the entropy against boundary rotation parameter $\varepsilon$ with $T=0.42$ and $q=0.07057$. The large black hole with $y_+=1$ is shown in the left panel, while in the right panel, two small branches with $y_+=0.1773$ and $y_+=0.0992$ are represented by red line and black dots respectively. For the large black hole, the entropy is a increasing function of boundary rotation parameter $\varepsilon$. The entropy approaches infinity as $\varepsilon\rightarrow2$, and we could not find charged deforming black hole solutions when $\varepsilon>2$. As for two small branches with a fixed temperature, the entropy decreases with the increase of boundary rotation parameter $\varepsilon$, and there exist solutions when $\varepsilon>2$. Furthermore, we also find another family of small black hole solutions, and in these solutions, the entropy increases with the increase of $\varepsilon$. ![The entropy as functions of boundary rotation parameter $\varepsilon$ for the temperature $T=0.42$ and the charge $q=0.07057$. *Left*: The entropy against boundary rotation parameter $\varepsilon$ for the large branch of black hole solutions $y_+=1$. *Right*: The entropy against boundary rotation parameter $\varepsilon$ for two small branches of black hole $y_+=0.1773$(black dots) and $y_{+}=0.0992$(red line). The vertical red dot lines represent $\varepsilon=2$.[]{data-label="42entropy"}](42largebranchentroy.pdf) ![The entropy as functions of boundary rotation parameter $\varepsilon$ for the temperature $T=0.42$ and the charge $q=0.07057$. *Left*: The entropy against boundary rotation parameter $\varepsilon$ for the large branch of black hole solutions $y_+=1$. *Right*: The entropy against boundary rotation parameter $\varepsilon$ for two small branches of black hole $y_+=0.1773$(black dots) and $y_{+}=0.0992$(red line). The vertical red dot lines represent $\varepsilon=2$.[]{data-label="42entropy"}](42smallentroy.pdf) To obtain the complete phase diagram of entropy for $\delta=1$, we investigate the whole region of temperature in terms of entropy. We show the entropy as functions of boundary rotation parameter $\varepsilon$ for different values of temperature $T$ with $\delta=1$ in Fig.$\ $\[deltaentropy\]. In the left panel, when we fix $q=0.07057$, there are two local extremums $T_{max}=0.4635$ and $T_{min}=0.2735$, the entropy of which are represented by red and green lines respectively. The two extremums divide the temperature into three regions: - Region A with $T>T_{max}$: There is only one value of horizon for a fixed temperature and the entropy increases with the increasing of boundary rotation parameter $\varepsilon$. The region A is indicated by the red area. - Region B with $T_{min}<T<T_{max}$: There are three values of horizon for a fixed temperature. For the large branch of black hole, the entropy increases with $\varepsilon$. Although these two small branches have different black hole horizons, they have same entropy which is a decreasing function of boundary rotation parameter $\varepsilon$. - Region C with $T<T_{min}$: There is only one horizon, but we could find two branches of entropy. The entropy increases with rotation parameter $\varepsilon$ at one branch, while it is a decreasing function of $\varepsilon$ in another branch. It is notable that when $T\leq T_{RN}\approx0.2599$, the two branches of entropy for one temperature join up. The region C is indicated by the blue area. ![The entropy as functions of boundary rotation parameter $\varepsilon$ for different values of temperature $T$ with $\delta=1$. *Left*: For $q=0.07057$, local maximum and minimum of temperature $T$ are equal to $0.4635$ and $0.2735$ represented by red and green lines respectively. The two extremums divide the phase diagram of entropy into three regions. *Right*: For $q=1.7068$, there is always one solution without extremum of temperature $T$. The vertical red dot lines represent $\varepsilon=2$. []{data-label="deltaentropy"}](entropy111.pdf) ![The entropy as functions of boundary rotation parameter $\varepsilon$ for different values of temperature $T$ with $\delta=1$. *Left*: For $q=0.07057$, local maximum and minimum of temperature $T$ are equal to $0.4635$ and $0.2735$ represented by red and green lines respectively. The two extremums divide the phase diagram of entropy into three regions. *Right*: For $q=1.7068$, there is always one solution without extremum of temperature $T$. The vertical red dot lines represent $\varepsilon=2$. []{data-label="deltaentropy"}](largeQentropy.pdf) In the right panel of Fig.$\ $\[deltaentropy\], we fix $q=1.7068>\frac{1}{6}$. There only exist one value of horizon for any temperature, but we could obtain two branches of entropy. The entropy increases with rotation parameter $\varepsilon$ at one branch, while it is a decreasing function of $\varepsilon$ in another branch. It is notable that when $T\leq T_{RN}$, the two branches of entropy could connect, which is similar to the region C in left panel. Similar to Subsection \[31\], we adjust $\delta<1$ to get three values of horizon and study the entropy for a fixed low temperature. In Fig.$\ $\[entropy\], we exhibit entropy as functions of boundary rotation parameter $\varepsilon$ for different values of temperature $T$ at $q=0.07057$. The minimal temperature $T=0.2735$ for $\delta=1$ is represented by black lines. For a fixed temperature $T\leq T_{min}\approx0.2599$, the large branch join up with two small branches, which form a set of lines. At each set, the line showing that entropy increases with the increasing of $\varepsilon$ describes large branch and corresponding solid line and dot line describe two small branches. From left to right, these sets of lines indicate $T= 0.2599, 0.2492, 0.2325,0.2257$ and $0.2104$, respectively. Similar to above results in the right panel of Fig.$\ $\[42entropy\], the two small branches have same entropy. When temperature is lower than $T_{min}$, the large branches also have solutions with $\varepsilon>2$. The entropy become infinity when $\varepsilon$ approaches to a maximum value of solutions. ![The entropy as functions of boundary rotation parameter $\varepsilon$ for $T\leq T_{min}=0.2735$ with $q=0.07057$. The Black lines indicates $T=T_{min}$ and the red vertical dash line indicates the $\varepsilon=2$.[]{data-label="entropy"}](deletaentropy.pdf){width="90.00000%"} Stability --------- In this subsection, we study the stability of deforming charged black hole solutions. Following the method provided in [@Markeviciute:2017jcp; @d1; @d2], we consider a free, massless and neutral scalar field perturbation to background and solve the Klein-Gordon equation $$\square\Phi=0 , \label{KG}$$ and we could decompose the scalar field as the following standard form $$\Phi=\hat{\Phi}_{\omega,m}(x,y)e^{-i\omega t+im\phi},$$ where the constant $\omega$ is the frequency of the complex scalar field and $m$ is the azimuthal harmonic index. Considering the ingoing Eddington-Finkelstein coordinates [@d2; @Berti:2009kk], the scalar field with the ans$\ddot{a}$tze of the black hole metric (\[key\]) could be decomposed into $$\Phi(t,x,y,\phi)=e^{-i\omega t}e^{i\omega \phi}y^{-i\frac{2\omega y_+}{1+3y_+^2}}(1-y^2)^3(1-x^2)^{|m|}\psi(x,y) ,$$ where the powers of $x$ and $y$ are chosen to make function $\psi(x,y)$ regular at the origin. The boundary conditions are imposed as follow: $$\left\{ \begin{aligned} &\partial_{x}\psi(x,y)=0, \quad x=\pm1,\\ &\partial_{y}\psi(x,y)=0, \quad y=0,\\ &-2iy_+\omega \psi(x,y)+(1+3y_+^2)\partial_y\psi(x,y)=0, \quad y=1. \end{aligned} \right.$$ ![ The real part of frequencies $\omega$ against the rotation parameter $\varepsilon$ of two small branches for different value of angular quantum number $m$ at $T=0.42$ and $q=0.07057$. The black horizontal line represents $Re$ $\omega=0$. The black vertical horizontal line represents $\varepsilon=2$.[]{data-label="QN"}](SandSSQuasinormalmode.pdf){width="80.00000%"} In Fig.$\ $\[QN\], we give the real part of quasinormal frequencies $\omega$ against the rotation parameter $\varepsilon$ of two small branches for different values of angular quantum number $m$. From top to bottom, these dot lines represent $m=5, 8, 10, 13$ and $16$ respectively. Similar to the above results of horizon geometric and entropy, these two small branches have equal quasinormal frequencies though the horizon radius of the bigger one is nearly twice as that of the smaller one. The real part of frequencies $Re$ $\omega$ is always positive when $m<13$. When $m\geq13$, $Re$ $\omega$ would appear a negative value with the increase of rotation parameter $\varepsilon$, which means we could obtain a stable deforming charged black hole solution with scalar condensation. Conclusions and Outlook {#Sec4} ======================= In this paper, we studied the conformal boundary of four-dimensional static asymptotically AdS solutions in Einstein-Maxwell gravity and constructed solutions of deforming charged AdS black hole. In contrast with the situations without charge, the charge $q$ could influence the extremums of temperature $T$ which divide the range of temperature into different regions according to the value of charge $q$. The number of horizons depends on the different regions of temperature $T$. Moreover, there exists no horizon when $T<T_{min}$ for $q=0$, but when we take charge $q\neq0$, there is at least one value of horizon for a fixed temperature. We also investigated physical properties for charged deforming AdS black holes, including the deformation of horizon, entropy and stability: - Deformation of horizon: In the region with three values of horizon for a fixed temperature, the deformation of horizon for large branch increases with the increasing of boundary rotation parameter $\varepsilon$, while that of small branches is a decreasing function of $\varepsilon$, which shows very similar results to the cases without charge. We also studied how the horizon deforms against the charge $q$ and found that the deformation of horizon became smaller as the charge $q$ increases. - Entropy: In the region with three values of horizon for a fixed temperature, with the increase of $\varepsilon$, the entropy of large branches increases, while that of small branches decreases. There also exist another set of unstable solutions of small branches, where the entropy increases with the increasing of $\varepsilon$. The entropy of large branch and small branches for a fixed temperature join up when temperature $T$ is lower than $T_{RN}$. It is worth noting that in the region with one value of horizon for a fixed temperature, we could find two families of solutions with same horizon radius, and they have different properties of entropy when the temperature $T<T_{RN}$. - Stability: We have studied the stability of scalar fields in the background of deforming charged AdS black holes, and found that when angular quantum number $m\geq13$, the real part of frequencies begins to appear negative values, which means scalar condensation. The most interesting finding in our research is that in the region with three values of horizon at one temperature, the two small branches for a fixed temperature have same numerical results, including deformation of horizon, entropy and stability though their horizon radii might vary many times. At present, we have studied the horizon geometry, entropy and stability of charged AdS black hole with differential rotation boundary. 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--- abstract: 'We study integral transforms mapping a function on the Euclidean space to the family of its integration on hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of the independent variables, and are imposed some symmetry with respect to the axes. These transforms are higher dimensional version of generalization of the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing conditions of functions.' address: 'College of Education, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan' author: - Hiroyuki Chihara title: 'Inversion of higher dimensional Radon transforms of seismic-type' --- [^1] Introduction {#section:introduction} ============ Let $n$ be a positive integer. Fix arbitrary $c=(c_1,\dotsc,c_n)\in\mathbb{R}$ and $\alpha_1,\dotsc,\alpha_n,\beta>1$. Set $\alpha=(\alpha_1,\dotsc,\alpha_n)$ for short. Let $$(x,y)=(x_1,\dotsc,x_n,y),\ (s,u)=(s_1,\dotsc,s_n,u) \in \mathbb{R}^{n+1} = \mathbb{R}^n\times\mathbb{R}$$ be independent variables of functions. We study three types of integral transforms $\mathcal{P}_{\alpha}f(s,u)$, $\mathcal{Q}_{\alpha}f(s,u)$ and $\mathcal{R}_{\alpha,\beta}f(s,u)$ of a function $f(x,y)$. These are the integration of $f(x,y)$ on the graph of functions of the fixed axis $x$. Their precise definitions are the following. Firstly, $\mathcal{P}_{\alpha}f(s,u)$ is defined by $$\begin{aligned} \mathcal{P}_{\alpha}f(s,u) & = \int_{\mathbb{R}^n} f \left( x, \sum_{i=1}^n s_i \lvert{x_i-c_i}\rvert^{\alpha_i} + u \right) dx \\ & = \int_{\mathbb{R}^n} f \left( x+c, \sum_{i=1}^n s_i \lvert{x_i}\rvert^{\alpha_i} + u \right) dx. \end{aligned}$$ $\mathcal{P}_{\alpha}f(s,u)$ is the integration of $f$ on a hypersurface $$\Gamma_\mathcal{P}(\alpha;s,u) = \left\{ \left( x, \sum_{i=1}^n s_i \lvert{x_i-c_i}\rvert^{\alpha_i} + u \right) \ : \ x\in\mathbb{R}^n \right\},$$ and $dx$ is not the standard volume of $\Gamma_\mathcal{P}(\alpha;s,u)$ induced by the Euclidean metric of $\mathbb{R}^{n+1}$. $\Gamma_\mathcal{P}(2,\dotsc,2;s,u)$ is a paraboloid if $s_1{\dotsb}s_n\ne0$. In particular, when $n=1$, $\mathcal{P}_2f(s,u)$ is called the parabolic Radon transform of $f$ in seismology. Set $$g_\alpha(x) = f \left( x+c, \sum_{i=1}^n s_i \lvert{x_i}\rvert^{\alpha_i} + u \right)$$ for short. Note that $$\begin{aligned} g(x) & = \frac{g(x)+g(x_1,\dotsc,x_{i-1},-x_i,x_{i+1},\dotsc,x_n)}{2} \\ & + \frac{g(x)-g(x_1,\dotsc,x_{i-1},-x_i,x_{i+1},\dotsc,x_n)}{2}\end{aligned}$$ for $i=1,\dotsc,n$, and the contribution of the second term of the right hand side is $0$. So, it is natural to assume that $$f(x_1,\dotsc,x_{i-1},-x_i,x_{i+1},\dotsc,x_n,y) = f(x,y). \label{equation:symmetry1}$$ for $i=1,\dotsc,n$. Secondly, $\mathcal{Q}_{\alpha}f(s,u)$ is defined by $$\begin{aligned} \mathcal{Q}_{\alpha}f(s,u) & = \int_{\mathbb{R}^n} f \left( x, \sum_{i=1}^n s_i (x_i-c_i)\lvert{x_i-c_i}\rvert^{\alpha_i-1} + u \right) dx \\ & = \int_{\mathbb{R}^n} f \left( x+c, \sum_{i=1}^n s_i x_i\lvert{x_i}\rvert^{\alpha_i-1} + u \right) dx.\end{aligned}$$ $\mathcal{Q}_{\alpha}f(s,u)$ is the integration of $f$ on a hypersurface $$\Gamma_\mathcal{Q}(\alpha;s,u) = \left\{ (x,y)\in\mathbb{R}^{n+1} \ : \ y = \sum_{i=1}^n s_i (x_i-c_i)\lvert{x_i-c_i}\rvert^{\alpha_i-1} + u \right\}$$ for $f$ and the measure is not the standard volume of $\Gamma_\mathcal{Q}(\alpha;s,u)$ induced by the Euclidean metric of $\mathbb{R}^{n+1}$. Thirdly, $\mathcal{R}_{\alpha,\beta}f(s,u)$ is defined by $$\begin{aligned} & \mathcal{R}_{\alpha,\beta}f(s,u) \\ = & \int_{\substack{\mathbb{R}^n \\ \sum s_i\lvert{x_i-c_i}\rvert^{\alpha_i}+u>0}} f \left( x, \left( \sum_{i=1}^n s_i \lvert{x_i-c_i}\rvert^{\alpha_i} + u \right)^{1/\beta} \right) \bigg/ \left( \sum_{i=1}^n s_i \lvert{x_i-c_i}\rvert^{\alpha_i} + u \right)^{1/\beta} dx \\ = & \int_{\substack{\mathbb{R}^n \\ \sum s_i\lvert{x_i}\rvert^{\alpha_i}+u>0}} f \left( x+c, \left( \sum_{i=1}^n s_i \lvert{x_i}\rvert^{\alpha_i} + u \right)^{1/\beta} \right) \bigg/ \left( \sum_{i=1}^n s_i \lvert{x_i}\rvert^{\alpha_i} + u \right)^{1/\beta} dx.\end{aligned}$$ $\mathcal{R}_{\alpha,\beta}f(s,u)$ is the integration of $f$ on a hypersurface $$\Gamma_\mathcal{R}(\alpha,\beta;s,u) = \left\{ (x,y)\in\mathbb{R}^{n+1} \ : \ \lvert{y}\rvert^\beta = \sum_{i=1}^n s_i \lvert{x_i-c_i}\rvert^{\alpha_i} + u \right\}$$ for $f$ satisfying $$f(x+c,-y)=f(x+c,y), \label{equation:symmetry2}$$ and the measure is not the standard volume of $\Gamma_\mathcal{R}(\alpha,\beta;s,u)$ induced by the Euclidean metric of $\mathbb{R}^{n+1}$. To resolve the singularity at $\sum s_i\lvert{x_i-c_i}\rvert^{\alpha_i}+u=0$, it is natural to assume that $f(x,0)=0$. $\Gamma_\mathcal{R}(2,\dotsc,2,2;s,u)$ is a quadratic hypersurface like hyperboloids if $s_1{\dotsb}s_n\ne0$. In particular, when $n=1$, $\mathcal{R}_{2,2}f(s,u)$ is called the hyperbolic Radon transform of $f$ in seismology. Here we recall the background of our transforms. We start with the transform on $\mathbb{R}^2$, that is, the integration on the plane curves. In the early 1980s, Cormack introduced the Radon transform of a family of plane curves and studied the basic properties in his pioneering works [@Cormack1981] and [@Cormack1982]. More than a decade later, Denecker, van Overloop and Sommen in [@Denecker1998] studied the parabolic Radon transform without fixed axis, in particular, the support theorem, higher dimensional generalization and etc. Recently, Moon established the inversion of the parabolic Radon transform $\mathcal{P}_2$ and the inversion of the hyperbolic Radon transform $\mathcal{R}_{2,2}$ respectively in his interesting paper [@Moon2016b]. He introduced some change of variables in $(x,y)\in\mathbb{R}^2$ so that the Radon transform of a family of plane curves became so-called the X-ray transform, that is, the Radon transform of a family of lines. More recently, replacing $x^2$ by some function $\varphi(x)$ in the parabolic Radon transform, Ustaogle developed Moon’s idea to try to obtain the inversion of more general Radon transforms on the plane in [@Ustaogle2017]. More recently, the author studied $\mathcal{P}_\alpha$, $\mathcal{Q}_\alpha$ and $\mathcal{R}_{\alpha,\beta}$ on $\mathbb{R}^2$, and obtained the inversion formulas for them in [@Chihara2019]. Those are the mathematical background of our transforms on the plane. For the scientific background of the parabolic Radon transform $\mathcal{P}_2$ and the hyperbolic transform $\mathcal{R}_{2,2}$, see [@Hampson1986] and the introduction of [@Moon2016b] for both of them, [@Maeland1998], [@Maeland2000] for the parabolic Radon transform, and [@Bickel2000] for the hyperbolic Radon transform respectively. Here we turn to the higher dimensional case $n\geqq2$. There are no mathematical results on our transforms so far. Unfortunately, however, our transforms with $n\geqq2$ have no scientific background at this time. Here we quote some works on the transforms which are integrations on hypersurfaces. Moon studied integral transforms over ellipsoids in [@Moon2014] and [@Moon2016a]. This arises in synthetic sperture radar (SAR), ultrasound reflection tomography (URT) and radio tomography. Recently, transforms on cones have been intensively studied. This models compton cameras, and is sometimes called cone transform or compton transform. See, e.g., [@Kuchment2016] anr references therein. The drawback of our transforms is that they treat only the graph of functions given by the sum of functions of one variable, and do not deal even with general hypersurfaces locally given by the graph of homogeneous functions, e.g., $y=x_1x_2$ for $n=2$ and $y=x_1x_2x_3$ for $n=3$. The aim of the present paper is to establish the inversion formulas for $\mathcal{P}_\alpha$, $\mathcal{Q}_\alpha$ and $\mathcal{R}_{\alpha,\beta}$. Here we introduce some function spaces to state our results. These function spaces consists of Schwartz functions on $\mathbb{R}^{n+1}$ satisfying some symmetries and vanishing conditions. The symmetries are natural for our transforms, and the vanishing conditions are used for justifying the change of variables for the reduction to the standard transform. We denote the set of all Schwartz functions on $\mathbb{R}^{n+1}$ by $\mathscr{S}(\mathbb{R}^{n+1})$. \[definition:functionspaces\] Fix $c=(c_1,\dotsc,c_n)\in\mathbb{R}^n$. Let $m=(m_1,\dotsc,m_n)$ be a multi-index of nonnegative integers. We define function spaces $\mathscr{S}_{c,m}(\mathbb{R}^{n+1})$, $\mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$ and $\mathscr{S}_{c,m}^\mathcal{R}(\mathbb{R}^{n+1})$ as follows. - $\mathscr{S}_{c,m}(\mathbb{R}^{n+1})$ is the set of all $f(x,y)\in\mathscr{S}(\mathbb{R}^{n+1})$ satisfying the following vanishing conditions $$\frac{\partial^k f}{\partial x_i^k}(x+c,y)\bigg\vert_{x_i=0}=0, \quad k=0,1,\dotsc,m_i,\ i=1,\dotsc,n.$$ - $\mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$ is the set of all $f(x,y)\in\mathscr{S}_{c,m}(\mathbb{R}^{n+1})$ satisfying the symmetry . - $\mathscr{S}_{c,m}^\mathcal{R}(\mathbb{R}^{n+1})$ is the set of all $f(x,y)\in\mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$ satisfying the symmetry and the vanishing condition $f(x+c,0)=0$. Recall $\alpha_1,\dotsc,\alpha_n,\beta>1$. Throughout the present paper we assume that the vanishing order $m_i$ at $x_i=c_i$ satisfies $m_i \geqq \alpha_i-2$ for all $i=1,\dotsc,n$. This condition guarantees the reduction to the standard Radon transform to work well. Our main results are the following. \[theorem:main\] Let $c=(c_1,\dotsc,c_n)\in\mathbb{R}^n$, and let $\alpha_1,\dotsc,\alpha_n,\beta>1$. Suppose that $m_i$ is a nonnegative integer satisfying $m_i \geqq \alpha_i-2$ for all $i=1,\dotsc,n$. - For any $f\in\mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$, if $n$ is odd, then $$\begin{aligned} f(x,y) & = \frac{4\cdot(-1)^{(n+1)/2}}{(4\pi)^{n+1}} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \nonumber \\ & \times \int_{\mathbb{R}^n} \left( \operatorname{vp} \int_{-\infty}^\infty \frac{\partial_u^n \mathcal{P}_{\alpha}f(s,u)}{y-\sum_{i=1}^ns_i\lvert{x_i-c_i}\rvert^{\alpha_i}-u} du \right) ds, \label{equation:inversion1} \end{aligned}$$ and if $n$ is even, then $$\begin{aligned} f(x,y) & = \frac{(-1)^{n/2}}{(4\pi)^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \nonumber \\ & \times \int_{\mathbb{R}^n} (\partial_u^n \mathcal{P}_{\alpha}f) \left( s, y-\sum_{i=1}^ns_i\lvert{x_i-c_i}\rvert^{\alpha_i} \right) ds. \label{equation:inversion2} \end{aligned}$$ - For any $f\in\mathscr{S}_{c,m}(\mathbb{R}^{n+1})$, if $n$ is odd, then $$\begin{aligned} f(x,y) & = \frac{4\cdot(-1)^{(n+1)/2}}{(2\pi)^{n+1}} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \nonumber \\ & \times \int_{\mathbb{R}^n} \left( \operatorname{vp} \int_{-\infty}^\infty \frac{\partial_u^n \mathcal{P}_{\alpha}f(s,u)}{y-\sum_{i=1}^ns_i(x_i-c_i)\lvert{x_i-c_i}\rvert^{\alpha_i-1}-u} du \right) ds, \label{equation:inversion5} \end{aligned}$$ and if $n$ is even, then $$\begin{aligned} f(x,y) & = \frac{(-1)^{n/2}}{(2\pi)^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \nonumber \\ & \times \int_{\mathbb{R}^n} (\partial_u^n \mathcal{P}_{\alpha}f) \left( s, y-\sum_{i=1}^ns_i(x_i-c_i)\lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) ds. \label{equation:inversion6} \end{aligned}$$ - For any $f\in\mathscr{S}_{c,m}^\mathcal{R}(\mathbb{R}^{n+1})$, if $n$ is odd, then $$\begin{aligned} f(x,y) & = \frac{4\cdot(-1)^{(n+1)/2}}{(4\pi)^{n+1}} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \lvert{y}\rvert \nonumber \\ & \times \int_{\mathbb{R}^n} \left( \operatorname{vp} \int_{-\infty}^\infty \frac{\partial_u^n \mathcal{R}_{\alpha,\beta}f(s,u)}{\lvert{y}\rvert^\beta-\sum_{i=1}^ns_i\lvert{x_i-c_i}\rvert^{\alpha_i}-u} du \right) ds, \label{equation:inversion3} \end{aligned}$$ and if $n$ is even, then $$\begin{aligned} f(x,y) & = \frac{(-1)^{n/2}}{(4\pi)^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \lvert{y}\rvert \nonumber \\ & \times \int_{\mathbb{R}^n} (\partial_u^n \mathcal{R}_{\alpha,\beta}f) \left( s, \lvert{y}\rvert^\beta-\sum_{i=1}^ns_i\lvert{x_i-c_i}\rvert^{\alpha_i} \right) ds. \label{equation:inversion4} \end{aligned}$$ We develop the method in [@Chihara2019] and prove Theorem \[theorem:main\]. The basic idea is the reduction to the standard Radon transform due to Moon in [@Moon2016b]. We begin with the basic facts on the standard Radon transform in Section \[section:radon\]. Section \[section:coordinatization\] is devoted to studying the coordinatization of the upper hemisphere by $s\in\mathbb{R}^n$. Next, we prepare some lemmas related with vanishing conditions in Section \[section:vanishing\]. Finally, we prove Theorem \[theorem:main\] in Section \[section:proofs\]. The standard Radon transform {#section:radon} ============================ We recall the definition of the standard Radon transform on $\mathbb{R}^{n+1}$ and the inversion formula. Let $\mathbb{S}^n$ be the unit sphre in $\mathbb{R}^n$ defined by $$\mathbb{S}^n = \{ \omega=(\omega_1,\dotsc,\omega_{n+1})\in\mathbb{R}^{n+1} \ : \ \lvert\omega\rvert^2 = \omega_1^2+\dotsb+\omega_{n+1}^2=1 \},$$ and let $\mathbb{S}^n_+$ be the upper hemisphere $$\mathbb{S}^n_+ = \{ \omega=(\omega_1,\dotsc,\omega_{n+1})\in\mathbb{S}^n \ : \ \omega_{n+1}>0 \}.$$ Set $$H(\omega,t) = \{ \zeta\in\mathbb{R}^{n+1} \ : \ \langle{\zeta,\omega}\rangle=t \}, \quad \omega\in\mathbb{S}^n,\ t\in\mathbb{R},$$ where $\langle\cdot,\cdot\rangle$ is the standard inner product of the Euclidean space. Note that $H(-\omega,-t)=H(\omega,t)$, and $$H(\omega,t) = \{\zeta+t\omega \ : \ \zeta \in H(\omega,0)\}$$ since $\langle{\zeta,\omega}\rangle-t=\langle{\zeta-t\omega,\omega}\rangle$. $H(\omega,t)$ is a hyperplane in $\mathbb{R}^{n+1}$ which is perpendicular to $\omega$ and is passing through $t\omega$. In particular, $H(\omega,0)$ is the orthogonal complement of $\{\omega\}$ in $\mathbb{R}^{n+1}$. The Radon transform of a function $F(\zeta)$ is defined by $$\mathcal{X}F(\omega,t) = \int_{H(\omega,t)} F(\zeta) dm(\zeta) = \int_{H(\omega,0)} F(t\omega+\zeta) dm(\zeta),$$ where $dm$ is the induced measure on the hyperplane from the Lebesgue measure on $\mathbb{R}^{n+1}$. Note that $\mathcal{X}F(-\omega,-t)=\mathcal{X}F(\omega,t)$. In the present paper, we deal with integrations on the graph of functions of the form $\langle{s,\xi}\rangle+u$ for the variables $\xi\in\mathbb{R}^n$ with some constants $(s,u)\in\mathbb{R}^n\times\mathbb{R}$. It is easy to see that $$\int_{\mathbb{R}^n} F(\xi,\langle{s,\xi}\rangle+u) d\xi = \frac{1}{\sqrt{1+\lvert{s}\rvert^2}} \mathcal{X}F \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right) \label{equation:radonF}$$ since $$\{ (\xi,\langle{s,\xi}\rangle+u) \ : \ \xi\in\mathbb{R}^n \} = H \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right), \quad dm=\sqrt{1+\lvert{s}\rvert^2}d\xi.$$ The inversion formula is as follows. \[theorem:inversionF\] For $F(\xi,\eta) \in \mathscr{S}(\mathbb{R}^{n+1})$, if $n$ is odd, then $$F(\xi,\eta) = \frac{2\cdot(-1)^{(n+1)/2}}{(2\pi)^{n+1}} \int_{\mathbb{S}^n_+} \left( \operatorname{vp} \int_{-\infty}^\infty \frac{\partial_t^n \mathcal{X}F(\omega,t)}{\langle(\xi,\eta),\omega\rangle-t} dt \right) d\omega,$$ and if $n$ is even, then $$F(\xi,\eta) = \frac{(-1)^{n/2}}{(2\pi)^n} \int_{\mathbb{S}^n_+} (\partial_t^n\mathcal{X}F) (\omega,\langle(\xi,\eta),\omega\rangle) d\omega,$$ where $d\omega$ is the induced measure on $\mathbb{S}^n$ from $\mathbb{R}^{n+1}$. For Theorem \[theorem:inversionF\], see, e.g., Corollary 2.6 and the Remark below in page 33 of Palamodov’s textbook [@Palamodov2004]. It is important to mention that Theorem \[theorem:inversionF\] holds for smooth functions $F(\xi,\eta)$ satisfying $F(\xi,\eta)=O\bigl((1+\lvert\xi\rvert+\lvert\eta\rvert)\bigr)^{-d}$ with some $d>n$, compactly suppred distributions, rapidly decaying Lebesgue measurable functions, and etc. See, e.g., [@Helgason2011] for the detail. Coordinatization of hemisphre {#section:coordinatization} ============================= We introduce a coordinatizarion $s\in\mathbb{R}^n$ of the upper hemisphre $\mathbb{S}^n_+$. The porlar coodinates $(\theta_1,\dotsc,\theta_n)\in(0,\pi)^n$ for the point $\omega\in\mathbb{S}^n_+$ is given by $$\omega_1=\cos\theta_1, \quad \omega_i=\sin\theta_1\dotsb\sin\theta_{i-1}\cdot\cos\theta_i \quad (i=1,\dotsc,n), \quad \omega_{n+1}=\sin\theta_1\dotsb\sin\theta_n,$$ and the volume form $d\omega$ given by $$d\omega = d\theta_1 \quad (n=1), \quad d\omega = \left( \prod_{i=1}^{n-1} \sin^{n-i}\theta_i \right) d\theta_1 \dotsb d\theta_n \quad (n=2,3,4,\dotsc)$$ is well-known. Since $\omega_1^2+\dotsb+\omega_{n+1}^2=1$, we introduce new coordinates $s=(s_1,\dotsc,s_n)\in\mathbb{R}^n$ of $\mathbb{S}^n_+$ defined by $$\omega = \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \quad\text{i.e.,}\quad (\omega_1,\dotsc,\omega_n,\omega_{n+1}) = \frac{(-s_1,\dotsc,-s_n,1)}{\sqrt{1+\lvert{s}\rvert^2}}. \label{equation:change222}$$ Note that $s$ moves in $\mathbb{R}^n$ if and only if $\theta$ moves $(0,\pi)^n$. Moreover we have $$s_i = - \frac{\cot\theta_i}{\sin\theta_{i+1}\dotsb\sin\theta_n} \quad (i=1,\dotsc,n-1), \quad s_n = - \cot\theta_n.$$ Elementary calculus yields $$\frac{\partial s_i}{\partial \theta_j} = 0 \quad (i=2,\dotsc,n,\ j<i),$$ $$\frac{\partial s_i}{\partial \theta_i} = \frac{1}{\sin^2\theta_i\cdot\sin\theta_{i+1}\dotsb\sin\theta_n} \quad (i=1,\dotsc,n-1), \quad \frac{\partial s_n}{\partial \theta_n} = \frac{1}{\sin^2\theta_n}.$$ Hence we have $$\frac{\partial(s_1,\dotsc,s_n)}{\partial(\theta_1,\dotsc,\theta_n)} = \det \begin{bmatrix} \dfrac{\partial s_1}{\partial \theta_1} & \ast & \dotsb & \ast \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ast \\ 0 & \dotsb & 0 & \dfrac{\partial s_n}{\partial s_n} \end{bmatrix} = \prod_{i=1}^n \frac{\partial s_i}{\partial \theta_i} = \left( \prod_{i=1}^n \sin^{i+1}\theta_i \right)^{-1},$$ $$\begin{aligned} d\omega & = \left( \prod_{j=1}^{n-1} \sin^{n-j}\theta_j \right) \left\lvert \frac{\partial(s_1,\dotsc,s_n)}{\partial(\theta_1,\dotsc,\theta_n)} \right\rvert^{-1} ds \nonumber \\ & = \left( \prod_{j=1}^{n-1} \sin^{n-j}\theta_j \cdot \prod_{i=1}^n \sin^{i+1}\theta_i \right) ds \nonumber \\ & = \left( \prod_{i=1}^n \sin\theta_i \right)^{n+1} ds = \frac{1}{(1+\lvert{s}\rvert^2)^{(n+1)/2}}\ ds. \label{equation:jacobian1} \end{aligned}$$ We also use change of variables $$(\omega,t) = \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right). \label{equation:change111}$$ In this case we have $$\begin{aligned} \frac{\partial(\theta_1,\dotsc,\theta_n,t)}{\partial(s_1,\dotsc,s_n,u)} & = \det \begin{bmatrix} \dfrac{\partial \theta_i}{\partial s_j} & \bm{0} \\ \bm{0} & \dfrac{\partial t}{\partial u} \end{bmatrix} = \frac{\partial(\theta_1,\dotsc,\theta_n)}{\partial(s_1,\dotsc,s_n)} \cdot \dfrac{\partial t}{\partial u} \\ & = \left( \prod_{i=1}^n \sin^{i+1}\theta_i \right) \cdot \frac{1}{\sqrt{1+\lvert{s}\rvert^2}}, \end{aligned}$$ $$d\omega dt = \frac{1}{(1+\lvert{s}\rvert^2)^{(n+2)/2}}\ dsdu. \label{equation:jacobian2}$$ Vanishing conditions {#section:vanishing} ==================== We prepare some lemmas related to vanishing conditions and symmetries. These lemmas are used for reducing our transforms to the standard Radon transform in the next section. We need the following lemma to make full use of the vanishing conditions. \[theorem:taylor\] - For $f(x,y) \in \mathscr{S}_{c,m}(\mathbb{R}^{n+1})$, $$\begin{aligned} & f(x+c,y) \nonumber \\ = & \frac{x_2^{m_2+1} \dotsb x_n^{m_n}}{m_2! \dotsb m_n!} \int_0^1 \dotsb \int_0^1 (1-t_2)^{m_2} \dotsb (1-t_n)^{m_n} \nonumber \\ \times & \frac{\partial^{m_2+\dotsb+m_n+n-1} f}{\partial x_2^{m_2+1} \dotsb \partial x_n^{m_n+1}}(x_1+c_1,t_2x_2+c_2,\dotsc,t_nx_n+c_n,y) dt_2 \dotsb dt_n, \label{equation:taylor1} \\ = & \frac{x_1^{m_1+1} \dotsb x_n^{m_n}}{m_1! \dotsb m_n!} \int_0^1 \dotsb \int_0^1 (1-t_1)^{m_1} \dotsb (1-t_n)^{m_n} \nonumber \\ \times & \frac{\partial^{m_1+\dotsb+m_n+n} f}{\partial x_1^{m_1+1} \dotsb \partial x_n^{m_n+1}}(t_1x_1+c_1,\dotsc,t_nx_n+c_n,y) dt_1 \dotsb dt_n. \label{equation:taylor2} \end{aligned}$$ - For $f(x,y) \in \mathscr{S}_{c,m}^\mathcal{R}(\mathbb{R}^{n+1})$, $$\begin{aligned} & f(x+c,y) \nonumber \\ = & \frac{x_2^{m_2+1} \dotsb x_n^{m_n} y}{m_2! \dotsb m_n!} \int_0^1 \dotsb \int_0^1 (1-t_2)^{m_2} \dotsb (1-t_n)^{m_n} \nonumber \\ \times & \frac{\partial^{m_2+\dotsb+m_n+n} f}{\partial x_2^{m_2+1} \dotsb \partial x_n^{m_n+1} \partial y}(x_1+c_1,t_2x_2+c_2,\dotsc,t_nx_n+c_n,\tau y) dt_2 \dotsb dt_n d\tau, \label{equation:taylor3} \\ = & \frac{x_1^{m_1+1} \dotsb x_n^{m_n} y}{m_1! \dotsb m_n!} \int_0^1 \dotsb \int_0^1 (1-t_1)^{m_1} \dotsb (1-t_n)^{m_n} \nonumber \\ \times & \frac{\partial^{m_1+\dotsb+m_n+n+1} f}{\partial x_1^{m_1+1} \dotsb \partial x_n^{m_n+1} \partial y}(t_1x_1+c_1,\dotsc,t_nx_n+c_n,\tau y) dt_1 \dotsb dt_n d\tau. \label{equation:taylor4} \end{aligned}$$ We can relpalce the role of $x_1$ in and by the other $x_i$, $i=2,\dotsc,n$. Here we porve and for $n=2$. The other parts can be proved in the same way. We omit the detail. Suppose $f(x_1,x_2,y) \in \mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^3)$. Since $$\frac{\partial^k f}{\partial x_2^k}(x_1+c_1,c_2,y)=0, \quad k=0,1,\dotsc,m_2,$$ Taylor’s formula gives $$\begin{aligned} f(x_1+c_1,x_2+c_2,y) & = \sum_{k=0}^{m_2} \frac{x_2^k}{k!} \frac{\partial^k f}{\partial x_2^k}(x_1+c_1,c_2,y) \nonumber \\ & + \frac{x_2^{m_2+1}}{m_2!} \int_0^1 (1-t_2)^{m_2} \frac{\partial^{m_2+1} f}{\partial x_2^{m_2+1}}(x_1+c_1,t_2x_2+c_2,y) dt_2 \nonumber \\ & = \frac{x_2^{m_2+1}}{m_2!} \int_0^1 (1-t_2)^{m_2} \frac{\partial^{m_2+1} f}{\partial x_2^{m_2+1}}(x_1+c_1,t_2x_2+c_2,y) dt_2. \label{equation:taylor101}\end{aligned}$$ This is for $n=2$. Since $$\frac{\partial^k f}{\partial x_1^k}(c_1,x_2+c_2,y)=0, \quad k=0,1,\dotsc,m_1$$ $$\frac{\partial^{k+l} f}{\partial x_1^k \partial x_2^l}(c_1,x_2+c_2,y)=0, \quad k=0,1,\dotsc,m_1, l=0,1,2,3,\dotsc.$$ If we use this with $l=m_2+1$, we have $$\begin{aligned} & \frac{\partial^{m_2+1} f}{\partial x_2^{m_2+1}}(x_1+c_1,t_2x_2+c_2,y) \\ = & \frac{x_1^{m_1+1}}{m_1!} \int_0^1 (1-t_1)^{m_1} \frac{\partial^{m_1+m_2+2} f}{\partial x_1^{m_1+1} \partial x_2^{m_2+1}}(t_1x_1+c_1,t_2x_2+c_2,y) dt_1. \end{aligned}$$ Substitute this into . Thus we obtain $$\begin{aligned} f(x_1+c_1,x_2+c_2,y) & = \frac{x_1^{m_1+1} x_2^{m_2+1}}{m_1! m_2!} \int_0^1\int_0^1 (1-t_1)^{m_1}(1-t_2)^{m_2} \\ & \times \frac{\partial^{m_1+m_2+2} f}{\partial x_1^{m_1+1} \partial x_2^{m_2+1}}(t_1x_1+c_1,t_2x_2+c_2,y) dt_1 dt_2, \end{aligned}$$ which is for $n=2$. Now we introduce functions defined by $f$, which is used for reducing our transforms to the standard Radon transform. For $f(x,y)$, set $$\begin{aligned} F_\alpha^\mathcal{P}(\xi,\eta) & = \begin{cases} \dfrac{2^n f(\xi_1^{1/\alpha_1}+c_1,\dotsc,\xi_n^{1/\alpha_n}+c_n,\eta)}{\alpha_1\dotsb\alpha_n \cdot \xi_1^{(\alpha_1-1)/\alpha_1}\dotsb\xi_n^{(\alpha_n-1)/\alpha_n}} &\ (\xi_1,\dotsc,\xi_n>0), \\ 0 &\ (\text{otherwise}), \end{cases} \\ F_\alpha^\mathcal{Q}(\xi,\eta) & = \frac{f(\xi_1\lvert{\xi_1}\rvert^{-1+1/\alpha_1}+c_1,\dotsc,\xi_n\lvert\xi_n\rvert^{-1+1/\alpha_n}+c_n,\eta)}{\alpha_1\dotsb\alpha_n \cdot \xi_1^{(\alpha_1-1)/\alpha_1}\dotsb\xi_n^{(\alpha_n-1)/\alpha_n}} \quad (\xi_1 \dotsb \xi_n\ne0), \\ F_{\alpha,\beta}^\mathcal{R}(\xi,\eta) & = \begin{cases} \dfrac{2^n f(\xi_1^{1/\alpha_1}+c_1,\dotsc,\xi_n^{1/\alpha_n}+c_n,\eta^{1/\beta})}{\alpha_1\dotsb\alpha_n \cdot \xi_1^{(\alpha_1-1)/\alpha_1}\dotsb\xi_n^{(\alpha_n-1)/\alpha_n} \eta^{1/\beta}} &\ (\xi_1,\dotsc,\xi_n, \eta>0), \\ 0 &\ (\text{otherwise}). \end{cases} \end{aligned}$$ Note that $F_\alpha^\mathcal{P}(\xi,\eta)=2^n F_\alpha^\mathcal{Q}(\xi,\eta)$ for $\xi_1,\dotsc,\xi_n>0$. \[theorem:change\] Suppose that $\alpha_1 \geqq m_i-2$ for all $i=1,\dotsc,n$. - For $f(x,y) \in \mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$, $$f(x,y) = \frac{1}{2^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) F_{\alpha}^\mathcal{P} (\lvert{x_1-c_1}\rvert^{\alpha_1},\dotsc,\lvert{x_n-c_n}\rvert^{\alpha_n},y), \label{equation:inverse1}$$ and for any $N>0$, there exists a constant $C_N>0$ such that $$\lvert{F_\alpha^\mathcal{P}(\xi,\eta)}\rvert \leqq C_N (1+\lvert\xi\rvert+\lvert\eta\rvert)^{-N}. \label{equation:decay1}$$ Moreover, when $\xi\in(0,\infty)^n$ tends to the boundary, $F_\alpha^\mathcal{P}(\xi,\langle{s,\xi}\rangle+u)$ has a finite limit for any $(s,u)\in\mathbb{R}^n\times\mathbb{R}$. - For $f(x,y) \in \mathscr{S}_{c,m}(\mathbb{R}^{n+1})$, $$\begin{aligned} f(x,y) & = \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \nonumber \\ & \times F_{\alpha}^\mathcal{Q} \bigl( (x_i-c_i)\lvert{x_1-c_1}\rvert^{\alpha_1-1}, \dotsc, (x_n-c_n)\lvert{x_n-c_n}\rvert^{\alpha_n-1}, y \bigr), \label{equation:inverse3} \end{aligned}$$ and for any $N>0$, there exists a constant $C_N>0$ such that $$\lvert{F_\alpha^\mathcal{Q}(\xi,\eta)}\rvert \leqq C_N (1+\lvert\xi\rvert+\lvert\eta\rvert)^{-N}. \label{equation:decay1}$$ Moreover, when $\sigma\xi\in(0,\infty)^n$ tends to the boundary, $F_\alpha^\mathcal{P}(\xi,\langle{s,\xi}\rangle+u)$ has a finite limit for any $(s,u)\in\mathbb{R}^n\times\mathbb{R}$. Here $\sigma=(\sigma_1,\dotsc,\sigma_n)\in\{\pm1\}^n$ and $\sigma\xi=(\sigma_1\xi_1,\dotsc,\sigma_n\xi_n)$. - For $f(x,y) \in \mathscr{S}_{c,m}^\mathcal{R}(\mathbb{R}^{n+1})$, $$f(x,y) = \frac{1}{2^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \lvert{y}\rvert \cdot F_{\alpha,\beta}^\mathcal{R} (\lvert{x_1-c_1}\rvert^{\alpha_1},\dotsc,\lvert{x_n-c_n}\rvert^{\alpha_n},\lvert{y}\rvert^\beta), \label{equation:inverse2}$$ and for any $N>0$, there exists a constant $C_N>0$ such that $$\lvert{F_{\alpha,\beta}^\mathcal{R}(\xi,\eta)}\rvert \leqq C_N (1+\lvert\xi\rvert+\lvert\eta\rvert)^{-N}. \label{equation:decay2}$$ Moreover, when $\xi\in(0,\infty)^n$ tends to the boundary, $F_{\alpha,\beta}^\mathcal{R}(\xi,\langle{s,\xi}\rangle+u)$ has a finite limit for any $(s,u)\in\mathbb{R}^n\times\mathbb{R}$. We prove only (i) here. (ii) and (iii) can be proved in the same way as (i). We omit the detail. Suppose that $f\in\mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$. Direct computation gives immediately. Set $C_0=2^n/\alpha_1\dotsb\alpha_n$ for short. It suffices to consider $F_\alpha^\mathcal{P}(\xi,\eta)$ only in $\xi\in(0,\infty)^n$. Note that implies that $$\begin{aligned} F_\alpha^\mathcal{P}(\xi,\eta) & = \frac{C_0}{m_1! \dotsb m_n!} \prod_{i=1}^n \xi_i^{(m_i+2-\alpha_i)/\alpha_i} \int_0^1\dotsb\int_0^1 (1-t_1)^{m_1}\dotsb(1-t_n)^{m_n} \nonumber \\ & \times \frac{\partial^{m_1+\dotsb+m_n+n} f}{\partial x_1^{m_1+1} \dotsb \partial x_n^{m_n+1}} (t_1\xi_1^{1/\alpha_1}+c_1,\dotsc,t_n\xi_n^{1/\alpha_n}+c_n,\eta) dt_1 \dotsb dt_n. \label{equation:taylor201}\end{aligned}$$ This shows that $F_\alpha^\mathcal{P}(\xi,\eta)$ is bounded for $\lvert\xi\rvert+\lvert\eta\rvert<1$ since $m_i+2-\alpha_i\geqq0$ for $i=1,\dotsc,n$. When $\lvert\xi\rvert+\lvert\eta\rvert\geqq1$, it follow that $\lvert\eta\rvert\geqq(\lvert\xi\rvert+\lvert\eta\rvert)/(\sqrt{n}+1)$ or $\xi_i\geqq(\lvert\xi\rvert+\lvert\eta\rvert)/(\sqrt{n}+1)$ for some $i=1,\dotsc,n$, say $i=1$. When $\lvert\eta\rvert\geqq(\lvert\xi\rvert+\lvert\eta\rvert)/(\sqrt{n}+1)\geqq1/(\sqrt{n}+1)$, shows . Note that implies that $$\begin{aligned} F_\alpha^\mathcal{P}(\xi,\eta) & = \frac{C_0}{\xi_1^{(\alpha_1-1)/\alpha_1} m_2! \dotsb m_n!} \prod_{i=2}^n \xi_i^{(m_i+2-\alpha_i)/\alpha_i} \int_0^1\dotsb\int_0^1 (1-t_2)^{m_2}\dotsb(1-t_n)^{m_n} \nonumber \\ & \times \frac{\partial^{m_2+\dotsb+m_n+n-1} f}{\partial x_2^{m_1+2} \dotsb \partial x_n^{m_n+1}} (\xi_1^{1/\alpha_1}+c_1,t_2\xi_2^{1/\alpha_2}+c_2,\dotsc,t_n\xi_n^{1/\alpha_n}+c_n,\eta) dt_2 \dotsb dt_n. \label{equation:taylor202}\end{aligned}$$ When $\xi_1\geqq(\lvert\xi\rvert+\lvert\eta\rvert)/(\sqrt{n}+1)\geqq1/(\sqrt{n}+1)$, shows . Combining the above all, we prove . We can prove the the exisistence and the finiteness of the limit of $F_\alpha^\mathcal{P}(\xi,\langle{s,\xi}\rangle+u)$ at the boundary of $(0,\infty)^n$ by using and . We omit the detail. Proof of Main Theorem {#section:proofs} ===================== We begin with computing $\mathcal{P}_{\alpha}f(s,u)$, $\mathcal{Q}_{\alpha}f(s,u)$ and $\mathcal{R}_{\alpha,\beta}f(s,u)$. \[theorem:compute\] - For $f(x,y) \in \mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$, $$\mathcal{P}_{\alpha}f(s,u) = \frac{1}{\sqrt{1+\lvert{s}\rvert^2}} \mathcal{X}F_\alpha^\mathcal{P} \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right), \label{equation:compute1}$$ $$\partial_u^n \mathcal{P}_{\alpha}f(s,u) = \frac{1}{(1+\lvert{s}\rvert^2)^{(n+1)/2}} (\partial_t^n \mathcal{X}F_\alpha^\mathcal{P}) \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right). \label{equation:compute2}$$ - For $f(x,y) \in \mathscr{S}_{c,m}(\mathbb{R}^{n+1})$, $$\mathcal{Q}_{\alpha}f(s,u) = \frac{1}{\sqrt{1+\lvert{s}\rvert^2}} \mathcal{X}F_\alpha^\mathcal{Q} \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right), \label{equation:compute5}$$ $$\partial_u^n \mathcal{Q}_{\alpha}f(s,u) = \frac{1}{(1+\lvert{s}\rvert^2)^{(n+1)/2}} (\partial_t^n \mathcal{X}F_\alpha^\mathcal{Q}) \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right). \label{equation:compute6}$$ - For $f(x,y) \in \mathscr{S}_{c,m}^\mathcal{R}(\mathbb{R}^{n+1})$, $$\mathcal{R}_{\alpha,\beta}f(s,u) = \frac{1}{\sqrt{1+\lvert{s}\rvert^2}} \mathcal{X}F_{\alpha,\beta}^\mathcal{R} \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right), \label{equation:compute3}$$ $$\partial_u^n \mathcal{R}_{\alpha,\beta}f(s,u) = \frac{1}{(1+\lvert{s}\rvert^2)^{(n+1)/2}} (\partial_t^n \mathcal{X}F_{\alpha,\beta}^\mathcal{R}) \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right). \label{equation:compute4}$$ Firstly, we prove (i). It suffices to prove . Using the symmetry , we have $$\mathcal{P}_{\alpha}f(s,u) = 2^n \int_{(0,\infty)^n} f\left(x+c,\sum_{i=1}^ns_ix_i^{\alpha_i}+u\right) dx.$$ We use the change of variables $x_i=\xi^{1/\alpha_i}$, $i=1,\dotsc,n$. Note that $$\frac{dx_i}{d\xi_i} = \frac{1}{\alpha_i} \cdot \frac{1}{\xi_i^{(\alpha_i-1)/\alpha_i}}, \quad x_i^{\alpha_i}=\xi_i.$$ By using this and , we deduce that $$\begin{aligned} \mathcal{P}_{\alpha}f(s,u) & = \int_{(0,\infty)^n} \frac{2^n f(\xi_1^{1/\alpha_1}+c_1,\dotsc,\xi_n^{1/\alpha_n}+c_n,\langle{s,\xi}\rangle+u)}{\alpha_1 \dotsb \alpha_n \cdot \xi_1^{(\alpha_1-1)/\alpha_1} \dotsb \xi_n^{(\alpha_n-1)/\alpha_n}} d\xi \\ & = \int_{(0,\infty)^n} F_\alpha^\mathcal{P}(\xi,\langle{s,\xi}\rangle+u) d\xi \\ & = \int_{\mathbb{R}^n} F_\alpha^\mathcal{P}(\xi,\langle{s,\xi}\rangle+u) d\xi \\ & = \frac{1}{\sqrt{1+\lvert{s}\rvert^2}} \mathcal{X}F_\alpha^\mathcal{P} \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right),\end{aligned}$$ which is . \(ii) can be proved in the same way as (i). We omit the detail. Secondly, we prove (iii). It suffices to prove . Using the symmetry , we have $$\mathcal{R}_{\alpha,\beta}f(s,u) = 2^n \int_{(0,\infty)^n} \frac{f\bigl(x+c,(\sum s_ix_i^{\alpha_i}+u)^{1/\beta}\bigr)}{(\sum s_ix_i^{\alpha_i}+u)^{1/\beta}} dx.$$ By using the change of variables $x_i=\xi^{1/\alpha_i}$, $i=1,\dotsc,n$, and , we deduce that $$\begin{aligned} \mathcal{R}_{\alpha,\beta}f(s,u) & = \int_{(0,\infty)^n} \frac{2^n f\bigl(\xi_1^{1/\alpha_1}+c_1,\dotsc,\xi_n^{1/\alpha_n}+c_n,(\langle{s,\xi}\rangle+u)^{1/\beta}\bigr)}{\alpha_1 \dotsb \alpha_n \cdot \xi_1^{(\alpha_1-1)/\alpha_1} \dotsb \xi_n^{(\alpha_n-1)/\alpha_n} \cdot (\langle{s,\xi}\rangle+u)^{1/\beta}} d\xi \\ & = \int_{(0,\infty)^n} F_{\alpha,\beta}^\mathcal{R}(\xi,\langle{s,\xi}\rangle+u) d\xi \\ & = \int_{\mathbb{R}^n} F_{\alpha,\beta}^\mathcal{R}(\xi,\langle{s,\xi}\rangle+u) d\xi \\ & = \frac{1}{\sqrt{1+\lvert{s}\rvert^2}} \mathcal{X}F_{\alpha,\beta}^\mathcal{R} \left( \frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}}, \frac{u}{\sqrt{1+\lvert{s}\rvert^2}} \right),\end{aligned}$$ which is . Finally we prove Theorem \[theorem:main\]. Firstly, we prove (i). Suppose that $f(x,y) \in \mathscr{S}_{c,m}^\mathcal{P}(\mathbb{R}^{n+1})$, and $m_i\geqq\alpha_i-2$ for all $i=1,\dotsc,n$. When $n$ is odd, by using the identity , the inversion formula of the standard Radon transform Theorem \[theorem:inversionF\], the change of the variables , and the identity in order, we deduce that $$\begin{aligned} f(x,y) & = \frac{1}{2^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) F_{\alpha}^\mathcal{P} (\lvert{x_1-c_1}\rvert^{\alpha_1},\dotsc,\lvert{x_n-c_n}\rvert^{\alpha_n},y) \\ & = \frac{4\cdot(-1)^{(n+1)/2}}{(4\pi)^{n+1}} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \\ & \times \int_{\mathbb{S}^n_+} \left( \operatorname{vp} \int_{-\infty}^\infty \frac{\partial_t^n \mathcal{X}F_\alpha^\mathcal{P}(\omega,t)}{\sum \omega_i\lvert{x_i-c_i}\rvert^{\alpha_i}+\omega_{n+1}y-t} dt \right) d\omega \\ & = \frac{4\cdot(-1)^{(n+1)/2}}{(4\pi)^{n+1}} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \\ & \times \int_{\mathbb{R}^n} \left( \operatorname{vp} \int_{-\infty}^\infty \frac{\sqrt{1+\lvert{s}\rvert^2}}{y-\sum s_i\lvert{x_i-c_i}\rvert^{\alpha_i}-u} \right. \\ & = \left. (\partial_t^n \mathcal{X}F_\alpha^\mathcal{P}) \left(\frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}},\frac{u}{\sqrt{1+\lvert{s}\rvert^2}}\right) \cdot \frac{1}{(1+\lvert{s}\rvert^2)^{(n+2)/2}} du \right) ds \\ & = \frac{4\cdot(-1)^{(n+1)/2}}{(4\pi)^{n+1}} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \\ & \times \int_{\mathbb{R}^n} \left( \operatorname{vp} \int_{-\infty}^\infty \frac{\partial_u^n \mathcal{P}_{\alpha}f(s,u)}{y-\sum s_i\lvert{x_i-c_i}\rvert^{\alpha_i}-u} du \right) ds,\end{aligned}$$ which is . When $n$ is even, by using the identity , the inversion formula of the standard Radon transform Theorem \[theorem:inversionF\], the change of the variables , and the identity in order, we deduce that $$\begin{aligned} f(x,y) & = \frac{1}{2^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) F_{\alpha}^\mathcal{P} (\lvert{x_1-c_1}\rvert^{\alpha_1},\dotsc,\lvert{x_n-c_n}\rvert^{\alpha_n},y) \\ & = \frac{(-1)^{n/2}}{(4\pi)^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \\ & \times \int_{\mathbb{S}^n_+} \partial_t^n \mathcal{X}F_\alpha^\mathcal{P} \left(\omega,\sum_{i=1}^n\omega_i\lvert{x_i-c_i}\rvert^{\alpha_i}+\omega_{n+1}y\right) d\omega \\ & = \frac{(-1)^{n/2}}{(4\pi)^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \\ & \times \int_{\mathbb{R}^n} \frac{1}{(1+\lvert{s}\rvert^2)^{(n+1)/2}} \cdot (\partial_t^n \mathcal{X}F_\alpha^\mathcal{P}) \left(\frac{(-s,1)}{\sqrt{1+\lvert{s}\rvert^2}},\frac{y-\sum s_i \lvert{x_i-c_i}\rvert^{\alpha_i}}{\sqrt{1+\lvert{s}\rvert^2}}\right) ds \\ & = \frac{(-1)^{n/2}}{(4\pi)^n} \left( \prod_{i=1}^n \alpha_i \lvert{x_i-c_i}\rvert^{\alpha_i-1} \right) \\ & \times \int_{\mathbb{R}^n} (\partial_u^n \mathcal{P}_{\alpha}f) \left( s, y-\sum_{i=1}^ns_i \lvert{x_i-c_i}\rvert^{\alpha_i} \right) ds,\end{aligned}$$ which is . Secondly, we prove (ii). Suppose that $f(x,y) \in \mathscr{S}_{c,m}(\mathbb{R}^{n+1})$, and $m_i\geqq\alpha_i-2$ for all $i=1,\dotsc,n$. When $n$ is odd, applying the inversion formula of the standard Radon transform Theorem \[theorem:inversionF\], the change of the variables and the identity in order, we obtain in the exactly same way as . When $n$ is even, applying the inversion formula of the standard Radon transform Theorem \[theorem:inversionF\], the change of the variables and the identity in order, , we obtain in the exactly same way as . We omit the detail. Finally, we prove (iii). Suppose that $f(x,y) \in \mathscr{S}_{c,m}^\mathcal{R}(\mathbb{R}^{n+1})$, and $m_i\geqq\alpha_i-2$ for all $i=1,\dotsc,n$. When $n$ is odd, applying the inversion formula of the standard Radon transform Theorem \[theorem:inversionF\], the change of the variables and the identity in order, we obtain in the exactly same way as . When $n$ is even, applying the inversion formula of the standard Radon transform Theorem \[theorem:inversionF\], the change of the variables and the identity in order, , we obtain in the exactly same way as . We omit the detail. [99]{} Bickel, S.H.: [*Focusing aspects of the hyperbolic Radon transform*]{}, Geophysics [**65**]{} (2000), pp.652–655. Chihara, H.: [*Inversion of seismic-type Radon transforms on the plane*]{}, arXiv:1910.02645. Cormack, A.M.: [*The Radon transform on a family of curves in the plane*]{}, Proc. Amer. Math. Soc., [**83**]{} (1981), pp.325–330. Cormack, A.M.: [*The Radon transform on a family of curves in the plane II*]{}, Proc. Amer. Math. Soc., [**86**]{} (1982), pp.293–298. Denecker, K., van Overloop, J., Sommen, F.: [*The general quadratic Radon transform*]{}, Inverse Problems [**14**]{} (1998), pp.615–633. Hampson, D.: [*Inverse velocity stacking for multiple elimination*]{}, Journal of the Canadian Society of Exploration Geophysics, [**22**]{} (1986), pp.44–55. Helgason, S.: “Integral Geometry and Radon Transforms”, Springer, 2011. Kuchment, P. and Terzioglu, F.: [*Three-dimensional image reconstruction from Compton camera data*]{}, SIAM J. Imaging Sci., [**9**]{} (2016), pp.1708–17125. Maeland, E.: [*Focusing aspects of the parabolic Radon transform*]{}, Geophysics [**63**]{} (1998), pp.1708–1715. Maeland, E.: [*An overlooked aspect of the parabolic Radon transform*]{}, Geophysics [**65**]{} (2000), pp.1326–1329. Moon, S.: [*On the determination of a function from an elliptical Radon transform*]{}, J. Math. Anal. Appl., [**416**]{} (2014), pp.724–734. Moon, S.: [*Inversion of the elliptical Radon transform arising in migration imaging using the regular Radon transform*]{}, J. Math. Anal. Appl., [**436**]{} (2016), pp.138–148. 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--- abstract: | For any real-analytic hypersurface $M\subset \CN$, which does not contain any complex-analytic subvariety of positive dimension, we show that for every point $p\in M$ the local real-analytic CR automorphisms of $M$ fixing $p$ can be parametrized real-analytically by their $\ell_p$ jets at $p$. As a direct application, we derive a Lie group structure for the topological group $\autMp$. Furthermore, we also show that the order $\ell_p$ of the jet space in which the group $\autMp$ embeds can be chosen to depend upper-semicontinuously on $p$. As a first consequence, it follows that that given any compact real-analytic hypersurface $M$ in $\CN$, there exists an integer $k$ depending only on $M$ such for every point $p\in M$ germs at $p$ of CR diffeomorphisms mapping $M$ into another real-analytic hypersurface in $\CN$ are uniquely determined by their $k$-jet at that point. Another consequence is the following boundary version of H. Cartan’s uniqueness theorem: given any bounded domain $\Omega$ with smooth real-analytic boundary, there exists an integer $k$ depending only on $\partial \Omega$ such that if $H\colon \Omega\to \Omega$ is a proper holomorphic mapping extending smoothly up to $\partial \Omega$ near some point $p\in \partial \Omega$ with the same $k$-jet at $p$ with that of the identity mapping, then necessarily $H={\rm Id}$. Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities. address: - | Universität Wien\ Fakultät für Mathematik\ Nordbergstrasse 15\ A-1090 Wien\ Austria - | Université de Rouen\ Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS\ Avenue de l’Université, B.P. 12\ 76801 Saint Etienne du Rouvray\ France author: - Bernhard Lamel - Nordine Mir bibliography: - 'bibliography\_oct04.bib' title: Parametrization of local CR Automorphisms by finite Jets and Applications --- [^1] [^1]: The first author was supported by the FWF, Projekt P17111.

--- abstract: 'In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian Systems through Liouville Arnold Theorem and Integrability of Linear Differential Equations through Differential Galois Theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system,' address: - 'Professor PAC, Instituto Superior de Formación Docente Salomé Ureña, Recinto Emilio Prud’Homme, Santiago de los Caballeros – Dominican Republic. Honorary Researcher, Facultad de Ciencias Básicas y Biomédicas, Universidad Simón Bolívar, Barranquilla – Colombia' - 'Professor, Department of Mathematics and Statistics,Universidad del Norte, Barranquilla - Colombia' author: - 'Primitivo Belén Acosta-Humánez' - Germán Jiménez Blanco title: 'Some tastings in Morales-Ramis Theory' --- Introduction ============ To understand the Morales-Ramis theory, we need to introduce two different notions of integrability, the integrability of Hamiltonian systems in Liouville sense and the integrability of linear differential equations in Picard-Vessiot sense. Integrability of Hamiltonian Systems ------------------------------------ From the previous section, let us consider a $n$ degrees of freedom hamiltonian $H$. The equations of the flow of the hamiltonian system, in a system of canonical coordinates, $x_1,\ldots,x_n,y_1,\ldots,y_n$, are written $$\dot x = \frac{\partial H}{\partial y}, \quad \dot y = - \frac{\partial H}{\partial x} ,$$ and they are known as *Hamilton equations*. We recall that the *Poisson bracket* between $f(x_1,x_2,y_1,y_2)$ and $g(x_1,x_2,y_1,y_2)$ is given by $$\{f,g\}=\sum_{k=1}^n\left({\partial f\over \partial y_k}{\partial g\over \partial x_k}-{\partial f\over \partial y_k}{\partial g\over \partial x_k}\right).$$ We say that $f$ and $g$ are in involution when $\{f,g\}=0$, also we say in this case that $f$ and $g$ commute under the Poisson bracket. In this way, we can write the Hamilton equations as follows: $$\dot x = \{H, x\}, \quad \dot y = \{ H, y\}.$$ A hamiltonian $H$ in $\mathbb{C}^{2n}$ is called *integrable in the sense of Liouville* if there exist $n$ independent first integrals of the hamiltonian system in involution. We will say that $H$ in integrable *by terms of rational functions* if we can find a complete set of integrals within the family of rational functions. Respectively, we can say that $H$ is integrable *by terms of meromorphic functions* if we can find a complete set of integrals within the family of meromorphic functions. The following examples can illustrate a procedure to obtain first integrals, see also [@audin] for further examples. [**Example.**]{} Consider the following hamiltonian, which we will proof that is integrable. $$H\left( q_{1},q_{2},p_{1},p_{2}\right) =\frac{1}{2}p_{1}^{2}+\frac{1}{2}% p_{2}^{2}-2q_{1}^{3}-6q_{1}q_{2}^{2}, \label{ex2g}$$the Hamilton equation for $H$ are. $$\begin{aligned} \overset{\cdot }{q}_{1} &=&p_{1}\qquad \overset{\cdot }{q}_{2}=p_{2} \\ \overset{\cdot }{p}_{1} &=&6q_{1}^{2}+6q_{2}^{2}\qquad \overset{\cdot }{p}% _{2}=12q_{1}q_{2}.\end{aligned}$$Setting $q_{1}=x_{1}-x_{2}\qquad q_{2}=x_{1}+x_{2}$, then $$\begin{aligned} \widehat{H}\left( x_{1},x_{2},p_{1}\left( y_{1},y_{2}\right) ,p_{2}\left( y_{1},y_{2}\right) \right) &=&\frac{1}{2}p_{1}^{2}\left( y_{1},y_{2}\right) +% \frac{1}{2}p_{2}^{2}\left( y_{1},y_{2}\right) -2\left( x_{1}-x_{2}\right) ^{3}-6\left( x_{1}-x_{2}\right) \left( x_{1}+x_{2}\right) ^{2} \\ &=&\frac{1}{2}p_{1}^{2}\left( y_{1},y_{2}\right) +\frac{1}{2}p_{2}^{2}\left( y_{1},y_{2}\right) -8x_{1}^{3}+8x_{2}^{3}.\end{aligned}$$From the Hamilton equations corresponding to $\widehat{H}$ we have $$\begin{aligned} \overset{\cdot }{x}_{1} &=&\frac{\partial \widehat{H}}{\partial y_{1}}=p_{1}% \frac{\partial p_{1}}{\partial y_{1}}+p_{2}\frac{\partial p_{2}}{\partial y_{1}} \\ \overset{\cdot }{x}_{2} &=&\frac{\partial \widehat{H}}{\partial y_{2}}=p_{1}% \frac{\partial p_{1}}{\partial y_{2}}+p_{2}\frac{\partial p_{2}}{\partial y_{2}},\end{aligned}$$since $$\begin{aligned} \overset{\cdot }{x}_{1}+\overset{\cdot }{x}_{2} &=&\overset{\cdot }{q}% _{2}=p_{2} \\ \overset{\cdot }{x}_{1}-\overset{\cdot }{x}_{2} &=&\overset{\cdot }{q}% _{1}=p_{1}\end{aligned}$$$$\begin{aligned} p_{1}\left( \frac{\partial p_{1}}{\partial y_{1}}+\frac{\partial p_{1}}{% \partial y_{2}}\right) +p_{2}\left( \frac{\partial p_{2}}{\partial y_{1}}+% \frac{\partial p_{2}}{\partial y_{2}}\right) &=&p_{2} \\ p_{1}\left( \frac{\partial p_{1}}{\partial y_{1}}-\frac{\partial p_{1}}{% \partial y_{2}}\right) +p_{2}\left( \frac{\partial p_{2}}{\partial y_{1}}-% \frac{\partial p_{2}}{\partial y_{2}}\right) &=&p_{1}.\end{aligned}$$A solution for the above system is $$\begin{aligned} p_{1} &=&\frac{y_{1}-y_{2}}{2} \\ p_{2} &=&\frac{y_{1}+y_{2}}{2},\end{aligned}$$then $$\begin{aligned} \widehat{H}\left( x_{1},x_{2},y_{1},y_{2}\right) &=&\frac{1}{2}\left( \frac{% y_{1}-y_{2}}{2}\right) ^{2}+\frac{1}{2}\left( \frac{y_{1}+y_{2}}{2}\right) ^{2}-8x_{1}^{3}+8x_{2}^{3} \\ &=&\frac{1}{4}y_{1}^{2}+\frac{1}{4}y_{2}^{2}-8x_{1}^{3}+8x_{2}^{3}.\end{aligned}$$For instance, a first integral corresponding to $\widehat{H}$ is $$\widehat{I}=\frac{1}{4}y_{1}^{2}-8x_{1}^{3}$$and therefore another first integral corresponding to $H$ is $$I=\frac{1}{4}\left( p_{1}+p_{2}\right) ^{2}-\left( q_{1}+q_{2}\right) ^{3}.$$In this way, we have that $H$ and $I$ are first integrals, independent and in involution due to $$\begin{aligned} \frac{d I}{d t} &=&\frac{1}{2}\left( p_{1}+p_{2}\right) \left( \overset{\cdot }{p}_{1}+\overset{\cdot }{p}_{2}\right) -3\left( q_{1}+q_{2}\right) ^{2}\left( \overset{\cdot }{q}_{1}+\overset{\cdot }{q}% _{2}\right) \\ &=&\frac{1}{2}\left( p_{1}+p_{2}\right) \left( 6q_{1}^{2}+6q_{2}^{2}+12q_{1}q_{2}\right) -3\left( q_{1}+q_{2}\right) ^{2}\left( p_{1}+p_{2}\right) \\ &=&3\left( p_{1}+p_{2}\right) \left( q_{1}+q_{2}\right) ^{2}-3\left( q_{1}+q_{2}\right) ^{2}\left( p_{1}+p_{2}\right) =0\end{aligned}$$ and $\{H, I\}=0$. We denote by $X_H$ the hamiltonian vector field, that is, the right-hand side of the hamilton equations. In a general way, we deal with non-linear hamiltonian systems. For suitability, without lost of generality, we can consider hamiltonian systems with two degrees of freedom, that is a hamiltonian $H$ in $\mathbb{C}^{4}$. Let $\Gamma$ be an integral curve of $X_H$, being $\Gamma$ parametrized by $\gamma\colon t\mapsto (q_1(t),q_2(t),p_1(t),p_2(t))$, the *first variational equation* (VE) along $\Gamma$ is given by, $$\left(% \begin{array}{c} \dot \xi_1 \\ \dot\xi_2\\ \dot \xi_3 \\ \dot\xi_4% \end{array}% \right) = \left(% \begin{array}{cccc} \frac{\partial^2 H}{\partial p_1\partial q_1}(\gamma(t)) & \frac{\partial^2 H% }{\partial p_1 \partial q_2}(\gamma(t)) & \frac{\partial^2 H}{\partial p_1^2}(\gamma(t)) & \frac{\partial^2 H% }{\partial p_1 \partial p_2}(\gamma(t))\\ \frac{\partial^2 H}{\partial p_2\partial q_1}(\gamma(t)) & \frac{\partial^2 H% }{\partial p_2 \partial q_2}(\gamma(t)) & \frac{\partial^2 H}{\partial p_2\partial p_1}(\gamma(t)) & \frac{\partial^2 H% }{\partial p_2^2}(\gamma(t))\\ - \frac{\partial^2 H}{\partial q_1^2}(\gamma(t)) & -\frac{% \partial^2 H}{\partial q_1 \partial q_2}(\gamma(t))&- \frac{\partial^2 H}{\partial q_1 \partial p_1}(\gamma(t)) & -\frac{% \partial^2 H}{\partial q_1 \partial p_2}(\gamma(t))\\- \frac{\partial^2 H}{\partial q_2 \partial q_1}(\gamma(t)) & -\frac{% \partial^2 H}{\partial q_2^2}(\gamma(t))&- \frac{\partial^2 H}{\partial q_2 \partial p_1}(\gamma(t)) & -\frac{% \partial^2 H}{\partial q_2 \partial p_2}(\gamma(t)) \end{array}% \right) \left(% \begin{array}{c} \xi_1 \\ \xi_2\\ \xi_3 \\ \xi_4% \end{array} \right).$$ Picard-Vessiot Theory. ---------------------- The Picard-Vessiot theory is the Galois theory of linear differential equations. In the classical Galois theory, the main object is a group of permutations of the roots, while in the Picard-Vessiot theory is a linear algebraic group. In the remainder of this paper we only work, as particular case, with linear differential equations of second order $$\label{LDE} y^{\prime \prime }+ay^{\prime }+by=0,\quad a,b\in \mathbb{C}(x). \tag{LDE}$$ Suppose that $y_1, y_2$ is a fundamental system of solutions of the differential equation. This means that $y_1, y_2$ are linearly independent over $\mathbb{C}$ and every solution is a linear combination of these two. Let $L = \mathbb{C}(x)\langle y_1, y_2 \rangle = \mathbb{C}(x)(y_1, y_2, y_1^{\prime }, y_2^{\prime })$, that is the smallest differential field containing to $\mathbb{C}(x)$ and $\{y_{1},y_{2}\}.$ The group of all differential automorphisms of $L$ over $\mathbb{C}(x)$ is called the [*Galois group*]{} of $L$ over $\mathbb{C}(x)$ and denoted by $% Gal(L/\mathbb{C}(x))$ or also by $Gal^L_{\mathbb{C}(x)}$. This means that for $\sigma\colon L\to L$, $\sigma(a^{\prime })=\sigma^{\prime }(a)$ and $% \forall a\in \mathbb{C}(x),$ $\sigma(a)=a$. If $\sigma \in Gal(L/\mathbb{C}(x))$ then $\sigma y_1, \sigma y_2$ is another fundamental system of solutions of the linear differential equation. Hence there exists a matrix $A= \begin{pmatrix} a & b \\ c & d% \end{pmatrix} \in GL(2,\mathbb{C})$, such that $$\sigma \begin{pmatrix} y_{1} \\ y_{2}% \end{pmatrix} = \begin{pmatrix} \sigma y_{1} \\ \sigma y_{2}% \end{pmatrix} =A \begin{pmatrix} y_{1} \\ y_{2}% \end{pmatrix} .$$ [**Theorem 1.**]{} The Galois group $G=Gal(L/\mathbb{C}(x))$ is an algebraic subgroup of $GL(2,% \mathbb{C})$. Moreover, the Galois group of a reduced linear differential equation $$\label{LDE} \xi^{\prime \prime }=r\xi,\quad r\in \mathbb{C}(x), \tag{RLDE}$$ is an algebraic subgroup $SL(2,\mathbb{C})$. Let $F\subset L$ be a differential field extension, and let $\eta$ be and element of $L$, then 1. $\eta$ is [*algebraic*]{} over $F$ if $\eta$ satisfies a polynomial equation with coefficients in $F$, i.e. $\eta$ is an algebraic function of one variable. 2. $\eta$ is [*primitive*]{} over $F$ if $\eta^{\prime }\in F$, i.e. $% \eta = \int f$ for some $f \in F$. 3. $\eta$ is [*exponential*]{} over $F$ if $\eta^{\prime }/\eta \in F$, i.e. $\eta = e^{\int f}$ for some $f \in F$. Algebraic, primitive and exponential functions are called Liouvillian functions. Thus a Liouvillian function is built up using algebraic functions, integrals and exponentials. In the case $F=\mathbb{C}(x)$ we get, for instance logarithmic, trigonometric functions, but not special functions such that the Airy functions. Solvability of LDE is expressed in terms of these function. This is one of the main results of Picard-Vessiot theory. [**Theorem 2.**]{} A linear differential equation is solvable integrable by terms of, Liouvillian functions, if and only if the connected component of the identity element of its Galois group is a solvable group. The Cauchy-Euler equation with Galois group in $SL(2,\mathbb{C})$ is given by $$y''={m(m+1)\over x^2}.$$ The integrability and Galois group of this equation has been studied by the first author in his PhD thesis, where is proved that the differential Galois group is abelian for all $m\in \mathbb{C}$. Morales Ramis Theory ==================== We want to relate integrability of hamiltonian systems to Picard-Vessiot theory. The following theorems treat this problem. [**Theorem 3 (Morales-Ramis, [@mora1]).**]{} Let $H$ be a Hamiltonian in $\mathbb{C}^{2n}$, and $\gamma$ a particular solution such that the has regular (resp. irregular) singularities at the points of $\gamma$ at infinity. Then, if $H$ is completely integrable by terms of meromorphic (resp. rational) functions, then the Identity component of Galois Group of the is abelian. To understand completely this technical result, it is required a formal study of concerning to differential Galois theory and Morales-Ramis theory. We can illustrat this theorem through the following examples, but only for a basic level. [**Example.**]{} We study the conditions of the previous theorem in the hamiltonian system given in equation , which is given by $$H\left( q_{1},q_{2},p_{1},p_{2}\right) =\frac{1}{2}p_{1}^{2}+\frac{1}{2}% p_{2}^{2}-2q_{1}^{3}-6q_{1}q_{2}^{2}.$$ The hamilton equations are:$$\begin{aligned} \overset{\cdot }{q_{1}} &=&p_{1},\qquad \overset{\cdot }{q_{2}}=p_{2} \\ \overset{\cdot }{p_{1}} &=&6q_{1}^{2}+6q_{1}^{2} \\ \overset{\cdot }{p_{2}} &=&12q_{1}q_{2}\end{aligned}$$taking the invariant plane $q_{2}=p_{2}=0.$ we have $\overset{\cdot \cdot }{% q_{1}}=6q_{1}$ a solution for this equation is $q_{1}\left( t\right) =\frac{1% }{t^{2}}$, and the variational equation is:$$\left[ \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{12}{t^{2}} & 0 & 0 & 0 \\ 0 & \frac{12}{t^{2}} & 0 & 0% \end{array}% \right] \left[ \begin{array}{c} \xi _{1} \\ \xi _{2} \\ \xi _{3} \\ \xi _{4}% \end{array}% \right] =\left[ \begin{array}{c} \overset{\cdot }{\xi _{1}} \\ \overset{\cdot }{\xi _{2}} \\ \overset{\cdot }{\xi _{3}} \\ \overset{\cdot }{\xi _{4}}% \end{array}% \right]$$then $12\xi _{1}=t^{2}\overset{\cdot \cdot }{\xi _{1}}$ , which corresponds to a Cauchy-Euler equation, thus, the Galois group is abelian due to the hamiltonian system is integrable. The following examples were taken from [@acbl]. [**Example.**]{} Consider the hamiltonian $$H=\frac{p_{1}^{2}+p_{2}^{2}}{2}-Q(q_{1})\frac{q_{2}^{2}}{2}+\beta (q_{1},q_{2})q_{2}^{3}, \label{ex3g}$$ where $Q(q_{1})$ is a polynomial and $\beta \left( q_{1},q_{2}\right) $ is a function of two variables with continuous partial derivative and $% \lim\limits_{q_{2}\rightarrow 0}\frac{\partial ^{j}\beta \left( q_{1},q_{2}\right) }{\partial q_{2}^{j}}<\infty \ \ 0\leq j\leq 2.$The hamilton equations are:$$\begin{aligned} \overset{\cdot }{q}_{1} &=&p_{1},\qquad \overset{\cdot }{q}_{2}=p_{2} \\ \overset{\cdot }{p}_{1} &=&Q^{^{\prime }}(q_{1})\frac{q_{2}^{2}}{2}-\frac{% \partial \beta (q_{1},q_{2})}{\partial q_{1}}q_{2}^{3} \\ \overset{\cdot }{p}_{2} &=&Q\left( q_{1}\right) q_{2}-\frac{\partial \beta (q_{1},q_{2})}{\partial q_{2}}q_{2}^{3}-3\beta \left( q_{1},q_{2}\right) \left( q_{2}\right) ^{2}\end{aligned}$$taking the invariant plane $q_{2}=p_{2}=0.$ we have $q_{1}\left( t\right) =at+b$ and the variational equation is:$$\left[ \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & Q\left( q_{1}\right) & 0 & 0% \end{array}% \right] \left[ \begin{array}{c} \xi _{1} \\ \xi _{2} \\ \xi _{3} \\ \xi _{4}% \end{array}% \right] =\left[ \begin{array}{c} \overset{\cdot }{\xi _{1}} \\ \overset{\cdot }{\xi _{2}} \\ \overset{\cdot }{\xi _{3}} \\ \overset{\cdot }{\xi _{4}}% \end{array}% \right]$$then $\xi _{3}=\overset{\cdot }{\xi _{1}},\qquad \xi _{4}=\overset{\cdot }{% \xi _{2}},\qquad Q\left( q_{1}\right) \xi _{2}=\overset{\cdot }{\xi _{4}}$ hence $Q\left( q_{1}\right) \xi _{2}=\overset{\cdot \cdot }{\xi_2}.$ If $Q\left( q_{1}\right) $ is a polynomial then the Galois group is not abelian, although in some case is solvable (see [@acbl]), hence the Hamiltonian System is not integrable by Morales-Ramis Theorem. [**Example.**]{} Consider the following hamiltonian $$H=\frac{p_{1}^{2}+p_{2}^{2}}{2}-\frac{\lambda _{4}}{(\lambda _{2}+2\lambda _{3}q_{1})^{2}}+\lambda _{0}-\lambda _{1}q_{2}^{2}-\lambda _{2}q_{1}q_{2}^{2}-\lambda _{3}q_{1}^{2}q_{2}^{2}+\beta (q_{1},q_{2})q_{2}^{3},$$Where $\lambda _{i}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ,\ \ $with $\lambda _{3}\neq 0$,   $\beta \left( q_{1},q_{2}\right) $ is a function of two variables with continuous partial derivative and $% \lim\limits_{q_{2}\rightarrow 0}\frac{\partial ^{j}\beta \left( q_{1},q_{2}\right) }{\partial q_{2}^{j}}<\infty \ \ 0\leq j\leq 2.$ The hamilton equations are:$$\begin{aligned} \overset{\cdot }{q}_{1} &=&p_{1},\qquad \overset{\cdot }{q}_{2}=p_{2} \\ \overset{\cdot }{p}_{1} &=&\frac{-4\lambda _{3}\lambda _{4}}{(\lambda _{2}+2\lambda _{3}q_{1})^{3}}+\lambda _{2}q_{2}^{2}+2\lambda _{3}q_{1}q_{2}^{2}-\frac{\partial \beta }{\partial q_{1}}\left( q_{1},q_{2}\right) q_{2}^{3} \\ \overset{\cdot }{p_{2}} &=&2\lambda _{1}q_{2}+2\lambda _{2}q_{1}q_{2}+2\lambda _{3}q_{1}^{2}q_{2}-\frac{\partial \beta }{\partial q_{2}}\left( q_{1},q_{2}\right) q_{2}^{3}-3\beta (q_{1},q_{2})q_{2}^{2}.\end{aligned}$$Taking $\ q_{2}=p_{2}=0$ and setting $H\left( q_{1},0,p_{1},0\right) =h$, we see that $$\begin{aligned} h &=&\frac{1}{2}p_{1}^{2}-\dfrac{\lambda _{4}}{(\lambda _{2}+2\lambda _{3}q_{1})^{2}}+\lambda _{0} \\ \overset{\cdot }{q_{1}} &=&p_{1}=\sqrt{2h+\dfrac{2\lambda _{4}}{(\lambda _{2}+2\lambda _{3}q_{1})^{2}}-2\lambda _{0}}.\end{aligned}$$Now, we pick $h=\lambda _{0}$, thus we have $$\begin{aligned} \frac{dq_{1}}{dt} &=&\sqrt{\dfrac{\lambda _{4}}{(\lambda _{2}+2\lambda _{3}q_{1})^{2}}} \\ \lambda _{2}q_{1}+\lambda _{3}q_{1}^{2} &=&\pm \sqrt{\lambda _{4}}t+c.\end{aligned}$$For instance the variational equation is$$\left[ \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{24\lambda _{3}^{2}\lambda _{4}}{(\lambda _{2}+2\lambda _{3}q_{1})^{4}} & 0 & 0 & 0 \\ 0 & 2\lambda _{1}+2\lambda _{2}q_{1}+2\lambda _{3}q_{1}^{2} & 0 & 0% \end{array}% \right] \left[ \begin{array}{c} \xi _{1} \\ \xi _{2} \\ \xi _{3} \\ \xi _{4}% \end{array}% \right] =\left[ \begin{array}{c} \overset{\cdot }{\xi _{1}} \\ \overset{\cdot }{\xi _{2}} \\ \overset{\cdot }{\xi _{3}} \\ \overset{\cdot }{\xi _{4}}% \end{array}% \right]$$then $$\left( 2\lambda _{1}+2\lambda _{2}q_{1}+2\lambda _{3}q_{1}^{2}\right) \xi _{2}=\overset{\cdot \cdot }{\xi _{2}} \label{L13}$$replacing in equation $\left( \ref{L13}\right) $ we have $$p\left( t\right) \xi _{2}=\overset{\cdot \cdot }{\xi _{2}} \label{L14}$$where $p\left( t\right) =2\lambda _{1}+2\left( \pm \sqrt{\lambda _{4}}% t+c\right) .$ and consequently the Galois group of $\left( \ref{L14}\right) $ is not abelian hence the hamiltonian system is not integrable. [**Example.**]{} Consider the following hamiltonian$$H=\frac{1}{2}p_{1}^{2}+\frac{1}{2}p_{1}^{2}+\frac{1}{% aq_{1}^{3}+bq_{1}^{2}q_{2}+cq_{2}^{3}}$$Where $a,b,c\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion .$ The hamilton equations are:$$\begin{aligned} \overset{\cdot }{q}_{1} &=&p_{1} \\ \overset{\cdot }{q}_{2} &=&p_{2} \\ \overset{\cdot }{p}_{1} &=&\frac{3aq_{1}^{2}+2bq_{1}q_{2}}{\left( aq_{1}^{3}+bq_{1}^{2}q_{2}+cq_{2}^{3}\right) ^{2}} \\ \overset{\cdot }{p_{2}} &=&\frac{bq_{1}^{2}+3cq_{2}^{2}}{\left( aq_{1}^{3}+bq_{1}^{2}q_{2}+cq_{2}^{3}\right) ^{2}}\end{aligned}$$taking the invariant plane $q_{1}=p_{1}=0$ we have $$\begin{aligned} \overset{\cdot }{p_{2}} &=&\frac{3}{cq_{2}^{4}}\qquad \overset{\cdot \cdot }{% q_{2}}=\frac{3}{cq_{2}^{4}} \\ q_{2}\left( t\right) &=&\left( \dfrac{-25}{2c}\right) ^{\dfrac{1}{5}}t^{% \dfrac{2}{5}}\end{aligned}$$$Z\left( t\right) =\left( 0,\left( \dfrac{-25}{2c}\right) ^{\dfrac{1}{5}}t^{% \dfrac{2}{5}},0,\dfrac{-5}{c\left( -\frac{25}{2c}\right) ^{\frac{4}{5}}t^{% \frac{3}{5}}}\right) .\ \ $the variational equation is $$\left[ \begin{array}{c} \overset{\cdot }{\xi _{1}} \\ \overset{\cdot }{\xi _{2}} \\ \overset{\cdot }{\xi _{3}} \\ \overset{\cdot }{\xi _{4}}% \end{array}% \right] =\left[ \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{-4b}{25t^{2}} & 0 & 0 & 0 \\ 0 & \frac{24}{-25t^{2}} & 0 & 0% \end{array}% \right] \left[ \begin{array}{c} \xi _{1} \\ \xi _{2} \\ \xi _{3} \\ \xi _{4}% \end{array}% \right]$$$\overset{\cdot }{q}_{1}=p_{1}\qquad \overset{\cdot }{p}_{1}=\frac{-4b}{% 25t^{2}}q_{1}$ then $\overset{\cdot \cdot }{q}_{1}=\frac{-4b}{25t^{2}}q_{1}$is a Cauchy-Euler equation, the Galois group is the identity, this group is abelian but we cannot state that the Hamiltonian system is integrable. The previous theorem of Morales-Ramis was extended by Morales-Ramis-Simó. [**Theorem 4. (Morales-Ramis-Simo, [@morasi]).**]{} Let $H$ be a Hamiltonian in $\mathbb{C}^{2n}$, and $\gamma $ a particular solution such that the has regular (resp. irregular) singularities at the points of $\gamma $ at infinity. Then, if $H$ is completely integrable by terms of meromorphic (resp. rational) functions, then the identity component of Galois Group of any linearised high order variational equation is abelian. The following example illustrate the way to compute the second order variational equation. Consider the Hamiltonian $$H=\frac{1}{2}p_1^2+\frac{1}{2}p_2^2+\frac{1}{2}a_1q_1^2+\frac{1}{2}a_2q_2^2+% \frac{1}{4}a_5q_1^4+\frac{1}{4}a_3q_2^4+\frac{1}{2}a_4q_1^2q_2^2,$$ where the Hamilton equations are given by $$\begin{array}{lll} \dot{q}_1 & = & p_1 \\ \dot{q}_2 & = & p_2 \\ \dot{p}_1 & = & -a_1q_1-a_5q_1^3-a_4q_1q_2^2 \\ \dot{p}_2 & = & -a_2q_2-a_3q_2^3-a_4q_1^2q_2.% \end{array}%$$ Taking as invariant plane $\Gamma=\{(q_1,q_2,p_1,p_2): q_2=p_2=0\}$ we obtain first variational equation $$\dot \xi^{(1)}=A(t)\xi^{(1)}, \quad A(t)= \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -a_1-3a_5q_1^2 & 0 & 0 & 0 \\ 0 & -a_2-a_4q_1^2 & 0 & 0% \end{pmatrix}% ,$$ where $\xi^{(1)}=(\xi^{(1)}_1,\xi^{(1)}_2,\xi^{(1)}_3,\xi^{(1)}_4)^T$ and $% q_1=q_1(t)$, being $(q_1(t),0,\dot q_1(t),0)$ a particular solution of the hamiltonian system over the invariant plane. The second variational equation is given by $$\dot \xi^{(2)}=A(t)\xi^{(2)}+f(t), \quad f(t)=% \begin{pmatrix} 0 \\ 0 \\ -3a_5(\xi^{(1)}_1)^2-a_4q_1(\xi^{(1)}_2)^2 \\ -2a_4q_1\xi^{(1)}_1\xi^{(1)}_2,% \end{pmatrix}%$$ where $\xi^{(2)}=(\xi^{(2)}_1,\xi^{(2)}_2,\xi^{(2)}_3,\xi^{(3)}_4)^T$. The following is our original contribution to this paper. [**[Proposition.]{}**]{} Assume the Hamiltonian system given by $$H={p_1^2+p_2^2\over 2}-{1 \over aq_1^m+bq_2^m}, \quad a\neq 0, \quad m>2.$$ The differential Galois group of the variational equation corresponding to the invariant plane $q_2=p_2=0$ and energy level $h=0$, is virtually abelian. Furthermore, the Galois group is independent of the choice of $a$ and $b$. *Proof.* The subsystem in the invariant plane is $$h={p_1^2\over 2}-{1 \over aq_1^m}, \quad a\neq 0, \quad m>2,$$ then we obtain a particular solution for $q_1$ given by $$q_1(t)=\left({m+2\over \sqrt{2a}t}\right)^{2\over m+2},$$ for instance the variational equation is given by $$\dot \xi=A(t)\xi, \quad A(t)= \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ {m(m+1)\over aq_1^{m+2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \quad \xi=(\xi_1,\xi_2,\xi_3,\xi_4)^T.$$ Thus, we arrive to the Cauchy-Euler equation $${\frac {d^{2}}{d{t}^{2}}}\xi_1 ={\frac {2m \left( m+1 \right) }{ \left( m+2 \right) ^{2}{t}^{2}}}\xi_1$$ and for instance the Galois group is always abelian. [9]{} P. Acosta-Humánez & D. Blázquez-Sanz, *Non-integrability of some Hamiltonians with rational potentials* DCDS-A, **10**, (2008), 265–293 V. Arnold, *Mathematical Methods of Classical Mechanics* (Graduate Texts in Mathematics, Vol. 60) M. Audin, *Les systèmes hamiltoniens et leur intégrabilité*, Cours Spécialisés, 8, Société mathématique de France & EDP Sciences, 2001 J. J. Morales-Ruiz, *Differential Galois Theory and Non-integrability of Hamiltonian Systems*, Progress in Mathematics 179, Birkhäuser, 1999. J. J. Morales-Ruiz and J. P. Ramis, * Galoisian obstructions to integrability of hamiltonian systems I,* Methods Appl. Anal. **8** (2001), no. 1, 33–95. J. J. Morales-Ruiz and J. P. Ramis, * Galoisian obstructions to integrability of hamiltonian systems II,* Methods Appl. Anal. **8** (2001), no. 1, 97–111. J. J. Morales-Ruiz, J. P. Ramis and C. Simó, *Integrability of hamiltonian systems and differential Galois groups of higher variational equations,* Ann. Sci. École Norm. Sup. (4) **40** (2007), no. 6, 845–884.

--- abstract: | In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the $s$th power of a positive definite operator having a discrete spectrum in ${\mathbb R}^+$. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter $s$. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the “control parameter” that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new class of identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter $s$ also the domain of definition, and thus the underlying function space, of the fractional operator changes. author: - 'Jürgen Sprekels[^1]' - 'Enrico Valdinoci[^2]' title: | A new type of identification problems:\ optimizing the fractional order\ in a nonlocal evolution equation --- Fractional operators, identification problems, first-oder necessary and second-order sufficient optimality conditions, existence, uniqueness, regularity. 49K21, 35S11, 49R05, 47A60. Introduction ============ Let $\Omega\subset {\mathbb R}^n$ be a given open domain and, with a given $T>0$, $Q:={\Omega}\times (0,T)$. We consider in ${\Omega}$ the evolution of a fractional diffusion process governed by the $s-$power of a positive definite operator ${\mathcal{L}}$. In this paper, we study, for a given $L\in(0,+\infty)\cup \{+\infty\}$, the following identification problem for fractional evolutionary systems: [[**(IP)**]{} Minimize the cost function]{} $$\begin{aligned} \label{cost} J(y,s):= \frac{1}{2} {\int_0^T\!\!\int_{\Omega}}\big|y(x,t)-y_Q(x,t)\big|^2\,dx\,dt \,+\,{\varphi}(s)\end{aligned}$$ with $s$ in the interval $(0,L)$, subject to the fractional evolution problem $$\begin{aligned} \label{ss1} & {{\partial_t}y\,+\,{\mathcal{L}}^s y\,=\,f \quad\mbox{in }\,Q,}\\[1mm] \label{ss2} & {\hspace*{5mm} y(\cdot,0)\,=\,y_0 \quad\mbox{in }\,{\Omega}.}\end{aligned}$$ In this connection, $y_Q\in L^2(Q)$ is a given target function, and ${\varphi}\in C^2(0,L)$ is a nonnegative penalty function satisfying $$\begin{aligned} \label{phi} \lim_{s\searrow 0}{\varphi}(s)=+\infty=\lim_{s\nearrow L}{\varphi}(s).\end{aligned}$$ Examples of penalty functions which fulfill are $$\label{EXA:PHI} \begin{split} &{\varphi}(s)=\frac{1}{s\,(L-s)}\qquad{\mbox{ for $s\in(0,L)$, if $L\ne+\infty$}}\\ {\mbox{and }} &{\varphi}(s)=\frac{e^s}{s}\qquad{\mbox{ for $s\in(0,L)$, if $L=+\infty$}}. \end{split}$$ The properties of the right-hand side $f$ and of the initial datum $y_0$ will be specified later. [Problem [**(IP)**]{} defines a class of identification problems which, to the authors’ best knowledge, has never been studied before. Indeed, while there exists a vast literature on the identification of coefficient functions or of right-hand sides in parabolic and hyperbolic evolution equations (which cannot be cited here), there are only but a few contributions to the control theory of fractional operators of diffusion type. In this connection, we refer the reader to the recent papers [@AO1], [@AO2], [@AO3] and [@Bors]. However, in these works the fractional operator was fixed and given a priori. In contrast to these papers, in our case the type of the fractional order operator itself, which is defined by the parameter $s$, is to be determined.]{} The fact that the fractional order parameter $s$ is the “control variable” in our problem entails a mathematical difficulty, namely, that with changing $s$ also the domain of ${\mathcal{L}}^s$ changes. As a consequence, in the functional analytic framework also the underlying solution space changes with $s$. From this, mathematical difficulties have to be expected. For instance, simple compactness arguments are likely not to work if existence is to be proved. In order to overcome this difficulty, we present in Section 4 (see the compactness result of Lemma 6) an argument which is based on Tikhonov’s compactness theorem. Another feature of the problem [**(IP)**]{} is the following: if we want to establish necessary and sufficient optimality conditions, then we have to derive differentiability properties of the control-to-state ($s\mapsto y$) mapping. A major part of this work is devoted to this analysis. In this paper, the fractional power of the diffusive operator is seen as an “optimization parameter”. This type of problems has natural applications. For instance, a biological motivation is the following: in the study of the diffusion of biological species (see, e.g., [@RO; @F12; @P13; @MV] and the references therein) there is experimental evidence (see [@W96; @H10]) that many predatory species follow “fractional” diffusion patterns instead of classical ones: roughly speaking, for instance, suitably long excursions may lead to a more successful hunting strategy. In this framework, optimizing over the fractional parameter $s$ reflects into optimizing over the “average excursion” in the hunting procedure, which plays a crucial role for the survival and the evolution of a biological population (and, indeed, different species in nature adopt different fractional diffusive behaviors). In this connection[^3], the solution $y$ to the state system , can be thought of as the spatial density of the predators (where the birth and death rates of the population are not taken into account here, but rather its capability of adapting to the environmental situation). In this sense, the minimization of $\,J\,$ is related to finding the “optimal” distribution for the population (for instance, in terms of the availability of resources, possibility of using favorable environments, distributions of possible preys, favorable conditions for reproduction, etc.). Differently from the existing literature, this optimization is obtained here by changing the nonlocal diffusion parameter $s$, where, roughly speaking, a small $s$ corresponds to a not very dynamic population and a large $s$ to a rather mobile one. The growth condition has to be understood against this biological background: in nature, neither a complete immobility of the individuals (i.e., the choice $s=0$) nor an extremely fast diffusion (observe that even the extreme case $s=L=+\infty$ is allowed in our setting) are likely to guarantee the survival of the species. In this connection, we may interpret the target function $y_Q$ as, e.g., the spatial distribution of the prey. To adapt their strategy, the predators must know these seasonal distributions a priori; however, this is often the case from long standing experience. We also remark that in nature the prey species in turn adapt their behavior to the strategy of the predators; it would thus be more realistic to consider a predator-prey system with two (possibly different) values of $s$. Such an analysis, however, goes beyond the scope of this work in which we confine ourselves to the simplest possible situation. The remainder of the paper is organized as follows: in the following section, we formulate the functional analytic framework of our problem and prove the basic well-posedness results for the state system , , as well as its differentiability properties with respect to the parameter $s$. Afterwards, in Section 3, we study the problem [**(IP)**]{} and establish the first-order necessary and the second-order sufficient conditions of optimality. Some elementary explicit examples are also provided, in order to show the influence of the boundary data and of the target distribution on the optimal exponent. The final section then brings an existence result whose proof employs a compactness result (established in Lemma 6), which is based on Tikhonov’s compactness theorem. Functional analytic setting and results for the solution operator ================================================================= The mathematical setting in which we work is the following: we consider an open and bounded domain $\Omega\subset{\mathbb R}^n$ and a differential operator ${\mathcal{L}}$ acting on functions mapping $\Omega$ into ${\mathbb R}$, together with appropriate boundary conditions. We generally assume that there exists a complete orthonormal system (i.e., an orthonormal basis) $\{e_j\}_{j\in{\mathbb N}}$ of $L^2(\Omega)$ having the property that each $e_j$ lies in a suitable subspace ${\mathcal{D}}$ of $L^2(\Omega)$, and such that $e_j$ is an eigenfunction of ${\mathcal{L}}$ with corresponding eigenvalue $\lambda_j\in{\mathbb R}$, for any ${j\in{\mathbb N}}$ (notice that in this way the boundary conditions of the differential operator ${\mathcal{L}}$ can be encoded in the functional space ${\mathcal{D}}$). In this setting, we may write, [for any $j\in{\mathbb N}$,]{} $$\mathcal{L} e_j =\lambda_j e_j {\mbox{ in }}\Omega,\quad e_j\in{\mathcal{D}}.$$ We also generally assume that $${\mbox{$\lambda_j{\geqslant}0$ for any~$j\in{\mathbb N}$.}}$$ The prototype of operator ${\mathcal{L}}$ that we have in mind is, of course, (minus) the Laplacian in a bounded and smooth domain $\Omega$ (possibly in the distributional sense), together with either Dirichlet or Neumann homogeneous boundary conditions (in these cases, for smooth domains, one can take, respectively, either ${\mathcal{D}}:=H^2(\Omega)\cap H^1_0(\Omega)$ or ${\mathcal{D}}:= H^2(\Omega)$). For any $v$, $w\in L^2(\Omega)$, we consider the scalar product $$\langle v, w\rangle := \int_\Omega v(x)\,w(x)\,dx.$$ In this way, we can write any function $v\in L^2(\Omega)$ [in the form]{} $$v =\sum_{j\in{\mathbb N}} \langle v, e_j\rangle\, e_j,$$ where the equality is indented in the $L^2(\Omega)$-sense, and, if $$v\in {\mathcal{H}}^1 := \Big\{ v\in L^2(\Omega):\,\, \{\lambda_j \,\langle v, e_j\rangle\}_{j\in{\mathbb N}}\in\ell^2 \Big\}$$ then $${\mathcal{L}} v = \sum_{j\in{\mathbb N}} \lambda_j \, \langle v, e_j\rangle\,e_j.$$ For any $s>0$, we define the $s$-power of the operator ${\mathcal{L}}$ in the following way. First, we consider the space $$\label{HD} {\mathcal{H}}^s := \left\{v\in L^2(\Omega):\,\,{\|v\|_{\mathcal{H}^s}<+\infty} \right\},$$ where we use the notation $$\label{Su-s-2s} \| v \|_{{\mathcal{H}}^s}:= \Bigl(\sum_{j\in{\mathbb N}} \lambda_j^{2s} \,\big|\langle v, e_j\rangle\big|^2\Bigr)^{1/2}.$$ Notice that the notation of the space ${\mathcal{H}}^s$ has been chosen in such a way that ${{\mathcal{H}}^s}$, for $s=1$, reduces to the space ${{\mathcal{H}}^1}$ that was introduced above. This notation is reminiscent of, but different from, the notation for fractional Sobolev spaces (roughly speaking, $s=1$ in our notation forces the Fourier coefficients to be in $\ell^2$ weighted by one power of the eigenvalues; in the case of second order operators this would correspond to Sobolev spaces of order two, rather than one, and this difference in the notation is the main reason for which we chose to use calligraphic fonts for our functional spaces). [We then set, for any $v\in{\mathcal{H}}^s$,]{} $$\label{Ls} {\mathcal{L}}^s v:= \sum_{j\in{\mathbb N}} \lambda_j^s \, \langle v, e_j\rangle\,e_j.$$ We are ready now to define our notion of a solution to the state system: given $y_0\in L^2(\Omega)$ and $f:\Omega\times[0,T]\to{\mathbb R}$ such that $f(\cdot,t)\in L^2(\Omega)$ for every $t\in[0,T]$, we say that $\,y:\Omega\times[0,T]\to{\mathbb R}\,$ is a solution to the state system , , if and only if the following conditions are satisfied: $$\begin{aligned} \label{SL1}&&\mbox{$y(\cdot, t)\in{\mathcal{H}}^s$ \,for any \,$t\in(0,T]$,}\\ \label{OV}&& \lim_{t\searrow0} \,\langle y(\cdot,t),\,e_j\rangle= \langle y_0,\,e_j\rangle \quad\mbox{for all }\,j\in{\mathbb N},\\ \label{SL2}&&\mbox{for every $j\in{\mathbb N}$, the mapping $(0,T)\ni t\mapsto \langle y(\cdot,t),\,e_j\rangle$ is } \\ &&\nonumber\mbox{absolutely continuous,}\\ \label{SL3}&&{\mbox{and it holds\,\, $\partial_t \langle y(\cdot,t),\,e_j\rangle +\lambda_j^s \langle y(\cdot,t),\,e_j\rangle =\langle f(\cdot,t),\,e_j\rangle, $}} \\ &&\nonumber\mbox{for every $j\in{\mathbb N}$ and almost every $t\in (0,T)$.}\end{aligned}$$ We remark that conditions , , and  are precisely the functional analytic translations of the functional identity in , . \[TH\] Suppose that $f:\Omega\times[0,T]\to{\mathbb R}$ satisfies $f(\cdot,t)\in L^2(\Omega)$, for every $t\in[0,T]$, as well as $$\label{L2-l2} \sum_{j\in{\mathbb N}} f_j^2\,<\,+\infty, \quad\mbox{where } \,f_j:=\sup_{\theta\in(0,T)} \big| \langle f(\cdot,\theta),e_j\rangle\big|\,.$$ Then the following holds true: [**(i)**]{} If $\,y_0\in L^2(\Omega),$ then there exists for every $s>0$ a unique solution $y(s):=y\,$ to the state system [, ]{} that fulfills the conditions [– $L^2(Q)$]{}. Moreover, with the control-to-state operator ${\mathcal{S}}:s\mapsto y(s)$, we have the explicit representation $$\label{sol1} {\mathcal{S}}(s)(x,t)=y(s)(x,t) = \sum_{j\in {\mathbb N}} y_j(t,s)\,e_j(x) \quad\mbox{a.\,e. in }\,Q,$$ where, for $j\in{\mathbb N}$ and $t\in [0,T]$, we have set $$\label{sol2} y_j(t,s):= \langle y_0,\,e_j\rangle\,e^{-\lambda_j^s t} +\int_0^t \langle f(\cdot,\tau),\,e_j\rangle \,e^{\lambda_j^s (\tau-t)}\,d\tau.$$ [**(ii)**]{} If $\,y_0\in \mathcal{H}^{s/2}$, then $$\label{regu} y(s)\in H^1(0,T;L^2(\Omega))\cap L^\infty(0,T;{\mathcal{H}}^{s/2})\cap L^2(0,T;{\mathcal{H}}^s)\,,$$ and $$\label{dty} \partial_t y(s)\,=\,\sum_{j\in{\mathbb N}} \partial_t y_j(\cdot,s)\,e_j\,.$$ Moreover, we have the estimate $$\begin{aligned} \label{la:sec2} &\| \partial_t y(s)\|_{L^2(Q)}^2\,+\,\|y(s)\|^2_{L^\infty(0,T;{\mathcal{H}^{s/2}})}\, +\,\| y(s)\|_{L^2(0,T;{\mathcal{H}}^s)}^2\,\\[1mm] &\nonumber {\leqslant}\,T\sum_{j\in{\mathbb N}}\sup_{\theta\in (0,T)}|\langle f(\cdot,\theta), e_j\rangle|^2\,+\, \|y_0\|_{{\mathcal{H}}^{s/2}}^2.\end{aligned}$$ [**Remark:**]{} We point out that formula  is of classical flavor and related to Duhamel’s Superposition Principle. In our setting, this kind of explicit representation is an auxiliary tool used to prove the regularity estimates with respect to the fractional parameter $s$ that will be needed later in this paper. [Proof of Theorem \[TH\]:]{} [**(i)**]{}: We first prove that the series defined in represents a function in $L^2(Q)$. To this end, we show that $\{\sum_{j=1}^n y_j(\cdot,s)\,e_j\}_{n\in{\mathbb N}}$ forms a Cauchy sequence in $L^2(Q)$. Indeed, we have, for every $n,p\in{\mathbb N}$, the identity $$\begin{aligned} \label{Cauchy1} &\left\|\mbox{$\sum_{j=1}^{n+p}y_j(\cdot,s)\,e_j\,-\,\sum_{j=1}^n y_j(\cdot,s)$}\,e_j\right\|^2_{L^2(Q)} \\ \nonumber &=\,\int_0^T\left\|\mbox{$\sum_{j=n+1}^{n+p} y_j(t,s)\,e_j$}\right\|^2_{L^2(\Omega)} dt \,=\,\int_0^T\sum_{j=n+1}^{n+p}|y_j(t,s)|^2 dt\,.\end{aligned}$$ Now, for any $\tau\in(0,t)$, we have that $ e^{\lambda_j^s (\tau-t)} {\leqslant}1$, since $\lambda_j{\geqslant}0$. Accordingly, $$\begin{aligned} && \left| \int_0^t \langle f(\cdot,\tau),\,e_j\rangle \,e^{\lambda_j^s (\tau-t)}\,d\tau\right| {\leqslant}\int_0^t \big| \langle f(\cdot,\tau),\,e_j\rangle\big| \,e^{\lambda_j^s (\tau-t)}\,d\tau \\ &&\qquad {\leqslant}\int_0^t \big| \langle f(\cdot,\tau),\,e_j\rangle\big| \,d\tau {\leqslant}T\sup_{\theta\in (0,T)} |\langle f(\cdot,\theta),e_j\rangle|.\end{aligned}$$ Thus, it follows from that for every $j\in{\mathbb N}$ and $t\in [0,T]$ it holds $$\begin{aligned} |y_j(t,s)|\,{\leqslant}\,|\langle y_0,e_j\rangle| \,+\,T\sup_{\theta\in (0,T)} |\langle f(\cdot,\theta),e_j\rangle|\,.\end{aligned}$$ Since $y_0\in L^2(\Omega)$, we have $\,\sum_{j\in{\mathbb N}}|\langle y_0,e_j\rangle|^2=\|y_0\|_{L^2(\Omega)}^2$, and it readily follows from that the sequence $\{ \sum_{j=1}^n \int_0^T |y_j(t,s)|^2 dt\}_{n\in{\mathbb N}}$ is a Cauchy sequence in ${\mathbb R}$, which proves the claim. Next, we observe that $$\begin{aligned} \label{estif} \sup_{\theta\in(0,T)} \| f(\cdot,\theta)\|_{L^2(\Omega)}^2 &=& \sup_{\theta\in(0,T)} \sum_{j\in{\mathbb N}} \big| \langle f(\cdot,\theta),e_j\rangle\big|^2 \\ \nonumber &{\leqslant}&\sum_{j\in{\mathbb N}} \sup_{\theta\in(0,T)}\big| \langle f(\cdot,\theta),e_j\rangle\big|^2,\end{aligned}$$ which is finite, thanks to . Consequently, $$\label{L2-finite} \begin{split} & \int_0^T\| f(\cdot,t)\|^2_{L^2(\Omega)}\,dt < +\infty\\ {\mbox{and }}\quad& \int_0^T\| f(\cdot,t)\|_{L^2(\Omega)}\,dt < +\infty. \end{split}$$ Now, we prove the asserted existence result by showing that the function $y(s)$, which is explicitly defined by , in the statement of the theorem, fulfills for every $s>0$ all of the conditions –. To this end, let $s>0$ be fixed. We set, for $j\in{\mathbb N}$ and $t\in [0,T]$, $$\begin{aligned} \label{V_W_DE} v_j(t,s):=\langle y_0,\,e_j\rangle\,e^{-\lambda_j^s t},\quad w_j(t,s):= \displaystyle\int_0^t \langle f(\cdot,\tau),\,e_j\rangle \,e^{\lambda_j^s (\tau-t)}\,d\tau.\end{aligned}$$ Since $y(s)\in L^2(Q)$, we conclude from and that for every $j\in{\mathbb N}$ and $t\in [0,T]$ it holds that $$\begin{aligned} \label{U2} \langle y(s)(\cdot,t),e_j\rangle\,& =\,\lim_{n\to\infty} \sum_{k=1}^n\langle y_k(t,s)\,e_k,e_j\rangle\\[1mm] \nonumber &=\,y_j(t,s)\,=\,v_j(t,s)+w_j(t,s).\end{aligned}$$ Moreover, for any $t\in(0,T]$, we set $$\kappa(t) := \sup_{r{\geqslant}0}\left(r e^{-rt}\right).$$ Notice that $\kappa(t)<+\infty$ for any $t\in(0,T]$, and $$\begin{aligned} \lambda_j^s \,| v_j(t,s)|&{\leqslant}&\lambda_j^s \big|\langle y_0,\,e_j\rangle\big|\,e^{-\lambda_j^s t}\,{\leqslant}\, \kappa(t)\, \big|\langle y_0,\,e_j\rangle\big|\,.\end{aligned}$$ Since $y_0\in L^2(\Omega)$, we therefore have $$\label{U3} \mbox{$\{\lambda_j^s\, v_j(t,s)\}_{j\in{\mathbb N}}\in\ell^2$, for any~$t\in(0,T]$.}$$ In addition, it holds that $$\begin{aligned} \lambda_j^s \,| w_j(t,s)|&{\leqslant}& \int_0^t \big|\langle f(\cdot,\tau),\,e_j\rangle\big|\,\lambda_j^s \,e^{\lambda_j^s (\tau-t)}\,d\tau\\ &{\leqslant}& \sup_{\theta\in (0,T)} \big|\langle f(\cdot,\theta),\,e_j\rangle\big| \, \int_0^t \lambda_j^s \,e^{\lambda_j^s (\tau-t)}\,d\tau \\ &=& \sup_{\theta\in (0,T)} \big|\langle f(\cdot,\theta),\,e_j\rangle\big|\,(1-e^{-\lambda_j^s t}) \\ &{\leqslant}& \sup_{\theta\in (0,T)} \big|\langle f(\cdot,\theta),\,e_j\rangle\big|,\end{aligned}$$ and we infer from  that also $\,\{\lambda_j^s\, w_j(t,s)\}_{j\in{\mathbb N}}\in\ell^2\,$, for any $t\in(0,T]$. Combining this with and , we see that also the sequence $ \{\lambda_j^s \, \langle y(s)(\cdot,t),e_j\rangle\}_{j\in{\mathbb N}}$ belongs to $\ell^2$, for any $t\in(0,T]$. Thus, by  and , we conclude that $\,y(s)(\cdot,t)\in {\mathcal{H}}^s\,$ for any $t\in(0,T]$, and this proves . Next, we point out that  follows directly from , and thus we focus on the proof of  and . To this end, fix $t\in(0,T)$. If $|h|>0$ is so small that $t+h\in (0,T)$, then we observe that $$\begin{aligned} \label{KH:8a8a-1} &w_j(t+h,s)-w_j(t,s)\\ &=\, e^{-\lambda_j^s(t+h) } \int_t^{t+h} \langle f(\cdot,\tau),\,e_j\rangle \,e^{\lambda_j^s \tau}\,d\tau\nonumber\\ &\quad +\big( e^{-\lambda_j^s h} -1\big) \int_0^{t} \langle f(\cdot,\tau),\,e_j\rangle \, e^{\lambda_j^s (\tau-t)} \,d\tau .\nonumber\end{aligned}$$ On the other hand, if we set $$g_j(t,s):= \langle f(\cdot,t),\,e_j\rangle \,e^{\lambda_j^s t},$$ then we have that $$\begin{aligned} \| g_j(\cdot,s)\|_{L^1(0,T)} &{\leqslant}& e^{\lambda_j^s T}\,\int_0^T \big|\langle f(\cdot,t),\,e_j\rangle\big|\,dt \\ &{\leqslant}& e^{\lambda_j^s T}\,\int_0^T \| f(\cdot,t)\|_{L^2(\Omega)}\,dt,\end{aligned}$$ which is finite, thanks to . Hence, $$g_j(\cdot,s)\in L^1(0,T),$$ and so $w_j(\cdot,s)$ is absolutely continuous, and, by the Lebesgue Differentiation Theorem (see e.g. [@KO] and the references therein), $$\begin{aligned} && \lim_{h\to0} \frac1{h}\int_t^{t+h} \langle f(\cdot,\tau),\,e_j\rangle \,e^{\lambda_j^s \tau}\,d\tau = \lim_{h\to0} \frac1{h}\int_t^{t+h}g_j(\tau,s)\,d\tau \\&&\qquad =g_j(t,s)=\langle f(\cdot,t),\,e_j\rangle \,e^{\lambda_j^s t},\end{aligned}$$ for almost every $t\in(0,T)$. From this and , we infer that $$\lim_{h\to0} \frac{ w_j(t+h,s)-w_j(t,s)}{h} = \langle f(\cdot,t),\,e_j\rangle -\lambda_j^s \int_0^{t} \langle f(\cdot,\tau),\,e_j\rangle \, e^{\lambda_j^s (\tau-t)} \,d\tau ,$$ for almost every $t\in(0,T)$. Since also $v_j(\cdot,s)$ is obviously absolutely continuous, we thus obtain that $y_j(\cdot,s)$ is absolutely continuous and thus differentiable almost everywhere in $(0,T)$, and we have the identity $$\begin{aligned} &&\partial_t\langle y(s) (\cdot,t),e_j\rangle\,=\,\partial_t y_j(t,s)\\ &&=\, -\lambda_j^s\,\langle y_0,\,e_j\rangle\,e^{-\lambda_j^s t}\,+\, \langle f(\cdot,t),\,e_j\rangle \,-\,\lambda_j^s \int_0^{t} \langle f(\cdot,\tau),\,e_j\rangle \, e^{\lambda_j^s (\tau-t)} \,d\tau \\[1mm] &&=\, -\lambda_j^s \,y_j(t,s)+ \langle f(\cdot,t),\,e_j\rangle\\[2mm] &&=\,-\lambda_j^s \,\langle y(s)(\cdot,t),e_j\rangle\,+\, \langle f(\cdot,t),\,e_j\rangle, \quad\mbox{for almost every }\,t\in (0,T)\,.\end{aligned}$$ This proves  and . As for the uniqueness result, we again fix $s>0$ and assume that there are two solutions $y(s), \tilde y(s)\in L^2(Q)$. We put $y^*(s):= y(s)-\tilde y(s)$, and, adapting the notation of , $y^*_j(t,s):=\langle y^*(s)(\cdot,t),e_j\rangle$, for $j\in{\mathbb N}$. Then, using , , and , we infer that for every $j\in{\mathbb N}$ the mapping $\,\,t\mapsto y^*_j(t,s)\,\,$ is absolutely continuous in $(0,T)$, and it satisfies $$\label{SP-01} \partial_t y^*_j(t,s) +\lambda_j^s \,y^*_j(t,s) =0 \quad\mbox{for almost every $\,t\in (0,T)$,}$$ as well as $$\label{SP-02} \lim_{t\searrow0} y^*_j(t,s)=0 .$$ Owing to the absolute continuity of $y^*_j(\cdot,s)$, we obtain (see, e.g., Remark 8 on page 206 of [@brezis]) that $y^*_j(\cdot,s)\in W^{1,1}(0,T)$, so that we can use the chain rule (see, e.g., Corollary 8.11 in [@brezis]). Thus, if we define $\,\zeta_j:=\ln\big(1+(y^*_j(\cdot,s))^2\big)$ and make use of , we have that $$\partial_t \zeta_j = \frac{2 y^*_j(\cdot,s)\, \partial_t y^*_j(\cdot,s)} {1+(y^*_j(\cdot,s))^2} = \frac{-2\lambda_j^s\, (y^*_j(\cdot,s))^2 }{1+(y^*_j(\cdot,s))^2}{\leqslant}0 \quad\mbox{a.\,e. in}\,(0,T).$$ Integrating this relation (see, e.g., Lemma 8.2 in [@brezis]), we find that, for any $t_1<t_2\in(0,T)$, $$\zeta_j(t_2){\leqslant}\zeta_j(t_1).$$ Thus, from , $$\zeta_j(t_2) {\leqslant}\lim_{t_1\searrow0} \zeta_j(t_1) = \lim_{t_1\searrow0} \ln\big(1+(y^*_j(t_1,s))^2\big) =\ln (1) =0,$$ for any $t_2\in(0,T)$. Since also $\zeta_j{\geqslant}0$, we infer that $\zeta_j$ vanishes identically, and thus also  $y^*_j(\cdot,s)$. This proves the uniqueness claim. It remains to show the validity of the claim [**(ii)**]{}. To this end, let again $s>0$ be fixed and assume that $y_0\in {\mathcal{H}}^{s/2}$, which means that $y_0\in L^2(\Omega)$ and $$\sum_{j\in{\mathbb N}} \lambda_j^s\,|\langle y_0,e_j\rangle|^2 < +\infty.$$ Now recall that $\partial_t y_j(t,s)+\lambda_j^s\,y_j(t,s)=\langle f(\cdot,t),e_j\rangle$, for every $j\in{\mathbb N}$ and almost every $t\in (0,T)$. Squaring this equality, we find that $$|\partial_t y_j(t,s)|^2\,+\,\lambda_j^s\,\frac d{dt} \,|y_j(t,s)|^2 \,+\,\lambda_j^{2s}\,|y_j(t,s)|^2\,=\, |\langle f(\cdot,t), e_j\rangle|^2\,,$$ and integration over $[0,\tau]$, where $\tau\in [0,T]$, yields that for every $j\in{\mathbb N}$ we have the identity $$\begin{aligned} &\int_0^\tau |\partial_t y_j(t,s)|^2 dt\,+\,\lambda_j^s\,|y_j(\tau,s)|^2\,+\,\int_0^\tau \lambda_j^{2s}\,|y_j(t,s)|^2 dt \\ &\nonumber =\,\lambda_j^s\,|\langle y_0,e_j\rangle |^2\,+\,\int_0^\tau| \langle f(\cdot,t), e_j\rangle|^2 dt\,,\end{aligned}$$ whence, for every $n\in {\mathbb N}\cup \{0\}$, $p\in{\mathbb N}$, and $\tau\in [0,T]$, $$\begin{aligned} \label{Cauchy2} &\int_0^\tau \sum_{j=n+1}^{n+p}\!\!|\partial_t y_j(t,s)|^2 dt\,+\sum_{j=n+1}^{n+p}\!\!\lambda_j^s\,|y_j(\tau,s)|^2 \,+\int_0^\tau\sum_{j=n+1}^{n+p}\!\! \lambda_j^{2s}\,|y_j(t,s)|^2 dt\\ &{\leqslant}\,\sum_{j=n+1}^{n+p}\lambda_j^s\,|\langle y_0,e_j\rangle|^2\,+\,T\sum_{j=n+1}^{n+p} \sup_{\theta\in (0,T)}\,|\langle f(\cdot,\theta),e_j\rangle|^2\,.\nonumber\end{aligned}$$ We remark that we exchanged here summations and integrals: since, up to now, we are only dealing with a finite summation, this exchange is valid due to the finite additivity of the integrals (in particular, we do not need here any fine result of measure theory). Now, using the same Cauchy criterion argument as in the beginning of the proof of [**(i)**]{}, we can therefore infer that the series $$\begin{aligned} \sum_{j\in{\mathbb N}} \partial_t y_j(\cdot,s)\,e_j, \quad \sum_{j\in{\mathbb N}} \lambda_j^{s/2} \,y_j(\cdot,s)\,e_j, \quad\mbox{and }\,\sum_{j\in{\mathbb N}} \lambda_j^s \,y_j(\cdot,s)\,e_j,\end{aligned}$$ are strongly convergent in the spaces $L^2(Q)$, $L^\infty(0,T;L^2(\Omega))$, and $L^2(Q)$, in this order. Consequently, we have $\,y(s)\in L^\infty(0,T;{\mathcal{H}}^{s/2})\cap L^2(0,T;\mathcal{H}^s)$, and also $\partial_t y(s)\in L^2(0,T;L^2(\Omega))$. We now show that holds true, where we denote the limit of series on the right-hand side by $z$. From the above considerations, we know that, as $n\to\infty$, $$\sum_{j=1}^n y_j(\cdot,s)\,e_j\,\to\,y(s), \quad \sum_{j=1}^n \partial_t y_j(\cdot,s)\,e_j\,\to\,z, \quad\mbox{strongly in }\,L^2(Q)\,.$$ Hence, there is a subsequence $\,\{n_k\}_{k\in{\mathbb N}}\subset{\mathbb N}\,$ such that, for every test function $\,\phi\in C^\infty_0(Q)$, $$\phi\,\sum_{j=1}^{n_k} y_j(\cdot,s)\,e_j\,\to\,\phi\,y(s), \quad \phi\,\sum_{j=1}^{n_k} \partial_t y_j(\cdot,s)\,e_j\,\to\,\phi\,z,\quad\mbox{as }\,k\to\infty,$$ pointwise almost everywhere in $Q$. Using Lebesgue’s Dominated Convergence Theorem and Fubini’s Theorem twice, we therefore have the chain of equalities $$\begin{aligned} &\int_0^T\!\!\int_\Omega \phi(x,t)\,z(x,t)\,dx\,dt\,=\,\lim_{k\to\infty}\int_\Omega\sum_{j=1}^{n_k} e_j(x) \int_0^T\!\!\phi(x,t)\,\partial_t y_j(t,s)\,dt\,dx\\ &=\,-\,\lim_{k\to\infty}\int_\Omega\sum_{j=1}^{n_k} e_j(x)\int_0^T\!\!\partial_t\phi(x,t)\,y_j(t,s)\,dt\,dx\\ &=\,-\int_0^T\!\!\int_\Omega\!\partial_t\phi(x,t)\,y(s)(x,t)\,dx\,dt\,,\end{aligned}$$ for every $\,\phi\in C^\infty_0(Q)$, that is, we have $\,z=\partial_t y(s)\,$ in the sense of distributions. Since $z\in L^2(Q)$, it therefore holds $\,y(s)\in H^1(0,T;L^2(\Omega))\,$ with $\,\partial_t y(s)=z$, as claimed. Finally, we obtain the estimate from choosing $n=0$ and letting $p\to\infty$ in , which concludes the proof of the assertion. [$\Box$]{} Next, we prove an auxiliary result on the derivatives of a function of exponential type that will play an important role in the subsequent analysis. To this end, we define, for fixed $\lambda>0$ and $t>0$, the real-valued function $$\label{DEF:E} E_{\lambda,t} (s):= e^{-\lambda^s t} \quad\mbox{for \,$s>0$},$$ and denote its first, second, and third derivatives with respect to $s$ by $E_{\lambda,t}' (s)$, $E_{\lambda,t}'' (s)$, and $E_{\lambda,t}''' (s)$, respectively. We have the following result. \[DER:1\] There exist constants $\,\widehat{C}_{i}>0$, $0{\leqslant}i{\leqslant}3$, such that, for all $\lambda>0$, $t\in (0,T]$, and $s>0$, $$\begin{aligned} &\big| E_{\lambda,t} (s)\big|\,{\leqslant}\,\widehat C_0, \quad \left|E_{\lambda,t}'(s)\right|\,{\leqslant}\,s^{-1}\,\widehat C_1 \,\big(1+|\ln(t)|\big),\\ &\left|E_{\lambda,t}''(s)\right|\,{\leqslant}\,s^{-2}\,\widehat C_2\,\big( 1+|\ln(t)|^2\big), \quad \left|E_{\lambda,t}'''(s)\right|{\leqslant}\,s^{-3}\,\widehat C_3\,\big(1+|\ln(t)|^3\big)\,.\end{aligned}$$ [Proof:]{} Obviously, we may choose $\widehat C_0=1$, and a simple differentiation exercise shows that the first three derivatives of $E_{\lambda,t}$ are given by $$\begin{aligned} &\nonumber E'_{\lambda,t}(s)=-\lambda^s\, t \,e^{-\lambda^s t}\,\ln (\lambda), \quad E''_{\lambda,t}(s)=\lambda^s\,t\,e^{-\lambda^s t}\left(\lambda^s t-1\right)\,(\ln (\lambda))^2,\\ &\nonumber E'''_{\lambda,t}(s)=\lambda^s\,t\,e^{-\lambda^s t}\left(3\lambda^s t-1-(\lambda^s t)^2\right) (\ln (\lambda))^3. \end{aligned}$$ Now, observe that $$\frac{\ln(\lambda^s t)-\ln (t)}{s} = \frac{\ln(\lambda^s)+\ln( t)-\ln(t)}{s}= \ln(\lambda) .$$ Accordingly, we may substitute for $\ln(\lambda)$ in the above identities to obtain that $$\begin{aligned} \label{tbwb-2} E_{\lambda,t}' (s) \,&= - s^{-1}\lambda^s t\,e^{-\lambda^s t}\, \big( \ln(\lambda^s t)-\ln(t)\big), \\ E_{\lambda,t}'' (s) \,&=s^{-2} \lambda^s t\,e^{-\lambda^s t}\,(\lambda^s t-1)\, \big( \ln(\lambda^s t)-\ln(t)\big)^2, \nonumber\\ E_{\lambda,t}'''(s)\,&=s^{-3}\lambda^s t\,e^{-\lambda^s t} \left(3 \lambda^s t-1-(\lambda^s t)^2\right) \big(\ln(\lambda^s t)-\ln(t)\big)^3\,.\nonumber \end{aligned}$$ Thus, we may consider $\,r:= \lambda^s t\,$ as a “free variable” in . Using the fact that $$|\ln(r)-\ln(t)|^k\,{\leqslant}\,2^k\left(|\ln(r)|^k+|\ln(t)|^k\right) \quad\mbox{for } \,1{\leqslant}k{\leqslant}3,$$ and introducing the finite quantities $$\begin{aligned} M_1 &:= \sup_{r>0} \left(r\,e^{-r}\,|\ln (r)|\right),\\ M_2 &:= \sup_{r>0} \left(r\,e^{-r}\right),\\ M_3 &:= \sup_{r>0} \left(r\,e^{-r}\, |r-1|\,4\,|\ln(r)|^2\right),\end{aligned}$$ $$\begin{aligned} M_4&:= \sup_{r>0} \left(r\,e^{-r}\, 4\,|r-1|\right),\\ M_5&:= \sup_{r>0} \left(r\,e^{-r}\left|3r-1-r^2\right|\,8\,|\ln(r)|^3\right),\\ M_6&:= \sup_{r>0} \left(r\,e^{-r}\,8\,\left|3r-1-r^2\right|\right),\end{aligned}$$ we deduce from  the estimates $$\begin{aligned} \big| E_{\lambda,t}' (s)\big|\,&{\leqslant}\, s^{-1} \big( M_1+M_2\,|\ln (t)|\big), \\ \big| E_{\lambda,t}'' (s)\big|\,&{\leqslant}\, s^{-2} \big( M_3+M_4\,|\ln (t)|^2\big),\\ \big|E_{\lambda,t}'''(s)\big|\,&{\leqslant}\,s^{-3}\big(M_5+M_6\,|\ln(t)|^3\big),\end{aligned}$$ whence the assertion follows. [$\Box$]{} We are now in the position to derive differentiability properties for the control-to-state mapping $\mathcal{S}$. As a matter of fact, we will focus on the first and second derivatives, but derivatives of higher order may be taken into account with similar methods. In detail, we have the following result: \[II O\] Suppose that that $f:\Omega\times[0,T]\to{\mathbb R}$ satisfies $f(\cdot,t)\in L^2(\Omega)$, for every $t\in[0,T]$, as well as the condition . Moreover, let $y_0\in L^2(\Omega)$. Then the control-to-state mapping $\mathcal{S}$ is twice Fréchet differentiable on $(0,+\infty)$ when viewed as a mapping from $(0,+\infty)$ into $L^2(Q)$, and for every ${\overline{s}}\in(0,+\infty)$ the first and second Fréchet derivatives $D_s\mathcal{S}({\overline{s}})\in\mathcal{L}({\mathbb R},L^2(Q))$ and $D^2_{ss}\mathcal{S}({\overline{s}})\in\mathcal{L}({\mathbb R},\mathcal{L}({\mathbb R},L^2(Q)))$ can be identified with the $L^2(Q)$–functions $$\begin{aligned} \label{deriS1} \partial_s y({\overline{s}})& \,:=\,\sum_{j\in{\mathbb N}}\partial_s y_j(\cdot,{\overline{s}})\,e_j,\quad \partial^2_{ss} y({\overline{s}})\,:=\,\sum_{j\in{\mathbb N}}\partial^2_{ss} y_j(\cdot,{\overline{s}})\,e_j\,,\end{aligned}$$ respectively. More precisely, we have, for all $h,k\in{\mathbb R}$, $$\label{deriS2} D_s\mathcal{S}({\overline{s}})(h)\,=\,h\,\partial_s y({\overline{s}})\quad\mbox{and }\, D^2_{ss}\mathcal{S}({\overline{s}})(h)(k)\,=\,h\,k\,\partial^2 _{ss}y({\overline{s}})\,.$$ Moreover, there is a constant $\widehat C_4>0$ such that for all ${\overline{s}}\in(0,+\infty)$ it holds that $$\begin{aligned} \label{estideri1} \left\| D_s\mathcal{S}({\overline{s}}) \right\|_{\mathcal{L}({\mathbb R},L^2(Q))}&\,=\,\|\partial_s y({\overline{s}})\|_{L^2(Q)}\,\,{\leqslant}\,\, \frac{\widehat C_4}{{\overline{s}}}\,,\\ \label{estideri2} \left \|D^2_{ss}\mathcal{S}({\overline{s}})\right\|_{\mathcal{L}({\mathbb R},\mathcal{L}({\mathbb R},L^2(Q)))}&\,=\, \|\partial^2_{ss} y({\overline{s}})\|_{L^2(Q)}\,{\leqslant}\,\frac{\widehat C_4}{{\overline{s}}^2}\,.\end{aligned}$$ [Proof:]{} Let ${\overline{s}}\in(0,+\infty)$ be fixed. We first show that the functions defined in do in fact belong to $L^2(Q)$. To this end, we first note that $$e^{\lambda_j^s (\tau-t)}\,{\leqslant}\,e^{\lambda_j^s(t-\tau)}\quad \mbox{for }\,0 {\leqslant}\tau < t,$$ and that for $1{\leqslant}k{\leqslant}3$ the functions $$\begin{aligned} \phi_k(t):= 1+\,|\ln(t)|^k,\quad \psi_k(t):=\int_0^t\!\!\left(1+|\ln(t-\tau)|^k\right)d\tau, \quad t\in (0,T],\end{aligned}$$ belong to $L^2(0,T)$. To check this fact, we use the substitution $\theta=-\ln(t)$ and we observe that $$\begin{aligned} && \int_0^T |\ln(t)|^{k}\,dt{\leqslant}\int_0^{T+1} |\ln(t)|^{k}\,dt \\&& \qquad {\leqslant}\int_0^{1} |\ln(t)|^{k}\,dt + \int_1^{T+1} |\ln(T+1)|^{k}\,dt \\&& \qquad= \int_1^{+\infty} \theta^{k}\,e^{-\theta}\,d\theta + T\,|\ln(T+1)|^{k}{\leqslant}C(k,T),\end{aligned}$$ for some $C(k,T)\in(0,+\infty)$. Accordingly, $$\int_0^T |\phi_k(t)|^2\,dt {\leqslant}4 \int_0^T \big(1+|\ln(t)|^{2k}\big)\,dt{\leqslant}4T +4C(2k,T),$$ hence $\phi_k\in L^2(0,T)$. Similarly, for any $t\in(0,T]$, $$\int_0^t |\ln(t-\tau)|^{k}\,d\tau = \int_0^t |\ln(\vartheta)|^{k}\,d\vartheta{\leqslant}\int_0^T |\ln(\vartheta)|^{k}\,d\vartheta{\leqslant}C(k,T),$$ and therefore $$|\psi_k(t)|{\leqslant}T+C(k,T),$$ which gives that $\psi_k\in L^\infty(0,T)\subset L^2(0,T)$, as desired. Next, we infer from  and Lemma \[DER:1\] that, for every $t\in (0,T]$, $j\in{\mathbb N}$, and $1{\leqslant}k{\leqslant}3$, the estimates $$\begin{aligned} &\left|\frac{\partial^k}{\partial s^k}\,v_j(t,{\overline{s}})\right|\,{\leqslant}\, |\langle y_0,e_j\rangle|\left| \frac{d^k}{d s^k}\,E_{\lambda_j,t}({\overline{s}}) \right| \,{\leqslant}\,\frac{\widehat C_k}{{\overline{s}}^{\,k}}\,\phi_k(t)\,|\langle y_0,e_j\rangle|\,,\\[1mm] &\left|\frac{\partial^k}{\partial s^k}\,w_j(t,{\overline{s}})\right|\,{\leqslant}\,\int_0^t |\langle f(\cdot,\tau), e_j\rangle|\,\left| \frac{d^k}{ds^k}\,E_{\lambda_j,\tau-t}({\overline{s}})\right| d\tau \\[2mm] &\hspace*{25mm}{\leqslant}\,\widehat C_k\,{\overline{s}}^{\,-k}\,\psi_k(t)\,\sup_{\theta\in (0,T)}\,|\langle f(\cdot,\theta), e_j\rangle|\,.\nonumber\end{aligned}$$ Therefore, recalling , we find that, for every $p\in{\mathbb N}$, $n\in{\mathbb N}\cup\{0\}$, and $1{\leqslant}k{\leqslant}2$, $$\begin{aligned} &\left\|\sum_{j=n+1}^{n+p}\frac{\partial^k}{\partial s^k} \,y_j(t,{\overline{s}})\,e_j\right\|^2 _{L^2(Q)} \,{\leqslant}\,\sum_{j=n+1}^{n+p}\int_0^T\left|\frac{\partial ^k}{\partial s^k}\,y_j(t,{\overline{s}})\right|^2 dt\\ &\,{\leqslant}\,2\sum_{j=n+1}^{n+p}\int_0^T\left|\frac{\partial ^k}{\partial s^k}\,v_j(t,{\overline{s}})\right|^2 dt \,+\,2\sum_{j=n+1}^{n+p}\int_0^T\left|\frac{\partial ^k}{\partial s^k}\,w_j(t,{\overline{s}})\right|^2 dt \nonumber\\ &\nonumber {\leqslant}\,2\,\widehat C_k^2\,{\overline{s}}^{\,-2k}\left( \int_0^T\phi_k^2(t)\,dt \sum_{j=n+1}^{n+p}|\langle y_0,e_j\rangle|^2\right.\\ &\nonumber\hspace*{10mm}\left.+ \int_0^T \psi_k^2(t)\,dt\,\sum_{j=n+1}^{n+p}\,\sup_{\theta\in (0,T)} \,|\langle f(\cdot,\theta),e_j\rangle|^2\right)\,\,\longrightarrow 0\,, \end{aligned}$$ as $n\to\infty$. The Cauchy criterion for series then shows the validity of our claim. Moreover, taking $n=0$ and letting $p\to\infty$ in the above estimate, we find that and are valid provided that holds true. It remains to show the differentiability results. To this end, let $0<|h|<{\overline{s}}/2$. Then $\frac 1{{\overline{s}}-|h|}<\frac 2 {{\overline{s}}}$, and, invoking Lemma 2 and Taylor’s Theorem, we obtain for all $j\in{\mathbb N}$ and $t\in (0,T]$ the estimates $$\begin{aligned} &\left|E_{\lambda_j,t}({\overline{s}}+h)-E_{\lambda_j,t}({\overline{s}})-h\,E_{\lambda_j,t}'({\overline{s}})\right| \,=\,\frac 1 2\,h^2\,\left|E_{\lambda_j,t}''(\xi_h)\right|\\[1mm] &\nonumber{\leqslant}\,\frac 1 2 \,\widehat C_2\,\xi_h^{-2}\,\phi_2(t)\,h^2 \,{\leqslant}\, 2 \,\widehat C_2\,{\overline{s}}^{\,-2}\,\phi_2(t)\,h^2,\\[2mm] &\left|E_{\lambda_j,t}'({\overline{s}}+h)-E_{\lambda_j,t}'({\overline{s}})-h\,E_{\lambda_j,t}''({\overline{s}})\right| \,=\,\frac 1 2\,h^2\,\left|E_{\lambda_j,t}'''(\eta_h)\right|\\ &{\leqslant}\,4\,\widehat C_3\,{\overline{s}}^{\,-3}\,\phi_3(t)\,h^2,\nonumber\end{aligned}$$ with suitable points $\,\xi_h,\eta_h\in ({\overline{s}}-|h|,{\overline{s}}+|h|)$. By the same token, $$\begin{aligned} &\int_0^t \left|E_{\lambda_j,\tau-t}({\overline{s}}+h)-E_{\lambda_j,\tau-t}({\overline{s}})- h\,E_{\lambda_j,\tau-t}'({\overline{s}})\right| d\tau\\ &\nonumber{\leqslant}\,2\,\widehat C_2\,{\overline{s}}^{\,-2}\int_0^t\phi_2(t-\tau)\,d\tau\,h^2,\\[2mm] &\int_0^t \left|E_{\lambda_j,\tau-t}'({\overline{s}}+h)-E_{\lambda_j,\tau-t}'({\overline{s}})- h\,E_{\lambda_j,\tau-t}''({\overline{s}})\right| d\tau\\ &\nonumber{\leqslant}\,4\,\widehat C_3\,{\overline{s}}^{\,-3}\int_0^t\phi_3(t-\tau)\,d\tau\,h^2\,.\end{aligned}$$ From this, we conclude that with suitable constants $K_i>0$, $1{\leqslant}i{\leqslant}4$, which depend on ${\overline{s}}$ but not on $0<|h|<{\overline{s}}/2$, $j\in{\mathbb N}$, and $t\in (0,T]$, we have the estimates $$\begin{aligned} \label{stime1} &\left|v_j(t,{\overline{s}}+h)-v_j(t,{\overline{s}})-h\,\partial_s v_j(t,{\overline{s}})\right|^2\,{\leqslant}\, K_1\,\phi_2^2(t) \,|\langle y_0,e_j\rangle|^2\,h^4,\\[1mm] \label{stime2} &\left|\partial_s v_j(t,{\overline{s}}+h)-\partial_s v_j(t,{\overline{s}})-h\,\partial^2_{ss} v_j(t,{\overline{s}})\right|^2 \\ &\nonumber{\leqslant}\, K_2\,\phi_3^2(t) \,|\langle y_0,e_j\rangle|^2\,h^4,\\[1mm] \label{stime3} &\left|w_j(t,{\overline{s}}+h)-w_j(t,{\overline{s}})-h\,\partial_s w_j(t,{\overline{s}})\right|^2\\ &\nonumber{\leqslant}\, K_3\int_0^T\!\!\phi_2^2(t) dt\,\sup_{\theta\in (0,T)} |\langle f(\cdot,\theta),e_j\rangle|^2\,h^4,\\[1mm] \label{stime4} &\left|\partial_s w_j(t,{\overline{s}}+h)-\partial_s w_j(t,{\overline{s}})- h\,\partial^2_{ss} w_j(t,{\overline{s}})\right|^2\\ &\nonumber {\leqslant}\, K_4\int_0^T\!\!\phi_3^2(t) dt\,\sup_{\theta\in (0,T)} |\langle f(\cdot,\theta),e_j\rangle|^2\,h^4\,.\end{aligned}$$ From and , we infer that there is a constant $K_5>0$, which is independent of $0<|h|<{\overline{s}}/2$, such that $$\begin{aligned} &\Bigl\|y({\overline{s}}+h)-y({\overline{s}})-h\,\sum_{j\in{\mathbb N}}\partial_s y_j(\cdot,{\overline{s}})\,e_j\Bigr\|^2_{L^2(Q)}\\ &\nonumber{\leqslant}\,\lim_{n\to\infty} \sum_{j=1}^n\int_0^T|y_j(t,{\overline{s}}+h)-y_j(t,{\overline{s}}) -h\,\partial_s y_j(t,{\overline{s}})|^2\,dt\\ &\nonumber{\leqslant}\,K_5\,\Bigl(\sum_{j\in{\mathbb N}}|\langle y_0,e_j\rangle|^2\,+\,\sum_{j\in{\mathbb N}} f_j^2\Bigr)\,h^4\,.\end{aligned}$$ Hence, $\mathcal{S}$ is Fréchet differentiable at ${\overline{s}}$ as a mapping from $(0,+\infty)$ into $L^2(Q)$, and the Fréchet derivative is given by the linear mapping $$h \mapsto D_s\mathcal{S}({\overline{s}})(h)\,=\,h\,\sum_{j\in{\mathbb N}}\partial_s y_j(\cdot,{\overline{s}})\,e_j,$$ as claimed. The corresponding result for the second Fréchet derivative follows similarly employing the estimates and . This concludes the proof of the assertion. [$\Box$]{} Optimality conditions ===================== In this section, we establish first-order necessary and second-order sufficient optimality conditions for the control problem [**(IP)**]{}. We do not address the question of existence of optimal controls, here; this will be the subject of the forthcoming section. We have the following result. \[CL3A\] Suppose that that $f:\Omega\times[0,T]\to{\mathbb R}$ satisfies $f(\cdot,t)\in L^2(\Omega)$, for every $t\in[0,T]$, as well as condition . Moreover, let $y_0\in L^2(\Omega)$ be given. Then the following holds true: [**(i)**]{} If ${\overline{s}}\in (0,L)$ is an optimal parameter for [**(IP)**]{} and $y({\overline{s}})$ is the associated (unique) solution to the state system – according to Theorem 1, then $$\begin{aligned} \label{necessary} \int_0^T\!\!\int_\Omega (y({\overline{s}})-y_Q)\,\partial_s y({\overline{s}})\,dx\,dt \,+\,\varphi'({\overline{s}})\,=\,0,\end{aligned}$$ where $\,\partial_s y({\overline{s}})$ is given by . [**(ii)**]{} If ${\overline{s}}\in (0,L)$ satisfies condition and, in addition, $$\begin{aligned} \label{sufficient} \int_0^T\!\!\int_\Omega\left[ (\partial_s y({\overline{s}}))^2\,+\,(y({\overline{s}})-y_Q)\,\partial^2_{ss} y({\overline{s}})\right]dx\,dt \,+\,\varphi''({\overline{s}})\,>\,0,\end{aligned}$$ where $\,\partial^2_{ss} y({\overline{s}})\,$ is defined in , then ${\overline{s}}$ is optimal for [**(IP)**]{}. [Proof:]{} By Theorem \[II O\], the “reduced” cost functional $\,s\mapsto \mathcal{J}(s):=J(y(s),s)\,$ is twice differentiable on $(0,L)$, and it follows directly from the chain rule that $$\begin{aligned} \mathcal{J}'({\overline{s}})\,&=\,\frac d{ds} \,J(y({\overline{s}}),{\overline{s}})\,=\,\partial_yJ(y({\overline{s}}),{\overline{s}})\circ D_s\mathcal{S}({\overline{s}}) + \partial_s J(y({\overline{s}}),{\overline{s}})\\ &=\,\int_0^T\!\!\int_\Omega (y({\overline{s}})-y_Q)\,\partial_s y({\overline{s}})\,dx\,dt \,+\,\varphi'({\overline{s}})\,.\end{aligned}$$ Moreover, $$\begin{aligned} \mathcal{J}''({\overline{s}})\,=\,\int_0^T\!\!\int_\Omega\left[ (\partial_s y({\overline{s}}))^2\,+\,(y({\overline{s}})-y_Q)\, \partial^2_{ss} y({\overline{s}})\right]\,dx\,dt \,+\,\varphi''({\overline{s}})\,.\end{aligned}$$ The assertions [**(i)**]{} and [**(ii)**]{} then immediately follow.[$\Box$]{} [**Remark:**]{} In our framework, optimizers ${\overline{s}}$ can be found by minimizing methods (see Theorem \[TH 5\]): in this setting, the conditions in assure that the optimal parameter ${\overline{s}}$ lies in the open interval $(0,L)$. Also, if $\varphi'(s)$ blows up near $0$ faster than $1/{s}$ (as it happens in the examples given in ), solutions of do not accumulate near $0$, since, by and , $$\left| \int_0^T\!\!\int_\Omega (y(s)-y_Q)\,\partial_s y(s)\,dx\,dt \right| {\leqslant}\| y(s)-y_Q \|_{L^2(Q)}\,\| \partial_s y(s)\|_{L^2(Q)}{\leqslant}\frac{C}{s},$$ for some $C>0$. [**Remark:**]{} It is customary in optimal control theory to formulate the first-order necessary optimality conditions in terms of a variational inequality (which encodes possible control constraints) and an adjoint state equation, while second-order sufficient condition also involve the so-called “$\tau$–critical cone” (see, e.g., the textbook [@TR]). In our situation, we can avoid these abstract concepts, since we have explicit formulas for the relevant quantities at our disposal. Indeed, in order to evaluate $\,y(s)\,,\,\partial_s y(s)\,,\,\partial_{ss}y(s)$, we can use the series representations given in and . In practice, this amounts to determining the eigenvalues $\,\lambda_j\,$ and the associated eigenfunctions $e_j$ up to a sufficiently large index $j$, and then to making use of the differentation formulas for the functions for $\lambda=\lambda_j$ that are provided at the beginning of the proof of Lemma \[DER:1\]. Using a standard technique (say, Newton’s method), we then can easily find an approximate minimizer of the cost functional. Also in the case that control constraints $-\infty< a{\leqslant}s{\leqslant}b <+\infty$ are to be respected, this strategy would still work to find interior minimizers $\bar s\in (a,b)$, while the value of the cost at $a$ and $b$ can also be calculated. [**Remark:**]{} We recall that in infinite dimensional setting conditions like are not necessarily sufficient conditions, see Example 3.3 in [@MR3311948]. On the other hand, this is the case in finite dimensions. To clarify Theorem \[CL3A\], we now present two simple explicit examples that outline the behavior of the optimal exponent ${\overline{s}}$ (recall  and ). To make the arguments as simple as possible, we assume that ${\varphi}$ is strictly convex and that the forcing term $f$ is identically zero (as a matter of fact, the functions ${\varphi}$ presented in  as examples fulfill also this convexity assumtpion). Notice that under these assumptions on ${\varphi}$ the function ${\varphi}$ has a unique critical point $s_0\in(0,+\infty)$, which is a minimum (see Figure \[CORCO\]). ![The natural cost function $\varphi$ and its derivative.[]{data-label="CORCO"}](PHI.pdf){height="6cm"} The examples are related to the fractional Laplacian in one variable, namely, the case of homogeneous Neumann data and the case of homogeneous Dirichlet data on an interval. We will see that, in general, the optimal exponent ${\overline{s}}$ differs from the minimum $s_0$ of ${\varphi}$ (and, in general, it can be both larger or smaller). In a sense, this shows that different boundary data and different target distributions $y_Q$ influence the optimal exponent ${\overline{s}}$ and its relation with the minimum $s_0$ for $\varphi$. [**Example 1.**]{} Consider as operator ${\mathcal{L}}$ the classical $\,-\Delta$ on the interval $(0,\pi)$ with homogeneous Neumann data. In this case, we can take as eigenfunctions $e_j(x):= c_j\,\cos(j\,x)$, where $c_j\in{\mathbb R}\setminus\{0\}$ is a normalizing constant, and $j=0,1,2,3,\dots$. The eigenvalue corresponding to $e_j$ is $\lambda_j=j^2$. Now let, with a fixed $j_0\in{\mathbb N}$, where $j_0>1$, and $\epsilon\in{\mathbb R}$, $$y_0(x):= 1 + \epsilon\, e_{j_0}(x) \quad\forall\,x\in [0,\pi].$$ Then it is easily verified that for every $s>0$ the unique solution to , is given by $$y(s)(x,t) = 1 + \epsilon \, e_{j_0}(x)\, e^{-j_0^{2s}\,t} \quad\forall\, (x,t)\in\overline{Q}.$$ We now make the special choice $y_Q(x,t):=1$ for the target function. We then observe that $$\partial_s y(s)(x,t) = -2\epsilon \,j_0^{2s}\, \ln(j_0)\, t\, e_{j_0}(x)\, e^{-j_0^{2s}\,t},$$ and therefore, using the substitution $\vartheta:=j_0^{2s}\,t$, $$\begin{aligned} && \int_0^T\!\!\int_\Omega (y(s)-y_Q)\,\partial_s y (s) \,dx\,dt \\ &=& -2\epsilon^2 \,j_0^{2s}\, \ln(j_0)\, \int_0^T\!\!\int_\Omega t\, e_{j_0}^2(x)\, e^{-2j_0^{2s}\,t} \,dx\,dt \\ &=& -2\epsilon^2 \,j_0^{2s}\, \ln(j_0)\, \int_0^T t\, e^{-j_0^{2s}\,t}\,dt \\ &=& -2\epsilon^2 \,j_0^{-2s} \,\ln(j_0)\, \int_0^{j_0^{2s}T} \vartheta\, e^{-2\vartheta}\,dt.\end{aligned}$$ As a consequence, condition  becomes, in this case, $$\label{CA-1} \varphi'({\overline{s}}) = 2\epsilon^2 \,j_0^{-2{\overline{s}}} \,\ln(j_0) \int_0^{j_0^{2{\overline{s}}}T} \vartheta\, e^{-2\vartheta}\,dt.$$ If $\epsilon=0$ (and when $j_0\to+\infty$), then the identity in  reduces to $\varphi'({\overline{s}}) =0$; that is, in this case the “natural” optimal exponent $s_0$ coincides with the optimal exponent ${\overline{s}}$ given by the full cost functional (that is, in this case the external conditions given by the exterior forcing term and the resources do not alter the natural diffusive inclination of the population). ![The optimal exponent ${\overline{s}}$ in Example 1.[]{data-label="FIG:NEUMANN"}](AVANTI.pdf){height="6cm"} But, in general, for fixed $\epsilon\ne0$ and $j_0>1$, the identity in  gives that $\varphi'({\overline{s}}) >0$. This, given the convexity of $\varphi$, implies that ${\overline{s}}>s_0$, i.e., the optimal exponent given by the cost functional is larger than the natural one (see Figure \[FIG:NEUMANN\]). [**Example 2.**]{} Now we consider as operator ${\mathcal{L}}$ the classical $-\Delta$ on the interval $(0,\pi)$ with homogeneous Dirichlet data. In this case, we can take as eigenfunctions $e_j(x):= c_j\,\sin(j\,x)$, where $c_j\in{\mathbb R}\setminus\{0\}$ is a normalizing constant, and $j=1,2,3,\dots$. The eigenvalue corresponding to $e_j$ is $\lambda_j=j^2$. ![The optimal exponent ${\overline{s}}$ in Example 2.[]{data-label="FIG:DIRI"}](DIETRO.pdf){height="6cm"} For fixed $j_0\in{\mathbb N}$ with $j_0{\geqslant}1$, and $\epsilon\in{\mathbb R}$, we set $$y_0(x) := \epsilon\, e_{j_0}(x)\quad\forall\,x\in [0,\pi]\,.$$ Then, for every $s>0$, the corresponding solution is given by $$y(s)(x,t) = \epsilon \,e_{j_0}(x)\, e^{-j_0^{2s}\,t} \quad\forall\,(x,t)\in\overline{Q}\,.$$ Now, let $y_Q(x,t):=\epsilon\, e_{j_0}(x)$ for $(x,t)\in Q$. We have $$\partial_s y(s)(x,t) = -2\epsilon \,j_0^{2s}\, \ln(j_0)\, t\, e_{j_0}(x)\, e^{-j_0^{2s}\,t},$$ and therefore, using the substitution $\vartheta:=j_0^{2s}\,t$, $$\begin{aligned} && \int_0^T\!\!\int_\Omega (y(s)-y_Q)\,\partial_s y (s) \,dx\,dt \\ &=& -2\epsilon^2\,j_0^{2s}\, \ln(j_0)\, \int_0^T\!\!\int_\Omega t\,e_{j_0}^2(x)\, \big( e^{-j_0^{2s}\,t} -1\big)\,e^{-j_0^{2s}\,t} \,dx\,dt \\ &=& -2\epsilon^2\,j_0^{2s}\, \ln(j_0)\, \int_0^T t\, \big( e^{-j_0^{2s}\,t} -1\big)\,e^{-j_0^{2s}\,t} \,dt \\ &=& -2\epsilon^2\,j_0^{-2s}\,\ln(j_0)\, \int_0^{j_0^{2s} T} \vartheta\, \big( e^{-\vartheta} -1\big)\,e^{-\vartheta} \,d\vartheta.\end{aligned}$$ So, in this case, condition  becomes $$\label{CA-2} \varphi'({\overline{s}}) = 2\epsilon^2\,j_0^{-2{\overline{s}}}\,\ln(j_0)\, \int_0^{j_0^{2{\overline{s}}} T} \vartheta\, \big( e^{-\vartheta} -1\big)\,e^{-\vartheta} \,d\vartheta.$$ If $\epsilon=0$ (and when $j_0\to+\infty$), then the identity in  reduces to $\varphi'({\overline{s}}) =0$, which boils down to ${\overline{s}}=s_0$. But if $\epsilon\ne0$ and $j_0{\geqslant}1$, then the identity in  gives that $\varphi'({\overline{s}}) <0$. By the convexity of $\varphi$, this implies that ${\overline{s}}<s_0$, i.e., the optimal exponent given by the full cost functional is in this case smaller than the natural one (see Figure \[FIG:DIRI\]). We observe that, in the framework of Examples 1 and 2, the effect of a larger $s$ is to “cancel faster” the higher order harmonics in the solution $y$; since these harmonics are related to “wilder oscillations”, one may think that the higher $s$ becomes, the bigger the smoothing effect is. In this regard, roughly speaking, a larger $s$ “matches better” with a constant target function $y_Q$ and a smaller $s$ with an oscillating one (compare again Figures \[FIG:NEUMANN\] and \[FIG:DIRI\]). We also remark that when $j_0{\geqslant}2$ in Example 2 (or if $\epsilon$ is large in Example 1), the solution $y$ is not positive. On the one hand, this seems to reduce the problem, in this case, to a purely mathematical question, since if $y$ represents the density of a biological population, the assumption $y{\geqslant}0$ seems to be a natural one. On the other hand, there are other models in applied mathematics in which the condition $y{\geqslant}0$ is not assumed: for instance, if $y$ represents the availability of specialized workforce in a given field, the fact that $y$ becomes negative (in some regions of space, at some time) translates into the fact that there is a lack of this specialized workforce (and, for example, non-specialized workers have to be used to compensate this lack). The use of mathematical models to deal with problems in the job market is indeed an important topic of contemporary research, see e.g. [@Stew] and the references therein. The models arising in the (short time) job market also provide natural examples in which the birth/death effects in the diffusion equations are negligible. Existence and a compactness lemma ================================= In this section, we establish an existence result for the identification problem [**(IP)**]{}. We make the following general assumption for the initial datum $y_0$: $$\label{initial} \sup_{s\in(0,L)} \|y_0\|_{{\mathcal{H}}^s}<+\infty.$$ [**Remark:**]{} We remark that the condition can be very restrictive if $L$ is large. Indeed, we obviously have $\,\lambda_j^{2s}{\leqslant}1\,$ for $\,\lambda_j{\leqslant}1$, and for $\,\lambda_j>1\,$ the function $\,\,s\mapsto \lambda_j^{2s}\,$ is strictly increasing. From this it follows that is certainly fulfilled for a finite $L$ if only $\|y_0\|_{{\mathcal{H}}^L}<+\infty$, that is, if $y_0\in {\mathcal{H}}^L$. For an example, consider the prototypical case when ${\mathcal{L}}=-\Delta$ with zero Dirichlet boundary condition. Then the choice $L=\frac 12$ leads to the requirement $y_0\in H^1_0(\Omega)$, while for the choice $L=1$ we must have $y_0\in H^2(\Omega)\cap H^1_0(\Omega)$: indeed, if $\,\{\lambda_j\}_{j\in{\mathbb N}}\,$ are the corresponding eigenvalues with associated orthogonal eigenfunctions $\{e_j\}_{j\in{\mathbb N}}$, normalized by $\|e_j\|_{L^2(\Omega)}=1$ for all $j\in{\mathbb N}$, then it is readily verified that the set $\{\lambda_j^{-1/2}e_j\}_{j\in{\mathbb N}}$ forms an orthonormal basis in the Hilbert space $\bigl(H^1_0(\Omega), \langle \cdot,\cdot\rangle_1\bigr)$ with respect to the inner product $\,\langle u,v\rangle_1:={\int_{\Omega}}\nabla u\cdot\nabla v\,dx$. Therefore, if $y_0\in H^1_0(\Omega)$, it follows from Parseval’s identity and integration by parts that $$\begin{aligned} +\infty\,&>\,\|y_0\|^2_{H^1_0(\Omega)}\,=\,\sum_{j\in{\mathbb N}} \left|\langle y_0,\lambda_j^{-1/2}\,e_j\rangle_1 \right|^2\\ &=\,\sum_{j\in{\mathbb N}}\frac 1{\lambda_j}\Bigl|{\int_{\Omega}}\nabla y_0\cdot\nabla e_j\,dx\Bigr|^2 \,=\,\sum_{j\in{\mathbb N}}\frac 1{\lambda_j}\Bigl|-{\int_{\Omega}}y_0\,\Delta e_j\,dx \Bigr|^2\\ &=\,\sum_{j\in{\mathbb N}}\lambda_j\,|\langle y_0,e_j\rangle|^2\,=\,\|y_0\|^2_{\mathcal{H}^{1/2}}\,.\end{aligned}$$ The case $L=1$ is handled similarly. It ought to be clear that with increasing $L$ the condition imposes ever higher regularity postulates on $y_0$. On the other hand, is obviously satisfied for every finite $L>0$ if $y_0$ belongs to the set of finite linear combinations of the eigenfunctions $\{e_j\}_{j\in{\mathbb N}}$, that is, on a dense subset of $L^2(\Omega)$. We now give sufficient conditions that guarantee the existence of a solution to the optimal control problem [**(IP)**]{}. \[TH 5\] Suppose that $f:\Omega\times[0,T]\to{\mathbb R}$ satisfies $f(\cdot,t)\in L^2(\Omega)$, for every $t\in[0,T]$, as well as condition . Moreover, let $y_0\in L^2(\Omega)$ satisfy the condition . If $\,\lambda_j\nearrow +\infty\,$ as $\,j\to +\infty$, then the control problem [**(IP)**]{} has a solution, that is, $\mathcal{J}$attains a minimum in $(0,+\infty)$. Before proving the existence result, we establish an auxiliary compactness lemma, which is of some interest in itself, since it acts between spaces with different fractional coefficients $s$. \[compactness\] Assume that the sequence $\{\lambda_k\}_{k\in{\mathbb N}}$ of eigenvalues of $\,\mathcal{L}\,$ satisfies $\,\lambda_k\nearrow +\infty$ as $k\to\infty$, and assume that the sequence $\,\{s_k\}_{k\in{\mathbb N}}\subset (0,+\infty)$ satisfies $\,s_k\to{\overline{s}}\,$ as $k\to\infty$, for some $\,{\overline{s}}\in(0,+\infty)\cup\{+\infty\}$. Moreover, let a sequence $\,\{y_k\}_{k\in{\mathbb N}}\,$ be given such that $\,y_k\in L^2(0,T;{\mathcal{H}^{s_k}})\,$ and $\,\partial_t y_k\in L^2(Q)$, for all $k\in{\mathbb N}$, as well as $$\begin{aligned} \label{CST} & \sup_{k\in{\mathbb N}}\Big( \| y_k \|_{L^2(Q)}\,+\, \| y_k \|_{L^2(0,T;{\mathcal{H}^{s_k}})}\Big)<+\infty, \quad \mbox{and }\;\, \\[1mm] &\nonumber \sup_{k\in{\mathbb N}}\| \partial_t y_k\|_{L^2(Q)}<+\infty .\end{aligned}$$ Then $\,\{y_k\}_{k\in{\mathbb N}}\,$ contains a subsequence that converges strongly in $\,L^2(Q)$. [Proof:]{} For fixed $j\in{\mathbb N}$, we define $$y_{k,j}(t):=\int_\Omega y_k(x,t)\,e_j(x)\,dx.$$ Notice that $$\begin{aligned} && \int_0^T |\partial_t y_{k,j}(t)|^2\,dt \,{\leqslant}\, \int_0^T\!\!\left( \int_\Omega |\partial_t y_k(x,t)|\,|e_j(x)|\,dx\right)^2 dt\\ &&\quad{\leqslant}\int_0^T\!\!\left( \int_\Omega |\partial_t y_k(x,t)|^2\,dx\right)dt \,=\, \| \partial_t y_k\|_{L^2(Q)}^2,\end{aligned}$$ which is bounded uniformly in $k$, thanks to . Hence, we obtain a bound in $H^1(0,T)$ for $y_{k,j}$, which is uniform with respect to $k\in{\mathbb N}$, for every $j \in{\mathbb N}$. Owing to the compactness of the embedding $H^1(0,T)\subset C^{1/4}([0,T])$, the sequence $\{y_{k,j}\}_{k\in{\mathbb N}}$ thus forms for every $j\in{\mathbb N}$ a compact subset $C_j$ of $C^{1/4}([0,T])$. Therefore, the infinite string $\big(\{y_{k,1}\}_{k\in{\mathbb N}},\, \{y_{k,2}\}_{k\in{\mathbb N}},\dots\big)$ lies in $C_1\times C_2\times\dots$, which, by virtue of Tikhonov’s Theorem, is compact in the product space $$C^{1/4}([0,T])\times C^{1/4}([0,T])\times\dots\,.$$ Hence, there is a subsequence (denoted by the index $k_m$), which converges in this product space to an infinite string of the form $\big(y^*_{1},y^*_2,\dots\big)$. More explicitly, we have that $y^*_j\in C^{1/4}([0,T])$, for any $j\in{\mathbb N}$, and $$\label{0s72uuJ} \lim_{m\to\infty} \| y_{k_m,j} -y^*_j\|_{C^{1/4}([0,T])}=0 \quad\mbox{for every $j\in{\mathbb N}$.}$$ We then define $$y^*(x,t):=\sum_{j\in{\mathbb N}} y^*_j(t)\,e_j(x)$$ and claim that $$\label{78HHA} y_{k_m}\to y^*\quad\mbox{strongly in}\,L^2(Q).$$ To prove this claim, we fix $\epsilon\in(0,1)$ and choose $j_*\in{\mathbb N}$ so large that $$\label{9wkUUa} {\mbox{$ \lambda_j{\geqslant}\epsilon^{-1}$ for any~$j{\geqslant}j_*$.}}$$ Then, by , we may also fix $m_*\in{\mathbb N}$ large enough, so that for any $m{\geqslant}m_*$ it holds that $$s_{k_m} {\geqslant}\min\left\{ 1,\,\frac{{\overline{s}}}{2}\right\}=:\sigma,$$ as well as $$\| y_{k_m,j} -y^*_j\|_{C^{1/4}([0,T])}{\leqslant}\frac{\epsilon}{j_*+1}\qquad {\mbox{ for every }}j<j_*.$$ Now, let $t\in(0,T)$ be fixed. Then, for any $m{\geqslant}m_*$, $$\begin{aligned} \label{78:8YUU12} & \| y^*(\cdot, t)-y_{k_m}(\cdot,t)\|^2_{L^2(\Omega)}\,=\, \sum_{j\in{\mathbb N}} |y^*_j(t) -y_{k_m,j}(t)|^2\\ &\nonumber{\leqslant}\, \sum_{{j\in{\mathbb N}}\atop{j<j_*}} |y^*_j(t) -y_{k_m,j}(t)|^2 \,+\,4\sum_{{j\in{\mathbb N}}\atop{j{\geqslant}j_*}}\big( |y^*_j(t)|^2 +|y_{k_m,j}(t)|^2\big)\\ &\nonumber{\leqslant}\, \epsilon +4\sum_{{j\in{\mathbb N}}\atop{j{\geqslant}j_*}}\big( |y^*_j(t)|^2 +|y_{k_m,j}(t)|^2\big).\end{aligned}$$ Moreover, by , for any $\ell\in{\mathbb N}$, $$\begin{aligned} \label{UI:0011} & \sum_{ {j\in{\mathbb N}}\atop{j_*{\leqslant}j{\leqslant}j_*+\ell}} |y_{k_m,j}(t)|^2 \,{\leqslant}\, \sum_{{j\in{\mathbb N}}\atop{j_*{\leqslant}j{\leqslant}j_*+\ell}} \epsilon^{2s_{k_m}} \lambda_j^{2s_{k_m}}|y_{k_m,j}(t)|^2\\[2mm] &\nonumber{\leqslant}\, \epsilon^{2\sigma}\, \|y_{k_m} \|_{{\mathcal{H}}^{s_{k_m}}}^2\,{\leqslant}\, \epsilon^{2\sigma}M,\end{aligned}$$ for some $M>0$, where the last inequality follows from . Hence, by virtue of , taking limit as $\,m\to\infty$, we obtain that $$\label{UI:0012} \sum_{ {j\in{\mathbb N}}\atop{j_*{\leqslant}j{\leqslant}j_*+\ell}} |y^*_{j}(t)|^2 {\leqslant}\epsilon^{2\sigma}M .$$ Therefore, letting $\ell\to\infty$ in  and , we find that $$\sum_{ {j\in{\mathbb N}}\atop{j{\geqslant}j_*}} |y_{k_m,j}(t)|^2 {\leqslant}\epsilon^{2\sigma}M\;\;{\mbox{ and }}\;\; \sum_{ {j\in{\mathbb N}}\atop{j{\geqslant}j_*}} |y^*_{j}(t)|^2 {\leqslant}\epsilon^{2\sigma}M.$$ Insertion of these bounds in  then yields that $$\| y^*(\cdot, t)-y_{k_m}(\cdot,t)\|_{L^2(\Omega)}^2 {\leqslant}\epsilon +8\epsilon^{2\sigma}M,$$ as long as $m{\geqslant}m_*$. By taking $\epsilon$ arbitrarily small, we conclude the validity of  and thus of the assertion of the lemma. [$\Box$]{} [Proof of Theorem \[TH 5\]:]{} The proof is a combination of the Direct Method with the regularity results proved in Theorem 1 and the compactness argument stated in Lemma 6. First of all, we observe that $\mathcal{J}(\frac L2)<+\infty$ if $0<L<+\infty$, while ${\mathcal{J}}( \frac 12)<+\infty$ if $L=+\infty$. Hence, owing to , we have $$0<\inf_{0<s< L} \mathcal{J}(s)<+\infty.$$ Now, we pick a minimizing sequence $\{s_k\}_{k\in{\mathbb N}}\subset (0,L)$ and consider, for every $k\in{\mathbb N}$, the (unique) solution $\,y_k:=\mathcal{S}(s_k)=y(s_k)$ to the state system , associated with $s=s_k$. We may without loss of generality assume that $${\mathcal{J}}(s_k){\leqslant}1+{\mathcal{J}}(s^*) \quad\forall \,k\in{\mathbb N},$$ where $s^*:=\frac L2$ if $L<+\infty$ and $\,s^*:=\frac 12$ otherwise. We then infer that $$\label{b1} \| y_k\|_{L^2(Q)}\,+\,\varphi(s_k)\,{\leqslant}\,C_1 \quad\forall\,k\in{\mathbb N},$$ where, here and in the following, we denote by $C_i$, $i\in{\mathbb N}$, constants that may depend on the data of the problem but not on $k$. In particular, by , the sequence $\,\{s_k\}_{k\in{\mathbb N}}\,$ is bounded, and we may without loss of generality assume that $s_k\to {\overline{s}}$ for some $\,{\overline{s}}\in(0,L)$. Also, by virtue of and , we obtain that $$\label{U8:TY:A} \| \partial_t y_k\|_{L^2(Q)}\,+\, \| y_k\|_{L^2(0,T;{\mathcal{H}}^s)}\,{\leqslant}\,C_2\,,$$ whence, in particular, $$\label{b2} \sum_{j\in{\mathbb N}} \int_0^T |\langle \partial_t y_k(\cdot,t), e_j\rangle|^2\,dt\,{\leqslant}\,C_3\quad\forall\,k\in{\mathbb N}.$$ Thus, using the compactness result of Lemma 6, we can select a subsequence, which is again indexed by $k$, such that there is some ${\overline{y}}\in H^1(0,T;L^2(\Omega))$ satisfying $$\begin{aligned} y_k&\to {\overline{y}}\quad\mbox{strongly in $L^2(Q)$ and pointwise a.\,e. in $Q$}, \\ \nonumber y_k&\to {\overline{y}}\quad\mbox{weakly in $H^1(0,T;L^2(\Omega))$}.\end{aligned}$$ Therefore, we can infer from that $$\label{b3} \sum_{j\in{\mathbb N}} \int_0^T |\langle \partial_t {\overline{y}}(\cdot,t), e_j\rangle|^2\,dt\,{\leqslant}\, C_3.$$ We now claim that ${\overline{y}}=y({\overline{s}})$, that is, that ${\overline{y}}$ is the (unique) solution to the state system associated with $s={\overline{s}}$. To this end, it suffices to show that ${\overline{y}}$ satisfies the conditions –, since then the claim follows exactly in the same way as uniqueness was established in the proof of Theorem 1; in this connection, observe that for this argument the validity of was not needed. To begin with, we fix $j\in{\mathbb N}$. We conclude from that it holds that $$\int_0^T |\partial_t \langle y_k(\cdot,t),e_j\rangle|^2\,dt\,{\leqslant}\,C_4\quad\forall \,k\in{\mathbb N}\,.$$ Hence, the sequence formed by the mappings $\,t\mapsto\langle y_k(\cdot,t),e_j\rangle$ is a bounded subset of $H^1(0,T)$. Hence, its weak limit, which is given by the mapping $\,t\mapsto \langle {\overline{y}}(\cdot,t),e_j\rangle$, belongs to $H^1(0,T)$ and is thus absolutely continuous, which implies that holds true for ${\overline{y}}$. Moreover, by virtue of the continuity of the embedding $H^1(0,T)\subset C^{1/2}([0,T])$, we can infer from the Arzelà–Ascoli Theorem that the convergence of the sequence $\,\{\langle y_k(\cdot,t),e_j\rangle\}_{k\in{\mathbb N}}\,$ is uniform on $[0,T]$. Therefore, to any fixed $\epsilon>0$ there exists some $\,k_\epsilon\in{\mathbb N}\,$ such that, for $\,k{\geqslant}k_\epsilon$, $$\begin{aligned} && \big| \langle {\overline{y}}(\cdot,t),e_j\rangle -\langle y_0,e_j\rangle \big|\\ &&{\leqslant}\, \big| \langle {\overline{y}}(\cdot,t),e_j\rangle -\langle y_k(\cdot,t),e_j\rangle \big| \,+\, \big| \langle y_k(\cdot,t),e_j\rangle -\langle y_0,e_j\rangle \big|\\ &&{\leqslant}\, \big| \langle y_k(\cdot,t),e_j\rangle -\langle y_0,e_j\rangle \big| \,+\,\epsilon.\end{aligned}$$ Hence, taking the limit in $t$, and then letting $\epsilon\searrow0$, we obtain that ${\overline{y}}$ fulfills . Now we use the fact that the mapping $t\mapsto \langle y_k(\cdot,t),e_j\rangle\,$ belongs to $H^1(0,T)$ to write  in the weak sense. We have, for any test function $\Psi\in C^\infty_0(0,T)$, $$\begin{aligned} && -\int_0^T \langle y_k(\cdot,t),\,e_j\rangle\,\partial_t\Psi(t)\,dt +\lambda_j^{s_k} \int_0^T\langle y_k(\cdot,t),\,e_j\rangle\,\Psi(t)\,dt \\&&\quad=\int_0^T\langle f(\cdot,t),\,e_j\rangle\,\Psi(t)\,dt.\end{aligned}$$ Passage to the limit as $k\to\infty$ then yields the identity $$\begin{aligned} && -\int_0^T \langle {\overline{y}}(\cdot,t),\,e_j\rangle\,\partial_t\Psi(t)\,dt +\lambda_j^{{\overline{s}}} \int_0^T\langle {\overline{y}}(\cdot,t),\,e_j\rangle\,\Psi(t)\,dt \\&&\quad=\int_0^T\langle f(\cdot,t),\,e_j\rangle\,\Psi(t)\,dt.\end{aligned}$$ This, and the fact that the mapping $\,t\mapsto \langle {\overline{y}}(\cdot,t),e_j\rangle\,$ belongs to the space $H^1(0,T)$, give  (recall, for instance, Theorem 6.5 in [@lieb]). In conclusion, it holds ${\overline{y}}=y({\overline{s}})$, and thus the pair $({\overline{s}},{\overline{y}})$ is admissible for the problem [**(IP)**]{}. By the weak sequential semicontinuity of the cost functional, ${\overline{s}}$ is a minimizer of $\mathcal{J}$. This concludes the proof of the assertion. [$\Box$]{} Acknowledgments {#acknowledgments .unnumbered} =============== EV was supported by ERC grant 277749 “EPSILON Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities” and PRIN grant 201274FYK7 “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”. [99]{} , [*A FEM for an optimal control problem of fractional powers of elliptic operators*]{}. Preprint arXiv:1406.7460v3 \[math.OC\] 18 April 2015. , [*A fractional space-time optimal control problem: analysis and discretization*]{}. 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American Mathematical Society, Providence, Rhode Island (2010). , [*Lévy flight search patterns of wandering albatrosses*]{}. Nature 381, 413-415 (1996). [^1]: Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, and Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany. [^2]: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany, and Dipartimento di Matematica Federigo Enriques, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy. [^3]: As a technical remark, we point out that, strictly speaking, in view of their probabilistic and statistical interpretations, many of the experiments available in the literature are often more closely related to fractional operators of integrodifferential type rather than to fractional operators of spectral type, and these two notions are, in general, not the same (see e.g. [@TWO]), although they coincide, for instance, on the torus, and are under reasonable assumptions asymptotic to each other in large domains (see e.g. Theorem 1 in [@NAZA3] for precise estimates). Of course, the problem considered in this paper does not aim to be exhaustive, and other types of operators and cost functions may be studied as well, and, in fact, in concrete situations different “case by case” analytic and phenomenological considerations may be needed to produce detailed models which are as accurate as possible for “real life” applications.

--- abstract: 'Recent years have seen an explosive research and development of nanoplasmonics in the visible and near-infrared (near-ir) frequency regions.[@Novotny_Hecht_2006_Principles_of_Nanooptics] One of the most fundamental effects in nanoplasmonics is nano-concentration of optical energy. Plasmonic nanofocusing has been predicted [@Phys_Rev_Lett_93_2004_Tapered_Plasmonic_Waveguides] and experimentally achieved. Nanoconcentration of optical energy at nanoplasmonic probes made possible optical ultramicroscopy with nanometer-scale resolution and ultrasensitive Raman spectroscopy.[@DiFabrizio_et_al_Nano_Lett_2008_Photonic_Nanoplasmonic_Device] It will be very beneficial for the fundamental science, engineering, environmental, and defense applications to be able to nano-concentrate terahertz radiation (frequency $1-10$ THz or vacuum wavelength $\lambda_0=300-30~\mathrm{\mu m}$). This will allow for the nanoscale spatial resolution for THz imaging[@Mittleman_et_al_Rep_Progr_Phys_2007_THz_Imaging] and introduce the THz spectroscopy on the nanoscale, taking full advantage of the rich THz spectra and submicron to nanoscale structures of many engineering, physical, and biological objects of wide interest: electronic components (integrated circuits, etc.), bacteria, their spores, viruses, macromolecules, carbon clusters and nanotubes, etc. In this Letter we establish the principal limits for the nanoconcentration of the THz radiation in metal/dielectric waveguides and determine their optimum shapes required for this nanoconcentration We predict that the adiabatic compression of THz radiation from the initial spot size of $R_0\sim \lambda_0$ to the final size of $R=100-250$ nm can be achieved with the THz radiation intensity increased by a factor of $\times10$ to $\times 250$. This THz energy nanoconcentration will not only improve the spatial resolution and increase the signal/noise ratio for the THz imaging and spectroscopy, but in combination with the recently developed sources of powerful THz pulses [@Hebling_et_al_high_power_THz_sources_2008] will allow the observation of nonlinear THz effects and a carrying out a variety of nonlinear spectroscopies (such as two-dimensional spectroscopy), which are highly informative. This will find a wide spectrum of applications in science, engineering, biomedical research, environmental monitoring, and defense.' author: - Anastasia Rusina - Maxim Durach - 'Keith A. Nelson' - 'Mark I. Stockman' title: Nanoconcentration of Terahertz Radiation in Plasmonic Waveguides --- There are existing approaches to deep subwavelength THz imaging and probing based on sharp tips irradiated by a THz source,[@Chen_Kersting_Cho_APL_2003_THz_Imaging_with_100_nm_resolution] adiabatically-tapered metal-dielectric waveguides[@Klein_et_al_JAP_2005_Metal_Dielectric_Antenna_for_THz_SNOM] similar to optical adiabatic concentrators, and nonlinear microscopic THz sources.[@Lecaque_Gresillon_Boccara_Opt_Expr_2008_THz_Emission_Microscopy] For the development of the THz nanotechnology, it is extremely important to understand spatial limits to which the THz radiation energy can be concentrated (nanofocused). A major challenge for the nanoconcentration of the electromagnetic energy in the THz region is the large radiation wavelength in vacuum or conventional dielectrics, $\lambda_0=30~\mathrm{\mu m}-300~\mathrm{\mu m}$, where the THz radiation can only be focused to the relatively very large regions of size $\sim \lambda_0/2$. The developed field of optical energy concentration, which is based on surface plasmon polaritons (SPPs), suggests that one of the ways to solving this problem is to employ the surface electromagnetic waves (SEWs). In the far infrared (ir), the dielectric permittivity of metals has large imaginary part which dominates over its negative real part.[@Ordal_fir_metal_dielectric_permittivities_ApplOpt_1983] This implies that SEWs propagating along a metal-dielectric flat interface in this frequency range, known as Sommerfeld-Zenneck waves,[@Sommerfeld_Ann_Phys_Chem_1899; @Zenneck_Ann_Phys_1907] are weakly bound to the surface[@Saxler_TSP_PRB_2004] and can hardly be used for the confinement of THz radiation. It has been suggested that periodically perforating flat surfaces of ideal metals with grooves or holes leads to the appearance of SEWs, which mimic (“spoof”) SPPs to be stronger bound to the surfaces[@Mills_Maradudin_roughness_PRB_1989; @pendry_spoof; @Garcia_Vidal_Planar_Spoof_JOA_2005] permitting a better control over the THz fields. It has been predicted that SPPs on an array of parallel grooves cut on the surface of a perfect conductor wire can be localized by adiabatic deepening of the grooves. At a point, where grooves are approximately a quarter of wavelength of light, the highest concentration is achieved to be on the order of tens of micrometers. This method restricts the localization point to a particular frequency, making the concentration very narrow-band. Also, the depth of a groove should be $\approx \lambda_0/4$, i.e., in the tens to hundred micron range, which precludes completely nanoscale devices. It is well known from microwave technology that the ideal-metal waveguides with smooth surfaces support TEM waves, where the electric field lines are either infinitely extended or terminate at the metal surfaces normally to them. The latter case requires the waveguide cross-section topology to be more than single-connected; an example may be a coaxial waveguide (“coax”). Such waveguides possess are very wide-band in frequency. The THz waveguides can be adiabatically tapered to concentrate energy. The idea of adiabatic energy concentration comes from ultramicroscopy and nanoplasmonics,[@Phys_Rev_Lett_93_2004_Tapered_Plasmonic_Waveguides] where it has been developed both theoretically and experimentally and used in ultrasensitive surface enhanced Raman spectroscopy.[@DiFabrizio_et_al_Nano_Lett_2008_Photonic_Nanoplasmonic_Device] Employing these ideas of the adiabatic concentration and using a tapered metal-dielectric waveguide, the THz spatial resolution achieved is $\sim 20~\mathrm{\mu m}$ across the entire THz spectrum.[@Klein_et_al_JAP_2005_Metal_Dielectric_Antenna_for_THz_SNOM] In this Letter, for the first time, we establish the fundamental limits and find the principles of designing the optimum and efficient metal/dielectric waveguides suitable for the THz nanofocusing. The specific examples are for the wide-band concentrators: a plasmonic metal wedge cavity and tapered coax waveguides, which are terminated by funnel-type adiabatic tapers. Such nano-concentrators along with the advent of high-power sources[@Hebling_et_al_high_power_THz_sources_2008; @Shen_et_al_high_power_THz_radiation_PRL_2007] and sensitive detectors[@Komiyama_et_al_THz_photon_detection_Nat_2000] of THz radiation, will open up an extremely wide range of possible THz applications, in particular, in material diagnostics, probe nanoimaging, biomedical applications etc. – cf. Refs.. Note that an alternative approach to the THz energy concentration using doped semiconductor tapers has also been proposed. However, the required heavy doping of the semiconductors may cause fast electron relaxation due to the collisions with the inflicted lattice defects and bring about high losses. Therefore, in this Letter we will pursue the adiabatic nanoconcentration of the THz radiation using metal/dielectric structures. Conventionally for THz and microwave regions, the metals are considered as ideal which is equivalent to neglecting their skin depth $l_s=\lambdabar_0/\mathrm{Re}\,\sqrt{-\varepsilon_m}$, where $\varepsilon_m$ is the permittivity of the metal (we take into accoint that in the THz region $\left|\varepsilon_m\right|\gg 1$), and $\lambdabar_0=c/\omega$ is the reduced wavelength in vacuum. It is true that in the THz region $l_s=30-60$ nm. i.e., $l_s\ll\lambda_0$. However, as we show below in this Letter, it is the finite skin depth, though as small as it is, that principally limits the ultimum localization size of the THz fields. For larger waveguides, the THz wave energy is localized mostly in the vacuum (dielectric) and its losses, which occur in the metal’s skin layer, are correspondingly small. The effective quality factor (or, figure of merit) of the waveguide, which shows how many periods the wave can propagate without significantly loosing its energy, can be estimated as $$Q\sim 2a/l_s~, \label{Q}$$ where $a$ is the characteristic minimum size of the waveguide; this estimate becomes a good approximation for a metal-dielectric-metal planar waveguide \[see below Eq. (\[slab\_wavenum\_1\])\]. When the waveguide size reduces to become on the order of the skin depth, $a\lesssim l_s $, the THz field is pushed into the metal, and the quality factor reduces to $Q\lesssim 1$, which implies strong losses. Qualitatively, this establishes the limit to the nanoconcentration: for upper THz region $a\gtrsim l_s\approx 30$ nm, while for the 1 THz frequency $a\gtrsim l_s\approx 60$ nm. These are the practical limits of the THz nanoconcentration for the noble metals (silver, gold, and platinum) and for aluminum. If one pursues the goal of creating enhanced local fields in a small region, but not necessarily to efficiently transfer the THz energy from the far field to the near field, then the apertureless SNOM approach, where a sharp metal or dielectric tip is irradiated by THz radiation, can, in principle, achieve even higher resolution.[@Keilmann_J_Biol_Phys_2003_Review] However, the efficiency of utilizing the THz energy of the source in this case will be extremely low; the stray, far-field THz energy may create a significant parasitic background. ![\[par\_plate\_guide.eps\] Geometry and properties of the THz TM mode in a parallel metal-slab waveguide. (a) Schematic of the waveguide. The width of the dielectric gap $a$ and the skin depth $l_s$ are indicated. (b) An instantaneous distribution of the longitudinal electric field $E_y$ along the propagation coordinate $y$ for $a=10~\mathrm{\mu m}$ and frequency 1 THz in a silver-vacuum-silver waveguide. (c) The same as in panel (b) but for $a=200$ nm. (d) Modal refraction index $n=k/k_0$ ($\mathrm{Re}\,n$ is denoted by the red line and $\mathrm{Im}\,n$ by the blue line) as a function of the waveguide width $a$. Dashed green line indicates the value of $n$ for the perfect conductor. Skin-depth value is shown by the vertical dashed line.](par_plate_guideA.eps){width=".48\textwidth"} Here and below in this Letter, we consider examples of the THz adiabatic nanoconcentration quantitatively, where the effect of the specific geometry will become apparent. Consider first a parallel plate waveguide that consists of a dielectric slab of thickness $a$ with dielectric permittivity $\varepsilon_d$ sandwiched between two thick metal plates (with thickness of at least a few $l_s$, i.e., greater than 200 nm in practical terms) \[see Fig.\[par\_plate\_guide.eps\](a)\]. The permittivity of the metal $\varepsilon_m$ in the THz region has a very large ($\gtrsim 10^6$) imaginary part that defines the very small skin depth $l_s\lesssim 100$ nm, which justifies the usual consideration of the metals as perfect conductors.[@Ordal_fir_metal_dielectric_permittivities_ApplOpt_1983] However, as we have already mentioned, for our purposes of the THz nanoconcentration, we need to take into account the field penetration into the metal, which makes the problem plasmonic. In this case, the propagating modes of the system are SPPs, which are TM modes characterized by the symmetry with respect to the reflection in the center plane. We will orient the coordinate system with its $z$ axis normal to the plane and the $y$ axis in the direction of propagation. The symmetric (even) modes have even field components $H_x$ and $E_z$ and odd $E_y$; the parity of the antisymmetric (odd) modes is opposite. From plasmonics it is known that the even modes have a larger fraction of their energy localized in the dielectric and the odd modes in the metal. Therefore, the even modes have much smaller damping and are, therefore, most suitable for the THz energy concentrations. The dispersion relation for the even modes is given in the Methods section as Eq. (\[slab\_disp\]). This equation can be much simplified and solved in a closed analytical form taking into account that we are interested in the subwavelength focusing, i.e., $a\lesssim \lambdabar$, where $\lambdabar=\lambdabar_0/\varepsilon_d$ is the reduced wavelength in the dielectric; also, in the entire THz region $l_s\ll\lambdabar$. This shows that there exists a small parameter in the problem $l_s a/\lambdabar^2\ll 1$ \[see also Eq. (\[applicability\])\], which allows one to solve analytically the dispersion relation (\[slab\_disp\]) obtaining the modal refraction index $n=k/k_0$, where $k$ is the THz wavevector, and $k_0=1/\lambdabar_0$, $$n=\sqrt{\varepsilon_d}\left(1+\frac{l_s (1+i)}{a}\right)^{1/2}\approx \sqrt{\varepsilon_d} \left(1+i\frac{l_s}{2a}\right)~, \label{slab_wavenum_1}$$ where the approximate equality is valid for not too tight nanofocusing, i.e., for $l_s\ll a$. From this, we can obtain the quality factor of the waveguide $Q=\mathrm{Re}\,n/\mathrm{Im}\,n=2a/l_s$, giving a quantitative meaning to the estimate (\[Q\]). Plasmonic effects (i.e. those of the finite skin depth) are illustrated in Fig. \[par\_plate\_guide.eps\] for silver-vacuum-silver waveguide and frequency of 1 THz. Panels (b) and (c) display the longitudinal electric field $E_y$ obtained by the exact solution of the Maxwell equations. Note that this field component is absent for the ideal conductor; here it is relatively small: on the order of $10^{-3}$ of the transverse field. Panel (b) illustrates the case of a relatively wide waveguide ($a=10~\mathrm{\mu m}$), where it is evident that the electric field is localized mostly in the dielectric region of the waveguide, and the extinction of the wave is small. In a sharp contrast, for a nanoscopic waveguide ($a=200$ nm) in panel (c), the electric field significantly penetrates the metal. In accord with our arguments, there is a very significant extinction of the fields as they propagate; the retardation effects are also evident: the lines of equal amplitude are at an angle relative to the normal ($z$) direction. The dependence of the modal refraction index on the thickness $a$ of the waveguide obtained from Eq. (\[slab\_wavenum\_1\]) is plotted in Fig.\[par\_plate\_guide.eps\] (d). This index increases as $a$ becomes comparable with the skin depth. While $\mathrm{Re}\,n$ and $\mathrm{Im}\,n$ increase by the same absolute amount, the quality factor $Q$, obviously, greatly decreases with decrease of $a$. The mode described above can be used for broadband energy concentration of THz waves. To introduce the THz nanoconcentration, consider a metal-dielectric-metal waveguide that is slowly (adiabatically) tapered off as a wedge, as illustrated in Fig.\[wedge.eps\] (a). Because of the adiabatic change of the parameters, a wave propagating in such a waveguide will adjust to it without reflection or scattering, just as it takes place in nanoplasmonic waveguides.[@Phys_Rev_Lett_93_2004_Tapered_Plasmonic_Waveguides] As a result, propagating it will concentrate its energy, conforming to the tapering of the waveguide. The corresponding solution can be obtained from the Maxwell equations using the Wentzel-Kramers-Brillouin (WKB) approximation, similarly to the nanoplasmonic case in the visible,[@Phys_Rev_Lett_93_2004_Tapered_Plasmonic_Waveguides] as described in the Methods section. The WKB approximation is applicable under the conditions that $$\delta=\left|d\left(\mathrm{Re}k^{-1}\right)/dy\right|\ll 1~,~~ \left| da/dy\right|\ll 1~, \label{adiabaticity}$$ where $\delta$ is the well-known adiabatic parameter describing how slowly the modal wavelength changes on a distance of its own, and $\left| da/dy\right|$ is a parameter describing how adiabatically the transverse size of the confined mode changes along the propagation coordinate. In the WKB approximation, the behavior of the dominating transverse field component $E_z$ as a function of the coordinate $y$ along the propagation direction is shown for the last $6~\mathrm{\mu m}$ of the propagation toward the edge in Fig. \[wedge.eps\] (b). There is a clearly seen spatial concentration of the energy and increase of the field as the wave is guided into the taper. The predicted behavior of the two components of electric field and the magnetic field for the last micron of the propagation is shown in panels (c)-(e). It apparently indicates the adiabatic concentration, without an appreciable loss of the intensity. The THz wave follows the waveguide up to the nanometric size. The red line in Fig. \[wedge.eps\] (f) indicates that the local intensity $I$ as the function of the thickness $a$ of the waveguide for $a<4~\mathrm{\mu m}$ increases significantly with $1/a$, in qualitative accord with the behavior expected for the negligibly low losses. This intensity reaches its maximum for $a=1.6~\mathrm{\mu m}$ and then starts to decrease as the losses overcome the adiabatic concentration. At smaller thicknesses, $a\lesssim 400$ nm, the intensity in Fig.\[wedge.eps\] (f) starts to increase again, which is unphysical. The reason is revealed by the behavior of the adiabatic parameter $\delta$ shown by the blue line: for $a\lesssim 400$ nm, $\delta$ becomes relatively large (comparable with 1), i.e., the adiabaticity is violated. This is due to the fact that the fraction of the THz field energy propagating in the metal is dramatically increased for $a\lesssim 400$ nm due to the constricted transverse extension of the dielectric in the waveguide. This causes a significant loss per wavelength $\lambda$, leading to a rapid change of the wave vector $k$, breaking down the adiabaticity. This constitutes a fundamental difference from the nanoplasmonic adiabatic concentration in the optical region where the adiabatic parameter is constant, and the adiabaticity holds everywhere including the vicinity of the tip.[@Phys_Rev_Lett_93_2004_Tapered_Plasmonic_Waveguides] ![\[wedge.eps\] Adiabatic concentration of THz field energy in a graded waveguide, where a dielectric wedge is surrounded by the thick silver layer. (a) Schematic of energy concentration, where $\theta$ is the wedge opening angle, the arrow indicates the direction of propagation of the THz wave, and the red highlights the area of the adiabatic concentration. The orientation of the coordinate system is shown in the inset. (b) An instantaneous distribution of the transverse electric field $E_z$ of the THz wave propagating and concentrating along the wedge waveguide for the last 6 mm of the propagation toward the edge. Note the difference in scales in the $z$ and $y$ directions. (c) An instantaneous spatial distribution of the transverse electric field $E_z$ close to the edge of the wedge, for the last $640~\mathrm{\mu m}$ of the propagation. (d) The same as (c) but for the longitudinal (with respect to the propagation direction) component of the field $E_y$. (e) The same as (c) but for the transverse component of the magnetic field $H_x$. The units of these field components are arbitrary but consistent between the panels. (f) Dependence of THz field intensity in the middle of waveguide on the dielectric gap width $a$ (the red line). The blue curve displays the dependence on $a$ of the adiabatic parameter $\delta$, scaled by a factor of 5. The values of $a$ indicated at the successive horizontal axis ticks differ by a factor of $10^{-1/2}$, i.e., by 5 dB. ](wedgeA.eps){width=".48\textwidth"} ![\[curved\_wedge.eps\] Terahertz energy concentration in adiabatically tapered curved-wedge waveguide. (a) Instantaneous distribution of the transverse component of the THz electric field $E_z$ (in the central plane $z=0$) as a function of the coordinate $y$ along the propagation direction for the last $400~\mathrm{\mu m}$ of the propagation. (b) The same as in panel (a) but for the longitudinal electric field component $E_y$. (c) The same as panel (a) but for the transverse magnetic field $H_x$. The units for the fields are arbitrary but consistent between the panels. (d) The THz field intensity $I$ (relative to the intensity $I_0$ at the entrance of the waveguide) as a function of the dielectric gap thickness $a$ is shown by the red line. The adiabatic parameter scaled by a factor of 10 as a function of $a$ is indicated by the blue line. The values of $a$ indicated at the horizontal axis ticks correspond to the values of $y$ at the ticks of panels (a)- (f). ](curved_wedgeA.eps){width=".48\textwidth"} To provide for the optimum guiding of the THz wave and its concentration on the nanoscale, the terminating (nanoscopic) part of the waveguide should be tapered slower, in a funnel-like manner. That is, one needs to decrease the grading $da/dy$ of the waveguide near the edge in order to keep the adiabaticity parameter $\delta=\left|d\left(\mathrm{Re}k^{-1}\right)/da\times da/dy\right|$ approximately constant and small enough to prevent the back-reflection. Because for the adiabatic grading (tapering), the derivative $d\left(\mathrm{Re}k^{-1}\right)/da$ does not depend on the grading (it is the same as for the plane waveguide) and is only a function of $a$, the equation $\delta=\delta(y)$ is a differential equation for the shape of the waveguide that can be easily integrated. This results in the dependence of the thickness $a$ on the longitudinal coordinate $y$ determined by a simple integral $$\mathrm{Re}\,n^{-1}(a)=k_0 \int\delta(y)dy~, \label{grading}$$ where $n(a)$ is the modal index defined in this case by Eq. (\[slab\_wavenum\_1\]), and $\delta(y)$ is the desired dependence of the adiabatic parameter along the waveguide, which is an arbitrary function of $y$ satisfying the adiabaticity conditions (\[adiabaticity\]). The geometry of an adiabatically-tapered end of the silver/vacuum waveguide found from Eq. (\[grading\]) and satisfying Eq.(\[adiabaticity\]) and the corresponding WKB solutions for the 1 THz fields are shown in Fig. \[curved\_wedge.eps\] (a)-(c). The optimum shape of the waveguide in this case is funnel-like, greatly elongated toward the edge. The nanoconcentration of the field is evident on panels (a)-(c), as well as its penetration into the metal for $a\lesssim 100$ nm. As these panels show quantitatively and the red curve on panel (d) qualitatively, the field intensity reaches its maximum at $a\approx 300$ nm where it is enhanced with respect to the field at the entrance to the funnel waveguide by a modest factor of 1.2. At the same time, the adiabatic parameter $\delta$ decreases toward the tip from 0.07 to 0.05, indicated the applicability of the WKB approximation everywhere. Note that the this funnel-shaped wedge, indeed, continues the linearly-graded wedge waveguide shown in Fig. \[wedge.eps\], which yields the enhancement factor of $\approx 8$ at $a=2~\mathrm{\mu m}$. Sequentially, these two waveguides provide the intensity enhancement by approximately $\times 10$ while compressing the THz wave to the thickness of $a=300$ nm and the enhancement by a factor of 3 for $a=100$ nm. Thus, true nanolocalization of THz radiation in one dimension (1d) is possible. The minimum transverse size of this nanolocalization is determined by the skin depth, as we have already discussed qualitatively in the introductory part of this Letter. The obtained 1d beam of the nanoconcentrated THz radiation may be used for different purposes, in particular as a source for the diffraction elements including the nanofocusing zone plates of the type introduced in Ref.. The two-dimensional (2d) concentration of the THz radiation can be achieved by using an adiabatically-tapered conical coax waveguide, whose geometry is illustrated in Fig. \[coax.eps\] (a). The central metal wire of radius $r$ is surrounded by a dielectric gap of the radial thickness $a$, which is enclosed by a thick ($\sim 200$ nm or thicker) outer metal shell. Both $r$ and $a$ are smooth functions of the longitudinal coordinate $y$, which describes the tapering of the coax toward the apex (tip) at $y=0$. The THz waves propagate from the wide end of the coax toward the apex, adiabatically following the tapering. In the spirit of WKB, for any particular $y$ the wave behavior for the tapered coax is the same as for a cylindrical coax with the values of $r$ and $a$ equal to the local values $r(y)$ and $a(y)$. ![\[coax.eps\] Geometry, modal index of refraction, and THz energy concentration in conically-tapered metal-dielectric waveguide. (a) Schematic of geometry and energy concentration. The central wire and the coax shell are shown along with the schematic of the THz energy concentration. (b) Dependence of modal refraction index $n$ in coaxial waveguide on the dielectric gap width $a$ for two central wire radii: $r=10~\mathrm{\mu m}$ and $r=60$ nm. The color coding of the lines is indicated. The dielectric in the gap is vacuum. (c) Instantaneous distribution of the radial THz electric field amplitude $E_\rho$ in the cross section of the coax for the last 3 mm of the propagation toward the tip. The amplitude of the field is color coded by the bar at the top of the panel. (d) Instantaneous distribution of the longitudinal THz electric field amplitude $E_y$ on the coordinate $y$ for the last $620~\mathrm{\mu m}$ of the propagation. (e) The same as (d) but for the transverse magnetic field $H_\varphi$. The units of these field components are arbitrary but consistent between the panels. (f) Dependence of THz field intensity in the middle of waveguide gap on the waveguide outer radius $R=r+a$ is shown in red. The blue curve displays the adiabatic parameter $\delta$ as a function of $R$, scaled by a factor of $10^2$. The values of $R$ indicated at the successive horizontal axis ticks differ by a factor of $10^{-1/2}$, i.e., by 5 dB. ](coaxA.eps){width=".48\textwidth"} ![\[curved\_coax.eps\] Adiabatic terahertz energy concentration in a self-similarly curved, funnel-shaped coaxial waveguide, where the metal is silver, and the dielectric in the gap is vacuum. The dielectric gap is between the pair of the neighboring curved lines, and the metal is everywhere else. (a) Instantaneous distribution of the radial (transverse) component $E_\rho$ of the electric field of the guided THz wave as a function of the propagation coordinate along the wedge $y$ for the last $600~\mathrm{\mu m}$ of the propagation. (b) The same for the longitudinal electric field component $E_y$. (c) The same for transverse magnetic field $H_\varphi$, whose lines form circles around the central metal wire. The units of these field components are arbitrary but consistent between the panels. (d) The THz intensity $I$ as a function of the waveguide radius $R$, displayed relative to the intensity $I_0$ at the beginning of the waveguide (red line). Adiabatic parameter $\delta$ multiplied by a factor of 10 as a function of $R$ (blue line). The values of the radius $R$ shown at the ticks correspond to those of $y$ shown in panels (a)-(c). ](curved_coax_self_simA.eps){width=".48\textwidth"} The dispersion relation for the coax waveguide that takes into account the plasmonic effects (i.e., the penetration of radiation into the metal and the concurrent losses) is obtained in the Methods section as Eq.(\[coax\_refraction\]). Calculated from this expression, the dependence of the modal refractive index $n=k(a,r)/k_0$ on the dielectric gap $a$ is displayed in Fig. \[coax.eps\] (b) for the frequency of 1 THz, silver as a metal, and vacuum in the dielectric gap. The results are shown for two values of the radius of the central wire: $r=10~\mathrm{\mu m}$ and $r=60~\mathrm{nm}$. As one can see, the real part of the modal index practically does not depend on $r$; it starts growing when $a$ decreases. The imaginary part of the index $n$ increases when the central wire thickness $r$ decreases, but this dependence is very weak. Both $\mathrm{Re}\,n$ and $\mathrm{Im}\,n$ grow dramatically for $r\lesssim l_s$. This is due to the penetration of the THz field into the metal, i.e., it is a plasmonic effect. The WKB solution for the radial field $E_\rho$ in the cross section of this coax waveguide is shown for the last $3~\mathrm{\mu m}$ of the propagation toward the tip in Fig. \[coax.eps\] (c). The adiabatic following and energy concentration are evident in this panel. The penetration into the metal of the tangential (to the metal surface) field components $E_y$ and $H_\varphi$ is noticeable in Figs.\[coax.eps\] (d) and (e). The intensity $I$ of the THz field (relative to the intensity $I_0$ at the entrance of the waveguide) as a function of the waveguide outer radius $R=r+a$ is shown by the red line in Fig.\[coax.eps\] (f). Dramatically, it shows the adiabatic nanoconcentration and the intensity increase by more than two orders of magnitude for the nanoconcentration from the waveguide radius $R=300~\mathrm{\mu m}$, where the THz radiation can be focused, toward $R=300$ nm. However, the dramatic increase of the adiabatic parameter $\delta$ \[plotted by the blue line in Fig. \[coax.eps\] (f)\] for $R\lesssim 1~\mathrm{\mu m}$ shows that these results can only be trusted for $R\gtrsim 1~\mathrm{\mu m}$. For the true 2d nanoconcentration of the THz radiation below this micron-scale radius, similar to the 1d case of the wedge, to preserve the adiabaticity, a funnel-like tapering is necessary. Generally, the tapering of the central wire and that of the outer metal shell do not need to be the same. However, we found that better results are obtained when it is the case, i.e., the waveguide is tapered-off self-similarly. In specific calculations, as everywhere in this Letter, we assume that the metal of the waveguide is silver, the dielectric is vacuum, and the frequency is 1 THz. Doing so, we have found the corresponding grading of the waveguide using Eq. (\[grading\]) and setting $\delta=0.05$, which is small enough to satisfy the adiabaticity very well. In this case, indeed, we have used the corresponding dispersion relation (\[coax\_refraction\]). The obtained shape of the waveguide is a strongly-elongated funnel, as shown in Fig. \[curved\_coax.eps\] (a)-(c). These figures display the THz fields that we have calculated in the WKB approximation for this waveguide. As one can see from these figures, within the last half micron of the propagation, the electric and magnetic fields of the THz wave efficiently follow the adiabatically curved waveguide. The penetration into the metal of the tangential (to the metal-dielectric interfaces) field components for $y< 400~\mathrm{\mu m}$ is evident in panels (b) and (c). The longitudinal electric field component $E_y$ is significantly localized in the central metal wire \[panel(b)\], which is a plasmonic effect. The dependence of the THz field intensity in the gap (relative to the intensity $I_0$ at the entrance of this funnel) on the total radius of the waveguide $R$ is shown in Fig. \[curved\_coax.eps\] (d) by the red line. In this case, the adiabatic concentration is very efficient. The intensity of the THz radiation increases by a factor of $\times 5$ when it is compressed from the initial radius of $R=1~\mathrm{\mu m}$ to the radius $R\approx 250$ nm. The penetration of the fields into the metal for smaller values of the radius $R$ (tighter confinement) causes losses that dominate over the effect of the concentration. Again, we remind that this funnel waveguide is a continuation and termination for the straigt cone that yields the field enhancement by $\times 50$ for $R=1~\mathrm{\mu m}$ \[see Fig. \[coax.eps\] (f)\]. Consecutively, these two waveguides (the initial cone continued and terminated by the funnel) are very efficient, adiabatically compressing the THz radiation from the initial radius $R=300~\mathrm{\mu m}$ to the radius $R=250$ nm increasing its intensity by a factor $\times 250$. Even for the final radius $R=100$ nm, the total THz intensity is increased by a factor of $\times 10$ (which is the products of factors $\times 50$ for the cone part and $\times 0.2$ for the funnel. Thus, the optimally graded plasmonic-metal 2d waveguide is very efficient in the concentration and guidance of the THz fields with the transverse radius of confinement $R\lesssim 100$ nm. To discuss the results, we have shown that the THz radiation can be concentrated to the $\sim 100$ nm transverse size in adiabatically graded plasmonic (metal/dielectric) waveguides. In the optimum adiabatically-graded, coaxial waveguide, which consists of the initial cone terminated with a funnel, the radiation of a 1 THz frequency whose wavelength is 300 $\mathrm{\mu m}$, can be compressed to a spot of 250 nm radius, where its intensity increases by a factor of $\times 250$. Even in the case of the extreme compression to a spot of the 100 nm radius, the THz intensity is enhanced by one order of magnitude with respect to the initial intensity of the $300~\mathrm{\mu m}$ spot at the entrance of the waveguide. The physical process that limits the extent of this spatial concentration is the skin effect, i.e., penetration of the radiation into the metal that causes the losses: the THz field penetrates the depth of $l_s=30-60$ nm of the metal, which by the order of magnitude determines the ultimum localization radius. The THz nanoconcentration predicted in this Letter for optimally-graded adiabatic plasmonic waveguides provides unique opportunities for THz science and technology, of which we will mention below just a few. The nanoconcentration of the THz radiation will provide the THz ultramicroscopy with a THz source of unprecedented spatial resolution and brightness. The increase of the THz intensity by two orders of magnitude along with the novel high-power THz sources[@Hebling_et_al_high_power_THz_sources_2008] would allow the observation of a wide range of electronic and vibrational nonlinear effects in metal, semiconductors, insulators, and molecules. These nonlinear THz phenomena can be used to investigate behavior of various materials in ultrastrong fields, for nonlinear spectroscopy (including the multidimensional spectroscopy), and for monitoring and detection of various environmental, biological, and chemical objects and threats such as single bacterial spores and viruses. Such applications will certainly be helped by very large absorption cross sections of various materials in the THz region. A distinct and significant advantage of the adiabatic nanofocusing is that the THz energy is mostly concentrated in the hollow region of the waveguide, whose size can be made comparable with the size of the objects of interest: in the range from 1 micron to 70 nm, which is a typical range for bacteria and their spores, and viruses. This will assure high sensitivity and low background for the objects that are confined inside these waveguides. Consider as a specific example the spectroscopy or detection of single particles, such as, e.g., anthrax spores, in the air. A sample containing the suspected nanoparticles in a gas, which can be air for the frequencies in the transparency windows, can be pumped through a THz waveguide, and the detection can be made for each particle in the gas separately on the basis of the two-dimensional nonlinear THz spectra that are expected to be highly informative for the detection and elimination of the false-positive alarms. Likewise, many other scientific, technological, environmental, and defense applications may become possible. Methods {#Methods} ======= Terahertz TM Wave in Finite-Conductivity Parallel Plate Waveguide ----------------------------------------------------------------- A parallel plate waveguide supports an even TM mode with wavenumber $k$, which satisfies the dispersion relation $$\tanh{\left(\frac{\kappa_d a}{2}\right)}= -\frac{\varepsilon_d\kappa_m}{\varepsilon_m\kappa_d} \label{slab_disp}$$ where $\kappa_d=\left(k^2-\varepsilon_d k_0^2\right)^{1/2}$, $\kappa_m=\left(k^2-\varepsilon_m k_0^2\right)^{1/2}$. In the terahertz range, $\varepsilon_m$ is mainly imaginary, where $\mathrm{Im}\,\varepsilon_m\gg 1$. Therefore, $\kappa_m\approx k_0\sqrt{-\varepsilon_m}=l_s^{-1} (1-i)$, where $l_s=1/\text{Re}{\kappa_m}=\sqrt{2}/(k_0 \sqrt{|\varepsilon_m|})$ is the metal skin depth, which is on the order of tens of nanometers. We also assume that $\kappa_d a\ll 1$, which is always the case for the mode under consideration because either this mode is close to the TEM mode where $k=k_0\varepsilon_d$, or the gap $a$ is thin enough. This leads to a closed expression for the index of refraction of the mode, which is Eq. (\[slab\_wavenum\_1\]). Using this, one can check that $\kappa_d a\approx \left(\varepsilon_d a l_s/\lambdabar^2\right)^{1/2}$. Consequently, the applicability condition of the approximation used is $$\left(\varepsilon_da l_s/\lambdabar^2\right)^{1/2}\ll 1~. \label{applicability}$$ This condition is satisfied for the realistic parameters of the problem. For instance, for the frequency $f=1$ THz, the skin depth for metals is $l_s\approx 60$ nm, while reduced wavelength is $\lambdabar=75~ \mathrm{\mu m}$. The condition (\[applicability\]) is well satisfied for $a\ll 100~\mathrm{\mu m}$, i.e., in the entire range of interest to us. Terahertz TM Wave in Finite-Conductivity Coaxial Waveguide ---------------------------------------------------------- Consider a coaxial waveguide (coax) with the inner wire radius $r$ and the outer radius $R=r+a$, where $a$ is the dielectric gap width. The characteristic relation for the TM modes of this waveguide has has the following form $$\begin{aligned} &\left(\frac{I_0(\kappa_d r)}{I_0(\kappa_m r)}- \xi\frac{I_1(\kappa_d r)}{I_1(\kappa_m r)}\right) \left(\frac{K_0(\kappa_d R)}{K_0(\kappa_m R)}- \xi\frac{K_1(\kappa_d R)}{K_1(\kappa_m R)}\right)= \label{coax_characteristic} \\\nonumber &\left(\frac{K_0(\kappa_d r)}{I_0(\kappa_m r)}+ \xi\frac{K_1(\kappa_d r)}{I_1(\kappa_m r)}\right) \left(\frac{I_0(\kappa_d R)}{K_0(\kappa_m R)}+ \xi\frac{I_1(\kappa_d R)}{K_1(\kappa_m R)}\right)~,\end{aligned}$$ where $I_{\nu}(x)$ and $K_\nu (x)$ are modified Bessel functions, and $\xi=\frac{\varepsilon_d \kappa_m}{\varepsilon_m \kappa_d}$. This equation is quadratic with respect to $\xi$ and can be written in the form $\alpha\xi^2+\beta\xi+\gamma=0$, where the coefficients $\alpha$, $\beta$, and $\gamma$ can be easily found by comparison to Eq. (\[coax\_characteristic\]) as combinations of the Bessel functions. It can obviously be resolved for $\xi$ yielding $$\frac{\beta\pm\sqrt{\beta^2-4\alpha\gamma}}{2\alpha}= -\frac{\varepsilon_d \kappa_m}{\varepsilon_m \kappa_d}~. \label{coax_disp}$$ In the THz region, only the mode with the minus sign in Eq. (\[coax\_disp\]) propagates. It can be treated in a manner similar to the mode described by Eq. (\[slab\_disp\]). The equation (\[coax\_disp\]) can be expanded over the small parameter $\kappa_d a\ll 1$, and the explicit form of the modal refraction index can be readily obtained as $$\begin{aligned} &n=\sqrt{\varepsilon_d} \left(1+\left(\frac{I_0(\kappa_m r)}{I_1(\kappa_m r)}+ \frac{K_0(\kappa_m R)}{K_1(\kappa_m R)}\right) \frac{l_s (1+i)}{2a}\right)^{1/2}~,~~ \label{coax_refraction}\end{aligned}$$ where $\kappa_m=l_s^{-1}(1-i)$. Similar to the wedge waveguide case, the applicability condition of this solution is given by Eq. (\[applicability\]). WKB Solution for the TM Wave in Graded Waveguide ------------------------------------------------ In the WKB approximation, a solution of the Maxwell equations can be represented as a wave with amplitude and phase that are slowly varying functions of $y$ on the scale of local wavelength. The behavior in the transverse direction $z$ is the same as for the non-graded system. The WKB solution is valid if the adiabatic parameter is small $$\delta=|\frac{d}{dy}\frac{1}{k(y)}|\ll 1~.~~ \label{adiabatic_parameter}$$ In the WKB approximation, the phase of the mode (eikonal) is given by an integral $$\phi(y)=k_0 \int n(y) dy~,~~ \label{wedge_eikonal}$$ where $n(y)$ is the local refraction index of the mode. The behavior of wave amplitude as a function of the propagation coordinate $y$ is found from the condition of flux conservation: $$v_g(y) \int_{-\infty}^{\infty} W(y,z) dz = \text{const}~,~~ \label{flux_conservation}$$ where $v_g(y)=\partial \omega/\partial k$ is the wave local group velocity, and $W(y,z)$ is energy density in the mode. This work was supported by grants from the Chemical Sciences, Biosciences and Geosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, a grant CHE-0507147 from NSF, and a grant from the US-Israel BSF. MIS is grateful to S. Gresillon for helpful remarks. Correspondence and requests for materials should be addressed to MIS (email: [email protected]) [33]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ** (, , ). , ****, (). , , , ****, (). , , , , , , ****, (). , , , ****, (). , ****, (). , , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , p. (). , , , ****, (). , , , , ****, (). , , , ****, (). , , , , , ****, (). , , , ****, (). , , , , , , , ****, (). , ****, (). , ****, (). , , , , , , ****, (). , ****, (). , , , ****, (). , , , p. (). , , , , ****, (). , ****, (). , , , , ****, (). , , , , , ****, (). , , , , , , , , ****, (). , , , , , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, ().

--- abstract: | The moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, with paying attention to the natural domains of these unbounded operators. The results are then applied to conveniently determine the moment operators of the Cartesian margins of the phase space observables. [**Keywords:**]{} moment operators, convolution, semispectral measure, phase space observables address: - 'Jukka Kiukas, Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland. At present: Institut für Mathematische Physik, TU Braunschweig, DE-38106 Braunschweig, Germany.' - 'Pekka Lahti, Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland' - 'Kari Ylinen, Department of Mathematics, University of Turku, FI-20014 Turku, Finland' author: - Jukka Kiukas - Pekka Lahti - Kari Ylinen title: Semispectral measures as convolutions and their moment operators --- Introduction ============ The increasingly accepted view of a quantum observable as a positive operator measure as opposed to the more traditional approach using only spectral measures has added a great deal to our understanding of the mathematical structure and foundational aspects of quantum mechanics. In many cases an observable that is not itself projection valued, nevertheless arises as an unsharp or smeared version of a spectral measure. One way to realize such a smearing is to convolve the spectral measure with a probability measure. In particular, the marginal observables of a phase space observable have such a structure. Phase space observables have several important applications in quantum mechanics, ranging from the theory of Husimi distributions in quantum optics to state tomography, phase space quantizations, and to the theory of approximate joint measurements of position and momentum, as highlighted, for instance, by the monographs [@Davies76; @Holevo82; @OQP95; @Schroeck96; @Leonhardt97; @Stulpe97; @Landsman98; @Ali00; @Holevo01; @Paris04; @Landsman07]. Optical implementations of such observables are also well understood, as described e.g. in a recent study [@JukkaVII], and their mathematical structure has been investigated with great detail [@Holevo79; @Werner84; @CDeV03; @JukkaII06]. The moments of measurement statistics of an observable are related to the moment operators of that observable in the same way as the outcome probabilities are related to the observable itself. In some cases, the moments may even carry the entire information on the observable [@Dvurecenskij2000; @JukkaVI]. Hence, it makes sense to study the moment operators of a semispectral measure obtained as a convolution, which is the aim of this paper. In Sect. \[preliminaries\], we give the technical lemmas needed for the main results. In particular, we discuss the difficulties in the integrability questions associated with convolutions of nonpositive scalar measures. In Sect. \[moments\], we consider the case of a general convolved semispectral measure by using the operator integral of [@LMY], and in Sect. \[phase\_space\], we work out the Cartesian marginal moment operators for a class of phase space observables. Preliminaries ============= To begin with, we recall the notion of the convolution of scalar measures, and we prove a lemma on their moment integrals. The *convolution* of two complex Borel measures $\mu,\nu:{\mathcal{B}}({\mathbb{R}})\to {\mathbb{C}}$ is the measure $\mu*\nu:{\mathcal{B}}({\mathbb{R}})\to {\mathbb{C}}$, defined by $$\mu*\nu(X)=(\mu\times \nu)(\{(x,y)\mid x+y\in X\}), \ \ X\in {\mathcal{B}}({\mathbb{R}}),$$ where $\mu\times \nu$ is the product measure defined on ${\mathcal{B}}({\mathbb{R}}^2)$, the Borel $\sigma$-algebra of ${\mathbb{R}}^2$ (see e.g. [@Dunford p. 648, Definition 8]). \[integrallemma\] Let $\mu,\nu:{\mathcal{B}}({\mathbb{R}})\to {\mathbb{C}}$ be two complex measures, and let $k\in {\mathbb{N}}$. - A Borel function $f:{\mathbb{R}}\to {\mathbb{C}}$ is $\mu*\nu$-integrable if $(x,y)\mapsto f(x+y)$ is integrable with respect to the product measure $\mu\times \nu:{\mathcal{B}}({\mathbb{R}}^2)\to {\mathbb{C}}$. In that case, $$\int f(x)\, d(\mu*\nu)(x) = \int f(x+y)\,d(\mu\times \nu)(x,y).$$ - The function $(x,y)\mapsto (x+y)^k$ is $\mu\times \nu$-integrable, if and only if $x\mapsto x^k$ is both $\mu$- and $\nu$-integrable. In that case, $$\int (x+y)^k d(\mu\times \nu)(x,y) = \sum_{n=0}^k \binom{k}{n} \left(\int x^{k-n}d\mu(x)\right) \left(\int y^nd\nu(y)\right).$$ Let $\phi:{\mathbb{R}}^2\to {\mathbb{R}}$ denote addition, i.e. $\phi(x,y)=x+y$. Write the product measure $\mu\times \nu$ in terms of the positive and negative parts of its real and imaginary parts: $$\mu\times \nu = \nu_1+i\nu_2=\nu_1^+-\nu_1^-+i(\nu_2^+-\nu_2^-),$$ where $\nu_i^{\pm}=\frac 12 (|\nu_i|\pm \nu_i)$. Then $$\label{sum} \mu*\nu(X) = \nu_1^+(\phi^{-1}(X))-\nu_1^-(\phi^{-1}(X))+i(\nu_2^+(\phi^{-1}(X))-\nu_2^-(\phi^{-1}(X))), \ \ X\in {\mathcal{B}}({\mathbb{R}}).$$ Assume now that $f\circ\phi$ is $\mu\times \nu$-integrable. Then $f\circ\phi$ is integrable with respect to each $\nu_i^{\pm}$, and $$\label{integral} \int f\circ \phi\, d(\mu\times \nu) =\int f\circ\phi \,d\nu_1^+-\int f\circ \phi \,d\nu_1^- +i(\int f\circ \phi \,d\nu_2^+-\int f\circ \phi \,d\nu_2^-).$$ Since the measures $\nu_i^{\pm}$ are positive, it follows that $f$ is integrable with respect to each induced measure $X\mapsto \nu_1^+(\phi^{-1}(X))$, and the corresponding integrals are equal (see e.g. [@Halmos p. 163]). Now and imply that $f$ is $\mu*\nu$-integrable, with $$\int f d(\mu*\nu) = \int f\circ \phi \,d(\mu\times \nu).$$ This proves (a). To prove (b), suppose first that $(x,y)\mapsto \phi(x,y)^k$ is $\mu\times \nu$-integrable. Since both measures $\mu$ and $\nu$ are finite, it follows from [@Dunford p. 193, Theorem 13] that $x\mapsto \phi(x,y)^k=(x+y)^k$ is $\mu$-integrable for $\nu$-almost all $y\in {\mathbb{R}}$. Take any such $y\in {\mathbb{R}}$. Now $x\mapsto |x+y|^k$ is also $|\mu|$-integrable, where $|\mu|$ denotes the total variation measure of $\mu$. There are positive constants $M$ and $K$ satisfying $$|x^k|\leq K|x+y|^k +M, \ \ x\in {\mathbb{R}}.$$ This implies that $x\mapsto |x|^k$ is $|\mu$|-integrable, and hence also $\mu$-integrable. It is similarly seen that $x\mapsto |x|^k$ is $\nu$-integrable. Suppose now that $x\mapsto x^k$ is both $\mu$- and $\nu$-integrable. Since $x\mapsto |x|^k$ is now $|\mu|$- and $|\nu|$-integrable, and these are finite positive measures, it follows that $x\mapsto |x|^l$ is $|\mu|$- and $|\nu|$-integrable for all $l\in {\mathbb{N}}$, $l\leq k$. Hence, $(x,y)\mapsto |x^ly^m|$ is $|\mu|\times |\nu|$-integrable for all $l,m\in {\mathbb{N}}$, $l\leq k$, $m\leq k$. Since $|x+y|^k\leq \sum_{n=0}^k \binom{k}{n} |x^{k-n}y^n|$, this implies that $(x,y)\mapsto |\phi(x,y)^k|=|(x+y)^k|$ is $|\mu|\times |\nu|$-integrable. But $|\mu|\times |\nu|=|\mu\times \nu|$ by [@Dunford p. 192, Lemma 11], so $x\mapsto \phi(x,y)^k$ is $\mu\times \nu$-integrable. The claimed formula follows now easily, since we have shown above that the equivalent integrability conditions imply that $(x,y)\mapsto x^ly^m$ is $\mu\times \nu$-integrable for all $l,m\in {\mathbb{N}}$, $l\leq k$, $m\leq k$. The converse implication in part (a) of the above lemma does not hold if the measures $\mu$ and $\nu$ are not assumed to be positive. This is the conclusion of the brief discussion we now enter. Denote $\phi(x+y)=x+y$ as before, and $\Sigma=\{\phi^{-1}(X)\,|\,X\in {\mathcal{B}}({\mathbb{R}})\}.$ Then $\Sigma$ is a $\sigma$-algebra (properly) contained in ${\mathcal{B}}({\mathbb{R}}^2)$. Let $\mu$ and $\nu$ be complex Borel measures on ${\mathbb{R}}$ and $\lambda_2$ their convolution. Lemma 8 in [@Dunford p. 182] states that the formula $\lambda_1(\phi^{-1}(X))=\lambda_2(X)$ gives a well-defined complex measure on $\Sigma$. (To be precise, the lemma requires the additional assumption that $\phi$ be surjective, but of course this holds in our situation.) The same lemma says that the total variations satisfy $|\lambda_1|(\phi^{-1}(X))=|\lambda_2|(X)$ for all $X\in {\mathcal{B}}({\mathbb{R}})$, and moreover for any $\lambda_2$-integrable Borel function $f:{\mathbb{R}}\to{\mathbb{C}}$, the composite function $f\circ \phi$ is $\lambda_1$-integrable, and the natural integral transformation formula holds. Obviously $\lambda_1$ is just the restriction of the product measure $\mu\times\nu$ to $\Sigma$. Any $\Sigma$-measurable function $g:{\mathbb{R}}^2\to{\mathbb{C}}$ which is $\mu\times\nu$-integrable, is integrable with respect to the restriction of the variation measure $|\mu\times\nu|$ to $\Sigma$. In the following example we see that this need not be the case if $g$ is just assumed to be $\lambda_1$-integrable. This phenomenon is at the root of the fact that the implication in Lemma \[integrallemma\] (a) cannot be reversed. \[example1\] We construct two discrete measures $\mu$ and $\nu$ supported by ${\mathbb{Z}}$. Let $\sum_{k=0}^\infty a_k$ be any convergent series with positive terms, and define $b_{2k}=b_{2k+1}=a_k$ for all $k=0,\,1,\,2,\dots$, and $b_k=0$ if $k\in {\mathbb{Z}}$, $k\leq -1$. We set $\mu(\{n\})=b_n$ for all $n\in{\mathbb{Z}}$. The discrete measure $\nu$ is defined by setting $\nu(\{n\})=(-1)^nb_{-n}$ for all $n\in {\mathbb{Z}}$. Then the convolution $\lambda=\mu*\nu$ is supported by ${\mathbb{Z}}$, and we have $\lambda(\{n\})=\sum_{j=-\infty}^\infty b_j(-1)^{n-j}b_{j-n}=\sum_{j=0}^\infty b_j(-1)^{n-j}b_{j-n}$. If $n$ is even, it follows that $\lambda(\{n\})=0$, since $b_{2k}=b_{2k+1}$. However, $c_n=\sum_{j=-\infty}^\infty |b_j(-1)^{n-j}b_{j-n}|>0$. We now define $f:{\mathbb{R}}\to{\mathbb{C}}$ by setting $f(2k)=c_{2k}^{-1}$ for all $k\in Z$ and $f(x)=0$ if $x\in{\mathbb{R}}\setminus 2{\mathbb{Z}}$. Then $\int_{\mathbb{R}}f(x)d\lambda(x)=0$, but the function $(x,y)\mapsto f(x+y)$ is not $|\mu\times\nu|$-integrable, since its integral with respect to $|\mu\times\nu|$ over any set $\{(x,y)\,|\,x+y=n\}$, $n\in 2{\mathbb{Z}}$, equals 1. To close this preliminary section, we recall the notion of an operator integral in the sense of [@LMY]. Let $\Omega$ be a set and ${\mathcal{A}}$ a $\sigma$-algebra of subsets of $\Omega$. Let $E:{\mathcal{A}}\to{L(\mathcal{H})}$ be a semispectral measure (normalized positive operator measure) taking values in ${L(\mathcal{H})}$, the set of bounded operators on a complex Hilbert space ${\mathcal{H}}$ ($\ne \{0\}$). Thus, for any ${\varphi},\psi\in{\mathcal{H}}$, the set function $X\mapsto E_{\psi,{\varphi}}(X):= \langle\psi\,|\, E(X){\varphi}\rangle$ is a complex measure. For any measurable function $f:\Omega\to{\mathbb{C}}$ we let $D(f,E)$ denote the set of those vectors ${\varphi}\in{\mathcal{H}}$ for which $f$ is $E_{\psi,{\varphi}}$-integrable for all $\psi\in{\mathcal{H}}$. The set $D(f,E)$ is a vector subspace of ${\mathcal{H}}$ and the formula $$\langle\psi\,|\, L(f,E){\varphi}\rangle =\int_\Omega f\,dE_{\psi,{\varphi}}, \quad {\varphi}\in D(f,E), \psi\in{\mathcal{H}},$$ defines a unique linear operator $L(f,E)$, with the domain $D(f,E)$. The set $\tilde D(f,E)=\{{\varphi}\in{\mathcal{H}}\,|\, \int |f|^2\,dE_{{\varphi},{\varphi}} <\infty \}$ is a subspace of $D(f,E)$, and we let $\tilde L(f,E)$ denote the restriction of $L(f,E)$ into $\tilde D(f,E)$. We recall that if $E$ is a spectral (projection valued) measure, then $\tilde D(f,E)=D(f,E)$ and the operator $L(f,E)$ is densely defined. We consider here only the cases where $(\Omega,{\mathcal{A}})$ is $({\mathbb{R}},{\mathcal{B}({\mathbb{R}})})$ or $({\mathbb{R}}^2,{\mathcal{B}}({\mathbb{R}}^2))$. Convolutions and their moment operators {#moments} ======================================= For any $X\in {\mathcal{B}}({\mathbb{R}})$, let $\chi_X$ denote the characteristic function of $X$. Recall that $\phi$ denotes the map $(x,y)\mapsto x+y$. The function $\chi_X\circ \phi$ is bounded and thereby integrable with respect to the product measure. Hence Lemma \[integrallemma\](a) and Fubini’s theorem give that the function $$y\mapsto \mu(X-y)=\int\chi_{X-y}(x)\, d\mu(x)=\int\chi_X(x+y)\, d\mu(x)$$ coincides almost everywhere with a Borel function, and $$\mu*\nu(X) = \int\left(\int \chi_X(x+y)\, d\mu(x)\right)\, d\nu(y) = \int_{\mathbb{R}}\mu(X-y)d\nu(y), \ \ X\in {\mathcal{B}}({\mathbb{R}}).$$ Let now $E:{\mathcal{B}}({\mathbb{R}})\to L({\mathcal{H}})$ be a semispectral measure, and let $\mu:{\mathcal{B}}({\mathbb{R}})\to [0,1]$ be a probability measure. Since the sesquilinear form $$({\varphi},\psi)\mapsto \int_{\mathbb{R}}\mu(X-y)\, dE_{\psi,{\varphi}}(y)$$ is clearly bounded, one can define $\mu*E:{\mathcal{B}}({\mathbb{R}})\to L({\mathcal{H}})$ via $\langle {\varphi}|(\mu*E)(X)\psi\rangle := \mu*E_{\psi,{\varphi}}(X)$, ${\varphi},\psi\in {\mathcal{H}}$. It follows from the monotone convergence theorem that $\mu*E$ is a semispectral measure. Denote $\mu[k]:=\int x^k\, d\mu(x)$, in case this integral exists (i.e. when $\int |x^k|\, d\mu(x)<\infty$). Let $E:{\mathcal{B}}({\mathbb{R}})\to L({\mathcal{H}})$ be a semispectral measure, and $\mu:{\mathcal{B}}({\mathbb{R}})\to [0,1]$ a probability measure. Then - $\tilde{D}(x^k,\mu*E)$ equals either $\tilde{D}(x^k,E)$ or $\{0\}$, depending on whether $\mu[2k]$ exists or not. In the former case, $$\tilde{L}(x^k,\mu*E) = \sum_{n=0}^k \binom{k}{n} \mu[k-n]\tilde{L}(x^n,E).$$ - If $\mu[k]$ exists, then $D(x^k,E)\subset D(x^k,\mu*E)$, and $$L(x^k,\mu*E) \supset \sum_{n=0}^k \binom{k}{n} \mu[k-n]L(x^n,E).$$ Since $(\mu*E)_{{\varphi},{\varphi}}= \mu*E_{{\varphi},{\varphi}}$ by definition, and these measures are positive, it follows from e.g. [@Halmos p. 163] that $x^{2k}$ is $(\mu*E)_{{\varphi},{\varphi}}$-integrable if and only if $(x,y)\mapsto (x+y)^{2k}$ is $\mu\times E_{{\varphi},{\varphi}}$-integrable. By Lemma \[integrallemma\] (b), this happens exactly when $\mu[2k]$ exists and ${\varphi}\in \tilde{D}(x^k,E)$. Hence, $\tilde{D}(x^k,\mu*E)$ equals either $\tilde{D}(x^k,E)$ or $\{0\}$, depending on whether $\mu[2k]$ exists or not. Suppose now that $\mu[2k]$ exists, and let ${\varphi}\in \tilde{D}(x^k,\mu*E)=\tilde{D}(x^k,E)$. Since this set is contained in $D(x^k,E)$, it follows that $x^k$ is $E_{\psi,{\varphi}}$-integrable for all $\psi\in {\mathcal{H}}$. Also, $\mu[k]$ clearly exists. Hence, according to Lemma \[integrallemma\] (b), $(x,y)\mapsto (x+y)^k$ is $\mu\times E_{\psi,{\varphi}}$-integrable for all $\psi\in {\mathcal{H}}$, so using both (a) and (b) of that lemma, we get $$\label{binomialform} \int x^k \, d(\mu*E)_{\psi,{\varphi}} = \int (x+y)^k\, d(\mu\times E_{\psi,{\varphi}})(x,y) = \sum_{n=0}^k \binom{k}{n} \mu[k-n]\int x^n \, dE_{\psi,{\varphi}}, \ \ \psi\in {\mathcal{H}}.$$ This completes the proof of (a). To prove (b), suppose that $\mu[k]$ exists, so that $x^k$ is $\mu$-integrable. Now if ${\varphi}\in D(x^k,E)$, then $x^k$ is also $E_{\psi,{\varphi}}$-integrable for any $\psi\in {\mathcal{H}}$. According to Lemma \[integrallemma\] (b), this implies that $(x,y)\mapsto (x+y)^k$ is $\mu\times E_{\psi,{\varphi}}$-integrable for all $\psi\in {\mathcal{H}}$, and using again also Lemma \[integrallemma\] (a), we see that $x^k$ (and thus also $x^n$ with $n\leq k$) is $\mu*E_{\psi,{\varphi}}$-integrable (i.e. $(\mu*E)_{\psi,{\varphi}}$-integrable) for all $\psi\in {\mathcal{H}}$, and the relation holds. But this means that we have proved (b). \[domainprop\] Let $E:{\mathcal{B}}({\mathbb{R}})\to L({\mathcal{H}})$ be a spectral measure, let $k\in {\mathbb{N}}$, and let $\mu:{\mathcal{B}}({\mathbb{R}})\to [0,1]$ be a probability measure such that $\mu[k]$ exists. Denote $A=L(x,E)$. Then $$L(x^k,\mu*E)= \sum_{n=0}^k \binom{k}{n} \mu[k-n]A^n, \ \ D(x^k,\mu*E)=D(A^k).$$ Moreover, $\tilde{D}(x^k,\mu*E)$ equals either $D(A^k)=D(x^k,\mu*E)$ or $\{0\}$, depending on whether $\mu[2k]$ exists or not. Since $E$ is a spectral measure, $A$ is selfadjoint, and $D(A^k) = D(x^k,E) = \tilde{D}(x^k,E)$, $L(x^k,E)=A^k$ for all $k\in {\mathbb{N}}$. According to the preceding proposition (b), $L(x^k,\mu*E)$ is a symmetric extension of the selfadjoint operator $\sum_{n=0}^k \binom{k}{n} \mu[k-n]A^n$. Thus these operators must be equal. The last claim follows immediately from part (a) of the preceding proposition. Let $E:B({\mathbb{R}})\to L({\mathcal{H}})$ be any spectral measure, and choose a probability measure $\mu$ such that $\mu[k]$ exists but $\mu[2k]$ does not. Then $L(x^k,\mu*E)$ is a densely defined selfadjoint operator, but $\tilde{D}(x^k,\mu*E)=\{0\}$. Consider then the following special case. For any positive operator $T$ of trace one, and a selfadjoint operator $A$ in ${\mathcal{H}}$, let $p_T^A:{\mathcal{B}}({\mathbb{R}})\to [0,1]$ be the probability measure defined by $p_T^A(X) = {{\rm Tr }}[TE^{A}(X)]$, where $E^A$ is the spectral measure of $A$. Let $A$ be a selfadjoint operator and $k\in {\mathbb{N}}$, such that $p_T^A[k]$ exists. According to e.g. [@KLY2005 Lemma 1] and [@KLY2006 Lemma 1], this happens exactly when $\sqrt{|A|}^k\sqrt{T}$ is a Hilbert-Schmidt operator. Under this condition, we then have, according to the preceding proposition, that $$L(x^k,p_T^{A}*E^B)=\sum_{n=0}^k \binom{k}{n} p_T^A[k-n]B^n, \ \ D(x^k,p_T^A*E^B)=D(B^k)$$ for any selfadjoint operator $B$. Moreover, $\tilde{D}(x^k,p_T^{A}*E^B)\neq \{0\}$ if and only if $p_T^A[2k]$ exists, or, equivalently, $A^k\sqrt{T}$ is a Hilbert-Schmidt operator. This stronger condition assures also that $p_T^A[k-n]={{\rm Tr }}[A^{k-n}T]$ in the above formula, the operators $A^{k-n}T$ being in the trace class. As an example, take $T=|\eta\rangle\langle \eta|$ with $\eta\in D(\sqrt{|A|})$ but $\eta\notin D(A)$. Then $L(x,p_{|\eta\rangle\langle\eta|}^{A}*E^B)=B$, since $\sqrt{|A|} \sqrt{|\eta\rangle \langle \eta|} = \sqrt{|A|}|\eta\rangle \langle \eta|$ is clearly a Hilbert-Schmidt operator. However, the square integrability domain is $\{0\}$, since $|A|\sqrt{|\eta\rangle\langle \eta|}$ is quite far from being Hilbert-Schmidt (its domain is $\{0\}$). Note also that now $p_{|\eta\rangle\langle \eta|}^A[1]$ is not equal to ${{\rm Tr }}[A|\eta\rangle\langle \eta|]$, since this trace is not even defined. Phase space observables {#phase_space} ======================= Let ${\mathcal{H}}= L^2({\mathbb{R}})$, and let $Q$ and $P$ be the selfadjoint position and momentum operators in ${\mathcal{H}}$, and $W(q,p)$, $(q,p)\in{\mathbb{R}}^2$, the corresponding Weyl operators. Consider now the phase space observable $E^T:{\mathcal{B}}({\mathbb{R}}^2)\to L({\mathcal{H}})$, $$E^T(Z)=\frac{1}{2\pi}\int_Z W(q,p)TW(q,p)^*\, dqdp,$$ with $T$ a positive operator of trace one. The Cartesian marginal measures $E^{T,x},E^{T,y}:{\mathcal{B}}({\mathbb{R}})\to L({\mathcal{H}})$ are defined by $E^{T,x}(X):=E^T(X\times {\mathbb{R}})$, $E^{T,y}(Y):=E^T({\mathbb{R}}\times Y)$. It is well known that they are equal to $p_T^{-Q}*E^Q$ and $p_T^{-P}*E^P$, respectively, see e.g. [@Davies76 Theorem 3.4.2]. According to the above discussion, we can thus determine the $k$th moment operators of the $x$- and $y$- margins, under the respective conditions that $p_T^{-Q}[k]$ and $p_T^{-P}[k]$ exist, or, equivalently, $\sqrt{|Q|}^k\sqrt{T}$ and $\sqrt{|P|}^k\sqrt{T}$ are Hilbert-Schmidt: Let $k\in {\mathbb{N}}$. - If $\sqrt{|Q|}^k\sqrt{T}$ is a Hilbert-Schmidt operator, then $$L(x^k,E^{T,x}) = \sum_{n=0}^k \binom{k}{n} (-1)^{k-n}p_T^{Q}[k-n]Q^n, \ \ D(x^k,E^{T,x})=D(Q^k).$$ - Part (a) holds also when ”$x$” and ”$Q$” are replaced by ”$y$” and ”$P$”. Under the square integrability condition that $Q^k\sqrt{T}$ (respectively $P^k\sqrt{T}$) be Hilbert-Schmidt, we get $p_T^Q[k-n]= {{\rm Tr }}[Q^{k-n}T]$ ($p_T^P[k-n]= {{\rm Tr }}[P^{k-n}T]$). According to the discussion in the preceding remark, a simple example where $L(x,E^{T,x})=Q$ but $\tilde{D}(x,E^{T,x})=\{0\}$, is obtained by taking $T=|\eta\rangle \langle \eta|$, where $\eta\in {\mathcal{H}}$ is a unit vector with $\int |x||\eta(x)|^2\, dx<\infty$, $\int x |\eta(x)|^2\, dx = 0$, and $\int x^2|\eta(x)|^2\, dx=\infty$. An additional problem with the moment operators $L(x^k,E^{T,x})$ and $L(x^k,E^{T,y})$ is their connection to the operators $L(x^k,E^T)$ and $L(y^k,E^T)$, which we have considered before (see [@KLY2005; @KLY2006]). By writing e.g. $E^{T,x}(X)=E^T(\pi_1^{-1}(X))$ where $\pi_1:{\mathbb{R}}^2\to {\mathbb{R}}$ is the coordinate projection $(x,y)\mapsto x$, we notice that a similar ”change of variables” argument as that in Lemma \[integrallemma\] gives $L(x^k, E^{T,x})\supset L(x^k,E^T)$. Now if $\sqrt{|Q|}^k\sqrt{T}$ is Hilbert-Schmidt, then we know from the above proposition that $L(x^k, E^{T,x})$ is a selfadjoint operator, a polynomial in $Q$. However, this does not determine $L(x^k,E^T)$; we can only say that it has $L(x^k, E^{T,x})$ as a selfadjoint extension. Consider then the square integrability domains. Since the measures involved are now positive, the ”change of variables formula” (see e.g. [@Halmos p. 163]) can be used to conclude that the restrictions are equal: $\tilde{L}(x^k, E^{T,x})= \tilde{L}(x^k,E^T)$. According to Proposition \[domainprop\], this operator is nontrivial exactly when $|Q|^k\sqrt{T}$ is Hilbert-Schmidt, in which case it is selfadjoint. This stronger condition then forces both the symmetric extensions $L(x^k, E^{T,x})$ and $L(x^k,E^T)$ to coincide with the restriction, and we recover Theorem 4 of [@KLY2006]. [**Acknowledgment.**]{} One of us (J.K.) was supported by the Emil Aaltonen Foundation and the Finnish Cultural Foundation. [99]{} E. B. Davies; Quantum Theory of Open Systems, Academic Press, London, 1976. A.S. 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--- abstract: 'A general information-theoretic framework for deriving physical laws is presented and a principle of informational physics is enunciated within its context. Existing approaches intended to derive physical laws from information-theoretic first principles are unified as special cases of this framework with the introduction of constraints dependent on the physical process of observation. Some practical, theoretical and epistemological implications of the validity of this approach are examined.' author: - Nisheeth Srivastava title: The Relativity of Theory --- Introduction {#sec:intro} ============ The consideration of information (subjectively, awareness) as a fundamental ontological entity is a view with an extremely illustrious pedigree in several Idealistic traditions of philosophy[^1]. In the past half century, this philosophical position has been adopted by some scientific/computational programs and specified to various degrees of rigor in various representational structures, e.g. [@whee90; @zuse70; @wolf02]. While many of these observers have designed different perspectives on the subject, the theme of a fundamental informational nature of reality runs through all their narratives. These are, in general terms, the philosophical approaches to informational physics. They begin with an ontological assumption regarding the nature of reality and attempt to find representational theories of sufficient expressiveness and precision to be able to make testable statements in the physical world. In a different epistemological context, beginning with Frieden [@fried90], there have been several sporadic efforts (see, e.g., [@fried04; @skal05; @lisi06]) made to re-derive the foundational principles of quantum mechanics with a probabilistic and/or information-theoretic basis, making no physical assumptions. This effort is philosophically appealing because of the nebulous understanding of the role of the observer and the observer’s effect on the wave function in the standard Copenhagen interpretation. Furthermore, since most theories of physics, including the Standard Model, are generally expressed as quantum field theories, an information-theoretic interpretation of QFT, if justifiable, directly affects the entire epistemological basis of physics. As a matter of fact, a controversial information-theoretic unification model has already been proposed [@fried04] by Frieden as an extension to his original approach [@fried90]. These constitute what we call the mathematical approaches to informational physics. They begin from an agnostic epistemic viewpoint, operating entirely within existing representational frameworks of probability and information and attempt to characterize the ontological assumptions necessary for their mathematical findings to be consistent with existing theories of reality in the same mathematical representation. In this brief note, we examine prospects of unifying the philosophical and mathematical currents of informational physics. We do so by formulating the problem of information optimization in a general mathematical setting. We then discuss how the simple selection of an information measure (frequentist probability) and its physical interpretation allows us to recover statistical mechanics from our general framework, an insight first gleaned by Jaynes [@jayn63]. We remark on the prospect of employing a similar approach to deriving quantum mechanics for large systems, viz., quantum field theories, an observation independently made by Lisi [@lisi06]. We further note how a different choice of information measure (Fisher Information) recovers Frieden’s derivation of Schrodinger’s equation [@fried91] as well as (within some interpretative latitude) his broader Extreme Physical Information framework [@fried04] of informational physics. Finally, using heuristic arguments, we suggest that this apparent unity offers us insight into both the nature of physical laws as well as the surprising comprehensibility of physics using mathematics [@wign60]. Information optimization {#sec:inf} ======================== Let ${\mathcal{S}}$ represent the set of possible outcomes in an isolated system under observation, where every outcome $s\in{\mathcal{S}}$ is a set of variables that sufficiently represent the system behavior for each outcome. For example, for a classical point particle (position $x$, momentum $p$) moving in a straight line, $\{x,p\}$ is a sufficient representation. In order to study changes in outcomes, we consider each outcome $s$ to be a continuous function of a set of parameters $t$. If we were to consider the problem of obtaining knowledge in the broadest epistemic sense, an observer would be ignorant of the membership of ${\mathcal{S}}$. To be able to make meaningful statements about information optimization, we require that an observer possess information about the existence of at least some $s\in{\mathcal{S}}$[^2]. The goal of the observer is two-fold. In the first place, he wishes to explain the relative frequency of occurrence of different outcomes of the system with a *theory*. We consider a good theory to be one which explains the relative frequencies of the occurrence of outcomes with a set of statements of a much lower Kolmogorov complexity [@grun05 Ch.1-2] than the set of statements required to describe the occurrence of outcomes. In the second, he would like to be able to make predictions regarding the behavior of unobserved outcomes of the system based on his observation of past outcomes. Note that these two aspects of *episteme* need not be related, e.g., the true theory regarding a fair coin,‘This is a random process’, implies that prediction becomes impossible, whereas predicting a coin will show up heads (or tails) because it has shown up heads thrice before is a process fraught with peril[^3]. For either (or both) of these purposes, the observer associates a level of certainty or uncertainty with each possible outcome of the system. Let us denote this with an information measure $f(s)$, the choice of which will determine the form of the solution to the information optimization problem. Furthermore, since multiple outcomes are feasible, the observer must construct an aggregate measure of certainty (or uncertainty). In general, we can denote this by $F:{\mathcal{S}}'\rightarrow{\mathbb{R}}$, where ${\mathcal{S}}'\subseteq{\mathcal{S}}$ and depends on the observer’s knowledge of the membership of the system. Now that the observer has quantified the extent of his knowledge/ignorance of the system, he must adjust these quantities to account for observations of system outcomes that he makes. The method of adjustment employed, which is essentially *inference* may be arbitrary, but a principled approach would be to employ *Bayesian* inference. While this might appear to constrain the generality of our informational framework, it must be noted that the Bayesian formalism has been shown to be quite generally applicable and along with the axioms of probability, appears to be quite fundamental to the way in which humans naturally think [@tene01]. Therefore, to assume that the system of updates that the observer employs will be Bayesian in nature does not appear to be overly restrictive. Note further that Bayesian inference has been shown to be equivalent [@cati06] to the principle of maximum relative entropy [@shor81], an extension of Jaynes’ famous original MaxEnt principle [@jayn63] applicable to uniform prior distributions to arbitrary priors. MaxEnt, therefore, is a special case of Bayesian inference, where one begins with an agnostic belief regarding one’s current state of knowledge. This, naturally, exposes the argument to the same opposition that proponents of MaxEnt face in general. It should therefore be emphasized that the MaxEnt assumption is not a crucial aspect of this framework and is used here to simplify our analysis in the examples that we consider. The general Bayesian argument (or its maximum relative entropy equivalent) constitute our general system of updates. Having defined our terms of reference, we can now state our informational physics hypothesis somewhat more formally as a principle assumed in all our subsequent derivations. The principle of informational physics states that *an observer’s task of developing a theory concerning the evolution of the system characterized by a space of outcomes ${\mathcal{S}}$ is equivalent to extremizing a measure of his information/ignorance $F$ constrained by what he *believes-he-knows*[^4] about the system. Known physical laws emerge as mathematically optimal solutions to this information extremization procedure.* Physical laws ============= Realistic information measures {#sec:meas} ------------------------------ While a plethora of information/uncertainty measures (see e.g. [@tops08]) can be used within the information framework defined in Section \[sec:inf\], the one that is most commonly used in learning theory is Shannon entropy with an underlying frequentist definition of probability, i.e., assuming $p[s]$ measures the frequency of occurrence of outcome $s$ in the space of known outcomes $s\in{\mathcal{S}}'$, the observer’s uncertainty $f(s)$ with respect to the outcome $s$ is given by $- p[s]\log p[s]$ and the aggregate uncertainty across all known paths $F({\mathcal{S}}')$ as $$F({\mathcal{S}}') = -\sum_{s\in{\mathcal{S}}'}{p[s]\log p[s]} = -\int_{s\in{\mathcal{S}}'}{p[s]\log p[s]~ds}. \label{eqn:entropy}$$ The $F$ obtained here is precisely the Shannon entropy of the system and is a measure of the uncertainty of the observer regarding the behavior of the system. Concomitantly, it is possible to define $F(s)$ as the information divergence of the observer’s hypothesis concerning the system to the best[^5] theory. Representing both the observer’s hypothesis and the target theory as distributions over the space of outcomes, we can use the standard definition of information divergence (alternatively Kullback-Leibler divergence) to obtain an aggregate information measure for the system, viz., $$F_{b}({\mathcal{S}}') = \int_{s\in{\mathcal{S}}'}{a[s]\log\frac{a[s]}{b[s]}~ds}, \label{eqn:dive}$$ where $b[s]$ is the observer’s hypothesis and $a[s]$ is the target theory. Note that in this case, minimizing the information measure would result in the observer finding a theory close to the optimal. Parenthetically, this recovers the MDI principle of model selection first proposed by Kullback and subsequently extended via the AIC criterion [@akai74] to statistical model selection [@burn02]. The third important information information measure we consider here is a variant of Fisher Information first proposed in [@fried90]\]. Frieden [@fried04] demonstrates that assuming $n$ independent measurements of outcomes as well as shift invariance in the system, it is possible to construct an information measure of the form $$F(s) = 4\int_{s\in{\mathcal{S}}'}{\sum_{n}{\frac{\partial q}{\partial s}\cdot\frac{\partial q}{\partial s}}~ds}, \label{eqn:fisher}$$ where $p[s] = q^2[s]$. As we mention above, these are by no means the only information measures available. A wide variety of distance (e.g. Levenshtein, Hellinger, Dudley) and divergence measures (e.g., the Bregman family of divergences) can be used to replace (\[eqn:dive\]) [@tops08]. Much work has been done to generalize entropy measures e.g., Tsallis entropies, and multiple definitions of probability abound. Thus, there are several information measures that may be used in place of the ones we have considered[^6]. The optimization problem and its interpretation must necessarily remain the same. There is thus, much flexibility in this framework to allow for physical theories to be constructed using different representations of information. Statistical and quantum mechanics {#sec:shannon} --------------------------------- From the arguments presented in Section \[sec:inf\], it is evident that the task of constructing a physical theory for observations made on a physical system is equivalent to solving an inference problem given constraints on the observation process. Working with the entropic formulation of $F$ as defined in (\[eqn:entropy\]), we now define minimal constraints on the inference problem as observed in the process of physical measurements. For physical systems, it appears appropriate to conjecture that the probabilities $p[s]$ of all known outcomes should sum to one, .i.e., the observer should believe that he knows ${\mathcal{S}}$ (when, in fact, he only knows ${\mathcal{S}}'$). Thus, the observer *believes-he-knows* $$\sum_{s\in{\mathcal{S}}'}p[s(t)] = \int_{s\in{\mathcal{S}}'}{p[s]~ds} = 1. \label{eqn:exist}$$ Furthermore, in order to develop his theory, the observer, given multiple observations, *believes-he-knows* how the system has been behaving. Mathematically, this means that he *believes-he-knows* the expected value $\langle A\rangle$ of the *history* $A[s]$ of the physical system given multiple observations, where the *history* may be defined in any mathematical manner that takes the past behavior of the system corresponding to outcome $s$ into account. That is, $$\bar{A} = \langle A\rangle = \int_{s\in{\mathcal{S}}'}{A[s]p[s]~ds}. \label{eqn:measure}$$ It is instructive to note here that the choice of information measure is independent of the physical constraints imposed, and that the physical constraints imposed are consequent to the process of observation, not the physical system being measured. Recovering the optimal theory for the system using (\[eqn:entropy\]) as an information measure and (\[eqn:exist\]), (\[eqn:measure\]) as informational constraints is now equivalent to extremizing, $$\begin{aligned} F' &= - \int_{s\in{\mathcal{S}}'}{p[s]\log p[s]~ds} + \lambda\left(1 - \int_{s\in{\mathcal{S}}'}{p[s]~ds}\right) + \nu\left(\bar{A} - \int_{s\in{\mathcal{S}}'}{p[s]A[s]~ds}\right),\\ &= \lambda + \nu\bar{A} - \int_{s\in{\mathcal{S}}'}{(p[s]\log p[s] + \lambda p[s] + \nu p[s]A[s])~ds}.\end{aligned}$$ Differentiating with respect to $p[s]$ and setting the derivate to zero gives us$$\int_{s\in{\mathcal{S}}'}{(\log p[s] + 1 + \lambda + \nu \bar{A}[s])~ds} = 0,$$ which gives us an expression for $p[s]$ of the form $$p[s] = \frac{1}{Z}~e^{-\nu \bar{A}[s]}. \label{eqn:solve}$$ Differentiating with respect to $\lambda$, $Z = 1/e^{-1 - \lambda}$ which gives us, $$Z = \int_{s\in{\mathcal{S}}'}{e^{-\nu \bar{A}[s]}~ds}. \label{eqn:partition}$$ Similarly, we can determine $\nu$ by solving the equation $$-\frac{\partial}{\partial\nu}~\log Z = \bar{A}. \label{eqn:constt}$$ Now, note that Z, as defined in (\[eqn:partition\]) has the structure of a partition function, while $\nu$ plays the role of an intensive variable (does not depend on the cardinality of ${\mathcal{S}}'$). The physical interpretation of this partition function may be inferred from the context of the domain of the physical system. In the case of classical statistical ensembles, we specify the *history* of the system in a configuration $s$ to simply be the energy in that particular state and take the intensive variable to be the familiar $k_B T$. By doing so, we recover Jaynes’ information-theoretic formulation of thermodynamics in the form of the well-known statistical canonical ensemble $$Z = \sum_{states}{e^{-\frac{1}{k_B T}E[state]}}.$$ As Lisi points out, the formal similarity of the quantum partition function to the statistical canonical ensemble implies that it can be derived along similar lines. Following Lisi [@lisi06], we interpret the notion of *history*, in the case of a quantum ensemble, as the action $S$ corresponding to a particular *path* as understood in Feynman’s path integral formulation. The intensive variable is taken as $\imath\hbar$. By doing so, we recover the quantum partition function[^7], $$Z = \sum_{paths}{e^{-\frac{1}{\imath\hbar}S[path]}}.$$ Thus, we see that using an entropic formulation of uncertainty about the possible outcomes of a physical system allows us to derive partition functions that, afforded reasonable physical interpretations, allow us to recover existing physical laws and explain the similarity in their formal structure with a deeper informational hypothesis. Extreme Physical Information ---------------------------- In Section \[sec:shannon\], we have seen that using Shannon entropy as a measure of uncertainty allows us to recover existing physical laws for macroscopic ensembles. Beginning from the observation that a large number of physical laws are phrased in the form of second order differential equations, Frieden [@fried04] has suggested the possibility of deriving these laws through a general framework called Extreme Physical Information (EPI) which operates on a principle of Fisher Information extremization. In this Section, we examine the use of a Fisher Information based measure in our general informational framework and find that our approach replicates the basic mathematical structure of EPI without having to make any of its potentially questionable metaphysical assumptions[^8]. Recall that the standard definition of Fisher Information is of the form $$I = \int{\left(\frac{\partial\log p(y|{\boldsymbol{\theta}})}{\partial{\boldsymbol{\theta}}}\right)^2 p(y|{\boldsymbol{\theta}})~dy},$$ where $p(y|{\boldsymbol{\theta}})$ is some interpretation (we assume frequentist) of the probability of $y$ conditioned on ${\boldsymbol{\theta}}$. Fisher Information measures the amount of information that the random variable $y$ carries about the unknown parameter ${\boldsymbol{\theta}}$. In the context of physical systems, we take $y$ to represent the value of physical measurements and $\theta$ to be the *true*[^9] value of the physical quantity being measured. Geometrically speaking, the informativeness in this context is measured as a function of the steepness of the probability density function near the maximum likelihood estimate of the unknown parameter. Thus, unlike Shannon entropy, using Fisher Information as an uncertainty measure allows us to take local structure in the definition of the support of the space of outcomes ${\mathcal{S}}$ into account. Finally, note that since Fisher Information, like Kullback-Leibler divergence, measures informativeness in comparison to an ideal theory, finding a good physical theory in this estimation framework is equivalent to minimizing Fisher Information over the relevant domain. Without having to resort to an observer-Nature information game (observer maximizes information, Nature maximizes error), as posited in, e.g., [@fried04; @tops08], we recover the basic mathematical principle of EPI. In general, the unknown parameter ${\boldsymbol{\theta}}$ as well as the measurement ${\mathbf{y}}$ may be a $V$-dimensional vector, e.g., the position of a point particle in $\mathbb{R}^V$. In that case we obtain a $V\times V$ Fisher Information matrix. In physical measurements, the dimensions of the unknown parameter will often be orthogonal in the sense that the MLEs will be independent. Thus, the off-diagonal elements of the Fisher Information matrix will be zero, and the trace will be a comprehensive measure of the informativeness of the estimation. It may also be necessary to account for the effect of multiple observations ${\mathbf{y}}^{(n)}: n = 1\cdots N$ on the informativeness of the estimation procedure for the unknown parameter ${\boldsymbol{\theta}}_v^{(n)}$ at the $n^{th}$ instance. Following Frieden, the sum of the traces of the $V\times V$ matrix corresponding to each observation can be shown to be an upper bound on the Stam information of the system and is thus a bound on the capacity of the estimation procedure to convey information about the measured quantity. Also, from the assumption of independence, the joint probability $p({\mathbf{y}}|{\boldsymbol{\theta}})$ decomposes into a product of marginals $\prod_{n=1}^{N}{p_n({\mathbf{y}}^{(n)}|{\boldsymbol{\theta}}^{(n)})}$, resulting in the following simplification, $$\begin{aligned} F &= \sum_{n=1}^{N}\int{p(y|\theta)\sum_{v=1}^{V}\left(\frac{\partial\log p({\mathbf{y}}|{\boldsymbol{\theta}})}{\partial{\boldsymbol{\theta}}_{v}^{(n)}}\right)^2~d{\mathbf{y}}^{(n)}},\\ &= \sum_{n=1}^{N}\int{\frac{1}{p_n}\sum_{v=1}^{V}\left(\frac{\partial p_n}{\partial{\boldsymbol{\theta}}_v^{(n)}}\right)^2~d{\mathbf{y}}^{(n)}.}\end{aligned}$$ Finally, we assume shift invariance for the measurement process, i.e., the error in measurement ${\mathbf{x}}^{(n)} = {\mathbf{y}}^{(n)} - \theta^{(n)}$ is independent of the value of ${\boldsymbol{\theta}}$. Then, $p_n({\mathbf{y}}^{(n)}|{\boldsymbol{\theta}}^{(n)}) = p_n({\mathbf{x}}_n)$. To recapitulate, we have shown that it is possible to define a scalar information measure based on Fisher information that depends entirely on the fluctuations in the measurement process. By replacing the probability density $p_n$ with the ‘real’ probability amplitude $q_n$ such that $p_n = q_n^2$ and suppressing the index $v$ in the notation (assuming measurements are made with no preferred dimension), we get $$F = 4\int{\sum_{n}\nabla q_n\cdot\nabla q_n~d{\mathbf{x}}_n}, \label{eqn:frieden}$$ which is equivalent to (\[eqn:fisher\]) with some additional physical interpretation of the outcome space ${\mathcal{S}}$ attached. Specifically, whereas (\[eqn:fisher\]) measures informativeness in a general space of outcomes $s\in{\mathcal{S}}$, (\[eqn:frieden\]) measures informativeness corresponding to a physical measurement process where the form and the physical interpretation of the outcome ${\mathbf{x}}= {\mathbf{y}}- {\boldsymbol{\theta}}$ is well-specified. Since physical laws dealing with idealized measurements are largely concerned with this specific class of outcomes, we use (\[eqn:frieden\]) in the remainder of this Section. Using the principle of informational physics, deriving a physical law using (\[eqn:frieden\]) is equivalent to minimizing the information measure $F$ under constraints on the process of measurement. We formulate these constraints following our treatment of the Shannon entropy case in Section \[sec:shannon\]. Since the information measure $F$ has no outcome-specific probabilistic interpretation, we will not have a density function normalization constraint analogous to (\[eqn:exist\]). Analogous to (\[eqn:measure\]), however, we assume here that the observer, given multiple observations, *believes-he-knows* how the system has been behaving. In this context, this translates into the assumption that the observer’s empirical estimate of the expected *history* of all possible outcomes of the system is accurate, i.e., $$\bar{A} = \langle A\rangle = \int{A({\mathbf{x}}) p({\mathbf{x}}) d{\mathbf{x}}}.$$ Thus, the informational quantity to be minimized takes the form, $$F' = 4\int{\sum_{n}\nabla q_n\cdot\nabla q_n~d{\mathbf{x}}_n} + \lambda\left(\bar{A} - \int{A({\mathbf{x}}) p({\mathbf{x}}) d{\mathbf{x}}}\right), \label{eqn:epi}$$ which is structurally equivalent to the $K = I - J$ Lagrangian of EPI [@fried04]. Comparing this derivation of the EPI Lagrangian with that described in [@fried90] in the specific context of the derivation of the time-independent Schrodinger’s equation and subsequently explicated in [@fried04] to encompass a general framework that results in information-theoretic derivations of the Dirac, Klein-Gordon and Maxwell’s equations, we see that our informational framework not only subsumes the mathematical structure of EPI, but also motivates it in a far more convincing manner by eliminating the need for defining a bound information functional [@fried90] as well as the requirement of taking a Fourier transform of the original information measure [@fried04] to do so. Thus, we find that we have removed a significant number of the *ad hoc* procedural assumptions of EPI while recovering its mathematical structure as a special representational implementation of a more general informational physics framework. As an example of the application of the EPI method to deriving physical laws, we can briefly interpret Frieden’s derivation of Schrodinger’s equation (in one-dimension) starting from (\[eqn:epi\]). The history $\bar{A}$ is assigned a physical interpretation of kinetic energy of the particle (potential energy would appear as a uniform background energy to an observer concerned solely with measuring the position of the particle). Recall that Schrodinger’s equation measures probability using complex probability amplitudes. These may be constructed from real probability amplitudes $q_n(x)$ $$\psi_{n} = \frac{1}{\sqrt{N}}(q_{2n-1} + \imath q_{2n}), n = 1\cdots N/2,$$ with no loss of generality. Since the kinetic energy over N measurements is being considered, we work with the quantity $N\langle E_{kin}\rangle$. Following [@fried04], the quantity $\langle E_{kin}\rangle$ can be expressed as an expectation in terms of the probability distribution $ p(x) = \sum_n \psi_n^*\psi_n$. Substituting in (\[eqn:epi\]) and working with the single-dimension case for simplicity of analysis, we obtain $$\begin{aligned} F' &= 4N\int{\sum_n\left|\frac{d\psi_n(x)}{dx}\right|^2~dx} + \lambda\left(\bar{E}_{kin} - C\int{E_{kin}\sum_n |\psi_n(x)|^2~dx}\right),\\ &= 4N\int{\sum_n\left|\frac{d\psi_n(x)}{dx}\right|^2~dx} - \lambda C\int{[W - V(x)]\sum_n |\psi_n(x)|^2~dx} + \lambda\bar{E}_{kin}.\end{aligned}$$ Applying the Euler-Lagrange equation $$\frac{d}{dx}~\frac{\partial \mathcal{L}}{\partial\psi'} = \frac{\partial\mathcal{L}}{\partial\psi},$$ with $\mathcal{L} = F'$, we get the solution, $$\psi_n^{''}(x) + \frac{\lambda C}{4}[W - V(x)]\psi_n(x) = 0, \quad n = 1,\cdots N/2. \label{eqn:schrodinger}$$ Setting $\lambda C = \frac{8m}{\hbar^2}$ recovers a physically meaningful Schrodinger’s equation without time dependence. The general EPI approach has been used in several physical, biological and economics applications over the past decade with varying degrees of success. However, it has not acquired mainstream acceptance as a consequence of the arbitrary formulaic manner in which it has been presented and the *ad hoc* metaphysical assumptions required to phrase it. It is hoped that the derivation of the formal structure of EPI with our sparse set of assumptions might occasion a re-evaluation of its utility. Discussion {#sec:discuss} ========== Here, we summarize and discuss all the assumptions that we make in the process of deriving physical laws from the assumption of the ontological primacy of information (subjectively, awareness). - [**Complete system description:** In the first instance, deriving physical laws appears to require that $s\in{\mathcal{S}}\Rightarrow s\in S'$, since the probability assignments $p[s]$ to outcomes $s$ and all subsequent calculations are likely to be flawed irretrievably otherwise. This distinguishes physical laws from general statistical inference where ${\mathcal{S}}'\subset{\mathcal{S}}$. It must be clarified here that the use of the term *complete* to refer to the specification of set ${\mathcal{S}}$ is a slight misnomer, since we appear to be attaching ontological reality to ${\mathcal{S}}$ independent of the observer. This is partially true. While we could redress this concern by allowing a superset of ${\mathcal{S}}$ to replace ${\mathcal{S}}$ (with ${\mathcal{S}}$ becoming yet another ${\mathcal{S}}'$ in the process) whenever new information about the space of outcomes arises, such an approach allows for an unintuitive infinite regress. It is felt that it is simpler to allow ${\mathcal{S}}$ to have some ontological reality by considering it to be the space of outcomes observable to a universal observer with (or holding the potential for)*complete* knowledge. The ontological realism of this observer itself is a separate (and philosophically important) question, but one that need not be addressed for the mathematical consistency of our framework.]{} - [**Bayesian inference:** We have assumed that observers update their beliefs about the possibility of various outcomes occurring by observing the system’s behavior over multiple instances. While any other system of inference would be equally acceptable, for our demonstration of the emergence of physical laws, we have selected Bayesian inference as the specific model. It would be very surprising, in the author’s opinion, were another consistent system of inference to result in results radically different from the Bayesian approach. Thus, we feel that it is sufficiently general to require no additional justification.]{} - [**Agnosticism about outcomes:**]{} Since we have worked throughout our exposition with either the MaxEnt specialization of Bayesian inference or an equivalent Fisher Information minimization, we have assumed that observers do not assume any knowledge unjustified by observations. - [**Existence assumption:** In the case where individual probabilities can be assigned to system outcomes in a physically interpretable manner (Section \[sec:shannon\]) we have assumed that observers constructing physical laws believe they know of all possible outcomes that constitute the domain of the system.]{} - [**Measurements are accurate on average:** In all cases, we have assumed that observers believe they have estimated the behavior of the system accurately through a series of measurements upon various outcomes of the system.]{} - [**Physical constants:** Where appropriate, we have substituted physically meaningful quantities for scaling parameters and variables (e.g. Boltzmann’s constant, Planck’s constant). We do not claim any *a priori* reason for these substitutions, but suggest that their introduction in non-informational physics derivations tend to be no less arbitrary. Unfortunately, the informational physics framework does not appear to be any closer than standard formulations in solving the *fine-tuning*[^10] problem.]{} Subjectively speaking, this set of assumptions appears to be quite reasonable, sparse enough to allow the informational physics principle to be enunciated, yet not so sparse as to require suspension of disbelief in order to accept the principle’s implications. In plain language, the informational physics principle, in view of these assumptions, is suggesting the following: take a set of observers with complete knowledge of the range of behavior of a phenomenon under observation and give them sufficient observations to estimate some quantity that summarizes the past behavior of all outcomes of the system. While trying to infer a concise statement that would allow them to (a) summarize their findings and (b) make predictions about the system’s behavior, they will all find the same mathematical structures as statements of physical laws, a circumstance that emerges from the nature of the process of observation and no ontological reality apart from the observer’s perception. Future prospects {#sec:prospects} ---------------- Much of the circumspection that arises with respect to informational physics derives from the insubstantial and intangible nature of its constituent categories. That is, it is far simpler to think in terms of electrons than bits, since one can measure the former’s effects in the physical realm. Almost definitionally, the material (dualistic) definition of physics is far more amenable to reductionist manipulation than the informational (non-dualistic) definition. It is almost certain that this will remain a critical bottleneck to the emergence of informational physics as a mainstream discipline. However, we contend that demonstrating the consistency and duality of informational physics with respect to traditional physics is still not an enterprise without value. We examine and motivate future efforts along this line of inquiry in the following three contexts: - [**Practical implications:**]{} From the practical point of view, the informational view of physics argues for a reappraisal of developments in approximation techniques in the light of the existence and continual development of statistical learning algorithms for different domain representations in the machine learning community. At the very least, the inference method can guide selection of the appropriate class of analytical methods appropriate for a particular problem domain. The first component of using the informational framework in physically realistic setting is *domain specification*, which is equivalent to specifying an interesting and non-trivial domain of outcomes ${\mathcal{S}}$ of corresponding to physical measurements (for either single particle or ensemble settings). The critical element in performing inference, once the domain ${\mathcal{S}}$ has been specified, is the imposition of an appropriate measure to generate meaningful $p[s]$ corresponding to $s\in{\mathcal{S}}$, which should be a problem of some interest to measure theoreticians. The rest of the analysis can proceed using well known methods in both the statistics and computational learning literature. - [**Theoretical implications:**]{} Much has been made of the necessity of reaching a deeper understanding of the seemingly counter-intuitive foundations of quantum mechanics. Furthermore, in recent years, there has been a backlash in some segments of the theoretical physics community against the predominance of mathematical manipulation detached from reality. Informational physics carries the potential of remedying both these problems, as well as scope for a unification framework potentially more aesthetically satisfying than any action string theory might hope to derive. In the case of quantum mechanics, the irreconcilability of local realism with quantum predictions is trivially resolved by allowing quantum histories to emerge subjectively (an interpretation which is compatible with relational quantum mechanics [@rove96]). Furthermore, observer-dependent wave function collapse can now be interpreted as an update of outcome probabilities based on information acquisition. The mathematics of the informational framework of physics proceeds from well-understood and motivated assumptions regarding the nature of reality, which places it favorably in contrast to some more mathematically sophisticated but less simply interpretable systems. - [**Epistemological implications:**]{} As our final point, we ask the following related questions: what are the implications of an informational basis of reality as well as knowledge about reality (the laws of physics being a subset consisting of statements regarding ideal scenarios) on (a) the prospect of unification, (b) the prospect of understanding the nature of physical laws and (c) the prospect of understanding the nature of the comprehensibility and predictability of reality? Considering these questions one by one, it is not evident that the informational framework offers much greater hope for unification in the traditional sense of the term. While the informational framework is extremely general, the interpretability of its conclusions in the physical domain cannot occur arbitrarily. Should advances in *domain specification* be feasible, the mathematical power of the informational framework will exceed that of the standard system. Until then, unification will continue to be equally inaccessible to either paradigm, though somewhat less philosophically necessary for the former than the latter. Regarding the prospect of understanding the nature of physical laws, the informational framework claims a nearly complete resolution of this problem by showing that they emerge as special cases of a general inference procedure. The substitution of variables with specific values however, leaves it open to the fine-tuning criticism, as addressed earlier. The substitution of variables with quantities that appear to have specific physical interpretations leaves the framework open to the criticism of non-falsifiability. Simply put, the informational argument can retrieve existing laws and potentially find epistemologically-motivated approximations for them, but it cannot find completely new laws, since the variables that emerge would have no defined physical interpretation. Since it becomes impossible to construct a scenario where the informational framework constructs a law that is ‘false’, the problem of non-falsifiability arises. This is a strong argument, and one that we are not in a position to refute at the moment. It should, therefore, be borne in mind while considering the plausibility of the informational physics hypothesis. To answer our final question, we consider the promise that informational physics holds in answering Wigner’s question [@wign60]. An assumption that reality is foundationally based in information (subjectively, awareness) dovetails quite well with the evidence we have presented in favor of the hypothesis that knowledge-acquisition about system outcomes is the foundation of physical laws. Since the informational physics principle declaims against the ontological reality of observer-independent systems, it explains Wigner’s observation regarding the unreasonable effectiveness of mathematics as a descriptor of physical reality in the following way: observers construct laws of physics based on their information extremization apparatus (calculus of variations etc.). All observers who make the assumptions described in Section \[sec:discuss\] will discover the same mathematical (and cognitive) statements as physical laws. This will make the relevant mathematics appear to be unreasonably effective in describing what all the observers *believe-they-know*[^11]. Thus, notwithstanding the occasional ambiguity that must necessarily ensue in treating with slippery concepts like information and awareness while attempting to describe reality, there appear to be strong physical, mathematical and epistemological reasons for further investigating the informational physics hypothesis. Establishing the validity of this hypothesis is a necessary step in facilitating a meaningful discussion of the broader consideration of awareness being the primal ontological entity. [30]{} Akaike, Hirotugu (1974). “A new look at the statistical model identification”. IEEE Transactions on Automatic Control 19 (6): 716Ð723 Burnham, K. P. and Anderson D. R. 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Second Edn. Caticha A., Giffin A., 2006. Updating probabilities. MaxEnt 2006, 26th International Workshop on Bayesian Inference and Maximum Entropy Methods. Frieden, B.R., 1990, Fisher information, disorder, and the equilibrium distributions of physics. Phys. Rev. A 41, 4265 - 4276 Frieden, B.R., 1991. Fisher information and the complex nature of the Schršdinger wave equation. Found. Phy. Vol 21 Issue 7, 757-771 Frieden, B.R., 2nd edn. “Science from Fisher Information” (Cambridge Univ. Press, 2004). GrŸnwald P., 2005. “A Tutorial Introduction to the Minimum Description Length Principle”. Advances in Minimum Description Length: Theory and Applications, MIT Press. Jaynes, E. T., 1963, ‘Information Theory and Statistical Mechanics,’ (905Kb) in Statistical Physics, K. Ford (ed.), Benjamin, New York, p. 181 Lisi, A.G., 2006. Quantum mechanics from a universal action reservoir. arXiv:physics/0605068 C Rovelli, 1996. ÒRelational Quantum MechanicsÓ, International Journal of Theoretical Physics, 35, 1637 John E. Shore, Rodney W. Johnson, 1981. Properties of cross-entropy minimization. IEEE Transactions on Information Theory 27(4): 472-482 Skala L., Kapsa V., 2005. From probabilities to quantum and classical mechanics, Physica E 29:1-2, 119-128. Tenenbaum J. B., Griffiths T.L., 2001. Generalization, similarity, and Bayesian inference. Behavioral and Brain Sciences, 24 pp. 629-641 Topsoe F., 2008, “On the generation of measures of entropy, divergence and complexity”. 2nd International Conference on Entropy, EPFL, Lausanne. Wolfram, Stephen, 2002. A New Kind of Science, Wolfram Media, Inc. Wheeler, J. A. 1999. Information, physics, quantum: the search for links. In Feynman and Computation: Exploring the Limits of Computers, A. J. Hey, Ed. Perseus Books, Cambridge, MA, 309-336. E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Comm. in Pure and Appl. Math 13 (Wiley, N.Y., 1960) Zuse, K., 1970. Calculating Space, MIT Technical Translation AZT-70-164-GEMIT, Massachusetts Institute of Technology (Project MAC), Cambridge, Mass. 02139 [^1]: In the Western traditions, see e.g., Neoplatonism, Kant’s Transcendental Idealism and Berkeley’s Immaterialism. *Advaita* and *Zen* are good examples from the East. Sufism and Gnosticism are good representatives from the Semitic traditions. [^2]: This is equivalent to saying that no meaningful discussion of a coin-tossing experiment can commence unless the subject knows it has two outcomes. He need not know that the two outcomes occur with equal probability. [^3]: As a quick visit to the nearest roulette table at one’s friendly neighborhood casino would readily demonstrate! [^4]: The problem of distinguishing between *believe-he-knows* and *knows* is one that we will not address in this paper. [^5]: This notion can be formalized in terms of compactness, predictability, some combination thereof or in some other way. [^6]: We feel, however, that such specific efforts may not be very useful in practice. A more general view of information, on the other hand, would lack the mathematical rigor of our construction. [^7]: Since it is possible to subsume all possible quantum actions in the informational framework we have discussed in Section \[sec:inf\], it is tempting to declare victory for informational physics here. However, since we do not have a characterization of the class of actions that have physical interpretations, we are far from being in a position to do so. [^8]: In particular, we remove the necessity of an observer-Nature information game, the concept of ‘bound’ information and the necessity of Fourier transforms as a descriptor of symmetry [^9]: In the informational context, ‘true’ is equivalent to the value predicted by the best theory. [^10]: Why do the universal physical constants have the values they do? [^11]: Drawing a parallel to Plato’s Theory of Forms is irresistible!

--- abstract: 'In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by Poisson noise. A proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution and comparison to prior methods are also reported.' bibliography: - 'references.bib' title: 'Inverse Problems with Poisson noise: Primal and Primal-Dual Splitting[^1]' --- Inverse Problems, Poisson noise, Duality, Proximity operator, Sparsity. Introduction {#sec:intro} ============ Linear inverse problems in presence of Poisson noise have attracted less interest in the literature than their Gaussian counterpart, presumably because the noise properties are more complicated to handle. Such inverse problems have however important applications in imaging such as restoration (e.g. deconvolution in medical and astronomical imaging), or reconstruction (e.g. computerized tomography). For instance, the well-known Richardson-Lucy has been proposed for deconvolution. The RL algorithm, however, amplifies noise after a few iterations, which can be avoided by introducing regularization. In [@Dey2004], the authors presented a Total Variation (TV)-regularized RL algorithm, and [@Starck2006] advocated a wavelet-regularized RL algorithm. In the context of Poisson linear inverse problems using sparsity-promoting regularization, a few recent algorithms have been proposed. For example, [@Dupe2009c] stabilize the noise and proposed a family of nested schemes relying upon proximal splitting algorithms (Forward-Backward and Douglas-Rachford) to solve the corresponding optimization problem. The work of [@Chaux2009] is in the same vein. However, nested algorithms are time-consuming since they necessitate to sub-iterate. Using the augmented Lagrangian method with the alternating method of multipliers algorithm (ADMM), which is nothing but the Douglas-Rachford splitting applied to the Fenchel-Rockafellar dual problem, [@Figueiredo2010] presented a deconvolution algorithm with TV and sparsity regularization. This scheme however necessitates to solve a least-square problem which can be done explicitly only in some cases. In this paper, we propose a framework for solving linear inverse problems when the observations are corrupted by Poisson noise. In order to form the data fidelity term, we take the exact Poisson likelihood. As a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of atoms. The solution to the inverse problem is cast as the minimization of a non-smooth convex functional, for which we prove well-posedness of the optimization problem, characterize the corresponding minimizers, and solve them by means of primal and primal-dual proximal splitting algorithms originating from the realm of non-smooth convex optimization theory. Convergence of the algorithms is also shown. Experimental results and comparison to other algorithms on deconvolution are finally conducted. Notation and terminology {#sec:notation .unnumbered} ------------------------ Let $\Hc$ a real Hilbert space, here a finite dimensional vector subspace of $\mathbb{R}^n$. We denote by $\norm{.}$ the norm associated with the inner product in $\Hc$, and $\Id$ is the identity operator on $\Hc$. $\norm{.}_p, p \geq 1$ is the $\ell_p$ norm. $\vx$ and $\va$ are respectively reordered vectors of image samples and transform coefficients. We denote by $\ri \Cc$ the relative interior of a convex set $\Cc$. A real-valued function $f$ is coercive, if $\lim_{\norm{\vx} \to +\infty}f\parenth{\vx}=+\infty$, and is proper if its domain is non-empty $\dom f = \{ x\in\Hc \mid f(x) < +\infty \} \neq \emptyset$. $\Gamma_0(\Hc)$ is the class of all proper lower semicontinuous (lsc) convex functions from $\Hc$ to $(-\infty,+\infty]$. We denote by $\opnorm{\mathbf{M}}= \max_{\vx \neq 0} \frac{\norm{\mathbf{M}\vx}}{\norm{\vx}}$ the spectral norm of the linear operator $\mathbf{M}$, and $\kerm(\mathbf{M}):=\{x \in \Hc: \mathbf{M}x=0, x \neq 0\}$ its kernel. Let $x \in \Hc$ be an $\sqrt{n}\times\sqrt{n}$ image. $x$ can be written as the superposition of elementary atoms $\varphi_\gamma$ parameterized by $\gamma \in \mathcal{I}$ such that $x = \sum_{\gamma \in \mathcal{I}} \alpha_\gamma \varphi_\gamma = \Phb \va,\quad \abs{\mathcal{I}} = L, ~ L\ge n$. We denote by $\Phb: \Hc' \to \Hc$ the dictionary (typically a frame of $\Hc$), whose columns are the atoms all normalized to a unit $\ell_2$-norm Problem statement {#sec:problem-statement} ================= Consider the image formation model where an input image of $n$ pixels $\vx$ is indirectly observed through the action of a bounded linear operator $\Hmb: \Hc \to \Kc$, and contaminated by Poisson noise. The observed image is then a discrete collection of counts $\vy=(\vy[i])_{1 \le i \le n}$ which are bounded, i.e. $\vy \in \ell_{\infty}$. Each count $y[i]$ is a realization of an independent Poisson random variable with a mean $(\Hmb \vx)_i$. Formally, this writes in a vector form as [$$\label{eq:2} \vy \sim \Pc(\Hmb \vx)~.$$ ]{} The linear inverse problem at hand is to reconstruct $\vx$ from the observed count image $\vy$. A natural way to attack this problem would be to adopt a maximum a posteriori (MAP) bayesian framework with an appropriate likelihood function the distribution of the observed data $\vy$ given an original $\vx$ reflecting the Poisson statistics of the noise. As a prior, the image is supposed to be economically (sparsely) represented in a pre-chosen dictionary $\Phb$ as measured by a sparsity-promoting penalty $\Psi$ supposed throughout to be convex but non-smooth, e.g. the $\ell_1$ norm. From the probability density function of a Poisson random variable, the likelihood writes: [$$\label{eq:7} p(y|x) = \prod_i \frac{((\Hmb x)[i])^{y[i]} \exp\left(-(\Hmb x)[i]\right)}{y[i]!}~.$$]{} Taking the negative log-likelihood, we arrive at the following data fidelity term: [$$\begin{aligned} \label{eq:9} f_1\ &: \eta \in \mathbb{R}^n \mapsto \sum_{i=1}^n f_{\mathrm{poisson}}(\eta[i]), \\ \text{if } y[i] > 0,\quad f_{\mathrm{poisson}}(\eta[i]) &= \begin{cases} -y[i] \log(\eta[i]) + \eta[i] & \text{if } \eta[i] > 0,\\ +\infty & \text{otherwise,} \end{cases} \nonumber \\ \text{if } y[i] = 0,\quad f_{\mathrm{poisson}}(\eta[i]) &= \begin{cases} \eta[i] & \text{if } \eta[i] \in [0,+\infty), \\ +\infty & \text{otherwise.} \end{cases} \nonumber\end{aligned}$$ ]{} Our aim is then to solve the following optimization problems, under a synthesis-type sparsity prior[^2], [$$\label{eq:11} \begin{gathered} \tag{$\Pm_{\eqi,\psi}$} \argmin_{\va\in\Hc'} J(\va), \\ J\ :\ \va \mapsto {f_1\circ\Hmb\circ\Phb(\va)} + \eqi \Psi(\va) + \imath_{\Cc} \circ\Phb(\va)~. \end{gathered}$$ ]{} The penalty function $\Psi : \va \mapsto \sum_{i=0}^{L} \psi_i(\va[i])$ is positive, additive, and chosen to enforce sparsity, $\eqi > 0$ is a regularization parameter and $\imath_{\Cc}$ is the indicator function of the convex set $\Cc$. In our case, $\Cc$ is the positive orthant since we are fitting Poisson intensities, which are positive by nature. From the objective in , we get the following, \[prop:objectives\] [ ]{}\ (i) $f_1$ is a convex function and so are $f_1 \circ \Hmb$ and $f_1\circ\Hmb\circ\Phb$. (ii) $f_1$ is strictly convex if $\forall i \in \{1,\ldots,n\}, y[i] \ne 0$. $f_1\circ\Hmb\circ\Phb$ remains strictly convex if $\Phb$ is an orthobasis and $\kerm(\Hmb) = \emptyset$. (iii) Suppose that $(0,+\infty) \cap \Hmb\left([0,+\infty)\right) \neq \emptyset$. Then $J \in \Gamma_0(\Hc)$. Well-posedness of {#sec:char-solut} ------------------ Let $\Mc$ be the set of minimizers of problem . Suppose that $\Psi$ is coercive. Thus $J$ is coercive. Therefore, the following holds: [ ]{}\ (i) Existence: has at least one solution, i.e. $\Mc\ne\emptyset$. (ii) Uniqueness: has a unique solution if $\Psi$ is strictly convex, or under (ii) of Proposition \[prop:objectives\]. Iterative Minimization Algorithms {#sec:sparse-iter-deconv} ================================= Proximal calculus ----------------- We are now ready to describe the proximal splitting algorithms to solve . At the heart of the splitting framework is the notion of proximity operator. \[def:1\] Let $F \in \Gamma_{0}(\Hc)$. Then, for every $x\in\Hc$, the function $y \mapsto F(y) + \norm{x-y}^{2}/2$ achieves its infimum at a unique point denoted by $\prox_{F}x$. The operator $\prox_{F} : \Hc \to \Hc$ thus defined is the *proximity operator* of $F$. Then, the proximity operator of the indicator function of a convex set is merely its orthogonal projector. One important property of this operator is the separability property: \[lem:decomp\] Let $F_k \in \Gamma_0(\Hc),\ k \in \{1,\cdots,K\}$ and let $G : (x_k)_{1\le k\le K} \mapsto \sum_k F_k(x_k)$. Then $\prox_{G} = (\prox_{F_k}) _{1 \le k \le K}$. The following result can be proved easily by solving the proximal optimization problem in Definition \[def:1\] with $f_1$ as defined in , see also [@Combettes2007a]. \[lem:prpois\] Let $y$ be the count map (i.e. the observations), the proximity operator associated to $f_1$ (i.e. the Poisson anti log-likelihood) is, $$\label{eq:3} \prox_{\beta f_1} \vx = \left( \frac{\vx[i] - \beta + \sqrt{(\vx[i] -\beta)^2 + 4\beta \vy[i]}}{2} \right)_{1\le i \le n}~.$$ We now turn to $\prox_{\eqi\Psi}$ which is given by Lemma \[lem:decomp\] and the following result: \[th:3\] Suppose that $\forall~ i$: (i) $\psi_i$ is convex even-symmetric, non-negative and non-decreasing on $\mathbb{R}^+$, and $\psi_i(0)=0$; (ii) $\psi_i$ is twice differentiable on $\mathbb{R}\setminus \{0\}$; (iii) $\psi_i$ is continuous on $\mathbb{R}$, and admits a positive right derivative at zero ${\psi_i^{'}}_+(0) = \lim_{h\to 0^+} \frac{\psi_i(h)}{h} > 0$. Then, the proximity operator $\prox_{\delta\psi_i}(\beta) = \hat{\va}(\beta)$ has exactly one continuous solution decoupled in each coordinate $\beta[i]$ : $$\label{eq:10} \hat{\va}[i] = \begin{cases} 0 & \text{if } \abs{\beta[i]} \le \delta{\psi_i^{'}}_+(0)\\ \beta_i-\delta\psi_i^{'}(\hat{\va}[i]) & \text{if } \abs{\beta[i]} > \delta{\psi_i^{'}}_+(0) \end{cases}$$ Among the most popular penalty functions $\psi_i$ satisfying the above requirements, we have $\psi_i(\va[i]) = \abs{\va[i]}, \forall ~ i$, in which case the associated proximity operator is soft-thresholding, denoted $\mathrm{ST}$ in the sequel. Splitting on the primal problem {#sec:primal-method} ------------------------------- ### Splitting for sums of convex functions Suppose that the objective to be minimized can be expressed as the sum of $K$ functions in $\Gamma_0(\Hc)$, verifying domain qualification conditions: [$$\label{eq:sum} \argmin_{x \in \Hc} ~ \left(F(x) = \sum_{k=1}^K F_k(x)\right)~.$$ ]{} Proximal splitting methods for solving are iterative algorithms which may evaluate the individual proximity operators $\prox_{F_k}$, supposed to have an explicit convenient structure, but never proximity operators of sums of the $F_k$. Splitting algorithms have an extensive literature since the 1970’s, where the case $K=2$ predominates. Usually, splitting algorithms handling $K > 2$ have either explicitly or implicitly relied on reduction of to the case $K = 2$ in the product space $\Hc^K$. For instance, applying the Douglas-Rachford splitting to the reduced form produces Spingarn’s method, which performs independent proximal steps on each $F_k$, and then computes the next iterate by essentially averaging the individual proximity operators. The scheme described in [@Combettes2008] is very similar in spirit to Spingarn’s method, with some refinements. [**[Parameters:]{}**]{} The observed image counts $y$, the dictionary $\Phb$, number of iterations $N_{\mathrm{iter}}$, $\mu > 0$ and regularization parameter $\eqi > 0$.\ [**[Initialization:]{}**]{}\ $\forall i \in \{1,2,3\},\quad p_{(0,i)} = (0,0,0)^\mathrm{T}$. $z_0 = (0,0,0)^\mathrm{T}$.\ [**[Main iteration:]{}**]{}\ [**[For]{}**]{} $t=0$ [**[to]{}**]{} $N_{\mathrm{iter}}-1$, - (Lemma \[lem:prpois\]): $\xi_{(t,1)}[1] = \prox_{\mu f_1/3}(p_{(t,1)}[1])$. - (Lemma \[th:3\]): $\xi_{(t,1)}[2] = \prox_{\mu \eqi\Psi/3}(p_{(t,1)}[2])$. - : $\xi_{(t,1)}[3] = \Prj_{{\Cc}}(p_{(t,1)}[3])$. - (Lemma \[th:prjli\]): $\xi_{(t,2)} = \Prj_{{\ker \Lmb_1}}(p_{(t,2)}), \xi_{(t,3)} = \Prj_{{\ker \Lmb_2}}(p_{(t,3)})$. - Average the proximity operators: $\xi_{t} = (\xi_{(t,1)} + \xi_{(t,2)} + \xi_{(t,3)})/3$. - Choose $\theta_t\in]0,2[$. - Update the components: $\forall i \in \{1,2,3\},\quad p_{(t+1,i)} = p_{(t,i)} + \theta_t (2\xi_{t} - z_{t} - \xi_{(t,i)})$. - Update the coefficients estimate: $z_{t+1} = z_t + \theta_t(\xi_t - z_t)$. [**[End main iteration]{}**]{}\ [**[Output:]{}**]{} Reconstructed image $x^{\star}=z_{N_{\mathrm{iter}}}[0]$. ### Application to Poisson noise inverse problems Problem is amenable to the form , by wisely introducing auxiliary variables. As involves two linear operators ($\Phb$ and $\Hmb$), we need two of them, that we define as $\vx_1 = \Phb\va$ and $\vx_2 = \Hmb\vx_1$. The idea is to get rid of the composition of $\Phb$ and $\Hmb$. Let the two linear operators $\Lmb_1 =[ \Id \quad 0 \quad -\Phb]$ and $\Lmb_2 = [ -\Hmb \quad \Id \quad 0]$. Then, the optimization problem can be equivalently written: [$$\begin{gathered} \label{eq:1} \argmin_{(\vx_1,\vx_2,\va) \in \Hc\times\Kc\times\Hc'} \underbrace{f_1(\vx_2) + \imath_{\Cc}(\vx_1) + \eqi\Psi(\va)}_{G(\vx_1,\vx_2,\va)} + \\ \imath_{\ker \Lmb_1}(\vx_1,\vx_2,\va) + \imath_{\ker \Lmb_2}(\vx_1,\vx_2,\va)~.\end{gathered}$$ ]{} Notice that in our case $K=3$ by virtue of separability of the proximity operator of $G$ in $x_1$, $x_2$ and $\alpha$; see Lemma \[lem:decomp\]. The proximity operators of $F$ and $\Psi$ are easily accessible through Lemma \[lem:prpois\] and \[th:3\]. The projector onto the positive orthant $\Cc$ is also trivial. It remains now to compute the projector on $\ker \Lmb_i$, $i=1,2$, which by well-known linear algebra arguments, is obtained from the projector onto the image of $\Lmb_i^*$. \[th:prjli\] The proximity operator associated to $\imath_{\ker \Lmb_i}$ is $$\begin{gathered} \label{eq:10a} \Prj_{\ker \Lmb_i} = \Id - \Lmb_i^* (\Lmb_i \circ \Lmb_i^*)^{-1} \Lmb_i~. \end{gathered}$$ The inverse in the expression of $\Prj_{\ker \Lmb_1}$ is $(\Id + \Phb\circ\Phi^\mathrm{T})^{-1}$ can be computed efficiently when $\Phb$ is a tight frame. Similarly, for $\Lmb_2$, the inverse writes $(\Id + \Hmb\circ\Hmb^*)^{-1}$, and its computation can be done in the domain where $\Hmb$ is diagonal; e.g. Fourier for convolution. Finally, the main steps of our primal scheme are summarized in Algorithm \[algo:deconv\]. Its convergence is a corollary of [@Combettes2008]\[Theorem 3.4\]. Let $(z_t)_{t\in\mathbb{N}}$ be a sequence generated by Algorithm \[algo:deconv\]. Suppose that Proposition \[prop:objectives\]-(iii) is verified, and $\sum_{t\in\mathbb{N}} \theta_t(2-\theta_t) = +\infty$. Then $(z_t)_{t\in\mathbb{N}}$ converges to a (non-strict) global minimizer of . Splitting on the dual: Primal-dual algorithm {#sec:primal-dual-method} -------------------------------------------- Our problem can also be rewritten in the form, [$$\begin{gathered} \label{eq:5} \argmin_{\va \in \Hc'} F\circ\Kmb(\va) + \eqi\Psi(\va)\end{gathered}$$ ]{} where now $\Kmb = \begin{pmatrix} \Hmb\circ\Phb\quad \\ \Phb \end{pmatrix}$ and $F : (\vx_1,\vx_2) \mapsto f_1(x_1) + \imath_{\Cc}(x_2)$. Again, one may notice that the proximity operator of $F$ can be directly computed using the separability in $\vx_1$ and $\vx_2$. Recently, a primal-dual scheme, which turns to be a pre-conditioned version of ADMM, to minimize objectives of the form was proposed in [@Chambolle2010]. Transposed to our setting, this scheme gives the steps summarized in Algorithm \[algo:deconv2\]. Adapting the arguments of [@Chambolle2010], convergence of the sequence $(\va_t)_{t\in\mathbb{N}}$ generated by Algorithm \[algo:deconv2\] is ensured. Suppose that Proposition \[prop:objectives\]-(iii) holds. Let $\zeta = \opnorm{\Phb}^2(1+\opnorm{\Hmb}^2)$, choose $\tau > 0$ and $\sigma$ such that $\sigma\tau\zeta < 1$, and let $(\va_t)_{t\in\mathbb{R}}$ as defined by Algorithm \[algo:deconv2\]. Then, $(\va)_{t\in\mathbb{N}}$ converges to a (non-strict) global minimizer at the rate $O(1/t)$ on the restricted duality gap. Discussion ---------- Algorithm \[algo:deconv\] and \[algo:deconv2\] share some similarities, but exhibit also important differences. For instance, the primal-dual algorithm enjoys a convergence rate that is not known for the primal algorithm. Furthermore, the latter necessitates two operator inversions that can only be done efficiently for some $\Phb$ and $\Hmb$, while the former involves only application of these linear operators and their adjoints. Consequently, Algorithm \[algo:deconv2\] can virtually handle any inverse problem with a bounded linear $\Hmb$. In case where the inverses can be done efficiently, e.g. deconvolution with a tight frame, both algorithms have comparable computational burden. In general, if other regularizations/constraints are imposed on the solution, in the form of additional proper lsc convex terms that would appear in , both algorithms still apply by introducing wisely chosen auxiliary variables. [**[Parameters:]{}**]{} The observed image counts $y$, the dictionary $\Phb$, number of iterations $N_{\mathrm{iter}}$, proximal steps $\sigma > 0$ and $\tau > 0$, and regularization parameter $\eqi > 0$.\ [**[Initialization:]{}**]{}\ $\va_0 = \bar{\va}_0 = 0$ $\xi_0 = \eta_0 = 0$.\ [**[Main iteration:]{}**]{}\ [**[For]{}**]{} $t=0$ [**[to]{}**]{} $N_{\mathrm{iter}}-1$, - (Lemma \[lem:prpois\]): $\xi_{t+1} = (\Id - \sigma \prox_{f_1/\sigma})(\xi_{t}/\sigma + \Hmb\circ\Phb \bar{\va}_{t})$. - : $\eta_{t+1} = (\Id - \sigma\Prj_{{\Cc}})(\eta_{t}/\sigma + \Phb\bar{\va}_{t})$. - (Lemma \[th:3\]): $\va_{t+1} = \prox_{\tau \eqi\Psi} \left(\va_t - \tau\Phb^\mathrm{T}\left(\Hmb^*\xi_{t+1} + \eta_{t+1}\right)\right)$. - Update the coefficients estimate: $\bar{\va}_{t+1} = 2\va_{t+1} - \va_t$ [**[End main iteration]{}**]{}\ [**[Output:]{}**]{} Reconstructed image $x^{\star}=\Phb\va_{N_{\mathrm{iter}}}$. Experimental results {#sec:results} ==================== Our algorithms were applied to deconvolution. In all experiments, $\Psi$ was the $\ell_1$-norm. Table \[tab:maesky\] summarizes the mean absolute error (MAE) and the execution times for an astronomical image, where the dictionary consisted of the wavelet transform and the PSF was that of the Hubble telescope. Our algorithms were compared to state-of-the-art alternatives in the literature. In summary, flexibility of our framework and the fact that Poisson noise was handled properly, demonstrate the capabilities of our approach, and allow our algorithms to compare very favorably with other competitors. The computational burden of our approaches is also among the lowest, typically faster than the PIDAL algorithm. Fig. \[fig:plot\] displays the objective as a function of the iteration number and time (in s). We can clearly see that Algorithm 2 converges faster than Algorithm 1. ![Objective function in function if iterations (left) and times (right).[]{data-label="fig:plot"}](plot-itera "fig:"){width="0.4\linewidth"} ![Objective function in function if iterations (left) and times (right).[]{data-label="fig:plot"}](plot-times "fig:"){width="0.4\linewidth"} Conclusion {#sec:conclusion} ========== In this paper, we proposed two provably convergent algorithms for solving the Poisson inverse problems with a sparsity prior. The primal-dual proximal splitting algorithm seems to perform better in terms of convergence speed than the primal one. Moreover, its computational burden is lower than most comparable of state-of-art methods. [^1]: Submitted to ICIP 2011 on the 01/21/11. [^2]: Our framework and algorithms extend to an analysis-type prior just as well, though we omit this for obvious space limitation reasons.

--- abstract: - 'The historical premise of adaptive control was to control uncertain systems, while simultaneously learning the system parameters and providing robustness to uncertainties. Rudolf Kalman was the first to coin the term “self-tuning controller” in 1958 by introducing optimal linear-quadratic regulator (LQR) with explicit identification of parameters. The field of adaptive control since then witnessed tremendous developments, capturing different classes of nonlinear systems, including presence of unmodeled dynamics, switching models, hybrid systems and other singularities. The main architectures were inspired by inverse Lyapunov design, ensuring asymptotic stability in the presence of system uncertainties and disturbances. Recent developments in $\mathcal L_1$ adaptive control filled the last gap of explicitly introducing robustness into the problem formulation, leading to a framework with apriori guaranteed robustness, transient and steady-state specifications. Despite these vast developments, the issue of learning the system dynamics and/or uncertainties remained unresolved, as the typical estimation schemes in all these adaptive architectures require persistency of excitation (PE) type assumption on reference signals to ensure parameter convergence. Such requirement is unacceptable in safety-critical applications, rendering the conventional Lyapunov-based adaptive control architectures incomplete, if parameter/system identification is to be addressed simultaneously with robustness and transient specifications. The data explosion of the last two decades and the spurge of developments in machine learning inspired new developments in adaptive control through integration of Gaussian processes for modeling system uncertainties. This paper explores $\mathcal L_1$ adaptive controller using Gaussian processes for modeling uncertainties. Gaussian processes provide a framework for sample-efficient learning and have been exhaustively explored in machine learning literature. Since $\mathcal L_1$ adaptive controller has guaranteed robustness and transient tracking, it ensures [*safe learning with less data*]{} within the pre-specified system’s state bounds. On the other hand, the learned dynamics has the potential to impact the robustness/performance trade-off in $\mathcal L_1$ adaptive controllers, leading to less conservative designs.' - author: - | \ Mechanical Science and Engineering, University of Illinois at Urbana Champaign, Urbana, IL-61801\ Mechanical Science and Engineering, University of Illinois at Urbana Champaign, Urbana, IL-61801 bibliography: - 'thoughts.bib' title: 'Bayesian $\mathcal{L}_1$ Adaptive Control ' --- List of keywords List of keywords Introduction ============ Robust adaptive control methods were historically developed to control systems in the presence of uncertainties. . On the other hand, the last two decades have witnessed the explosion of data that has revolutionized the industry of autonomous systems. Tools from machine learning have been used . In many of these instances, proofs of stability have not been prioritized, yet having an impressive demonstration was the main objective to show the power of data-driven approaches born within computer science . As the flying technologies prove their societal benefits and dare to penetrate into our everyday lives, stability and robustness of data-driven controlled systems take a center stage. The fundamental question to be answered is often reduced to safe learning with robustness guarantees. In this paper we explore the L1 adaptive control architecture with Bayesian learning. L1 adaptive controller was developed over the last decade with the objective of addressing robustness guarantees for a class of architectures with fast estimation rates . It has since led to powerful implementations on subscale commercial jet at NASA , Learjet and F16 at Edwards AFB , and many other success stories . Over the last two years it has been explored within NASA’s Learn-To-Fly framework, wherein a real-time system identification tools box of NASA is integrated across the flight envelope to continuously update the model parameters and enable autonomous flight without intensive wind-tunnel testing . In this paper, we assume no availability of such real-time system identification toolbox and resort to data-driven approaches to understand the system dynamics and learn other uncertainties whenever possible. Since the predictor in L1 adaptive control architecture affords to incorporate the learned knowledge about the uncertainties in systematic way [^1], we explore here Gaussian processes as a modeling tool within the predictor. Gaussian processes have the advantage of being sample-efficient , which enables to expand the class of uncertainties to which L1 can be robust. We provide partial answers to the two questions raised above. We demonstrate through simulations that within state-feedback L1 architectures one can expand the class of uncertainties retaining same performance guarantees of L1. More research needs to be pursued in output-feedback case to obtain similar claims. We also consider a new formulation for system dynamics, which affords to use Gaussian processes for learning, while L1 controller for ensuring robustness. Problem Statement: Multirotor Angular Rates =========================================== Consider the following uncertain system representing the angular rate dynamics for a multirotor \[eqn:rate\_dynamics\] $$\begin{aligned} \dot{x}(t) = & -J^{-1}(x(t) \times J x(t)) + B_m \left(f(x(t))+u(t)\right), \quad x(0) = x_0,\\ y(t) =& x(t), \end{aligned}$$ where $x(t) \in \mathbb{R}^3$ is the measured state representing the angular rates in the body frame, $J$ is the known moment of inertia, $B_m = J^{-1}$, $f:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ represents the **model uncertainty**, and $u(t) \in \mathbb{R}^3$ is the control input representing the body-frame torques. ***Goal:*** Combine Bayesian learning and $\mathcal{L}_1$-adaptive control to generate input $u(t)$ which satisfies desired performance and robustness criteria while ensuring the stability of the closed-loop plant. Bayesian Learner ================ The Bayesian learner consists of a Gaussian Process Regression (GPR) based algorithm. The learning algorithm uses measurements of $x$ and $u$ from the closed-loop plant to generate $f_\mu$, which is the estimates of $f$. To be precise, this is the mean value of the posterior Gaussian distribution produced by the GPR algorithm. Using the predictive distribution, the learning algorithm updates the estimate based on the ‘quality’ of the learned estimates. The details of the algorithm, and what we mean by the ‘quality’ of learning will be defined later. For now, we only consider the output of the learning algorithm which is the learned model-parameters defined as $$\label{eqn:control_design:M} \mathcal{M}(x(t),t)=\hat{f}(x(t),t),$$ where $\hat{f}(x(t),t)$ is piecewise static in time and is defined as $$\label{eqn:control_design:update_parameters} \hat{f}(x(t),t) = f_{\mu,i}(x(t)), \quad t \in [T_{L_i},T_{L_{i+1}}), \quad T_{L_i} \in \mathcal{T}, \quad i \in \mathbb{N},$$ where $\mathcal{T}=\left\{T_{L_1},T_{L_2},\dots,T_{L_k},\dots \right\}$ is the discrete index-set comprised of the time instances when the learning algorithm updates the model parameters. Moreover, $f_{\mu,i}(x(t))$ denotes the model parameter over the temporal interval $[T_{L_i},T_{L_{i+1}})$. Note that the Bayesian learned will update the model parameters at a rate much slower than the control rate. This is due to the fact that the GPR algorithm has computational complexity $O(N^3)$, where $N\in \mathbb{N}$ is the size of the collected data. Therefore, w.l.o.g. we assume that each $T_{L_i} \in \mathcal{T}$ is bounded away from zero. Additionally, due to the computational complexity, it is clear that $T_{L_{i+1}} > T_{L_i}$, for all $i \in \mathbb{N}$. Furthermore, the Bayesian learned will choose to update only when the uniform error bounds over compacts sets has reduced by some pre-specified tolerance. This way, **only the improved knowledge of uncertainties are provided.** Control Design ============== Let us begin by decomposing the control input as $$u(t) = u_{BL}(t) + u_{ad}(t),$$ where $u_{BL}$ is the baseline control and $u_{ad}$ is the adaptive input. We now present the architecture for the design of these inputs. The architecture illustrated in Figure \[fig:architecture\] (). ![Eruptions on Enceladus contributing to Saturn’s E ring.[]{data-label="fig:architecture"}](figures/L1_GP_architecture.pdf){width="100.00000%"} Baseline Control Design ----------------------- The goal of the baseline is to inject the desired dynamics and pre-filter the reference command $r(t)$. The baseline input is given by $$\label{eqn:baseline_control} u_{BL}(t) = B_m^{-1}(A_m x(t) + K_g r(t)) + x(t) \times J x(t),$$ where $K_g$ is a feed-forward gain. Substituting into  produces the following partially closed-loop dynamics $$\label{eqn:adaptive_closed_loop} \dot{x}(t) = A_m x(t) + K_g r(t) + B_m \left(f(x(t)) + u_{ad}(t) \right), \quad x(0) = x_0.$$ $\mathcal{L}_1$ - $\mathcal{GP}$: Adaptive Control Design --------------------------------------------------------- The $\mathcal{L}_1$ - $\mathcal{GP}$ controller has three components: i) state-predictor, ii) adaptation law, and iii) the control law. **State-Predictor:** The state-predictor is given by $$\dot{\hat{x}}(t) = A_m \hat{x}(t) + K_g r(t) + B_m \left(C_L(s)\hat{\eta}(s) + u_{ad}(t) + \hat{\sigma}(t) \right), \quad \hat{x}(0) = x_0,$$ where $C_L(s)$ is a low-pass filter which we define as the *learning-based filter*, $\hat{\eta}(s)$ is the Laplace transform of the Bayesian learner output $\hat{f}(x(t),t)$, and $\hat{\sigma}(t)$ is the adaptive element. The role of each of the components will be discussed later. The adaptation law that governs the evolution of the adaptive element $\hat{\sigma}$ is chosen to the piecewise constant-in-time law which attempts to cancel uncertainties at a sampling rate of $T_s$. **Control Law:** The $\mathcal{L}_1$ - $\mathcal{GP}$ control law is given by $$\label{eqn:adaptive_law} u_{ad}(s) = -C_L(s)\hat{\eta}(s) -C(s)\hat{\sigma}(s),$$ where recall that $\hat{\eta}(s)$ is the Laplace transform of $\hat{f}(x(t),t)$, and $C(s)$ is a low-pass filter. A discussion on the architecture: --------------------------------- Note the role of the low-pass filter $C_L(s)$ here. It is responsible to introduce the learned dynamics $\hat{f}$ into the the system. At the update instances $t \in \mathcal{T}$, $\hat{f}$ will have discrete jumps in its values. These jumps will be relatively higher at the initial update instances as the the Bayesian learner evolves from zero-knowledge of $f$, to enough knowledge that an update is triggered. Thus, if $\hat{f}$ is allowed to enter the system unfiltered, it will cause $u_{ad}$ to behave as a high-gain input in-turn destroying the robustness of the closed loop system. Therefore, $C_L(s)$ introduces $\hat{f}$ smoothly and slowly so as to maintain robustness. We design $C_L(s)$ with an adaptive bandwidth $\omega_L(t)$ whose bandwidth is chosen inversely proportional to the uniform prediction error bound $\|f - \hat{f}\|$ over the update instances $t \in \mathcal{T}$ and smoothly climbs up to the maximum allowable bandwidth. This way, a large input is introduced slowly into the system while a relatively incremental improvement enters the system via a relatively higher-bandwidth low-pass filter. The maximum allowable bandwidth of $C_L(s)$ is set to be the bandwidth of $C(s)$. [**I mentioned that I do not like the idea of adaptive filter bandwidth. Please limit the first results to the case of having L1 with fixed bandwidth, and try to show the GP benefits here.**]{} Now, observe the dynamics of the prediction error state $\Tilde{x} = \hat{x} - x$ which is given by $$\dot{\Tilde{x}}(t) = A_m \Tilde{x}(t) + B_m \left(\Tilde{\eta}(s) + \hat{\sigma}(s) \right), \Tilde{x}(0) = x_0,$$ where $\Tilde{\eta}(s) = C_L(s)\hat{\eta}(s) - \eta(s)$, where $\eta(s)$ is the Laplace transform of $f(x(t))$. It is evident that the adaptive element compensates for $\Tilde{\eta}$, which is the model uncertainties after the learned dynamics $\hat{f}$ cancels $f$ within the bandwidth of the filter $C_L(s)$. Note that $C(s)$ needs to have a high-enough bandwidth to ensure the stability without learning. Therefore, the bandwidth of $C(s)$ will be high if $f$ admits a large local Lipschitz constant and vice-versa. The role of the low-pass filter $C(s)$ is designed to prevent high-gain components due to the adaptive element $\hat{\sigma}$ to enter the system. Therefore, both $C_L(s)$ and $C(s)$ are tasked with ensuring the robustness of the closed-loop system. Now as $\Tilde{\eta}$ becomes smaller as $\hat{f} \rightarrow f$ and $\hat{\eta}$ cancels $\eta = f$ within the variable bandwidth of $C_L(s)$, $\hat{\sigma}$ also becomes smaller in the sense of its local Lipschitz constant. In this scenario, using $\dot{\hat{\sigma}}$, we can roll-back the bandwidth of $C(s)$ to obtain more robustness. We refer to the control where the bandwidth of $C(s)$ remains fixed as $\mathcal{L}_1$ - $\mathcal{GP}$, and where the bandwidth of $C(s)$ is rolled back as a function of $\dot{\hat{\sigma}}$ as the $\mathcal{L}_1$ - $\mathcal{GP}$ *adaptive*. Simulations =========== For the simulations, we set $J = \text{diag}\{0.011,0.011,0.021\}$. We consider the following two sets of non-linear uncertainties $$f_1(x(t)) = \begin{bmatrix} 0.5 \sin(x_1(t)) \\ 0.01 \cos(x_3(t)) \\ 0.5 \left(\sin(x_1(t)) + \cos(x_2(t)) \right) \end{bmatrix}, \quad f_2(x(t)) = \begin{bmatrix} 0.01\left( x_1(t)^2 + x_3(t)^2 + 3\sin(x_1(t)) \right) \\ 0.01\left(x_3(t)x_2(t) + x_1(t)^2 \right) \\ 0.01\left( x_3(t)^2 + \sin(x_1(t)) + \cos(x_2(t)) \right). \end{bmatrix}$$ Note that even though these uncertainties are scaled down to be small, since they enter the dynamics through $B_m = J^{-1}$, these uncertainties have a considerable effect on the overall dynamics. $\mathcal{L}_1$ - $\mathcal{GP}$ adaptive control simulations ------------------------------------------------------------- We consider the system response to step and sinusoidal reference systems with either $f_1$ or $f_2$ as uncertainties is the system. The filter bandwidth for $C_L(s)$ has a lower bound of $0.001$ radians/s (rps) and an upper bound of $10$ rps. For the filter $C(s)$, we begin with a bandwidth of $80$ rps, and heuristically reducing it to $60$ and $20$ rps after $60s$ and $100s$, respectively. The Bayesian learner collects data at $1$ Hz and sends model updates at $0.1$ Hz. The results for the step and sinusoidal references for uncertainties $f_1$ and $f_2$ are presented in Figures \[fig:Evo\_2\] - \[fig:Evo\_1\]. -- -- -- -- -- -- -- -- $\mathcal{L}_1$ vs. $\mathcal{L}_1$ - $\mathcal{GP}$ (adaptive) performance comparison -------------------------------------------------------------------------------------- We now illustrate the performance of the controllers as measured by $\|x(t) - x_{id}(t)\|$, where $x_{id}$ is the state of the ideal system in response to the reference signal $r(t)$. The results of $\|x(t) - x_{id}(t)\|$ for $\mathcal{L}_1$ vs $\mathcal{L}_1$ - $\mathcal{GP}$ and $\mathcal{L}_1$ - $\mathcal{GP}$ adaptive are presented in Figures \[fig:perf\_1\] - \[fig:perf\_2\]. -- -- -- -- -- -- -- -- **A few points of note:** The performance difference between $\mathcal{L}_1$ - $\mathcal{GP}$ (fixed bandwidth of $C(s)$) and $\mathcal{L}_1$ - $\mathcal{GP}$ adaptive (bandwidth of $C(s)$ rolled-back over time) is marginal, if any. This can be attributed to the fact that $\hat{f}$ filtered via $C_L(s)$ compensates for $f$ considerably well since the Bayesian learner learns $f$. As a consequence, over the various model-updates the role of $\hat{\sigma}$ diminishes. Another point is the **relatively low sampling rate** $T_s = 1e-2$. If $T_s$ were considerably higher, we would indeed not see much improvement over each model update. The reason is that with a higher sampling rate $T_s$, $\hat{\sigma}$ can compensate for the model-uncertainties over each sampling period. Therefore, these illustrations should be seen as the Bayesian learning helping to recover performance which would only be possible with a higher $T_s$ and thus, a higher requirement of available computational hardware. Furthermore, even with a higher rate $T_s$, we would recover the same performance as illustrated in the figures, the key observation is that the Bayesian learner actually learns the model uncertainties, something which is not possible with just $\mathcal{L}_1$. Therefore, another viewpoint is that $\mathcal{L}_1$, with performance limited only by the computational hardware, allows us to **safely** learn the dynamics of the system which can the be used for other purposes like baseline re-design (NASA’s L2F) or for the purposes of planning (DDP or MPPI). $\mathcal{L}_1$ vs. $\mathcal{L}_1$ - $\mathcal{GP}$ (adaptive) time-delay margin comparison -------------------------------------------------------------------------------------------- We now present the numerical time-delay analysis for $\mathcal{L}_1$, $\mathcal{L}_1$ - $\mathcal{GP}$, and $\mathcal{L}_1$ - $\mathcal{GP}$ adaptive. The results are illustrated in Figure \[fig:time\_delay\]. ![Amount of time-delay $\tau_d$ (in mS) accepted by the system before going unstable. The $x$-axis indicates the the number of model updates performed by the Bayesian learner after which the control input is delayed.[]{data-label="fig:time_delay"}](figures/time_delay_margins.eps){width="100.00000%"} **Observations:** The time-delay margin for $\mathcal{L}_1$ and $\mathcal{L}_1$ - $\mathcal{GP}$ are the same. This is not surprising since with both these control schemes, we have a fixed bandwidth of $C(s)$. However, we observe that with $\mathcal{L}_1$ - $\mathcal{GP}$ adaptive, the time-delay margin improves as a function of the number of model updates. This is attributed to the fact that as $\hat{f}$ improves, the role of $\hat{\sigma}$ diminishes and allows us to roll back the bandwidth of $C(s)$, thus the improved time-delay margin. Therefore, the key observation here is that Bayesian learning can help improve the robustness of the closed-loop scheme. $\mathcal{L}_1$ vs. $\mathcal{L}_1$ - $\mathcal{GP}$: Adaptive to Non-Adaptive ------------------------------------------------------------------------------ As a final test, we plot the evolution of the adaptive element $\hat{\sigma}$ for $\mathcal{L}_1$ and $\mathcal{L}_1$ - $\mathcal{GP}$. This is illustrated in Figure \[fig:adaptive\_element\]. ![Evolution of the $\mathcal{L}_1$ adaptive element as a function of number of model updates.[]{data-label="fig:adaptive_element"}](figures/Adaptive_element.eps){width="100.00000%"} This illustration supports our intuition that as $\hat{f} \rightarrow f$, $\hat{\sigma} \rightarrow 0$. Thus, as the model knowledge improves, the control transitions from adaptive to non-adaptive. [^1]: ref to L1CSM 2011 article, where we explain how the delays and saturation’s were integrated into the predictor; it is ok to have a fat footnote here

--- abstract: 'Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement. We offer a fully analytical explanation of the observed parametric resonance, based on a Mathieu-type analysis of the non-polynomial Schr[ö]{}dinger equation. The theoretical prediction for the pattern periodicity versus the driving frequency is directly compared with the experimental data, yielding good qualitative and quantitative agreement between the two. These results are corroborated by direct numerical simulations of both the one-dimensional non-polynomial Schr[ö]{}dinger equation and of the fully three-dimensional Gross-Pitaevskii equation.' author: - 'Alexandru I. Nicolin' - 'R. Carretero-González' - 'P.G. Kevrekidis' date: 'Submitted to [*Phys. Rev. A*]{}, October 2007.' title: ' **Faraday waves in Bose-Einstein condensates**' --- Introduction ============ Pattern formation in driven systems is an important direction of current research that influences many fields ranging from hydrodynamics to biophysics and from nonlinear optics to reaction kinetics; see [@cross] for a comprehensive review of the topic. Some of the oldest and most well-known forms of such phenomena are the so-called Faraday patterns, stemming from the classical studies of Faraday in 1831 [@far], who studied the behavior of “groups of particles \[placed\] upon vibrating elastic surfaces" and (in the appendix of his much-celebrated paper) the dynamics of “fluids in contact with vibrating surfaces." Faraday’s experiment became a classical example of pattern formation, whereby the uniform state loses its stability against spatially modulated waveforms, whose dominant length-scale is determined by the intrinsic properties of the system (such as the dominant wavelength of the instability) and is only weakly dependent on boundary or initial conditions. In the past few years, there has been an increasing literature about observing phenomena of the above type in driven superfluids. Bose-Einstein condensates [@book1; @book2] offer perhaps the ideal playground for such experiments, since their experimental tunability permits to create such parametric resonance phenomena in a multiplicity of ways. One such way is by driving the magnetic trap confining the system, as was proposed in Ref. [@perez]. In the same spirit of modulating the confinement of the system to observe parametric resonances, the works of Refs. [@dalfovo1; @dalfovo2], motivated by the experiments of Ref. [@stoferle], considered a periodic modulation of an optical lattice potential; on the other hand, the later work of Ref. [@dalfovo3] focused on modulating in time the confinement potential in a toroidal trap. Another recent suggestion was to produce a parametric drive by means of periodically modulating the scattering length [@staliunas], which is directly related to the prefactor of the effective nonlinearity (due to the mean-field inter-particle interactions) of the system. This can be achieved by means of Feshbach resonances [@feshbach]. These theoretical propositions motivated the very recent experimental implementation of the Faraday waves in Bose-Einstein condensates in the work of Ref. [@engels]. The actual realization of the spatially modulated patterns arose in a somewhat different way than was proposed in the above studies, a way which is very close, however, to the spirit of the theoretical suggestion of Ref. [@staliunas_second] (see also the more recent consideration of a similar problem from a quantum point of view in Ref. [@kagan]). In particular, in Ref. [@engels], an elongated cigar-shaped condensate was used where the [*transverse*]{}, strong confinement directions were periodically modulated in time, while the weaker longitudinal direction confinement was time-independent. The parametric excitation at the driving frequency was recognized as being responsible for exciting oscillations at half the driving frequency, which is the main resonance also observed in Faraday’s experiments. Subsequently, based on this insight and the dispersion relation of longitudinal collective modes presented in Refs. [@eng13; @eng14], a relation was derived (and convincingly compared to the experimental results in a quantitative manner) for the resulting pattern periodicity versus the transverse driving frequency. The aim of the present paper is to provide a complete analysis of the instability from first principles and to obtain a fully analytical prediction that can be used for a detailed comparison with the experimental results and numerical results obtained from the 1D model reduction as well as full 3D simulations at the mean-field level. The principal feature which allows us to provide a detailed quantitative analysis of the system is the fact that for cigar-shaped condensates (such as the ones used in the experiment of Ref. [@engels]) there is a quantitatively accurate description in the form of the non-polynomial Schr[ö]{}dinger equation (NPSE) derived in Ref. [@npse] (provided that the transverse direction stays close to its ground state, which is approximately the case in the experiments of Ref. [@engels]). The remarkable feature of the NPSE is that in the resulting partial differential equation (PDE) for the longitudinal description of the condensate, the transverse frequency enters explicitly and hence provides, in this setting, the explicit parametric drive that will lead to the observed spatially modulated patterns. To elucidate this feature, we will perform a linear stability analysis of the spatially uniform states in this effectively one-dimensional setting. This naturally leads to a Mathieu equation. Then, one can use the theory of the Mathieu equation to identify the most unstable mode and the wavenumber (and associated wavelength) of its spatial periodicity. This, in turn, allows us to obtain a fully analytical expression for the pattern periodicity as a function of the driving frequency that can be directly compared with the experiment. We believe that this approach yields both qualitative and quantitative insight on the experiment and on the nature of the relevant phenomena. Our presentation will be structured as follows. In Section II, we develop our analytical approach and directly compare it with the experimental findings. In Section III, we complement our theoretical findings with numerical results simulating both the full 3D experimental setting and comparing it with results of its 1D analog, namely the NPSE equation. Finally, in Section IV, we summarize our findings and present some interesting directions for future work. Analytical Considerations ========================= Our starting point will be the standard mean-field model for Bose-Einstein condensates (BEC), namely the three-dimensional Gross-Pitaevskii (GP) equation [@book1; @book2] $$i\hbar\frac{\partial\psi}{\partial t}=\left[-\frac{\hbar^{2}}{2m}\nabla^{2} +V({\bf r})+gN\left|\psi\right|^{2}\right]\psi, \label{GPE}$$ where $\psi$ denotes the BEC wavefunction (normalized to unity), $N$ represents the number of atoms and $g=4 \pi \hbar^2 a_s/m$ is proportional to the scattering length $a_s$ of the interatomic interactions; the external potential confining the atoms is given by: $$\begin{aligned} V(r,z) & = & \frac{1}{2}m\omega_{r}^{2}r^{2}+U(z),\end{aligned}$$ where $r^2=x^2+y^2$, $\omega_r$ is the transverse trapping frequency, and $U(z)$, in the case of Ref. [@engels], represents a far weaker parabolic, longitudinal, potential (with a magnetic trap frequency less than 5% of the transverse frequency), which will therefore be neglected for the purposes of the (main analytical portion of the) present work. Following the approach of Ref. [@npse], the three-dimensional wave function can be decomposed into a radial and a longitudinal part as $$\psi({\bf r},t)=\phi(r,t;\sigma(z,t))f(z,t),\label{decomposition_var}$$ where the radial component is taken as $$\phi(r,t;\sigma(z,t))= \frac{\exp\left[-r^{2}/(2\sigma^{2}(z,t))\right]} {\sqrt{\pi}\sigma(z,t)}, \label{gaussian_ansatz_var}$$ with a spatially and temporally variable width characterized by $\sigma(z,t)$. It should be noted here that the radial profiles of high-density cigar-shaped condensates (as the one in Ref. [@engels]) are closer to the Thomas-Fermi regime than to the Gaussian one. While this introduces an element of approximation to the calculation below, the adjustable nature of the parameter $\sigma(z,t)$ (which is always larger than the transverse oscillator length by a spatially dependent factor that depends on the longitudinal wavefunction; see Ref. [@npse]) and the comparison that we will report below between our theory and the physical experiment render this approximation a reasonable one for the purposes of predicting the wavelength of the resulting pattern. We note in passing that it appears to be an interesting open problem to perform a derivation similar to that of Ref. [@npse], under the assumption of a transverse Thomas-Fermi wavefunction profile. Employing the standard variational recipe of Ref. [@npse], and neglecting the longitudinal potential $U(z)$, one obtains the following effective PDE, the so-called non-polynomial Schr[ö]{}dinger equation (NPSE), describing the longitudinal wavefunction: $$\begin{aligned} i\hbar\frac{\partial f}{\partial t} =\left[-\frac{\hbar^{2}}{2m} \frac{\partial^{2}}{\partial z^{2}} +\hbar\omega_{r}\frac{1+3\, a_{s}N\left|f\right|^{2}}{\sqrt{1+2\, a_{s}N\left|f\right|^{2}}}\right]f. \label{salasnich_eq}\end{aligned}$$ In accordance with the experimental setup of Ref. [@engels], the transverse frequency is modulated by $$\omega_{r}(t)=\omega_{r,0}\cdot(1+\epsilon\sin(\omega t)),$$ where $\omega_{r,0}$ is the reference tranverse frequency, and $\epsilon$ and $\omega$ are, respectively, the amplitude and frequency of the modulation. Then, the spatially homogeneous solution is given by $$f_{0}(t)=A\exp\left[-ic\left(t-\epsilon\frac{\cos(\omega t)}{\omega}\right)\right],\label{homogeneous_sol_var}$$ where $$c=\frac{\omega_{r,0}}{\sqrt{1+2\, a_{s}NA^{2}}}\left( 1+3\, a_{s}NA^{2}\right)\label{constant_var}$$ and $A$ is a positive constant. The numerical value of $A$ is computed from the normalization $ \int\!\!\int\!\!\int |\psi(r,z,t)|^2 \,d{\bf r} = 1. $ Computing the integral one has that $A=\sqrt{1/2L}$, where we assume that the condensate extends between $-L$ and $L$; i.e., we are assuming here that the condensate is in a box rather than in a very weak magnetic trap in the longitudinal direction. The validity of this assumption for the present phenomenology is verified both a priori (due to the very weak nature of the longitudinal confinement in comparison to the much stronger transverse confinement and its modulation) and a posteriori (from the comparison with the experimental results). Faraday patterns appear in this context due to a modulational instability along the (longitudinal) $z$-axis. To examine the modulational stability of uniform patterns in the $z$-direction, we use the ansatz $$f(t)=f_{0}(t)\left[1+(u(t)+iv(t))\cos(kz)\right]. \label{perturb_var}$$ Inserting this ansatz into Eq. (\[salasnich\_eq\]) and linearizing the ensuing equations yields a Mathieu-type equation for the perturbation: $$\frac{d^{2}u}{d{\tau}^{2}}+\left(a(k,\omega)+b(k,\omega)\sin(2{\tau})\right)u=0,\label{Mathieu_var}$$ where $a$ and **$b$** are given by Eqs. (\[a\_var\]) and (\[b\_var\]) and $\tau={\omega}t/2$. $$\begin{aligned} a(k,\omega)&=&\frac{k^{2}}{2 \hbar {\pi} m^{2}\omega^{2}}\frac{6 \pi a_{s}^{2}\hbar^{2}mN^{2} \omega_{r,0}+a_{s}N2\hbar^{3}k^{2}\pi\sqrt{L^{2}+La_{s}N} +2\hbar^{3}k^{2}\pi\sqrt{L^{4}+L^{3}a_{s}N}+2gLm^{2}N\omega_{r,0}}{a_{s}N\sqrt{L^{2}+La_{s}N}+\sqrt{L^{4}+L^{3}a_{s}N}}, \label{a_var} \\[2.0ex] b(k,\omega)&=&\frac{k^{2}\omega_{r,0}\epsilon N}{2\hbar\pi m\omega^{2}}\frac{2gmL + 6 a_{s}^{2}\hbar^{2}N\pi}{a_{s}N\sqrt{L^{2}+La_{s}N}+\sqrt{L^{4}+L^{3}a_{s}N}}. \label{b_var}\end{aligned}$$ As is commonly known, Mathieu equation exhibits an intricate stability chart comprising tongues of both stable and unstable solutions [@book_of_McLachlan]. Due to the periodic potential a generic solution takes the form $u(t)=e^{i\mu t}g(t)$, where $g(t)$ has the same periodicity as $\sin(2 \tau)$ (according to the Floquet-Bloch theorem), and $\mu$ is a complex exponent taken as $\mu=\mu_{1}+i\mu_{2}$, where both $\mu_{1}$ and $\mu_{2}$ are real numbers. Determining the most unstable mode (which is the one that is expected to be observed experimentally) amounts to finding the $\omega(k)$ curve corresponding to the critical exponent with the most negative imaginary part. While this is usually a complicated task, it can be shown that for small positive values of $b$ it amounts to $$a(k,\omega)\approx1.\label{final_eq_var}$$ This conclusive property can be seen both numerically and analytically. ![Imaginary part of the critical exponent $\mu$ as function of $a$ for $b=0.25$. Notice the main lobe centered around $a=1$. For the first lobe, the difference between the approximation (\[critical\_exponent\]) and the numerics is smaller than the thickness of the line. The second lobe, shown in the inset, is so much smaller than the first one that they can hardly be seen on the same scale.[]{data-label="fig_ce"}](crit_exp.eps){width="8cm"} Investigating $\mu_{2}$ numerically [@mathematica] one finds that (for small positive values of $b$) it consists of a set of symmetrical lobes centered around $a=n^{2}$, where $n$ is a positive integer, the one centered around $a=1$ being substantially larger than the other ones (see Fig. \[fig\_ce\]). These lobes correspond to the unstable regions. Inspecting these lobes, it is transparent that the most unstable mode corresponds to $a \approx 1$. One can also argue for the validity of Eq. (\[final\_eq\_var\]) by analytical means. There is a class of partly-forgotten approximate formulas for $\mu$ as a function of $a$ and $b$ stemming from celestial mechanics (see Ref. [@critical_exponent] for the main results). A convenient formula for our purpose is $$\mu=\frac{1}{\pi}\arccos\left[\cos\left(\pi\sqrt{a}\right)+\frac{\pi b^{2}\sin\left(\pi\sqrt{a}\right)}{16\sqrt{a}\left(a-1\right)}+\mathcal{O}\left(b^{4}\right)\right],\label{critical_exponent}$$ which describes accurately the first lobe of $\mu_{2}$ for small values of $b$. In order for $\mu$ to have an imaginary part the argument of $\arccos$ must be larger than one (in absolute value). Naturally, identifying the most unstable mode amounts to finding the extremum value of $$\cos\left(\pi\sqrt{a}\right)+\frac{\pi b^{2}\sin\left(\pi\sqrt{a}\right)}{16\sqrt{a}\left(a-1\right)}.$$ To leading order in $b$ this corresponds to $a=1$. Finally, numerically solving Eq. (\[final\_eq\_var\]), one readily obtains $2 \pi/k$, which represents the spacing of adjacent maxima, herein called $\cal{S}$, as a function of the driving frequency $\omega$ of the transverse confinement $\omega_{r}$. Neglecting the $k^4$ term in Eq. (\[final\_eq\_var\]), which is numerically small under typical experimental conditions, one has that $$\begin{aligned} k&=&\omega\frac{m^{1/2}}{\omega_{r,0}^{1/2}\hbar^{1/2}} (2a_{s}\rho)^{-1/2}(1+2a_{s}\rho)^{3/4} \nonumber \\[1.0ex] &&\times (4+6a_{s}\rho)^{-1/2}, \label{eq_disp_anal}\end{aligned}$$ where $\rho=N/2L$ is the density of the condensate. It is important to note that the above formula for $k$ \[Eq. (\[eq\_disp\_anal\])\] has been obtained by assuming a homogeneous condensate with constant density $\rho=\rho_0$. This approximation yields a spacing between adjacent maxima of $$S_0 = \frac{2 \pi}{k}, \label{S0}$$ where $k$ is computed using Eq. (\[eq\_disp\_anal\]) with $\rho=\rho_0$. However, the density of the condensate in the considered system is not homogeneous. To account for this inhomogeneity, as a first-order approximation, one uses the fact that, typically for the cases under consideration, the density of the condensate varies on a space scale much larger than that of the observed patterns and thus we can extend Eq. (\[eq\_disp\_anal\]) with the density being space dependent: $\rho=\rho(x)$. In fact, the density of the condensate can be approximated in this slowly-varying limit by the so-called Thomas-Fermi (TF) approximation: $$\rho(x)=3\frac{L^{2}-x^{2}}{4L^{3}}, \label{rhoTF}$$ when $-L<x<L$ and $\rho(x)=0$ otherwise. Therefore, it is possible to approximate the average spacing by taking a spatially averaged $\bar{k}$ for $k$ in Eq. (\[eq\_disp\_anal\]) using $\rho(x)$ given by the TF approximation in Eq. (\[rhoTF\]): $${\cal S}_{1}=\frac{2\pi}{\bar{k}}, \label{S1}$$ where $$\bar{k}=\frac{1}{2L}\int_{-L}^{L}k(x)dx.$$ Alternatively, one could average the spacing directly by using the expression $${\cal S}_{2}=\frac{1}{2L}\int_{-L}^{L}\frac{2\pi}{k(x)}dx, \label{S2}$$ where, again, $k(x)$ is given by Eq. (\[eq\_disp\_anal\]) with the Thomas-Fermi density $\rho(x)$ of Eq. (\[rhoTF\]). ![(Color online) Average spacing of adjacent maxima of the longitudinal patterns as a function of the transverse driving frequency. Blue diamond dots depict the experimental data of Ref. [@engels] (the experimental error bar depicted here only takes into account the error bar in the pixel size of the recorded experimental image). Dashed black line, solid red line and blue dashed-dotted line correspond, respectively, to spacings computed using $S_0$, $S_1$ and $S_2$. Green empty circles correspond to spacings extracted from the 1D NPSE numerics by the averaged spacing method. Pink squares correspond to the spacings extracted from the full 3D numerics using the FFT method and its associated error bars (see text for details). Notice that for $\omega/2\pi$ close to 160.5 Hz the theoretical prediction is far from the experimental data. The radial breathing mode excited at these frequencies cannot be captured by the NPSE but is well captured by the 3D numerics.[]{data-label="fig_e1"}](plot_fft_peakCN.ps){width="8cm"} The above expressions for the spacing $S$ provide the analytical prediction of the present study that can be readily compared quantitatively with the experimental results of Ref. [@engels]. This comparison can be seen in Fig. \[fig\_e1\], where the theoretical predictions for $S_0$, $S_1$ and $S_2$ defined above are adapted to the $^{87}$Rb experiments of Ref. [@engels], with $\omega_{r,0}/(2\pi)=160.5$ Hz, the condensate length $2 L = 180~\mu$m and $N=5 \times 10^5$ atoms [@cond_length]. We observe a very good qualitative and a good quantitative agreement between the theoretical predictions and the experimental result, solidifying our expectation that Eq. (\[eq\_disp\_anal\]) captures accurately and in a fully analytical way the observed phenomenology of the experiment of Ref. [@engels]. Is it worth mentioning that $S_1$ and $S_2$ are closer to the experimental data since they account for the inhomogeneity of the cloud. Notice as well that our analytical prediction shows deviations from the experimental data at low frequencies, where the spatial periods of the Faraday waves are comparable with the length of the condensate. At larger frequencies we have good agreement between the theoretical curve and the experimental data, as the periods of the Faraday waves are substantially smaller than the spatial extent of the cloud. The main sources of slight disparity between the theoretical predictions and the experimental results can be traced in the transverse Gaussian (as opposed to Thomas-Fermi) profile and the fact that the analysis cannot directly incorporate the weak longitudinal trapping potential (see the discussion above). These will be further clarified below, through the comparison with the direct numerical simulation results of both the NPSE as well as the full 3D GP equation. ![Faraday pattern formation for the 1D NPSE. The top panel depicts the growth rate $G$ ($L^2$ norm of the deviation from the initial density) of the pattern. The middle two rows display the Faraday patterns at the times indicated (see vertical lines in top panel) where the left subpanels show the density profile while the right subpanels show the deviation from the initial density profile. The bottom panel depicts the space-time evolution of the density. This case corresponds to the experiment in Ref. [@engels], namely, a cloud of $N=5 \times 10^5$ $^{87}$Rb atoms, trapped by $\{\omega_{r,0},\omega_z\}/(2\pi)=\{160.5,7\}$ Hz with a 20% modulation of the radial confinement at a frequency $\omega/(2\pi)=321$ Hz. []{data-label="fig1d1"}](1dcuts321.ps "fig:"){width="7cm" height="5cm"}\ ![Faraday pattern formation for the 1D NPSE. The top panel depicts the growth rate $G$ ($L^2$ norm of the deviation from the initial density) of the pattern. The middle two rows display the Faraday patterns at the times indicated (see vertical lines in top panel) where the left subpanels show the density profile while the right subpanels show the deviation from the initial density profile. The bottom panel depicts the space-time evolution of the density. This case corresponds to the experiment in Ref. [@engels], namely, a cloud of $N=5 \times 10^5$ $^{87}$Rb atoms, trapped by $\{\omega_{r,0},\omega_z\}/(2\pi)=\{160.5,7\}$ Hz with a 20% modulation of the radial confinement at a frequency $\omega/(2\pi)=321$ Hz. []{data-label="fig1d1"}](1dtop321.jpg.ps "fig:"){width="8cm"} Numerical Results ================= One-Dimensional Numerics on the NPSE ------------------------------------ Having completed the linear stability analysis, let us now turn to full numerical simulation to investigate the instability onset and the emergence of the relevant Faraday patterns. We have simulated the NPSE (\[salasnich\_eq\]) for the experimental conditions described in Ref. [@engels]. Specifically, in Fig. \[fig1d1\], we show the formation of the Faraday pattern for a condensed cloud of $N=5 \times 10^5$ $^{87}$Rb atoms contained in a magnetic trap with frequencies $\{\omega_{r,0}/(2\pi),\omega_z/(2\pi)\}=\{160.5,7\}$ Hz where the radial trap frequency has a modulation of 20% ($\epsilon=0.2$, which is within the typical range of experimentally used modulations) and a frequency $\omega/(2\pi)=321$ Hz corresponding to the resonant oscillation frequency of the radial breathing mode (i.e., $\omega \approx 2\omega_{r,0}$) [@string1; @perez2]. As it can be observed from the figure, the Faraday pattern grows exponentially until it is clearly visible in the density space-time evolution (bottom panel) after about 125 ms. It is reassuring that the NPSE is successful in capturing the Faraday pattern with the same wavelength of 10–11 $\mu$m as the experiment of Ref. [@engels]. On the other hand, we have found that the NPSE cannot capture the right time for the development of the instability. The NPSE results take about 125 ms (i.e., about 40 periods of the modulation drive) for the Faraday pattern to be visible while, in the experiments of Ref. [@engels], some of the patterns are clearly visible after some 10 periods of the drive. ![Same as in Fig. \[fig1d1\] for a (stronger) 40% modulation of the radial confinement at a (slower) driving frequency $\omega/(2\pi)=150$ Hz. []{data-label="fig1d2"}](1dcuts150.ps "fig:"){width="7cm" height="5cm"}\ ![Same as in Fig. \[fig1d1\] for a (stronger) 40% modulation of the radial confinement at a (slower) driving frequency $\omega/(2\pi)=150$ Hz. []{data-label="fig1d2"}](1dtop150.jpg.ps "fig:"){width="8cm"} A possible explanation for the discrepancy of the Faraday pattern growth between NPSE and the experiments may lie in the size of the initial perturbation, that will eventually seed the Faraday pattern. We have used various amplitudes for the initial perturbation after we obtained the steady state solution to Eq. (\[salasnich\_eq\]) by imaginary time relaxation. In the results presented in this work we used an initial perturbation with an amplitude randomly chosen in an interval 0.001 times the local density. We also tried larger perturbations, up to ten times larger, and the effect is to accelerate the appearance of the patterns (results not shown here), however we were unable to see distinguishable patterns earlier than 30–35 driving periods for the above setting. Another effect that needs to be taken into account is the amplitude of the modulation drive. While in the experiments of Ref. [@engels] Faraday patterns, for the resonant frequency $\omega/(2\pi)=321$ Hz, quickly formed for even small drive amplitudes (less than 4%, i.e., $\epsilon<0.04$), our numerical results using the NPSE always needed a much larger drive amplitude ($\epsilon=0.2$ in the results of Fig. \[fig1d1\]). Therefore, it is clear that the NPSE, although clearly able to capture the nature (wavelength) of the Faraday pattern, it is unable to predict the growth rate of the instability. The reason for this shortcoming stems from the fact that the transverse component of the wavefunction in Eq. (\[gaussian\_ansatz\_var\]) is considered to be at its [*ground state*]{} at [*all times*]{}. This is quite a strong assumption considering that the cloud presents impact oscillator-type dynamics for its radial width [@engels] (cf. top panel of Fig. \[fig3d1\] below). This will be verified when we relax the radial wavefunction profile in the 3D simulations shown below. ![(Color online) Method of averaging spacings in the inhomogeneous cloud for the NPSE model. The left and right columns of plots correspond, respectively, to $\omega/(2\pi)=200$ Hz and $\omega/(2\pi)=320$ Hz. The top row of panels shows the density deviation $\Delta f$ from the initial density profile. The middle row of panels depicts with dots the measured spacing between maxima of $\Delta f$ inside the cloud. The averaged spacing is shown with the red horizontal line. The bottom row of panels shows the FFT of the density $|f|^2$. The vertical lines depict the estimated window of the frequencies contained in the Faraday pattern due to the non-homogeneity in the spacings (see middle row). []{data-label="spacings_NPSE"}](spacings_NPSE.ps){width="8.5cm"}   ![(Color online) Method of averaging spacings in the inhomogeneous cloud for the NPSE model. The left and right columns of plots correspond, respectively, to $\omega/(2\pi)=200$ Hz and $\omega/(2\pi)=320$ Hz. The top row of panels shows the density deviation $\Delta f$ from the initial density profile. The middle row of panels depicts with dots the measured spacing between maxima of $\Delta f$ inside the cloud. The averaged spacing is shown with the red horizontal line. The bottom row of panels shows the FFT of the density $|f|^2$. The vertical lines depict the estimated window of the frequencies contained in the Faraday pattern due to the non-homogeneity in the spacings (see middle row). []{data-label="spacings_NPSE"}](FFT_NPSE.ps "fig:"){width="8.05cm"} The results presented in Fig. \[fig1d1\] correspond to the most pattern-forming sensitive case since the drive of the radial frequency is tuned to resonate with the natural breathing frequency of the radial mode. For other driving frequencies, the growth of the Faraday pattern is less pronounced. This is demonstrated in Fig. \[fig1d2\], where we use the same parameter values as in Fig. \[fig1d1\] but changed the driving frequency to $\omega/(2\pi)=150$ Hz and we doubled its amplitude ($\epsilon=0.4$). For this out-of-resonance frequency, the Faraday pattern takes longer to form and even a drive with twice the amplitude takes longer to seed the pattern (the pattern is not visible until approximately 140ms). Nonetheless, it is interesting that this out-of-resonance case only takes about 20 periods to manifest itself. To summarize the results of the 1D NPSE simulations and to compare them with our analytical prediction for the spacing of the ensuing Faraday patterns, we proceed to measure the averaged spacing in the simulations. The method relies on computing the spacing between maxima on the density deviation $\Delta f$ from the initial profile (see top row in Fig. \[spacings\_NPSE\]). A couple of examples of the dependence of the spacing inside the cloud are depicted in Fig. \[spacings\_NPSE\] together with their average, from now on denoted as $\bar{S}_N$. It is clear from these examples that at the center of the cloud, where the density is larger, the spacing is larger than at the periphery of the cloud where the density is lower. The averaged spacing $\bar{S}_N$ was computed as a function of the driving frequency and it is depicted by the green empty circles in Fig. \[fig\_e1\]. As it is clear from Fig. \[fig\_e1\], the spacing $S_1$ \[computed using the analytical expression for the spacing given in Eq. (\[eq\_disp\_anal\]) with a spatially averaged density on the TF approximation\] and the averaged spacing $\bar{S}_N$ from the NPSE dynamics are in good agreement (and the relevant approximation of using Eq. (\[eq\_disp\_anal\]) together with Eq. (\[rhoTF\]) to represent the effects of the longitudinal potential is a fairly accurate one). It is important to mention that the non-homogeneity of the spacings induces an inherent uncertainty in the quantification of the associated spacing for a particular Faraday pattern. The associated window of spacings contained in the Faraday pattern can also be seen from the fast Fourier transform (FFT) spectrum of the density. In the bottom panels of Fig. \[spacings\_NPSE\] we depict the FFT spectrum of the density for a couple of cases with their respective frequency windows (see below for further elaboration on this effect). Three-Dimensional Numerics -------------------------- In order to more accurately model the Faraday pattern formation, we used direct numerical simulations of the Gross-Pitaevskii equation (\[GPE\]). The numerics consist of integrating Eq. (\[GPE\]) in cylindrical coordinates. The choice of cylindrical symmetry instead of full 3D numerics is justified by the fact that the radial direction does not develop azimuthal instabilities as it can be observed in the experiments of Ref. [@engels] (and it was also verified by additional tests runs of the full 3D equation on a coarser grid). The main challenge in numerically integrating Eq. (\[GPE\]) stems from the impact oscillator-type dynamics of the radial profile that is driven at resonance. These oscillations produce two numerically challenging effects: (a) they bring most of the atoms close to $r=0$ requiring an extremely fine grid, and (b) as the cloud accelerates during the impact oscillations, the wavefunction oscillates, in space, very rapidly (although the density does not) again requiring a very fine grid. Therefore, although (a) could be circumvented by a grid refinement around $r=0$, challenge (b) requires a fine $r$-grid where the cloud is traveling fastest and this happens on a large portion of the domain. Thus a fine grid needs to be implemented throughout the (radial) $r$-direction, and as a (numerical scheme stability) consequence a very small time step is also required. For the (longitudinal) $z$-direction, it suffices to have enough points to accurately capture the Faraday pattern whose wavelength is quite manageable. In the 3D simulations shown in Figs. \[fig3d1\] and \[fig3d2\] we were able to accurately integrate Eq. (\[GPE\]) for about 7 cycles of the drive with a grid of 2001$\times$401 points in the $(r,z)$-plane with a finite difference scheme in space with 4–5$^{\rm th}$ Runge Kutta in time with a time step of $0.00025$ (in adimensionalized units). The initial condition used in the simulations (i.e., the ground state of the condensate) was obtained, as for the 1D case, by imaginary time relaxation. ![Faraday pattern from the $(r,z)$ 3D simulations. This case corresponds to the experiment in Ref. [@engels], namely, a cloud of $N=5 \times 10^5$ $^{87}$Rb atoms, trapped by $\{\omega_{r,0},\omega_z\}/(2\pi)=\{160.5,7\}$ Hz with a 20% modulation of the radial confinement at a driving frequency $\omega/(2\pi)=321$ Hz. The top panel depicts the transverse radius of the cloud displaying impact-oscillator behavior (thin vertical lines depict the times of the snapshots shown in Fig. \[fig3d2\]). The second panel depicts the growth of the Faraday pattern while the bottom two rows depict the $r$-integrated density profiles (left subpanels) and their deviation from the initial profile (right subpanels). []{data-label="fig3d1"}](impact_M500KB.ps "fig:"){width="6.8cm" height="1.75cm"} ![Faraday pattern from the $(r,z)$ 3D simulations. This case corresponds to the experiment in Ref. [@engels], namely, a cloud of $N=5 \times 10^5$ $^{87}$Rb atoms, trapped by $\{\omega_{r,0},\omega_z\}/(2\pi)=\{160.5,7\}$ Hz with a 20% modulation of the radial confinement at a driving frequency $\omega/(2\pi)=321$ Hz. The top panel depicts the transverse radius of the cloud displaying impact-oscillator behavior (thin vertical lines depict the times of the snapshots shown in Fig. \[fig3d2\]). The second panel depicts the growth of the Faraday pattern while the bottom two rows depict the $r$-integrated density profiles (left subpanels) and their deviation from the initial profile (right subpanels). []{data-label="fig3d1"}](3dcuts_M500K.ps "fig:"){width="7cm" height="5cm"} ![Development of the Faraday pattern in our 3D numerical simulations. Shown are the snapshots of the $y$-integrated profile density (i.e., the observable in the experiments) for the same data as in Fig. \[fig3d1\] at the times indicated. []{data-label="fig3d2"}](zint_M500K_t17.jpg.ps "fig:"){width="8cm" height="3.3cm"}\ ![Development of the Faraday pattern in our 3D numerical simulations. Shown are the snapshots of the $y$-integrated profile density (i.e., the observable in the experiments) for the same data as in Fig. \[fig3d1\] at the times indicated. []{data-label="fig3d2"}](zint_M500K_t18.jpg.ps "fig:"){width="8cm" height="3.3cm"}\ ![Development of the Faraday pattern in our 3D numerical simulations. Shown are the snapshots of the $y$-integrated profile density (i.e., the observable in the experiments) for the same data as in Fig. \[fig3d1\] at the times indicated. []{data-label="fig3d2"}](zint_M500K_t19.jpg.ps "fig:"){width="8cm" height="3.3cm"}\ ![Development of the Faraday pattern in our 3D numerical simulations. Shown are the snapshots of the $y$-integrated profile density (i.e., the observable in the experiments) for the same data as in Fig. \[fig3d1\] at the times indicated. []{data-label="fig3d2"}](zint_M500K_t20.jpg.ps "fig:"){width="8cm" height="3.3cm"}\ ![Development of the Faraday pattern in our 3D numerical simulations. Shown are the snapshots of the $y$-integrated profile density (i.e., the observable in the experiments) for the same data as in Fig. \[fig3d1\] at the times indicated. []{data-label="fig3d2"}](zint_M500K_t21.jpg.ps "fig:"){width="8cm" height="3.3cm"}\ ![Development of the Faraday pattern in our 3D numerical simulations. Shown are the snapshots of the $y$-integrated profile density (i.e., the observable in the experiments) for the same data as in Fig. \[fig3d1\] at the times indicated. []{data-label="fig3d2"}](zint_M500K_t22.jpg.ps "fig:"){width="8cm" height="3.3cm"} Figures \[fig3d1\] and \[fig3d2\] depict the Faraday pattern arising from the $(r,z)$ Gross-Pitaevskii simulation for a cloud of $N=5 \times 10^5$ $^{87}$Rb atoms driven at resonance ($\omega/(2\pi)=321$ Hz) by a modulation amplitude of 20%, corresponding to the experiments of Ref. [@engels]. Depicted in Fig. \[fig3d2\] is the $y$-integrated (top) view which is what is measured in the experiments. Clearly observable are the well separated fringes of the forming Faraday pattern with an approximate spacing of about 8.5 $\mu$m. Furthermore, in contrast with the NPSE simulations, the Faraday instability develops more rapidly (see second panel in Fig. \[fig3d1\]) and it is clearly observable after only 6–7 periods of the drive. As mentioned above, the instability sets in much more rapidly in the full 3D system than in the 1D NPSE reduction, as expected based on the previous discussion. It is worth mentioning that in the 3D simulations we did not introduce a perturbation to the initial condition to seed the Faraday patterns since the inherent numerical noise was capable of starting the pattern. ![(Color online) FFT analysis for the data from the experiments of Ref. [@engels]. Left and right column correspond, respectively, to a driving frequency of $\omega/(2\pi)=140$ Hz and $\omega/(2\pi)=300$ Hz. The top row depicts the images from the experiment. The second row depicts the corresponding FFTs and the third row the FFTs integrated over the $y$ direction. []{data-label="FFTexp"}](FFT_expA.jpg.ps.jpg.ps "fig:"){width="8.5cm"}\ -0.2cm ![(Color online) FFT analysis for the data from the experiments of Ref. [@engels]. Left and right column correspond, respectively, to a driving frequency of $\omega/(2\pi)=140$ Hz and $\omega/(2\pi)=300$ Hz. The top row depicts the images from the experiment. The second row depicts the corresponding FFTs and the third row the FFTs integrated over the $y$ direction. []{data-label="FFTexp"}](FFT_expB.ps "fig:"){width="8.7cm"} ![(Color online) FFT analysis from the 3D numerics for $\omega/(2\pi)=321$ Hz. The top two panels depict, respectively, the Faraday pattern and its FFT. The third panel depicts the FFT integrated over the $y$ direction. The bottom panel depicts the same as the third panel but the FFT is computed after adding a 5% noise to the density to simulate the experimental noise in the picture. The vertical lines in the last panel represent our estimated window of possible frequencies for the Faraday pattern (see error bars in the pink square points in Fig. \[fig\_e1\]). []{data-label="FFTnum"}](FFT_numA.jpg.ps "fig:"){width="6.5cm"}\  ![(Color online) FFT analysis from the 3D numerics for $\omega/(2\pi)=321$ Hz. The top two panels depict, respectively, the Faraday pattern and its FFT. The third panel depicts the FFT integrated over the $y$ direction. The bottom panel depicts the same as the third panel but the FFT is computed after adding a 5% noise to the density to simulate the experimental noise in the picture. The vertical lines in the last panel represent our estimated window of possible frequencies for the Faraday pattern (see error bars in the pink square points in Fig. \[fig\_e1\]). []{data-label="FFTnum"}](FFT_numB.ps "fig:"){width="6.6cm"} To validate our full 3D numerics we have computed the average spacing (using the FFT method, see below) for three different cases of the driving frequency. The results are depicted by the pink squares in Fig. \[fig\_e1\]. As it is evidenced from the figure, the 3D numerical simulations accurately reproduce the spacing from the experiments in Ref. [@engels] including the case when the external driving frequency is [*half*]{} of the resonant frequency (see left-most pink square point in Fig. \[fig\_e1\]). In Ref. [@engels], the sudden drop in the spacing around $\omega/(2\pi)=160$ Hz, i.e., half of the resonant frequency, was attributed to excitation of the radial breathing mode. However, our numerics suggest that this is not the case and that this is due to the fact that we are driving a sub-harmonic that excites the resonance. It is important to note that the data for the experiment in Ref. [@engels] has a significant variability in the spacing values. This natural experimental variability has many potential sources. Here, we would like to focus on the error generated by the width of the frequency peak used to measure the spacing through the fast Fourier transform (FFT). In the experiments of Ref. [@engels] (see blue diamond dots in Fig. \[fig\_e1\]), the spacing of the Faraday pattern was measured by computing the FFT of the integrated density image and extracting the spatial frequency of the dominant peak. The error bars depicted in Fig. \[fig\_e1\] for the experimental data [*only*]{} take into account the error bar in the pixel size of the experimental snapshots (and not the variability due to the width of the FFT peak, see below). In Fig. \[FFTexp\] we present a couple of examples (for $\omega/(2\pi)=140$ Hz and $\omega/(2\pi)=300$ Hz) of the Faraday patterns observed in the experiments of Ref. [@engels]. Also in the figure, we depict the FFT of the data (second row) as well as its $y$-integrated counterpart (bottom row). As it is clear from the figure, there is a dominant spatial frequency that can be measured in order to extract the associated spacing of the Faraday pattern. Nonetheless, it is important to note that the dominant peak in the FFT spectrum has a width that indicates an [*interval*]{} of spacings that make up the original Faraday pattern. The width of this peak gives an indication of spatial variability of the pattern spacing: the wider the spectral peak the more spatial variability there exists in the pattern spacing. We have observed the same phenomenology when using our 3D numerical data. A typical example of the FFT analysis of our 3D numerical data is presented in Fig. \[FFTnum\]. We have checked that our numerical data is able to reproduce the behavior of the FFT analysis of the experimental data. For a better comparison with experiments we added a 5% random noise to the numerically computed density so as to emulate the noise in the experiment (see bottom panel of Fig. \[FFTnum\]). In order to extract the average spacing, $S_{\rm 3D}$, from the 3D numerics we computed the position of the dominant peak in the FFT (see pink square points in Fig. \[fig\_e1\]). As it is clear from the integrated FFT curves (see bottom two panels in Fig. \[FFTnum\]), the Faraday pattern contains a dominant peak with a respective finite width. The width of the peak indicates the presence of a [*range*]{} of spatial frequencies (instead of a single one) and thus we can associate an error bar to the average spacing by taking the width of the dominant peak as the variability in spatial frequencies. This variability has been incorporated in the average spacing of 3D numerical data in Fig. \[fig\_e1\] by means of the vertical error bars for the pink square points. In the same spirit, the actual error bar in the experiment should be slightly larger to also account for variability of the spatial frequency due to the width of the computed FFT peaks from the experimental data. Within this variability that, based on Figs. \[FFTexp\] and \[FFTnum\], clearly exists both in the experimental and the numerical (3D) data, we can conclude that our theoretical results are in very good agreement with both their experimental and their numerical counterparts. Conclusions =========== In this communication, we have revisited the topic of Faraday waves and corresponding resonances in Bose-Einstein condensates. In particular, we have focused on the experimentally relevant case where the transverse confinement is periodically modulated in time. We have used the non-polynomial Schr[ö]{}dinger equation as a tool that permits to present in an explicit form this transverse modulation in an effective longitudinal equation describing the dynamics of the condensate wavefunction. Then, a subsequent modulational stability analysis permitted us to examine the stability of spatially uniform states in this transversely driven, yet effectively one-dimensional setting. This analysis, leading to a Mathieu equation, combined with the well-established theory of the latter equation allowed us to identify the dominant mode of the instability. This, in turn, led us to extract an explicit analytical formula that allows for this mode’s wavenumber (and hence its wavelength which is directly associated with the pattern periodicity and hence is an experimentally observable quantity) as a function of the driving frequency of the transverse confining potential. Direct comparison of the fully analytical result with the experimental observations confirmed the accuracy of our approach. These analytical and experimental results were also corroborated by numerical computations both within the framework of the one-dimensional NPSE equation, as well as for the case of the fully 3D Gross-Pitaevskii equation. The similarities of the two regarding the instability length scale and the differences of the two in connection to the instability growth rate have been accordingly highlighted. These computations have allowed us to validate the quality of our theoretical approximations and give a detailed comparison between theory, numerics and experiment. There are numerous interesting possibilities that this experiment presents for future studies. One of them is to consider the predominantly two-dimensional case of pancake-shaped condensates, where, depending on the driving frequency, square or rhombic patterns may form. In that setting too, the effective wave equations of Ref. [@npse] may allow to perform the modulational stability analysis and obtain a quantitative handle on the dominant unstable mode. On the other hand, modulations of all three spatial dimensions of the confining potential would be of interest in their own right. In the latter case, while reductions of the type used herein would not be relevant, nevertheless the dynamics may still be analytically describable upon appropriate assumptions, such as, e.g., the quadratic phase assumption of Ref. [@perez], by means of coupled, nonlinear ordinary differential equations such as Eqs. (12) of Ref. [@perez]. Understanding these settings in more quantitative detail, both analytically, numerically and experimentally, is under current examination and will be reported in future publications. Acknowledgments {#acknowledgments .unnumbered} =============== We are extremely thankful to Peter Engels for providing numerous fruitful discussions, for experimental data for Figs. \[fig\_e1\] and \[FFTexp\], and for carefully reading this manuscript and suggesting numerous corrections and additions. AIN kindly acknowledges the help of Lisbeth Dilling, the librarian of the Niels Bohr Institute, on the history of Eq. (\[critical\_exponent\]). 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F[é]{}lix Tisserand, *Trait[é]{} de m[é]{}canique c[é]{}leste*, 1894, vol III (see the first chapter on how to derive equation (\[critical\_exponent\])); G. W. Hill, *Acta Mathematica* **VIII**, 1886 (reprinted in *Collected Mathematical Works*, vol I, p. 255); E. L. Ince, *Monthly Notices of the Royal Astronomical Society* **LXXV**, 1915, p. 436; J. C. Adams, *Collected Scientific Papers*, vol I, p. 186, vol II, pp. 65, 86; E. T. Whittaker, *Proceedings of the Edinburgh Mathematical Society* **XXXII**, 1914, p. 75. The length of the condensate is computed assuming a Thomas-Fermi density profile both radially and longitudinally. For the setup described in Ref. [@engels] one obtains a length of 180 microns, i.e., $L=90\mu$m. S. Stringari, Phys. Rev. Lett. [**77**]{}, 2360 (1996). V. M. Perez-Garcia, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. Lett. [**77**]{}, 5320 (1996).

--- abstract: 'We present measurements of the true masses and orbital inclinations of the two Earth-mass planets in the PSR B1257+12 system, based on the analysis of their mutual gravitational perturbations detectable as microsecond variations of the arrival times of radio pulses from the pulsar. The 6.2-millisecond pulsar, PSR B1257+12, has been regularly timed with the Arecibo telescope since late 1990. Assuming the standard pulsar mass of 1.4 $M_{\odot}$, the derived masses of planets B and C are 4.3$\pm$0.2$M_{\oplus}$ and 3.9$\pm$0.2$M_{\oplus}$, respectively. The corresponding orbital inclinations of 53$^{\circ}\pm$4$^{\circ}$ and 47$^{\circ}\pm$3$^{\circ}$ (or 127$^{\circ}$ and 133$^{\circ}$) imply that the two orbits are almost coplanar. This result, together with the known near 3:2 resonance between the orbits of the two planets, strongly supports the hypothesis of a disk origin of the PSR B1257+12 planetary system. The system’s long-term stability is guaranteed by the low, Earth-like masses of planets B and C.' author: - Maciej Konacki - Alex Wolszczan title: Masses and orbital inclinations of planets in the PSR B1257+12 system --- Introduction ============ The first extrasolar planetary system consisting of three planets orbiting a neutron star, the 6.2-ms radio pulsar PSR B1257+12, has been systematically observed since the time of its discovery [@Wol:92::; @Wol:94::; @Wol:00::]. Pulsar planets comprise the only known system of terrestrial-mass planets beyond the Sun. They exhibit a striking dynamical similarity to the inner solar system [@Maz:95::]. The pulsar planets have also provided the first demonstration of orbital near resonance in an extrasolar planetary system with the resulting, measurable gravitational perturbations [@Wol:94::; @Kon:99::]. In the recent years, a variety of resonances and commensurabilities have been detected in planetary systems around normal stars [@Nel:02::; @Fis:03::]. A microsecond precision of the pulsar timing offers a possibility to detect additional, even lower mass planets around PSR B1257+12 and other neutron stars observable as radio pulsars [@Wol:97::] An accurate knowledge of masses and orbital inclinations of the PSR B1257+12 planets is critically important for our understanding of the origin and evolution of this unique planetary system. In this case, the near 3:2 mean motion resonance (MMR) between planets B and C in the pulsar system and the existence of detectable gravitational perturbations between the two planets [@Ras:92::; @Mal:92::; @Pea:93::; @Wol:94::; @Kon:99::] provide the mechanism to derive their masses without an a priori knowledge of orbital inclinations. An approximate analytical model which includes the effect of gravitational interactions between planets B and C has been published by [@Mal:93::]. [@Kon:00::] have developed a new semi-analytical model in which perturbations between the two planets are parametrized in terms of the two planetary masses and the mutual orientation of the orbits with a sufficient precision to make a practical application of this approach feasible. Using the simulated data, they have demonstrated that the planet masses and hence their orbital inclinations can be derived from a least-squares fit of this model to the pulse times-of-arrival (TOA) measurements spanning a sufficiently long period of time. In this paper, we describe the results of putting this model to a practical test by applying it to the TOA measurements of PSR B1257+12 made with the 305-m Arecibo telescope over a 12-year period between 1990 and 2003. In Section 2, we present a brief summary of the timing model published by [@Kon:00::]. Section 3 contains the description of Arecibo timing measurements of PSR B1257+12 and the details and results of data analysis. Consequences of our successful determination of masses and orbital inclinations of planets B and C in the PSR B1257+12 system and future prospects for new pulsar planet detections are discussed in Section 4. The timing model ================ As the first approximation, the orbits of the PSR B1257+12 system can be described in terms of the sum of the Keplerian motions of its planets, in which case the direct and indirect gravitational interactions between planets are negligible. However, as predicted by [@Ras:92::], [@Mal:92::], and [@Pea:93::], the 3:2 commensurability of the orbital periods of planets B and C does lead to observable deviations from the simple Keplerian model [@Wol:94::; @Kon:99::]. In the classical approach, such departures from the Keplerian dynamics can be described by means of the so-called osculating orbital elements by invoking the Keplerian orbital elements that are no longer constant but they change in time due to mutual interactions between planets. For PSR B1257+12 timing, the difficulty with this approach is that only a radial component of the spatial motion of the pulsar can be measured and a complete information on the orientation of orbits of the planets is not available. Below, we summarize an approach developed by [@Kon:00::] that addresses this problem and allows to accurately calculate the osculating elements of planetary orbits in the presence of gravitational perturbations. In the new timing model, we define the TOA variations $\Delta\tau(t)$ as a sum of the two components $$\Delta\tau(t) = \Delta\tau_{kep}(x_j^0,e_j^0,\omega_j^0,P_j^0,T_{pj}^0,t) + \delta\tau_{int}(\Delta x_j,\Delta e_j,\Delta \omega_j,\Delta P_j, \Delta T_{pj},t)$$ Here, $\Delta\tau_{kep}$ describes the TOA variations due to the Keplerian part of the motion and is a function of the instantaneous values of the Keplerian elements of planets at the moment $t_0$. For a $j$-th planet, these are the projected semi-major axis of the pulsar orbit $x_j^0$, eccentricity $e_j^0$, longitude of the periastron $\omega_j^0$, orbital period $P_j^0$, and time of the periastron passage $T_{pj}^0$, respectively. The second term in Eqn. (1), $\delta\tau_{int}$, describes the TOA variations caused by changes in the osculating orbital elements, $\Delta x_j,\Delta e_j, \Delta \omega_j,\Delta P_j, \Delta T_{pj}$. These variables are functions of masses of the pulsar and the planets and of the relative geometry of the orbits. In the model, the masses are expressed in terms of two parameters, $\gamma_B = m_B/M_{psr}$, and $\gamma_C = m_C/M_{psr}$, where $m_B,m_C,M_{psr}$ are the masses of the two planets and the pulsar, respectively. As shown in [@Kon:00::], geometry of the orbits can be described in terms of a relative position of the node of the orbits, $\tau$, if their relative inclination is small ($I \le 10^{\circ}$). The parameter $\tau$ is related to the longitudes of the ascending nodes, $\Omega_B,\Omega_C$, and the orbital inclinations $i_B,i_C$ through the following set of equations [see also Fig. 1 in @Kon:00::] $$\label{tau} \begin{array}{l} \cos (I/2) \sin (\tau/2) = \sin ((\Omega_C - \Omega_B)/2) \cos ((i_C + i_B)/2),\\ \cos (I/2) \cos (\tau/2) = \cos ((\Omega_C - \Omega_B)/2) \cos ((i_C - i_B)/2) \end{array}$$ As long as the above small-angle approvimation is valid, the parameters $\gamma_B,\gamma_C$, and $\tau$ represent an accurate description of the time evolution of the osculating orbital elements. Consequently, Eqn. (1) can be rewritten as $$\Delta\tau(t) = \Delta\tau_{kep}(x_j^0,e_j^0,\omega_j^0,P_j^0,T_{pj}^0,t) + \delta\tau_{int}(\gamma_B,\gamma_C,\tau,t)$$ to define, in general terms, a modified timing model which includes familiar quantities, ${x_j^0,e_j^0,\omega_j^0,P_j^0,T_{pj}^0}$, to parametrize Keplerian orbits and introduces three additional parameters, $\gamma_B$, $\gamma_C$, and $\tau$, to account for the perturbations between planets B and C. The correctness of this approach has been verified by extensive simulations described by [@Kon:00::]. Observations and data analysis ============================== PSR B1257+12 has been observed with the Arecibo telescope since its discovery in 1990 [@Wol:90::]. In the years 1990-1994, before the Arecibo upgrade, the pulsar had been timed at 430 MHz and 1400 MHz with the Princeton Mark-III backend that utilizes two 32-channel filterbanks [for a description see @Stin:92::]. Starting in 1994, just before the beginning of the Arecibo upgrade, the pulsar had also been timed with a 128-channel filterbank-based Penn State Pulsar Machine [PSPM; @Cad:97::] at 430 MHz. These observations were very useful in determining the timing offsets between the Mark-III and the PSPM data sets and in eliminating another offset that was found between the data acquired with the PSPM before and after the Arecibo upgrade. Observations with the PSPM were resumed after the upgrade in November 1997. Since then, the pulsar has been systematically timed at 430 MHZ and 1400 MHz at 3-4 week intervals. A more detailed description of the data acquisition and the TOA measurement process can be found in [@Wol:00::]. The timing model included the pulsar spin and astrometric parameters, Keplerian elements of the orbits of planets A, B, and C, and the three variables $\gamma_B$, $\gamma_C$, and $\tau$ introduced to parametrize perturbations between planets B and C, as described above. A propagation delay and its long-term decline due to the varying line-of-sight electron density were parametrized in terms of the dispersion measure (DM) and its first three time derivatives. Low-amplitude DM variations on the timescales of hundreds of days have been removed by means of direct measurements of local DM values averaged over consecutive 3 month intervals. The most recent version of the timing analysis package TEMPO (see [http://pulsar.princeton.edu/tempo]{}), was modified to incorporate this model and to least-squares fit it to the observed topocentric TOAs. The final best-fit residuals for daily-averaged TOAs are characterized by a $\sim$3.0 $\mu$s rms noise which is consistent with a predicted value of $\sim$2.0 $\mu$s based on the observing parameters and the system performance [e.g. @Fos:90::]. The residuals for three fits to data involving different sets of parameters are shown in Fig. 1 and the model parameters for the final fit of the full timing model are listed in Tables 1 and 2. The new timing model for PSR B1257+12 offers further improvement of the accuracy of the determination of the standard pulsar and planetary parameters and, most importantly, it includes highly significant values for the three perturbation-related parameters, $\gamma_B$, $\gamma_C$, and $\tau$ (Fig. 2). From $m_B=\gamma_B M_{psr},m_C=\gamma_C M_{psr}$, one obtains the masses of planets B and C to be 4.3$\pm$0.2$M_{\oplus}$ and 3.9$\pm$0.2$M_{\oplus}$, respectively, using the canonical pulsar mass, $M_{psr}$ = 1.4 $M_{\odot}$. Since the scatter in the measured neutron star masses is small [@Tho:99::], it is unlikely that a possible error in the assumed pulsar mass would significantly affect these results. Because of the $\sin(i)$ ambiguity, there are four possible sets of the orbital inclinations for the planets B and C: $(53^{\circ},47^{\circ})$, $(127^{\circ},133^{\circ})$ corresponding to the difference in the ascending nodes $\Omega_C-\Omega_B\approx0^{\circ}$ (relative inclination $I\approx6^{\circ}$), and $(53^{\circ},133^{\circ})$, $(127^{\circ},47^{\circ})$, corresponding to the difference in the ascending nodes $\Omega_C-\Omega_B\approx180^{\circ}$ (relative inclination $I\approx174^{\circ}$). Obviously, in both cases the planets have nearly coplanar orbits, but in the latter one, their orbital motions have opposite senses. Because our numerical simulations of the system’s dynamics show that this situation leads to distinctly different perturbative TOA variations that are not observed, only the first two sets of the orbital inclinations, 53$^{\circ}\pm$4$^{\circ}$ and 47$^{\circ}\pm$3$^{\circ}$ or 127$^{\circ}$ and 133$^{\circ}$ are plausible. This implies that the two planets move in nearly coplanar orbits in the same sense. In addition, with the known value of $\tau$ (Table 2), one obtains $\Omega_C-\Omega_B\approx3^{\circ}$ or $\Omega_C-\Omega_B\approx-3^{\circ}$ from equation . Since it is reasonable to assume that the inner planet A is in the same plane, its mass given in Table 2 has been calculated for orbital inclination of 50$^{\circ}$. Although the formal errors of the orbital inclinations allow their relative inclination, $I$, to be as high as $\sim\!13^{\circ}$, such a departure from the model assumption of $I\leq 10^{\circ}$ would have little effect on the best-fit masses of the planets [@Kon:00::]. Discussion ========== The results described in this paper demonstrate that, under special circumstances created by the existence of measurable gravitational perturbations between planets B and C in the PSR B1257+12 system, it is possible to determine their true masses and orbital inclinations. A near 3:2 MMR between the orbits of the two planets and the fact that they are nearly coplanar imply that the pulsar system has been created as the result of a disk evolution similar to that invoked to describe planet formation around normal stars [@Bos:03::]. This represents a firm observational constraint which requires that any viable theory of the origin of the pulsar planets provides means to create a circumpulsar disk of matter that survives long enough and has a sufficient angular momentum to enable planet formation. Continuing timing observations of PSR B1257+12 will eventually settle the problem of a fourth, more distant planet (or planets) around it [@Wol:00::] and provide further constraints on the origin and evolution of this planetary system. Another important consequence of the determination of true masses of planets B and C is the implied long-term stability of the pulsar system. This problem has been investigated by [@Ras:92::] and [@Mal:92::], who have established that the two planets would have to be as massive as 2-3 Jupiter masses to render the system dynamically unstable on a $10^4-10^5$ yr timescale. Obviously, the measured, terrestrial masses of the planets (Table 2) are much too low to create such a condition. In fact, this conclusion is not surprising, given another result of [@Mal:92::], who have calculated that, if the masses of the two planets were about 20-40$M_{\oplus}$, the system would be locked in the exact 3:2 MMR and the character of perturbations would be very different from the observed near-resonance configuration. The early theories of pulsar planet formation have been summarized by [@Pod:93::] and further discussed by [@Phi:93::]. More recently, [@Mil:01::] and [@Han:02::] have examined the conditions of survival and evolution of pulsar protoplanetary disks. They have concluded that an initially sufficiently massive ($>10^{28}$g) disk would be able to resist evaporation by the pulsar accretion flux and create planets on a typical, $\sim 10^7$-year timescale. A quick formation of a massive disk around the pulsar could, for instance, be accomplished by tidal disruption of a stellar companion [@Ste:92::; @Phi:93::] or, possibly, in the process of a white dwarf merger [@Pod:91::; @Liv:92::]. Both these processes, although entirely feasible, cannot be very common. In fact, with the exception of PSR B1257+12, no planetary companions have emerged from the precision timing of 48 galactic millisecond pulsars [@Lor:01::], implying their rarity, independently of the specific formation mechanism. Since the current evidence points to isolated millisecond pulsars as best candidates for a presence of planetary companions around them [@Wol:97::; @Mil:01::], new detections of such objects by the ongoing and future pulsar surveys will be very important. So far, only 10 solitary millisecond pulsars, including PSR B1257+12, have been discovered. This remains in a stark contrast with the sample of about 2000 solar-type stars that are included in the Doppler surveys for extrasolar planets [@Mar:03::]. Factors to be taken into account while designing pulsar planet searches must include the fact that such pulsars are less common and appear to be intrinsically even fainter than more typical, binary milisecond pulsars [@Bai:97::]. In addition, if the high space velocity of PSR B1257+12 ($\sim$300 km s$^{-1}$, Table 1) and the fact that it has planets were causally connected, such objects would spend most of the time near turnover points of their galactic orbits, which would make them difficult to detect. Altogether, it appears that further improvement of the statistics of neutron star planetary systems may be a lengthy process, even if they are similar to those established for the occurence of giant planets around normal stars [@Mar:03::]. M. K. is a Michelson Postdoctoral Fellow and is partially supported by the Polish Committee for Scientific Research, Grant No. 2P03D 001 22. A. W. is supported by the NASA grant NAG5-4301 and by the NSF under grant AST-9988217. The authors thank the referee, Renu Malhotra, for insightful comments on the manuscript. Bailes, M. et al. 1997, , 481, 386 Boss, A. P. 2003, ASP Conf. Ser. 294: Scientific Frontiers in Research on Extrasolar Planets, 269 Cadwell, B. J. 1997, Ph.D. Thesis Camilo, F., Thorsett, S. E., & Kulkarni, S. R. 1994, , 421, L15 Fischer, D. A. et al.  2003, , 586, 1394 Foster, R. S. & Backer, D. C. 1990, , 361, 300 Hansen, B. M. S. 2002, ASP Conf. Ser. 263: Stellar Collisions, Mergers and their Consequences, 221 Konacki, M., Maciejewski, A. J., & Wolszczan, A. 1999, , 513, 471 Konacki, M., Maciejewski, A. J., & Wolszczan, A. 2000, , 544, 921 Livio, M., Pringle, J. E., & Saffer, R. A. 1992, , 257, 15P Lorimer, D. R. 2001, Living Reviews in Relativity, 4, 5 Malhotra, R., Black, D., Eck, A., & Jackson, A. 1992, , 356, 583 Malhotra, R. 1993, , 407, 266 Marcy, G. W., Butler, R. P., Fischer, D. A., & Vogt, S S. 2003, ASP Conf. Ser. 294: Scientific Frontiers in Research on Extrasolar Planets, 1 Mazeh, T. & Goldman, I. 1995, , 107, 250 Miller, M. C. & Hamilton, D. P. 2001, , 550, 863 Nelson, R. P. & Papaloizou, J. C. B. 2002, , 333, L26 Peale, S. J. 1993, , 105, 1562 Phinney, E. S. & Hansen, B. M. S. 1993, ASP Conf. Ser.  36: Planets Around Pulsars, 371 Podsiadlowski, P., Pringle, J. E., & Rees, M. J. 1991, , 352, 783 Podsiadlowski, P. 1993, ASP Conf. Ser.  36: Planets Around Pulsars, 149 Rasio, F. A., Nicholson, P. D., Shapiro, S. L., & Teukolsky, S. A. 1992, , 355, 325 Shklovskii, I. S. 1970, Soviet Astronomy, 13, 562 Stevens, I. R., Rees, M. J., & Podsiadlowski, P. 1992, , 254, 19P Stinebring, D. R., Kaspi, V. M., Nice, D. J., Ryba, M. F., Taylor, J. H., Thorsett, S. E., & Hankins, T. H. 1992, Review of Scientific Instruments, 63, 3551 Taylor, J. H. & Cordes, J. M. 1993, , 411, 674 Thorsett, S. E. & Chakrabarty, D. 1999, , 512, 288 Wolszczan, A. 1990, , 5073, 1 Wolszczan, A. & Frail, D. A. 1992, , 355, 145 Wolszczan, A. 1994, Science, 264, 538 Wolszczan, A. 1997, Celestial Mechanics and Dynamical Astronomy, 68, 13 Wolszczan, A. et al.  2000, , 528, 907 [lc]{} Right ascension, $\alpha$ (J2000)& 13$^h$00$^m$03.$\!\!^s$5767(1)\ Declination, $\delta$ (J2000)& 12$^\circ$40$^\prime$56.$\!\!^{\prime\prime}$4721(3)\ Proper motion in $\alpha$, $\mu_{\alpha}$ (mas/yr)& 45.50(4)\ Proper motion in $\delta$, $\mu_{\delta}$ (mas/yr)& -84.70(7)\ Period, $P$ (ms)& 6.21853194840048(3)\ Period derivative, $\dot P$ (10$^{-20}$)& 11.43341(4)\ Epoch (MJD)& 49750.0\ Dispersion measure, $DM$ (cm$^{-3}$pc)& 10.16550(3)\ $\dot{DM}$ (cm$^{-3}$pc/yr)& -0.001141(7)\ $\ddot{DM}$ (cm$^{-3}$pc/yr$^2$)& 0.000121(3)\ $DM^{(3)}$ (cm$^{-3}$pc/yr$^3$)& 0.000011(1)\ [*DM distance*]{}, $d$ (kpc) & 0.6(1)\ [*Transverse velocity*]{}, $V_t$ (km/s) & 273(45)\ [*Kinematic correction*]{}, $\dot{P}_k$ ($10^{-20}$) & 8(3)\ [*Characteristic age*]{}, $t_c$ (Gyr) & 3(3)\ [*Surface magnetic field*]{}, $B_s$ ($10^8$ G) & 5(2)\ [lccc]{} Projected semi-major axis, $x^0$ (ms)& 0.0030(1) & 1.3106(1) & 1.4134(2)\ Eccentricity, $e^0$& 0.0 & 0.0186(2) & 0.0252(2)\ Epoch of pericenter, $T_p^0$ (MJD)& 49765.1(2) & 49768.1(1) & 49766.5(1)\ Orbital period, $P_b^0$ (d)& 25.262(3) & 66.5419(1) & 98.2114(2)\ Longitude of pericenter, $\omega^0$ (deg)& 0.0 & 250.4(6) & 108.3(5)\ [*Mass*]{} ($M_{\oplus}$)& 0.020(2) & 4.3(2) & 3.9(2)\ [*Inclination, solution 1*]{}, $i^0$ (deg) & ... & 53(4) & 47(3)\ [*Inclination, solution 2*]{}, $i^0$ (deg) & ... & 127(4) & 133(3)\ [*Planet semi-major axis*]{}, $a_{p}^0$ (AU) & 0.19 & 0.36 & 0.46\ $\gamma_B$ (10$^{-6}$) & & 9.2(4) &\ $\gamma_C$ (10$^{-6}$) & & 8.3(4) &\ $\tau$ (deg)& & 2.1(9) &\

--- abstract: 'Reference heavy-flavour cross sections at HERA and LHC have been computed following different theoretical approaches and the results have been compared.' author: - 'O. Behnke$^1$, M. Cacciari$^2$, M. Corradi$^3$, A. Dainese$^4$, H. Jung$^5$, E. Laenen$^6$, I. Schienbein$^7$, H. Spiesberger$^8$' title: 'Benchmark cross sections for heavy-flavour production' --- [*coordinators: M. Corradi, A. Dainese*]{} \[sec:benchmarks\] Introduction ============ This section presents a comparison of cross sections for HERA and LHC calculated according to different theoretical approaches. Different programs were used to calculate the same reference cross sections, using, as far as possible, the same input parameters and a consistent method to evaluate uncertainties. In this way it is possible to identify processes and kinematical regions in which different approaches give the same answer and regions where they differ. Unified criteria to evaluate the theoretical uncertainty should also allow to understand what approach is expected to be more precise. Moreover these calculations, which incorporate up-to-date parameters and PDF parametrisations, can be used as a reference for experiments and for further theoretical predictions. The cross sections presented here, are available in computer-readable format from the web page `http://www-zeus.desy.de/~corradi/benchmarks`, where figures in color can also be found. Programs ======== A list of the programs used for the cross section calculations is given below. For further details see the references and the theoretical review on heavy quark production in these proceedings. - [MNR]{}[@Mangano:1991jk] is a fixed-order (FO) NLO program for heavy-flavour hadro-production, it was used for LHC cross sections; - [FMNR]{}[@Frixione:1993dg; @Frixione:1994dv] is an extension of the previous program to photoproduction, it was used for photoproduction at HERA; - [HVQDIS]{}[@Harris:1997zq; @Smith:1996ts] is a FO-NLO program for heavy-flavour production in deep-inelastic scattering (DIS), it has been used for DIS at HERA; - [FONLL]{}[@Cacciari:2001td; @Cacciari:1998it] provides matched massive-massless calculations with NLO accuracy and resummation of large $p_T$ logarithms. It is available for hadro- and photo-production and was used for HERA photoproduction and LHC cross sections; - [GM-VFNS]{}[@Kniehl:2004fy; @Kniehl:2005de; @Kniehl:2005mk; @Kramer:2003jw] is a calculation in the generalised massive variable flavour number scheme. It has been used for charmed hadron $p_T$ spectra at LHC and in photoproduction at HERA; - [Cascade 1.2009]{}[@Jung:2001hx] is a full Monte Carlo program based on unintegrated parton densities and $K_T$ factorisation. It has been used to calculate cross sections for Photoproduction and DIS at HERA and for LHC; - [Rapgap 3]{}[@Jung:1993gf] is a multi-purpose MC program for $ep$ collisions, it implements heavy-flavour production through the boson-gluon-fusion process $\gamma^* g \rightarrow Q \bar{Q}$ at leading order. It has been used for DIS at HERA. Both [Cascade]{} and [Rapgap]{} use [Pythia]{}[@Sjostrand:2001yu] routines for fragmentation. Parameter program central value lower/upper ----------------------- --------------------- ------------------------------------------- --------------------------------------------------- $\Lambda^5_{\rm QCD}$ all 0.226 GeV fix $m_c$ all 1.5 GeV 1.3/1.7 GeV $m_b$ all 4.75 GeV 4.5/5.0 GeV p-PDF all-[Cascade]{} CTEQ6.1[@Stump:2003yu] MRST2002[@Martin:2002aw]/Alekhin[@Alekhin:2002fv] [Cascade]{} CCFM A0 - $\gamma$-PDF FMNR, FONLL AGF[@Aurenche:1994in] GRV[@Gluck:1991jc] $f(c\rightarrow D^*)$ all 0.235 fix $c$ fragmentation: (F)MNR,HVQDIS Pet.[@Peterson:1982ak] $\epsilon_c=0.021$ $0.002$/$0.11$ FONLL BCFY $r=0.1$ $0.06/0.135$ GM-VFNS [@Kniehl:2005de] - [Cascade, Rapgap]{} Pet. $\epsilon_c=0.075$ Def./$\epsilon_c=0.05$ $b$ fragmentation: (F)MNR,HVQDIS Pet. $\epsilon_b=0.001$ $0.0002$/$0.004$ FONLL Kart. $\alpha=29.1$ $25.6/34.0$ [Cascade, Rapgap]{} Pet. $\epsilon_b=0.002$ Def./$\epsilon_b=0.005$ : \[t:parameters\] The table shows input parameter to the different programs with the corresponding lower and upper values used for the uncertainty: $\Lambda_{\rm QCD}$, the quark masses, the proton and photon parton densities, the fraction of c quarks decaying into a $D^*$ meson, and the parameters used for fragmentation. The fragmentation form are abbreviated to Pet. for Peterson, Kart. for Kartvelishvili, Def. for the default [Pythia]{} fragmentation Parameters and uncertainties ============================ The different calculations were compared using the same input parameters and, where possible, with total uncertainty bands computed in a consistent way. The total uncertainty band includes the effect of the uncertainty on the input parameters and on the missing higher orders in the perturbative expansion. Perturbative uncertainty ------------------------ The perturbative uncertainty was obtained by varying the renormalisation and factorisation scales independently in the range $0.5\mu_0 < \mu_F, \mu_R< 2\mu_0 $, while keeping $1/2<\mu_R/\mu_F<2$, were $\mu_0$ is the nominal value, typically set to the transverse mass $p_T^2+m_Q^2$ or to $4m^2+Q^2$ in the DIS case. The largest positive and negative variations were taken as the perturbative uncertainty band. Input parameters ---------------- The uncertainty from the input parameters was obtained by varying each parameter around the central value. An effort was made within the working group to find the best central value and uncertainty for the input parameters. The values used for the perturbative parameters $\Lambda_{\rm QCD}^5$, $m_c$, $m_b$ as well as the parton distribution functions (PDF) for the proton and for the photon are reported in Table \[t:parameters\]. For practical reasons, rather than using the full treatment of the PDF uncertainty, few different parametrisations were tried and it was checked that the choice of the PDF set always gives a small contribution to the total uncertainty band. In the case of [Cascade]{}, the CCFM A0 parametrisation was used as the central value while the PDF parametrisations A0+ and A0-, obtained from fits to DIS data with different renormalisation scales, were used in conjunction with the variation of the renormalisation scale. Since the different programs have different perturbative contents, different parameters for the non-perturbative fragmentation function were used. The values were chosen in order to correspond to the same average fragmentation in $e^+e^-$ collisions as explained in the section on heavy quark fragmentation in these proceedings. Table \[t:parameters\] reports the fragmentation form and the corresponding parameters used in the different programs. In the FONLL calculation for charm, the BCFY[@Braaten:1994bz] fragmentation parameter $r$ was varied in conjunction with the variation of the charm mass since different values of $r$ are obtained from $e^+e^-$ data for different $m_c$[@Cacciari:2005uk]. Similarly for beauty, the Kartvelishvili[@Kartvelishvili:1977pi] parameter $\alpha$ was varied in conjunction with the variation of the $b$ mass\cite{}. For GM-VFNS, the fragmentation functions and fractions were taken from [@Kniehl:2005de]. The total uncertainty band was obtained from the sum of the uncertainties added in quadrature coming from the parameter variations and the perturbative uncertainty. Results ======= HERA Photoproduction -------------------- The results for HERA Photoproduction are given as $ep$ cross-sections for $0.2<y<0.8$ ($y$ is the Bjorken variable while $Y$ is the rapidity in the laboratory frame) and $Q^2<1$ GeV$^2$. The beam energies have been set to $E_e=27.52$ GeV, $E_p=920$ GeV with the proton beam going in the positive rapidity direction. Figure \[f:phpc\] shows the differential cross sections as a function of the charm quark transverse momentum (a) and pseudorapidity (b). In (c) and (d) the same cross sections are given for the charmed $D^*$ meson. A meaningful comparison can be performed only for the hadron variables, which are the real physical observables, since the quark level may be defined differently in different approaches. The FO calculation (FMNR) shows a large uncertainty ($\sim 60\%$) at the hadron level due to the related uncertainty on the fragmentation parameters. The resummed programs FONLL and GM-VFNS have much smaller uncertainty and are within the FMNR uncertainty band. The central values from FMNR and FONLL coincide at low transverse momenta. GM-VFNS, instead, tends to grow unphysically at low $p_T(D^*)$. As can be seen in (c), the quark-level disagreement between FO (FMNR) and FONLL calculations is consistently removed at the hadron-level. The unintegrated-PDF Monte Carlo [Cascade]{} tends to be above the other calculations, in particular at large $p_T$. In the case of beauty (Fig. \[f:phpb\]) the uncertainty bands are smaller ($\sim 20\%$ for FMNR), [Cascade]{} and FMNR are in good agreement. Due to the large $b$ mass, the resummed calculation FONLL (not shown) is expected to be similar to the fixed-order one (FMNR). For both beauty and charm, FMNR and FONLL show a shoulder at positive rapidities (b, d) due to the “hadron-like” component of the photon that is not present in [Cascade]{}. Figure \[f:breakphp\] shows the different components of the FMNR uncertainty band for charm and beauty. The uncertainties for quark production are typically dominated by the perturbative scale uncertainty with the exception of the low transverse momentum region ($p_T \sim m_Q$) where the uncertainty from the quark-mass can dominate. For hadron production, the fragmentation dominates the FMNR uncertainty at large $p_T$. The PDF uncertainty was found to be small. Resummed calculation have smaller uncertainty bands due to the smaller perturbative and fragmentation contributions at large $p_T$. HERA DIS -------- Heavy quark production in DIS is not available in the matched massive-massless approach (except for total cross sections). Therefore the DIS comparison was limited to the FO-NLO program HVQDIS, the unintegrated-PDF MC [Cascade]{} and the [Rapgap]{} Monte Carlo. The DIS cross sections at HERA are reported as $d\sigma/d\log_{10}(x)$ for different bins of $Q^2$ and are intended at the Born level, without electroweak corrections. Figure \[f:disc\] shows, for each $Q^2$ bin, the inclusive charm cross-section, the cross section for observing a $D^{*}$ meson in the “visible” range $p_T(D^{*})>1.5$ GeV, $|Y(D^{*})|<1.5$ and for observing a muon in the range $p_T(\mu)>3$ GeV, $|Y(\mu)|<2$. The three calculations are compatible at intermediate values of $x$ ($\sim 10^{-3}$). At large $x$ and low $Q^2$, [Cascade]{} and [Rapgap]{} drop to zero much faster than HVQDIS. At low $x$ [Rapgap]{} is significantly larger than HVQDIS while both are within the uncertainty band given by [Cascade]{}. A similar behavior is seen for beauty (Fig. \[f:disb\]). The uncertainty on HVQDIS, not given here, is expected to be small ($\sim 10 - 20\%$ for beauty [@Carli:2003cx]). The high-$x$ discrepancy between HVQDIS and the other two calculations seems therfore to be beyond the program uncertainties and deserves further investigations. LHC --- For LHC, we computed the cross sections in $pp$ collisions at $\sqrt{s}=14$ TeV. Figures \[f:lhcc\] and \[f:lhcb\] show the single inclusive cross sections as a function of $p_T$, at quark (upper panels) and hadron (lower panels) level, for charm and beauty, respectively. Two rapidity intervals are considered: $|Y|<2.5$, approximately covering the acceptance of the barrel detectors of ATLAS ($|\eta|<2.5$), CMS ($|\eta|<2.5$), and ALICE ($|\eta|<0.9$); $2.5<|Y|<4$, approximately covering the acceptance of LHCb ($2<\eta<5$) and of the ALICE muon spectrometer ($2.5<\eta<4$). For charm, we compare the fixed-order NLO results from MNR to the results from the [Cascade]{} event generator, from the GM-VFNS calculation and from the FONLL calculation. The agreement is in general good, in particular in the low-$p_T$ region; at high-$p_T$ [Cascade]{} predicts a larger cross section than the other calculations, especially at forward rapidities. The FONLL central prediction is in agreement with that of the FO NLO calculation at low $p_T$, while deviating from it at high $p_T$, where it gives a smaller cross section. For beauty, we compare FO NLO (MNR), FONLL and [Cascade]{}. Again, there is agreement at low $p_T$, where, as expected, the FONLL result coincides with the MNR result. At high $p_T$, both [Cascade]{} and FONLL predict a larger cross section than the MNR central values, but all models remain compatible within the theoretical uncertainties. At forward rapidities, for beauty as for charm, [Cascade]{} gives a significantly larger cross section than MNR. Figure \[f:breaklhc\] shows the breakdown of the uncertainties for hadron production as obtained with MNR. The perturbative component dominates at LHC. Only the fragmentation component for charm hadron production becomes comparable in size to the perturbative one at large $p_T$. $Q$-$\bar{Q}$ correlations -------------------------- The azimuthal separation between the two heavy quarks $\Delta\phi(Q\bar{Q})$ and the transverse momentum of the quark-antiquark system $p_T(Q\bar{Q})$ are particularly sensitive to higher-order effects since at leading order their distributions are delta functions peaked at $\Delta\phi(Q\bar{Q})=\pi$ and $p_T(Q\bar{Q})=0$. The distribution of these variables is therefore a direct probe of QCD radiation and is well suited for comparing different calculations. Figures \[f:lhcdphic\] and \[f:lhcdphib\] show the heavy-quark pair $p_T$ distribution and the quark-antiquark relative azimuthal angle distribution for charm and beauty at LHC, respectively. For both distributions, the two quarks of the pair are required to have $|Y|<2.5$; also minimum $p_T$ selections are applied to mimic the effect of realistic experimental cuts ($p_T^Q>3$ GeV and $p_T^{\bar Q}>6$ GeV). In the region near $\Delta\phi(Q\bar{Q})=\pi$ and $p_T(Q\bar{Q})=0$, where the cancellation of soft and collinear divergencies occur, the fixed-order NLO calculation gives an unphysical negative cross section with next to a large positive peak. A larger binning would be needed to average this behavior and produce a more physical results. The [Cascade]{} MC, has a more realistic behavior. Both calculations have a non-zero value at $\Delta\phi(Q\bar{Q})=0$ related to “gluon-splitting” events. A similar result was found for HERA as shown in Figure \[f:phpdphi\]. This kind of distribution is expected to be well described by programs that merge NLO matrix elements to the parton-shower MC approach such as MC@NLO [@Frixione:2003ei]. Conclusions =========== Heavy-flavour cross sections for HERA and LHC, obtained with fixed-order NLO programs, with matched massive/massless calculations and within the $K_T$-factorisation approach have been compared. Similar results are found for photoproduction at HERA and for the LHC. As expected the resummed calculations were found to be compatible with the fixed-order results but have smaller uncertainties at large $p_T$. Resummed calculations for charm in two different schemes (GM-VFNS and FONLL) are anyway somewhat incompatible both at HERA and LHC, suggesting that their uncertainty may be underestimated. The $K_T$-factorisation program [Cascade]{} predicts larger cross sections than the other approaches at large $p_T$ at LHC and for charm at HERA. The comparison for DIS was limited to FO-NLO and a MC program with leading order matrix elements. Large discrepancies, which deserve further investigations, were found in this case. A comparison with experimental data would be needed for further understanding of the quality of the available calculations.

--- author: - | Ayres Freitas\ University of Chicago, Chicago, IL 60637, USA\ HEP Division, Argonne National Laboratory, Argonne, IL 60439, USA\ Institut für Theoretische Physik, Universität Zürich, 8057 Zürich, Switzerland\ E-mail: - | Caroline Milsténe\ Fermi National Accelerator Laboratory, Batavia, IL 60510-500, USA\ E-mail: - | Michael Schmitt\ Northwestern University, Evanston, IL 60208, USA\ E-mail: - | Andre Sopczak\ Lancaster University, Lancaster LA1 4YB, United Kingdom\ E-mail: title: A Method for the Precision Mass Measurement of the Stop Quark at the International Linear Collider --- Introduction ============ An experiment at the International Linear Collider (ILC) will be able to make many precise measurements from which particle properties, and ultimately, the outlines of a particle physics model may be inferred. Due to the high statistical precision expected at the ILC, the optimization of the systematic errors is of particular importance. We have studied one specific example, namely, the extraction of the mass of an hypothetical stop squark from cross-section measurements near threshold. We have devised a method which reduces most systematic uncertainties and leads to a potentially very accurate measurement of the stop squark mass. This method, however, is general and could be applied to other particles produced in an ${e^+e^-}$ collider. The method relies on the comparison of production rates at two different center-of-mass energies, and knowledge of how the cross-section varies as a function of $\sqrt{s}$ and the particle mass. In simple terms, one measures the yield at an energy close to the pair-production threshold, which will be very sensitive to the particle mass, and then at a much higher energy, which has little sensitivity. The ratio of these two yields retains sensitivity to the mass, and at the same time is [*insensitive*]{} to many potential systematic effects We have chosen the case of a light scalar top squark with a mass not much higher than the mass of the lightest neutralino since production of this particle was already extensively studied in an ILC context [@stop; @stopsLC]. It was concluded that a conventional approach to the measurement of the stop squark mass culminated in an uncertainty of about ${\Delta m_{\tilde{t}_1}}= 1.2{\,\, \mathrm{GeV}}$ [@stop; @heavyq]. The new method improves substantially on this result, and for a similar scenario, we conclude that the uncertainty will be ${\Delta m_{\tilde{t}_1}}= 0.42$ GeV. For this analysis, we have performed realistic simulations of the signal and backgrounds, and used two techniques to separate the signal from the background. The first technique is based on conventional selection cuts, while the second employs an improved Iterative Discriminant Analysis (IDA) [@ida]. Furthermore, the hadronization of the stop has been included and we have carefully studied the systematic uncertainties arising from this and other sources. There are theoretical motivations for studying a light stop squark with a mass close to the neutralino mass. Specifically, we evoke a scenario within the Minimal Supersymmetric extension of the Standard Model (MSSM) which is able to explain the dark matter density of the universe as well as the baryon asymmetry through the mechanism of electroweak baryogenesis. The existence of dark matter has been firmly established by various observations, most notably by the measurements of the cosmic microwave background radiation by the Wilkinson Microwave Anisotropy Probe (WMAP) [@wmap] and the studies conducted by the Sloan Digital Sky Survey [@Tegmark:2003ud]. The known properties of dark matter suggest that it consists of primordial weakly-interacting massive particles. Within the context of supersymmetry, the best candidate is the lightest neutralino, ${\tilde{\chi}^0}_1$, which is generically the lightest supersymmetric particle, and would be stable if $R$-parity is conserved. Another well-established fact which poses a great puzzle for particle physics is the apparent asymmetry between the amount of matter and anti-matter in the universe. There are several competing theoretical explanations for the origin of this baryon asymmetry. One of these relies on asymmetries generated during the electroweak phase transition. The hypothesized mechanism is not viable within the Standard Model (SM), but is possible within the context of supersymmetry. In fact, requiring that the correct baryon asymmetry is generated at the electroweak phase transition places strong constraints on the parameter space of the MSSM [@CQW; @EWBG; @Carena:1997ki; @EWBG2]. In particular, the lightest scalar top squark ${{\tilde{t}}}_1$ must not be heavy, satisfying the bound ${m_{\tilde{t}_1}}{\,\raisebox{-.1ex}{$_{\textstyle <}\atop^{\textstyle\sim}$}\,}140$ GeV with concomitant bounds on the mass of the Higgs boson [@Carena:1997ki; @EWBG2]. Furthermore, this particle is predominantly of the right-handed chirality state. A small mass difference between the stop and the lightest neutralino can help to bring the dark matter relic density into the proper range due to co-annihilation between the stop and the neutralino. For this mechanism to be effective, the typical mass difference is rather small, ${m_{\tilde{t}_1}}- {m_{\tilde{\chi}^0_{1}}} {\,\raisebox{-.1ex}{$_{\textstyle <}\atop^{\textstyle\sim}$}\,}30{\,\, \mathrm{GeV}}$ [@Balazs:2004bu]. The dominant decay mode of the stop is ${{\tilde{t}}}_1 \to c\,{\tilde{\chi}^0}_1$, resulting in a final state with two soft charm jets and missing energy. Previous studies [@stop; @stopsLC] have shown that clean samples of such events can indeed be isolated at the ILC. This paper is organized as follows. The next section explains the ratio-of-yields method in detail. Section \[sec:ana\] describes the tools and methods used for simulating the relevant processes and the detector, as well as two methods for selecting a clear stop signal. Section \[sec:exp\] is devoted to a discussion of the experimental systematics, followed by Section \[sec:th\] which explores theoretical uncertainties. The last section reports the results for this specific channel, and shows the implications for future calculations of dark matter relic densities based in particle physics, specifically, supersymmetry. We comment briefly on the potential of the IDA method to discover this stop quark at $\sqrt{s} = 500{\,\, \mathrm{GeV}}$. Conclusions follow. Method {#sec:method} ====== One way to measure the stop mass would be through kinematic distributions of its final state products. However, jet energies are difficult to measure precisely, especially when the jets are not energetic. Furthermore, the radiation of gluons and the hadronization of the stop quarks complicate the kinematics in ways that are difficult to predict and model accurately. These effects make a precise stop mass measurement from kinematic distributions rather difficult [@stopsLC]. Alternatively, one can extract the stop mass and mixing angle from measurements of the cross-section. For example, it has been shown that using measurements with two different beam polarization at one center-of-mass energy, both the stop mass and mixing angle can be inferred with good accuracy [@stopsLC]. For light stop quarks with masses ${\cal O}(100 {\,\, \mathrm{GeV}})$, the typical achievable precision is ${\Delta m_{\tilde{t}_1}}\sim 1 {\,\, \mathrm{GeV}}$. However, this technique is limited by substantial systematic uncertainties on the measurement of the total cross-section, in particular the modeling of stop hadronization and the resulting uncertainties in the selection efficiency. We propose a new method which reduces the impact of these systematic uncertainties, and which we describe in this section in general terms. While our explication is based on the case of a light stop, the method could be applied to other particles. (See, for example, Ref. [@snumass] for a discussion of the sensitivity to unknown branching ratios.) The original presentation of this method concerned Higgs production at a future $\gamma\gamma$-collider [@gagaHiggs]. We want to extract the mass ($M_X$) of a particle from measurements of its production cross sections. In order to obtain the best result, two issues must be considered: 1. optimization of the energy and luminosity for the minimum statistical error, and 2. reduction of systematic uncertainties. The method described here seeks to address both issues in the best possible way. The error on the extracted mass ($\Delta M_X$) relates to the cross-section measurement error ($\Delta\sigma$) through $$\label{eq:dMX} \Delta M_X = \left| {\frac{d\sigma}{d\, M_X}} \right|^{-1} \Delta\sigma .$$ It is important to keep in mind that the statistical component in $\Delta\sigma$ depends on $\sigma$. For particles pair-produced mainly in the $s$-channel, the tree-level cross section depends on the mass through the phase space, which usually shows up as factors of the velocity of the particle: $ \beta = \sqrt{ 1 - (M_X/E_b)^2 } $ where $E_b = \sqrt{s}/2$ is the beam energy – hence, the maximum energy the given particle can have. For the pair-production of scalar particles, $\sigma \propto \beta^3/s$, and for fermions, $\sigma \propto \beta/s$. These simple rules can be modified by radiative corrections, and by beam energy spread, but the basic picture does not change dramatically. We can use this to frame the discussion of the statistical error. It is instructive to minimize $\Delta M_X$ as in Eq. (\[eq:dMX\]) with a simple Ansatz $\sigma = \sigma_0\beta^3/s$. We imagine that $M_X$ is already known approximately, and we want to select the beam energy at which to run the linear collider such that $\Delta M_X$ is minimized, for a given integrated luminosity ${{\cal{L}}}$ and selection efficiency ${\epsilon}$. One easily finds that $\Delta M_X = (s^2/12\sigma_0 M\beta) \Delta\sigma$. Ignoring systematic uncertainties, one might naively expect that $\Delta\sigma$ is proportional to $\sqrt{{N_{\mathrm{ev}}}}$, where ${N_{\mathrm{ev}}}= \sigma{\epsilon}{{\cal{L}}}$ is the number of selected events, which gives us $\Delta M_X \propto \sqrt{\beta}$. This surprising result indicates that zero uncertainty on the mass is obtained at the point at which the signal cross-section vanishes. The fallacy comes in assuming that $\Delta\sigma$ is proportional to $\sqrt{{N_{\mathrm{ev}}}}$, which certainly does not apply as ${N_{\mathrm{ev}}}\rightarrow 0$, even in the absence of background. The transition from a region in which the cross-section is being measured ($\Delta\sigma \propto \sqrt{{N_{\mathrm{ev}}}}$) to a region in which an upper limit is being set (${N_{\mathrm{ev}}}{\,\raisebox{-.1ex}{$_{\textstyle <}\atop^{\textstyle\sim}$}\,}3$) is discussed clearly in Ref. [@FeldmanCousins]. One must construct a confidence belt in the $(M_X,\sigma)$ plane, for a given confidence level — 68% would be appropriate for a measurement. In the present case, this belt will depend on $\sigma(M_X)$, as well as on ${\epsilon}{{\cal{L}}}$. When the expected value for ${N_{\mathrm{ev}}}$ becomes too small, there is no upper bound on $M_X$, and Eq. (\[eq:dMX\]) clearly does not apply. In addition, an account of background estimates, of experimental uncertainties on ${\epsilon}$ and ${{\cal{L}}}$, and theoretical uncertainties on $\sigma(M_X)$ would require that one does not collect data right at threshold, but rather at a point which provides a robust signal somewhat above threshold. Equation (\[eq:dMX\]) still provides a useful guide in the region above threshold. We carried out a Feldman-Cousins confidence-belt construction, and obtained the statistical uncertainty $\Delta M_X$ as a function of the difference $\sqrt{s}/2-M_X$, [*i.e.*]{}, the energy above threshold. Figure \[f:dMXbelt\] shows the result, based on the simple assumed cross-section $\sigma = \sigma_0\beta^3/s$, and approximate values for ${\epsilon}$ and ${{\cal{L}}}$ corresponding to the analysis described in Section \[sec:cut\]. As seen in Fig. \[f:dMXbelt\], the uncertainty on the mass, $\Delta M_X$, really does decrease as $\sqrt{s}/2\rightarrow M_X$, since the sensitivity of $\sigma(M_X)$ to $M_X$ improves more rapidly than the relative precision on the cross-section, $\Delta\sigma/\sigma$, worsens. We turn now to a more realistic case. The theoretical cross section as a function of $\sqrt{s}$ is shown in Fig. \[cs\], for two stop masses ($122.5{\,\, \mathrm{GeV}}$ and $123.5{\,\, \mathrm{GeV}}$). We include QED radiative effects, as described in Section \[sec:th\]. Following the scenario discussed in Ref. [@stop], we consider $M_X \approx 123{\,\, \mathrm{GeV}}$, and beam polarizations of 80% for the electron, and $-60$% for the positron. If we want to use a measurement of the cross section to distinguish these two masses, then clearly the threshold region is the most sensitive. This corresponds to maximizing $d \sigma/d\, M_X$, which will minimize $\Delta M_X$ according to Eq. (\[eq:dMX\]). The lower plot on the left side of Fig. \[cs\] zooms in on the threshold region, to show how much the cross-section differs for two different hypothetical stop masses, and the lower plot on the right shows this difference relative to the cross-section for ${m_{\tilde{t}_1}}= 123{\,\, \mathrm{GeV}}$. Recall the relation of the cross section to experimental quantities: $$\sigma = \frac{{N_{\mathrm{ev}}}- B}{\epsilon \, {{\cal{L}}}} .$$ In a real analysis, $B$, $\epsilon$ and ${{\cal{L}}}$ all carry systematic uncertainties, which must be assessed and taken into account. An ‘optimal’ analysis will keep these to a minimum. Usually the most difficult component in the systematic error comes from the efficiency and acceptance. An absolute cross section requires knowledge of the absolute efficiency, which, in the case of the ${\tilde{t}_1}$ search described in Ref. [@stop], involves charm-tagging as well as the hadronization and fragmentation of the ${\tilde{t}_1}$ and $c$-quark. While a large sample of ${e^+e^-}\rightarrow{\tilde{t}_1}{\tilde{t}_1^*}$ events will allow one to tune Monte Carlo models, and other Standard Model processes may provide large samples of $c$-jets for measuring efficiencies for $c$-tagging, it may be useful to have a method which is relatively insensitive to these sources of systematic uncertainties. The common step toward reducing systematic uncertainties from the efficiency is to work with ratios of cross sections. This also can reduce uncertainties from the luminosity measurement, and potentially, from the background and theoretical signal cross-section as well. We propose to measure the yield of signal events close to threshold, which will be very sensitive to $M_X$, and compare it to the yield near the peak of the excitation curve, which will be insensitive to $M_X$ (see Fig. \[cs\]). We define the observable $$\label{eq:Y} Y(M_X,{\sqrt{s_{\mathrm{th}}}}) \equiv \frac{{N_{\mathrm{th}}}-{B_{\mathrm{th}}}}{{N_{\mathrm{pk}}}-{B_{\mathrm{pk}}}} = \frac{\sigma_{{{\tilde{t}}}}({\sqrt{s_{\mathrm{th}}}})}{\sigma_{{{\tilde{t}}}}({\sqrt{s_{\mathrm{pk}}}})} \cdot \frac{{\epsilon_{\mathrm{th}}}}{{\epsilon_{\mathrm{pk}}}} \cdot \frac{{{\cal{L}}_{\mathrm{th}}}}{{{\cal{L}}_{\mathrm{pk}}}}$$ where ${N_{\mathrm{th}}}$ and ${B_{\mathrm{th}}}$ are the numbers of selected events and estimated background events for ${\sqrt{s_{\mathrm{th}}}}$ near threshold, and ${N_{\mathrm{pk}}}$, ${B_{\mathrm{pk}}}$ are the same quantities for ${\sqrt{s_{\mathrm{pk}}}}$ near the peak of the excitation curve. Anticipating the results of later sections, we have computed the observable $Y$ as a function of ${m_{\tilde{t}_1}}$, and displayed the result in Fig. \[F:y\_vs\_m\]. The slope of the line in Fig. \[F:y\_vs\_m\] depends on several factors, and one can attempt to optimize $Y$ in order to obtain the best measurement of ${m_{\tilde{t}_1}}$. The sensitivity of $Y$ to ${m_{\tilde{t}_1}}$ comes through $\sigma_{{{\tilde{t}}}}({\sqrt{s_{\mathrm{th}}}})$, so ${\sqrt{s_{\mathrm{th}}}}$ should be close to $2{m_{\tilde{t}_1}}$, as discussed above. Mindful of large theoretical and growing experimental uncertainties as ${\sqrt{s_{\mathrm{th}}}}\rightarrow2{m_{\tilde{t}_1}}$, we have selected ${\sqrt{s_{\mathrm{th}}}}= 260{\,\, \mathrm{GeV}}$, which is $14{\,\, \mathrm{GeV}}$ above the nominal threshold for a stop with ${m_{\tilde{t}_1}}= 123{\,\, \mathrm{GeV}}$. We find the peak cross-section occurs at ${\sqrt{s_{\mathrm{pk}}}}\approx 370{\,\, \mathrm{GeV}}$, but $\sqrt{s} = 500{\,\, \mathrm{GeV}}$ would also serve well. Reducing the statistical uncertainty on $Y$ to an absolute minimum would require maximizing the integrated luminosity at threshold, ${{\cal{L}}_{\mathrm{th}}}$, but in reality one would not run the ILC at ${\sqrt{s_{\mathrm{th}}}}= 260{\,\, \mathrm{GeV}}$ for very long, and in practice ${{\cal{L}}_{\mathrm{th}}}= 50~{{\mathrm{fb}^{-1}}}$ is already adequate. We assume ${{\cal{L}}_{\mathrm{pk}}}= 200~{{\mathrm{fb}^{-1}}}$. We computed the cross sections with the program [Calvin]{} [@calvin], which includes next-to-leading (NLO) order supersymmetric QCD corrections, and which was modified for this work to include resummed Coulomb corrections near threshold (see Section \[sec:th\]). For two common choices of beam polarization, the cross-sections are $$\begin{aligned} &P(e^-) = -80\% / P(e^+) = +60\%: & \sigma({\sqrt{s_{\mathrm{th}}}}) &= 17.4~{{\mathrm{fb}}}& \sigma({\sqrt{s_{\mathrm{pk}}}}) &= 72~{{\mathrm{fb}}}, \\ &P(e^-) = +80\% / P(e^+) = -60\%: & \sigma({\sqrt{s_{\mathrm{th}}}}) &= 77~{{\mathrm{fb}}}& \sigma({\sqrt{s_{\mathrm{pk}}}}) &= 276~{{\mathrm{fb}}}, \\ \end{aligned}$$ where $P<0$ stands for left-handed polarization and $P>0$ for right-handed polarization. We choose the second set of polarization values since it leads to a much better signal-to-background ratio. For the computation of the observable $Y$ depicted in Fig. \[F:y\_vs\_m\], we employed the results of the “cut-based” analysis described in Section \[sec:cut\]. The efficiencies at threshold and peak are ${\epsilon_{\mathrm{th}}}= 0.34$ and ${\epsilon_{\mathrm{pk}}}= 0.21$, and the total background cross-sections are $2.5~{{\mathrm{fb}}}$ and $10.3~{{\mathrm{fb}}}$, respectively. The strong variation of $Y$ with ${m_{\tilde{t}_1}}$ in Fig. \[F:y\_vs\_m\] indicates that a precise measurement of $Y$ will lead to a precise value for ${m_{\tilde{t}_1}}$. The shaded horizontal band corresponds to a 3% uncertainty on $Y$, resulting in ${\Delta m_{\tilde{t}_1}}= 0.2{\,\, \mathrm{GeV}}$, which would be far better than the result reported in Ref. [@stop]. We consider now the impact of systematic uncertainties on the observable $Y$, and Eq. (\[eq:Y\]) provides our starting point. For the event selection criteria described in Sections \[sec:cut\] and \[sec:ida\], the signal is much bigger than the background, so the main experimental uncertainties will come from ${\epsilon}$. The values for ${\epsilon}$ at threshold and on the peak come from Monte Carlo simulations of the signal process. Systematic errors arise when these simulations do not match reality perfectly. For example, the calibration of the calorimeter energy measurement for real data may be slightly different than is simulated, in which case the efficiency for a cut on the total visible energy ${E_{\mathrm{vis}}}$ as estimated from the simulation will be slightly incorrect. One can express the impact of this error on the efficiency as ${\epsilon^{\mathrm{true}}}= {\epsilon^{\mathrm{est}}}(1+\delta)$, so that $\delta$ is the [*relative*]{} shift in the efficiency. Then the impact on the observable $Y$ is simply $${Y^{\mathrm{true}}}= {Y^{\mathrm{est}}}\left( \frac{1+{\delta_{\mathrm{th}}}}{1+{\delta_{\mathrm{pk}}}} \right) \qquad \frac{{Y^{\mathrm{true}}}-{Y^{\mathrm{est}}}}{{Y^{\mathrm{true}}}} \approx {\delta_{\mathrm{pk}}}- {\delta_{\mathrm{th}}}.$$ Thus, if the systematic uncertainties ${\delta_{\mathrm{pk}}}$ and ${\delta_{\mathrm{th}}}$ are correlated, and if they have the same relative impact on ${\epsilon_{\mathrm{pk}}}$ and ${\epsilon_{\mathrm{th}}}$, the net effect on $Y$ will be zero, and there will be no error on ${m_{\tilde{t}_1}}$. For some systematic effects, the errors will be correlated, but of a different magnitude at the two energies, so that the cancellation $|{\delta_{\mathrm{pk}}}-{\delta_{\mathrm{th}}}|$ will not be complete. For other systematic effects, the errors will be uncorrelated, in which case there is no cancellation. Clearly the analysis should be designed in such a way as to take advantage of this cancellation. In practice, this means that the cuts should have a similar impact on the signal at both energies. For the present application, there is a large degree of cancellation, leading to a greatly reduced systematic uncertainty on the observable $Y$, and hence on ${m_{\tilde{t}_1}}$. The details are given in Section \[sec:exp\]. We proceed now to a detailed and realistic simulation, and the description of two fully-developed event selection methods. Event Selection and Analysis {#sec:ana} ============================ At an ${e^+e^-}$ collider, scalar top quarks would be produced in pairs, and decay to a $c$-quark and the lightest neutralino: $$e^+e^- \to \tilde{t}_1 \, \tilde{t}_1^* \to c {\tilde{\chi}^0}_1 \, \bar{c} {\tilde{\chi}^0}_1. \label{eq:signal}$$ The stop quarks live long enough to hadronize before decaying, so the final state signature consists of two charm quark jets, missing energy and possibly additional jets due to the hadronization process and gluon radiation. In the following sections, the method described in Section \[sec:method\] will be applied to the theoretical parameter point of Ref. [@heavyq]. The weak-scale MSSM parameters are $$\begin{aligned} m^2_{\rm\tilde{U}_3} &= -99^2{\,\, \mathrm{GeV}}, &\qquad m_{\rm\tilde{Q}_3} &= 4330{\,\, \mathrm{GeV}}, & m_{\rm\tilde{Q},\tilde{U},\tilde{D},\tilde{L},\tilde{R}_{1,2}} &= 10~{\mathrm{TeV}}, \\ M_1 &= 118.8{\,\, \mathrm{GeV}}, & M_2 &= 225{\,\, \mathrm{GeV}}, & |\mu| &= 225{\,\, \mathrm{GeV}}, \\ A_t &= -1100{\,\, \mathrm{GeV}}, & m_{\rm A^0} &= 800{\,\, \mathrm{GeV}}, & \phi_\mu &= 0.2, & \tan\beta &= 5. \end{aligned}$$ The corresponding tree-level masses are: $$\begin{aligned} && {m_{\tilde{t}_1}}&= 122.5{\,\, \mathrm{GeV}}, & m_{\tilde{t}_2} &= 4333{\,\, \mathrm{GeV}}, \\ {m_{\tilde{\chi}^0_{1}}} &= 107.2{\,\, \mathrm{GeV}}, & {m_{\tilde{\chi}^0_{2}}} &= 170.8{\,\, \mathrm{GeV}}, & {m_{\tilde{\chi}^0_{3}}} &= 231.2{\,\, \mathrm{GeV}}, & {m_{\tilde{\chi}^0_{4}}} &= 297.7{\,\, \mathrm{GeV}}, \\ && {m_{\tilde{\chi}^\pm_{1}}} &= 162.7{\,\, \mathrm{GeV}}, & {m_{\tilde{\chi}^\pm_{2}}} &= 296.2{\,\, \mathrm{GeV}}, \label{eq:masses} \end{aligned}$$ and the light stop state is almost completely right-chiral, $\cos\theta_{\tilde{t}} = 0.010$. As a result of the small stop-neutralino mass difference, the stop almost completely decays through the loop-induced process into a charm and neutralino, $\tilde{t} \to c \, {\tilde{\chi}^0}_1$. Due to the loop suppression of the decay, the stop is expected to hadronize before decaying. We have carried out realistic experimental simulations, and will present the analysis of relevant systematic effects. Simulation ---------- Both the signal and background events are generated with [Pythia 6.129]{} [@pythia]. The cross-sections for the signal process were computed with [Calvin]{} [@calvin] with some improvements as in Ref. [@slep]. The relevant background processes have been computed by adapting the Monte Carlo code used in Ref. [@slep] and by [Grace 2.0]{} [@grace], with cross-checks with [CompHep 4.4]{} [@comphep]. The simulation and cross-section calculations incorporated beamstrahlung for cold ILC technology as parameterized in the program [Circe 1.0]{} [@circe]. Table \[tab:xsec\] summarizes the predicted signal and background cross-sections. To avoid the infrared divergence of the two-photon background process, a cut on the minimal transverse momentum is applied, ${p_T}> 5$ GeV. Backgrounds from supersymmetric processes will be discussed below. Table \[tab:generated\] lists the numbers of events generated and equivalent luminosity based on the cross-sections in Table \[tab:xsec\]. Hadronization of the final state charm quark and the intermediate stop quark are a key issue in this study. The Lund string fragmentation model was used together with the Peterson fragmentation function [@fragfunc]. The stop fragmentation is simulated [@pythiastops] by labeling the stop quark as a stable particle in an intermediate step, and switching on the stop decay again after stop fragmentation. The modeling of the hadronization spectrum of the stop is described in Ref. [@kraan]. The dominant lightest stop hadron states are mesons composed of a stop and an up or down quark. The [Simdet]{} detector simulation [@simdet] was used, describing a typical ILC detector. The analysis used the [N-Tuple]{} tool [@ntuple], which incorporates jet-finding algorithms. In order to reduce the size of the ntuples, several pre-selection cuts were applied, as was done for the previous analysis [@stop]: $$\begin{aligned} 4 < {N_{\mathrm{tracks}}}< 50, \qquad {p_T}> 5{\,\, \mathrm{GeV}}, \cr |{\cos\theta_{\mathrm{thrust}}}| < 0.8, \qquad |p_L / p_{\mathrm{tot}}| < 0.9, \cr {E_{\mathrm{vis}}}< 0.75\sqrt{s}, \qquad m_{\mathrm{inv}} < 200 {\,\, \mathrm{GeV}}. \label{ntuplecuts}\end{aligned}$$ Most of these cuts have very little impact on the signal efficiency. Sequential-Cut Analysis {#sec:cut} ----------------------- Although Standard Model background processes are several orders of magnitude larger than the stop signal process, the background contributions can be reduced to an acceptable level by suitable selection cuts. This work follows the analysis of Ref. [@stop], but makes some adjustments to accommodate the stop fragmentation effects, and to take advantage of the cancellation of systematic uncertainties as discussed in Section \[sec:method\]. The event selection begins with some basic and common kinematic cuts based on global event quantities. The visible energy, ${E_{\mathrm{vis}}}$, must be less than $0.3\sqrt{s}$ to ensure a large missing-energy signature. It must be greater than $0.1\sqrt{s}$ to suppress the bulk of the two-photon events. Similarly, the number of reconstructed charged tracks must indicate real hadronic jets, so we require ${N_{\mathrm{tracks}}}\ge 5$. In order to suppress $W e\nu$ and ${q\bar{q}}$ signals, we place an upper bound ${N_{\mathrm{tracks}}}\le 25$ at threshold and ${N_{\mathrm{tracks}}}\le 20$ at peak. These cuts on ${N_{\mathrm{tracks}}}$ remove only a couple percent of the signal. We place one more kinematic and one topological cut to further reduce the backgrounds. The cuts values are carefully tuned to achieve a low systematic uncertainty for the observable $Y$, as well as a good background rejection. In practice, this means aiming to remove approximately the same amount of signal at the two center-of-mass energies, rather than achieving the highest signal efficiencies. In particular, the efficiency at ${\sqrt{s_{\mathrm{pk}}}}$ is relatively unimportant since we anticipate a large luminosity and a large signal cross-section there. The thrust value is useful for eliminating ${q\bar{q}}$ and two-photon events. As shown in Fig. \[f:marg\_thrust\], the thrust distribution for the signal is rather different at the two center-of-mass energies, so we require $0.77\le T\le 0.97$ at ${\sqrt{s_{\mathrm{th}}}}$ and $0.55\le T\le 0.90$ at ${\sqrt{s_{\mathrm{pk}}}}$. Similarly, the event ${p_T}$, calculated from all energy flow objects in the event, is crucial for eliminating the two-photon background. Our study indicates that a minimum cut ${p_T}> 15{\,\, \mathrm{GeV}}$ is needed. We tighten this cut to ${p_T}> 22{\,\, \mathrm{GeV}}$ at ${\sqrt{s_{\mathrm{pk}}}}$ in order to eliminate the same amount of signal events as are eliminated at ${\sqrt{s_{\mathrm{th}}}}$. Fig. \[f:marg\_pt\] shows that cutting at ${p_T}= 22{\,\, \mathrm{GeV}}$ at ${\sqrt{s_{\mathrm{pk}}}}$ places the cut at almost the same point in the ${p_T}$ distribution for both center-of-mass energies. An upper cut on ${p_T}$ helps reduce the $W e\nu$ background, so we require ${p_T}< 45{\,\, \mathrm{GeV}}$ at ${\sqrt{s_{\mathrm{th}}}}$ and ${p_T}< 50{\,\, \mathrm{GeV}}$ at ${\sqrt{s_{\mathrm{pk}}}}$, which again reflects our effort to minimize the systematic uncertainty. One might expect that the signal process (\[eq:signal\]) produces only two jets. However, additional soft jets can emerge from the stop hadronization process and also from the decay of the stop hadron. In order to maintain a high efficiency, and to avoid large systematic uncertainties from the modeling of the rate and characteristics of these extra jets, events with more than two jets should not be rejected. However, to suppress the background processes effectively, extra jets are allowed only when their energy falls below a certain cut-off value. To be specific, if there are more than two jets in an event, only two of the jets are allowed to have energies above $25{\,\, \mathrm{GeV}}$. In this paper, we refer to this requirement as the “extra-jet veto.” Furthermore, if there are more than three jets, the most energetic jet cannot be too energetic – its energy must be less than $35{\,\, \mathrm{GeV}}$. These cuts are useful against the troublesome $W e\nu$ backgrounds, especially at ${\sqrt{s_{\mathrm{pk}}}}$. Further substantial improvements of the signal-to-background ratio can be achieved by exploiting kinematic and topological correlations between the two $c$-quark jets. Therefore it is necessary to identify them from the plurality of jets, and for this we use charm tagging as realized using a neural network [@kuhlhiggs]. The neural network uses information about the vertex position of a jet based on a topological vertex finder, the impact parameter probability, the momenta of the associated tracks and the reconstructed mass. It has been optimized to single out charm jets with an energy that is typical for light stops, while rejecting light quark jets coming from $We\nu$ background. Each jet in an event is tested with the charm tagger, and a charm flag ${F^{(c)}_i}$ is set (ideally, ${F^{(c)}_i}= 1$ indicates a charm jet). First, if a displaced vertex is reconstructed, the jet is tagged positively with ${F^{(c)}_i}= 1$. A displaced vertex is found roughly 50% of the time for a charm jet, and less than 20% of the time for a light quark jet. If no such vertex is reconstructed, then the neural network is employed, which produces a charm flag value between zero and one, $0 \le {F^{(c)}_i}\le 1$. The output of the neutral network is shown in Fig. \[f:ctag\_ann\], for the second of the two charm-tagged jets. We consider the two jets in the event with the highest values of ${F^{(c)}_i}$, and require ${P_{\mathrm{c}}}\equiv {F^{(c)}_1}\times{F^{(c)}_2}> 0.6$, which is very effective at eliminating events with no charm-quark jets while retaining a high efficiency for signal events. In particular, the $W e\nu$ background is reduced by more than half. Fig. \[f:ctag\_product\] compares the quantity ${P_{\mathrm{c}}}$ for signal events and $We\nu$ background which have passed the kinematic event selection cuts. Since half of the $We\nu$ events have a genuine charm jet, it is the value of ${F^{(c)}_i}$ for the second jet which best distinguishes signal and background. A further substantial reduction can be obtained from cuts on the invariant mass of the the two best charm-tagged jets – we veto events in which that mass is consistent with the $W$-boson mass. The event selection cuts are summarized in Table \[tab:cuts\], for the two center-of-mass energies, ${\sqrt{s_{\mathrm{th}}}}= 260{\,\, \mathrm{GeV}}$ and ${\sqrt{s_{\mathrm{pk}}}}= 500{\,\, \mathrm{GeV}}$. These follow the pre-selection cuts listed in Eq. (\[ntuplecuts\]). Our estimates of the numbers of signal and background events surviving the cuts listed in Table \[tab:cuts\] are summarized in Table \[tab:nev\]. If, in a given channel, no simulated events remain after applying our cuts, we list an upper limit corresponding to one simulated event, and we count this amount in the total background estimate. As evident from the table, the background can be greatly reduced for ${\sqrt{s_{\mathrm{th}}}}$, resulting in a very good signal-to-background ratio. At ${\sqrt{s_{\mathrm{pk}}}}$, on the other hand, a large background from single-$W$ boson production is left. For unpolarized beams, the resulting signal-to-background ratio is $0.7$ While this would allow an unambiguous discovery of stop quarks (see Section \[S:discovery\]), it is not a very good basis for precision measurements of the stop mass. Fortunately, the signal-to-background ratio can be greatly improved by using polarized beams. With an essentially right-handed electron beam and left-handed positron beam, the signal is enhanced, while most backgrounds are substantially suppressed. As a result, the signal-to-background ratio at ${\sqrt{s_{\mathrm{pk}}}}= 500{\,\, \mathrm{GeV}}$ is improved from $0.7$ to $5.3$. We checked for possible supersymmetric backgrounds. The main concern is chargino pair production with the decay channel $\tilde{\chi}_1^+ \rightarrow {{\tilde{t}_1}}b$. We simulated a sample of these decays, consistent with our benchmark scenario, and found that the cuts listed in Table \[tab:cuts\] completely eliminate this background source. With the results listed in Table \[tab:nev\] for polarized beams, we can compute the observable $Y$ and its statistical error, obtaining $Y = 0.1082\pm 0.0034$ with a relative error of 3.1%. The corresponding stop quark mass would be $${m_{\tilde{t}_1}}= (122.5 \pm 0.19) {\,\, \mathrm{GeV}}\label{staterrcuts}$$ where the uncertainty depends on the slope, $dY/dM = -0.01755$, at $Y = 0.1082$. Without positron polarization, $P(e^+) = 0$, the precision of the measurement is reduced by roughly 20%, resulting in $\Delta Y / Y = 3.7$% and $\Delta{m_{\tilde{t}_1}}= 0.23{\,\, \mathrm{GeV}}$. Even in this case the statistical error is rather small. It should be recalled that the production cross-section is a strong function of the mixing angle, so the statistical error $\Delta{m_{\tilde{t}_1}}$ will also depend on it. In our reference scenario, the light stop eigenstate is almost completely composed of the partner of the right-handed stop, ${{\tilde{t}}}_1 \approx {{\tilde{t}}}_{\rm R}$, with the mixing angle $\cos\theta_{\tilde{t}} = 0.01$. While this scenario is preferred by electroweak precision data and the explanation of baryogenesis, an experimental analysis should consider all possible values for the stop mixing angle. For other values of $\cos\theta_{\tilde{t}}$, the production cross-section can change drastically, depending on the beam polarization. As concrete example, we consider two larger values of $\cos\theta_{\tilde{t}}$: $$\begin{aligned} \cos\theta_{\tilde{t}} &= 0.6: & \sigma_{\rm L,260} &= 52 \mbox{ fb}, & \sigma_{\rm L,500} &= 194 \mbox{ fb}, \\ && \sigma_{\rm R,260} &= 39 \mbox{ fb}, & \sigma_{\rm R,500} &= 148 \mbox{ fb}, \nonumber \\[1ex] \cos\theta_{\tilde{t}} &= 1.0: & \sigma_{\rm L,260} &= 169 \mbox{ fb}, & \sigma_{\rm L,500} &= 577 \mbox{ fb}, \\ && \sigma_{\rm R,260} &= 6.9 \mbox{ fb}, & \sigma_{\rm R,500} &= 30 \mbox{ fb}. \nonumber \end{aligned}$$ Here $\sigma_{{\rm L/R},E}$ stands for the stop production cross-section at center-of-mass energy $E{\,\, \mathrm{GeV}}$, and with beam polarization combinations $P(e^-) = -80\% /P(e^+) = +60\%$ and $P(e^-) = +80\% / P(e^+) = -60\%$, respectively. If the stop is dominantly left-chiral, with $|\cos\theta_{\tilde{t}}| > 0.5$, the production cross-section is substantially larger for left-handed electron and right-handed positron polarization, opposite to the situation for a right-chiral stop. Therefore, for large values of $|\cos\theta_{\tilde{t}}|$, it is better to use the beam polarizations $P(e^-) = -80\% / P(e^+) = +60\%$, even though one has to deal with much larger Standard Model background. The largest background, $e^+e^- \to W e \nu$, amounts to about $12800$ events at $\sqrt{s} = 500{\,\, \mathrm{GeV}}$ and ${{\cal{L}}}= 200~{{\mathrm{fb}^{-1}}}$ for this polarization. Nevertheless, due to large signal cross-sections, the resulting statistical error is still small, as summarized in Table \[tab:mixang\], which demonstrates that, for all values of the stop mixing angle, one can measure the stop mass with a statistical error better than $0.3{\,\, \mathrm{GeV}}$ using our method and an appropriate choice of beam polarization. Iterative Discriminant Analysis {#sec:ida} ------------------------------- A traditional, sequential-cut analysis was presented in the previous section. Often, more advanced multi-variate techniques can boost the sensitivity of a search. We investigated the efficacy of an Iterative Discriminant Analysis (IDA) for the purposes of measuring the stop quark mass based on the observable $Y$. The IDA method [@ida] is a modified Fisher Discriminant Analysis, the two main differences are the introduction of a non-linear discriminant function and iterations in order to enhance the separation of signal and background. Two IDA steps have been performed. In order to have two independent samples for the derivation of the IDA function and for the expected performance, the signal and background samples were divided into two equally-sized samples. For this analysis, the same kinematic variables and simulated event samples as in the cut-based analysis are used, including the charm tagging flags ${F^{(c)}_i}$. Before the multi-variable analysis is performed, cuts on the input variables, so-call “tail cuts,” are applied in order to improve the IDA performance. This is achieved by reducing the number of input events, and thus reducing the computational time. From the distributions of the input variables for the signal and background events, the IDA method calculates a separating surface in the multi-dimensional parameter space between signal and background events. The IDA output variable has a different shape for signal and background events, and therefore a cut on this variables is used to separate signal and background. In the first IDA step a cut is placed on this IDA output variable such that 99.5% of the signal efficiency are kept. The number of background events is largely reduced. From the smaller background sample and the 99.5% remaining signal events again a new IDA output variable is constructed. The cut on the IDA output variable in this second IDA step defines the signal efficiency and the corresponding number of background events. Different working points are possible: they are defined by choosing a certain signal efficiency and obtaining the corresponding number of background events. The working point was determined by the expected error on ${m_{\tilde{t}_1}}$. The results of the IDA method with stop fragmentation are shown in Fig. \[fig:ida2\] and Table \[tab:ida2\] expressed as number of expected background events for each contributing background process. As before, in the channels where no event is left after the signal selection, an upper limit corresponding to one simulated event is given in the table. The IDA method achieves a significantly more powerful discrimination between signal and background than the analysis with conventional cuts. When allowing similar background levels as for the cut-based analysis in Table \[tab:nev\], signal efficiencies of ${\epsilon_{\mathrm{th}}}= 0.387$ for ${\sqrt{s_{\mathrm{th}}}}= 260 {\,\, \mathrm{GeV}}$ and ${\epsilon_{\mathrm{pk}}}= 0.416$ for ${\sqrt{s_{\mathrm{pk}}}}= 500 {\,\, \mathrm{GeV}}$ are obtained. With the resulting event numbers given in Table \[tab:ida2\] for $P(e^-)/P(e^+) =$ +80%/$-$60%, the ratio quantity in eq.  amounts to $Y = 0.0648\pm 0.0018$ with a relative statistical error of $2.7\%$, translating into $${m_{\tilde{t}_1}}= (122.5 \pm 0.17) {\,\, \mathrm{GeV}}\label{staterrida}$$ where the uncertainty on the mass depends on the slope $dY/dM = -0.01052$. The higher signal efficiency and lower background achieved by the two-step IDA results in a slightly smaller statistical uncertainty ([*cf.*]{} Eq. (\[staterrcuts\])). Experimental Systematics {#sec:exp} ======================== The high signal efficiency and low backgrounds achieved in both the cut-based analysis (Section \[sec:cut\]) and the IDA (Section \[sec:ida\]) deliver an excellent statistical precision – $\Delta{m_{\tilde{t}_1}}< 0.2{\,\, \mathrm{GeV}}$. It remains to investigate systematic uncertainties, which were the dominant contribution in the previous analysis of Ref. [@stop]. We considered the following important sources of systematic errors: \[page:listsyst\] - detector calibration (energy scale) - charm tagging - hadronization / fragmentation - neutralino mass - luminosity measurement - beam energy spectrum - background estimate The first four sources pertain to the signal efficiency. We discuss these sources in detail in the context of the sequential-cut analysis detailed in Sec. \[sec:cut\] first, and then briefly report the results obtained from the IDA method described in Sec. \[sec:ida\]. Systematics for the Sequential-Cut Analysis {#sec:systcuts} ------------------------------------------- Many of the kinematic quantities used in these selections depend on a correct calibration of the calorimetry. Based on experience from LEP [@jeten], we assume an uncertainty of $1\%$ on the overall energy scale, which is rather pessimistic for a future ILC detector. We scaled simultaneously all kinematic quantities through a range of $\pm 6\%$ and observed correlated shifts in the overall selection efficiency at the two center-of-mass energies. In particular, the ${p_T}$ cut is sensitive to this kind of scale error, prompting us to tune the cut at ${\sqrt{s_{\mathrm{pk}}}}$ to achieve a minimal residual uncertainty for the [*ratio*]{} of efficiencies, as discussed in Section \[sec:cut\]. Figure \[f:scale\_eff\] shows how the selection efficiencies change as a function of the scale factor. Using our optimized ${p_T}$ cut shown in Fig. \[f:marg\_pt\], one sees a parallel behavior at threshold (upper solid line) and at peak (lower solid line). This leads to a very good cancellation for the ratio of efficiencies, as shown by the solid line in Fig. \[f:scale\_reldiff\]. If we had optimized for efficiency only, then we would have used nearly the same ${p_T}$ cuts at the peak as we use at threshold. However, this would have given a rather different dependence on the scale, as indicated by the dashed line in Fig. \[f:scale\_eff\], and therefore a much stronger dependence of the ratio of efficiencies on the scale, as shown by the dashed line in Fig. \[f:scale\_reldiff\]. With our best cuts, an uncertainty of $\pm 1\%$ on the calorimeter energy scale translates into an uncertainty of less than $0.6\%$ on the ratio of efficiencies. The efficiency for track reconstruction should be very high at an ILC detector. However, there is always an uncertainty in the value for that efficiency, which we took to be about $0.5\%$. We propagated this uncertainty to the cut on ${N_{\mathrm{tracks}}}$, since a random loss of tracks changes the shape of the distribution of ${N_{\mathrm{tracks}}}$. Since our cut is quite loose, however, the resulting uncertainty on the ratio of efficiencies is negligible. Knowledge of the efficiency for charm jets for a given cut on ${F^{(c)}_i}$ is not easy to obtain. Based on the work described in Ref. [@kuhlhiggs], we assumed an uncertainty of $0.5\%$ on the charm efficiency. Although one might expect this uncertainty to be correlated between the two center-of-mass energies, we assumed no strong correlation and assign an uncertainty of $0.5\%$ for the ratio of efficiencies. The observable $Y$ depends on the integrated luminosity at both center-of-mass energies. Traditionally, the luminosity is measured using Bhabha scattering, for which highly accurate theoretical cross-sections are available. The limiting systematic uncertainty for the LEP detectors comes from the acceptance of the luminosity calorimeters. Such an uncertainty would essentially cancel in the ratio of luminosities. Alternatively, one could define an effective luminosity through another clean QED process, such as ${e^+e^-}\rightarrow {\mu^+\mu^-}$, for which there is essentially no theoretical or experimental systematic effect. The precision of the ratio of luminosities would come from the statistical uncertainty on the number of ${\mu^+\mu^-}$ events recorded, which we estimate to be about $0.4\%$; this is the figure we use in this study. Apart from instrumental issues such as the energy scale, track reconstruction efficiency, charm tagging efficiency and the measurement of the integrated luminosity, the estimate of the signal efficiency will depend on the modeling of the signal itself. While the production of a pair of stop quarks is well understood and can be modeled accurately, the non-perturbative aspects of the formation of stop hadrons which then decay into two or more jets are more problematic[^1]. We have attempted to account for this fundamental difficulty by varying the parameter which controls the fragmentation function in our simulations. We used PYTHIA and the Peterson fragmentation function, with values of the fragmentation parameter reported by the OPAL Collaboration [@fraglep]. To be specific, we took $\epsilon_c = -0.031\pm 0.011$ and $\epsilon_{b} = -0.0050 \pm 0.0015$, and propagated $\epsilon_b$ according to the assumption that ${\epsilon_{\tilde{t}}}= \epsilon_b\, (m_b/m_{\tilde{t}})^2$ [@fragfunc; @stoplep]. We varied $\epsilon_c$ and ${\epsilon_{\tilde{t}}}$ independently, and measured the efficiencies at the two center-of-mass energies. The impact of varying $\epsilon_c$ is small. For variations of the stop quark fragmentation, however, we find that the variation of the efficiencies is rather different, so the desired cancellation of systematic uncertainties is not achieved. In fact, most of the systematic uncertainty comes from the cut on ${p_T}$, and has an [*opposite*]{} sense at ${\sqrt{s_{\mathrm{th}}}}$ and ${\sqrt{s_{\mathrm{pk}}}}$. The range in $\epsilon_b$ used in our simulations is quite broad. The more advanced measurements of $b$-quark fragmentation from ALEPH [@alephfrag] and OPAL [@opalfrag] give more constrained values: $\epsilon_b = -0.0031\pm 0.0006$ (ALEPH) and $\epsilon_b = -0.0041\pm 0.0004$ (OPAL), using rather different methodologies. On the basis of these measurements, one could argue that our assumed variation in $\epsilon_b$ is too large by a factor of two. Rather than relying on LEP measurements to predict stop quark fragmentation, we investigated the potential of ILC data to constrain the fragmentation. We already noted that most of the sensitivity to stop quark fragmentation comes from the cut on ${p_T}$; however, the change in the shape of the ${p_T}$ distribution is small. (The fact that the quantities chosen for cuts are insensitive to ${\epsilon_{\tilde{t}}}$ is a strong point of the analysis, of course.) We examined other kinematic quantities and found a few which exhibit clear changes in shape when we vary ${\epsilon_{\tilde{t}}}$. Four examples are shown in Fig. \[f:fragplots\]. The ${M_{\mathrm{vis}}}/\sqrt{s}$ distribution shows pronounced shifts as a function of ${\epsilon_{\tilde{t}}}$. Given an accumulation of a few $\times 10^4$ events at ${\sqrt{s_{\mathrm{pk}}}}$, one can show that the mean of this distribution alone would allow a differentiation of our three values $\epsilon_b = -0.0050\pm 0.0015$ at more than ten sigma (statistical uncertainty only). If the energy scale uncertainty were a problem, then one could normalize ${M_{\mathrm{vis}}}$ to ${E_{\mathrm{vis}}}$ — a clear distinction between the three distributions is visible near the peak of ${M_{\mathrm{vis}}}/{E_{\mathrm{vis}}}$. The energy of the third jet, when it exists, shows a good sensitivity to ${\epsilon_{\tilde{t}}}$. (Recall that the jets are ordered in decreasing energy.) Better, perhaps, is the smaller of the two di-jet invariant masses formed by combining this third jet with the first and second jets. Although these considerations are not equivalent to a full study of a possible measurement of the stop fragmentation, they do indicate that a good measurement should be possible, well beyond the extrapolation of LEP results on $\epsilon_b$ to ${\epsilon_{\tilde{t}}}$ and all the attendant assumptions behind such an extrapolation. On this basis, we judge that the uncertainty on the stop fragmentation would be no larger than one-fourth of the uncertainty obtained by comparing simulations with $\epsilon_b = -0.0035$, $-0.0050$ and $-0.0065$, which corresponds to $\Delta{\epsilon_{\tilde{t}}}= (2.5\times 10^{-6})/4 = 0.6\times 10^{-6}$. Another empirical quantity which induces an uncertainty on the selection efficiency is the mass of the neutralino, ${m_{\tilde{\chi}^0_{1}}}$. The mass difference ${m_{\tilde{t}_1}}-{m_{\tilde{\chi}^0_{1}}}$ clearly impacts the kinematic distributions, so the efficiency estimated from the simulation depends directly — and dramatically — on ${m_{\tilde{\chi}^0_{1}}}$. We simulated a sample with ${m_{\tilde{\chi}^0_{1}}} = 108.2{\,\, \mathrm{GeV}}$, which is one${\,\, \mathrm{GeV}}$ higher than our default value. The relative change in the selection efficiencies is roughly $10\%$. Since the changes are parallel, the ratio of efficiencies change by only $2.8\%$, once again illustrating the robustness of this method. Other studies have shown [@heavyq] that ${m_{\tilde{\chi}^0_{1}}}$ can be measured with an accuracy of $0.3{\,\, \mathrm{GeV}}$ or better, so we assign an uncertainty of $0.8\%$ due to the unknown neutralino mass. The predicted cross-sections depend on the beam energy and the beam energy spectrum. Due to beamstrahlung and other effects, the mean energy can be significantly lower than the peak value. While we used CIRCE for taking this fact into account, the question remains how well a program such as CIRCE can be validated using real data. This question has been addressed by several authors, using, for example, Bhabha scattering and radiative returns to the $Z$ pole [@beamspectrum]. The studies indicate that models for the spectrum and the beam energy can be constrained directly from the data to an accuracy on the order of $0.1{\,\, \mathrm{GeV}}$. We include this uncertainty as a direct uncertainty on ${m_{\tilde{t}_1}}$, but not on the observable $Y$. Finally, we must consider uncertainties on the estimated contributions from background processes. The SM backgrounds fall naturally in two categories: two-photon interactions, which are difficult to predict, and the others, which involve high-${p_T}$ electro-weak processes, for which direct theoretical calculations are reliable. We also consider background contributions from the production of other supersymmetric particles. Two-photon interactions cannot be fully described by perturbative QCD, and so phenomenological models are required [@twophoton_review]. These must be tuned to match real data, which is difficult due to the fact that most two-photon scattering events emit particles that are lost down the beam pipe. Parameters pertaining to the softest interactions are difficult to constrain; fortunately, such interactions are easily eliminated by our cuts on ${p_T}$, ${N_{\mathrm{tracks}}}$ and ${E_{\mathrm{vis}}}$. Many of the events coming in at higher ${p_T}$ can be described using models with a basis in perturbative QCD. The investigations of the photon structure functions by the LEP Collaborations illustrate the procedure of tuning parameters and confronting the models with real data, leading to interesting conclusions about the success of the various models [@twophoton_LEP; @twophoton_OPAL; @twophoton_LEPWG]. It is not straightforward to translate those conclusions into constraints on our two-photon background, although Figs. 19, 21 and 23 in the report from the LEP Working Group [@twophoton_LEPWG] and Figs. 5-7 in the OPAL paper [@twophoton_OPAL] are quite relevant for our study, and indicate that modeling the tails of the ${p_T}$ distribution at the 20% level should be possible. Assuming that the study of two-photon interactions would be greatly extended at the ILC, we assign a 20% uncertainty to the background estimate for two-photon interactions. The resulting relative uncertainty on the $Y$ observable is $0.8\%$. The dominant background is ${e^+e^-}\rightarrow We\nu$, according to Table \[tab:nev\] (and Table \[tab:ida2\]). A precise prediction of this background requires accurate measurements of this process combined with the calculation of higher-order radiative corrections. While a complete NLO calculation of that process is missing, a recent result for the related process of $W$ pair production [@wwnlo] suggests that a NLO calculation of $W e \nu$ is feasible within the next years with an error remaining well below $0.5\%$. The impact on $Y$ is negligible, on the order of $0.1\%$, relative. A summary of the experimental systematic uncertainties for the sequential-cut analysis is shown in Table \[tab:sys\]. A good cancellation of experimental systematics is obtained, except for the stop quark fragmentation uncertainty and the background estimation. The goal of the new method is therefore fairly well achieved with this set of sequential cuts. The implications for the measurement of the observable $Y$ and the inferred mass ${m_{\tilde{t}_1}}$ are discussed in Sec. \[sec:comb\]. Systematics for the Iterative Discriminant Analysis {#sec:systida} --------------------------------------------------- We evaluated the impact of the sources of systematics listed on page  in a manner similar to the methods of Sec. \[sec:systcuts\]. We scaled all kinematic inputs to the IDA according to an overall energy scale uncertainty. The systematic uncertainty from the number of tracks is assumed to be negligible. The variations in the charm and stop quark fragmentation functions were evaluated as before. The sensitivity to ${m_{\tilde{\chi}^0_{1}}}$ and the uncertainty on the background estimate were evaluated precisely as above. The luminosity uncertainty is, of course, the same as in the sequential-cut analysis. The resulting systematic uncertainties are listed in Table \[tab:sysida\]. We observe a much larger uncertainty coming from the scale uncertainty as compared to the sequential-cut analysis ([*cf.*]{} Table \[tab:sys\]). With multi-variate methods such as the IDA, it is difficult to ascertain what role any given quantity plays in the final output variable, so no dissection of the IDA to reveal the sensitivities to the energy scale is possible. Furthermore, one cannot tune the operation of the IDA in order to balance efficiencies for each quantity, as we did for thrust $T$ and event-${p_T}$ in the sequential-cut analysis. For this kind of precision measurement, it would appear that the better discrimination of signal and background provided by the IDA is of limited value in light of the larger and uncontrollable sensitivity to experimental sources of systematic uncertainty. Nonetheless, when performing a measurement with real data, one would welcome an alternative analysis in order to check the robustness and stability of the measurement. In Section \[S:discovery\], we show the power of the IDA in the discovery of a light stop quark. Theoretical Uncertainties {#sec:th} ========================= The inference of the stop mass from stop cross-section measurements requires precise theoretical calculations for the cross-sections. The stop production cross-section receives large corrections in particular from QCD gluon exchange between the final state stops. Near threshold, when the stop quarks are slowly moving, these effects become very large, which is the well-known Coulomb correction [@coulomb]. The NLO QCD corrections to stop production have been computed several years ago [@calvin] and it was found that the corrections range between about 10% at high energies and up to 100% near threshold. This shows that higher-order corrections are crucial. Over the last few years, sophisticated techniques have been developed for calculating top-pair production at NNLO [@topthr]. Near threshold, they include resummation of terms of order $O(\alpha_{\rm s}/v)$ for the low velocity $v$ of the top-quarks. For the production of scalar quarks, similar calculations are not yet done. However, one can use partial results to obtain a reliable estimate of the uncertainty of the NNLO corrections. At NNLO order, several contributions enter in the computation. The largest effect near threshold arises from the Coulomb correction. The impact of the Coulomb corrections is calculated through NNLO order [@clnnlo], including resummation via non-relativistic QCD. Technically, here the non-relativistic Schrödinger equation is used for computing the Coulomb effects [@schreq]. Similar to the case of top pair production, it is found that the NNLO term to stop pair production is of similar order of magnitude as the NLO term, [*i.e.*]{}, the perturbation series is converging rather slowly. From the behavior of the perturbation series and the remaining scale dependence, the size of the missing higher-order contributions is estimated to be around 7% at 260 GeV and 2.5% at 500 GeV. However, we want to point out that several improvements to this straightforward approach could be made. Besides the large Coulomb-type corrections of order $O(\alpha_{\rm s}/v)$, there are also potentially large logarithmic contributions $O(\log(\alpha_{\rm s}/v))$. They can be resummed with more sophisticated techniques, for instance velocity non-relativistic QCD [@vnrqcd]. Using the results of Ref. [@vnrqcd:scalar] for the NLO corrections to squark pair production, it is found that the uncertainty with respect to the NLO computation is reduced significantly. A similar improvement can be expected at the NNLO level. In addition, instead of directly computing the total cross-section near threshold, one can describe it through moments [@moments] that avoid the non-perturbative contribution of stopponium bound states that can form just below the nominal stop-pair threshold. With these refinements it is expected that the theoretical uncertainty can be brought down by a factor of two (however the actual calculation remains for the future). So here an uncertainty of 3.5% at 260 GeV and 1% at 500 GeV are assumed. Besides the QCD corrections, the electroweak corrections need to be considered. The NLO electroweak corrections have been computed [@ewnlo], and found to amount to several per-cent. While they need to be taken into account, the NNLO corrections are expected to be much less than 1%, with the exception of leading initial- and final-state QED corrections that can easily be resummed to higher orders. Combining the two errors, a total theoretical error of 4% at $260{\,\, \mathrm{GeV}}$ and 1.5% at $500{\,\, \mathrm{GeV}}$ can be assigned. Pessimistically, we add these two uncertainties linearly, and assign a theoretical uncertainty of $5.5\%$ for the quantity $Y$. Results and Implications {#sec:comb} ======================== We derive the expected measurement error on the stop quark mass and discuss the implications for particle physics predictions of the relic density of dark matter. We also discuss the luminosity needed to discover a light stop quark in this scenario, using the IDA method. Precision on the Stop Quark Mass -------------------------------- A final assessment of the achievable precision on the stop mass will be based on the statistical and all systematic uncertainties. Table \[tab:sum\] summarizes these uncertainties for the observable $Y$ defined by Eq. . One sees that the IDA method achieves a smaller statistical uncertainty on $Y$ at the cost of a larger experimental systematic uncertainty. It would be important, in a measurement with real data, to implement two methods as we have done here, and check the consistency of the results. The stop quark mass is inferred from the measured values of the observable $Y$ following the example described in Sec. \[sec:method\]. The differing efficiencies for the sequential-cut and IDA methods lead to different central values for $Y$ and for the slope $dY/dM$ at that point. The inferred uncertainties on the stop quark mass are summarized in Table \[tab:mstoperr\] and are similar for the two analyses. We conclude that the stop quark mass could be measured with an uncertainty of $\Delta{m_{\tilde{t}_1}}= 0.42{\,\, \mathrm{GeV}}$. We investigated the dependence of the measurement error on the integrated luminosity. There is very little change in the statistical uncertainty if we increase the luminosity on peak, but the variation with the luminosity at ${\sqrt{s_{\mathrm{th}}}}= 260{\,\, \mathrm{GeV}}$ is interesting – see Fig. \[F:errorlum\]. The experimental uncertainty is dominated by the statistical contribution, so a decrease in the luminosity from our assumed value of ${{\cal{L}}_{\mathrm{th}}}= 50~{{\mathrm{fb}^{-1}}}$ has a significant impact. On the other hand, the theoretical uncertainty is very large by comparison, so increasing ${{\cal{L}}_{\mathrm{th}}}$ hardly improves the total error on ${m_{\tilde{t}_1}}$. A luminosity in the range $30~{{\mathrm{fb}^{-1}}}< {{\cal{L}}_{\mathrm{th}}}< 80~{{\mathrm{fb}^{-1}}}$ would appear to be optimal, for this analysis. The dominant uncertainty comes from the theoretical calculation of the signal cross-section. As discussed in Sec. \[sec:th\], this uncertainty comes mainly from higher-order corrections which are not easily summed at threshold. The estimate of this theory error relies on present computational techniques and some expectations on how they might improve in the future. However, the progress in calculations of radiative corrections can not really be predicted, so the assumed value for the theoretical uncertainty at the time when ILC is running might well be somewhat different than the value reported in Table \[tab:sum\]. In particular, history has shown that people working on loop computations often overcame big problems with unexpected ingenuity, in order to be able to make most of precise measurements. Therefore, in the following, the combined error in Table \[tab:sum\] will be taken as a conservative estimate. If one were to set aside the theoretical error on the cross section, then the total experimental error is quite small, amounting to $3.5\%$–$4.3\%$ on $Y$. In this case, the error on the stop quark mass would be a little larger than $\Delta{m_{\tilde{t}_1}}= 0.2{\,\, \mathrm{GeV}}$. Comparison with Previous Results -------------------------------- A previous study investigated the potential of the ILC running at $\sqrt{s} = 500$ GeV to discover a light stop quark and measure its parameters [@stop]. It was assumed that $250~{{\mathrm{fb}^{-1}}}$ would be taken at two beam polarization combinations: $P(e^-)/P(e^+) = +80\%/-60\%$ and $-80\%/+60\%$. Measurements of the stop squark production cross sections at these two polarizations are sufficient to deduce the mixing angle and mass of the stop squark. A host of systematic uncertainties was considered, with the conclusions that the [*absolute*]{} cross-sections could be measured to $1.3\%$–$2.4\%$, dominated by experimental systematic uncertainties (the statistical uncertainty was $0.8\%$). Under the given theoretical scenario, the total error on the stop quark mass was estimated[^2] to be $\Delta{m_{\tilde{t}_1}}= 1.2$ GeV. Our theoretical scenario coincides with the one studied in Ref. [@stop], and the method proposed here leads to a total error on the stop quark mass that is more than two times smaller: $\Delta{m_{\tilde{t}_1}}= 0.42$ GeV, even though a much smaller integrated luminosity is assumed. This improvement is certainly valuable, and is quite helpful for the calculation of the relic density, as we discuss below. We would like to point out, however, that the basis for the experimental analysis in Ref. [@stop] differs significantly from what was used for the present analysis. In particular, the fragmentation of the stop squark and of the charm quark produced in its decay was not simulated in Ref. [@stop], leading to very different signal characteristics which are not realistic. For example, the number of jets was almost always two, which contrasts starkly with the present study in which typically one or two additional jets are found due to the process of fragmentation. Thus the requirement in Ref. [@stop] of exactly two jets, which is very effective at suppressing the simulated $W e\nu$ background, leads in practice to a very low efficiency and large sensitivity to the modeling of the fragmentation practice. Ref. [@stop] assumed a $1\%$ systematic uncertainty on the [*absolute*]{} efficiency due to fragmentation, which dominates the total uncertainty on the cross-section measurement. As we have seen in the present study, this is likely to be significantly underestimated. Furthermore, no serious assessment of the theoretical uncertainty on the cross-section was given in Ref. [@stop]; this uncertainty was assumed to be negligible. As we discuss in Section \[sec:th\] above, this assumption is not justified, and with existing techniques we estimate the theoretical uncertainty to be as large as $2.5\%$ at $\sqrt{s} = 500$ GeV, which we assume will be reduced to $\approx 1\%$ by the time the ILC is running. We evaluated the efficiency of the selection in Ref. [@stop] using the signal samples generated for the present study. We find that the requirement ${N_{\mathrm{jets}}}= 2$ delivers an efficiency of only $7.3\%$, to be compared to $\approx 18\%$ reported in [@stop]. This would increase the statistical uncertainty on the cross-section measurement from $0.8\%$ to $1.2\%$. If we relax the cut to ${N_{\mathrm{jets}}}\ge 2$, the efficiency climbs to $21\%$, though clearly the background would become too large with this cut. We evaluated the scale dependence and found it to be negligible, since there are relatively few energy-based cuts in [@stop]. We evaluated the systematics for the stop and charm fragmentation, and found a very large sensitivity to the fragmentation, on the order of $5\%$, due to the requirement that ${N_{\mathrm{jets}}}= 2$ only. We conclude that the uncertainty on the stop quark mass, ${\Delta m_{\tilde{t}_1}}= 1.2{\,\, \mathrm{GeV}}$, reported in Ref. [@stop] was underestimated, so that our present result $\Delta{m_{\tilde{t}_1}}= 0.42$ GeV represents a major step forward. Implications for Relic Density Calculation ------------------------------------------ Precise measurements of supersymmetric particle properties at the LHC and ILC can be used to compute the dark matter relic abundance so as to compare with cosmological observations. If stop-neutralino co-annihilation is relevant, as in the scenario studied here [@heavyq], it is important to measure the stop-neutralino mass difference very precisely. The extraction of the neutralino properties, in particular the lightest neutralino mass, is studied in detail in Ref. [@heavyq]. It is found that a high precision of $\Delta{m_{\tilde{\chi}^0_{1}}} \approx 0.3$ GeV for the lightest neutralino mass can be achieved at the ILC, and also the other neutralino parameter can be inferred rather well. The limiting factor in the accuracy of the dark matter estimation is therefore the precision of the measurement of the scalar top quark mass. As discussed in the previous section, an older study using cross-section measurements at $\sqrt{s} =500$ GeV found ${\Delta m_{\tilde{t}_1}}= 1.2$ GeV and $|\cos\theta_{\tilde{t}}| < 0.077$ for the stop mass and mixing angle [@stop; @heavyq]. Based on these expected experimental results, the relic dark matter density is computed with the codes described in Ref. [@Balazs:2004bu; @morr]. Fig. \[fig:dm\] shows the result of a scan over the MSSM parameter space. The scattered gray dots indicate the region allowed by the collider experimental uncertainty, as a function of the measured stop mass. The horizontal bands depict the relic density as measured by WMAP [@wmap] with one and two standard deviation errors. Here, $\Omega_{\rm CDM}$ is the ratio of the dark matter energy density to the critical density $\rho_c = 2H_0^2/(8\pi G_{\rm N})$, with the Hubble constant $H_0 = h\times 100$ km/s/Mpc and Newton’s constant $G_{\rm N}$. At the 1$\sigma$ level, the astrophysical observations lead to $0.103 < \Omega_{\rm CDM} h^2 < 0.116$. With a stop mass measurement error of $m_{\tilde{t}_1} = (122.5 \pm 1.2) {\,\, \mathrm{GeV}}$, the relic density can be predicted to $0.082 < \Omega_{\rm CDM} h^2 < 0.139$ at the 1$\sigma$ level. With the new result of this work, $\Delta{m_{\tilde{t}_1}}= 0.42{\,\, \mathrm{GeV}}$, the relic density can be computed much more precisely, yielding the result $0.096 < \Omega_{\rm CDM} h^2 < 0.124$. This precision is very comparable to the direct WMAP measurement, as indicated by the black dots in Fig. \[fig:dm\]. As pointed out above, the estimate of a stop mass error of ${\Delta m_{\tilde{t}_1}}= 0.42{\,\, \mathrm{GeV}}$ is based on a rather conservative evaluation of systematic errors. In particular, this value is dominated by the conjectured theory error on the prediction of signal and background cross-sections. If on the other hand, with progress in calculation methods, the theory error could be reduced to a sub-dominant level, the remaining statistical and systematic experimental errors would give a stop mass error of ${\Delta m_{\tilde{t}_1}}= 0.24{\,\, \mathrm{GeV}}$ for the cut-based analysis and ${\Delta m_{\tilde{t}_1}}= 0.27{\,\, \mathrm{GeV}}$ for the IDA. The amelioration of the prediction for the dark matter relic density due to this improvement in stop mass precision is illustrated in Fig. \[fig:dm\]. For this accuracy of the stop mass measurement, the uncertainty of the dark matter prediction becomes limited due to the expected experimental errors in the lightest neutralino mass and mixing angles, which we have taken from Ref. [@heavyq]. As a result, taking an error of ${\Delta m_{\tilde{t}_1}}= 0.24{\,\, \mathrm{GeV}}$ for the stop mass, we find $0.099 < \Omega_{\rm CDM} h^2 < 0.121$, which is only a small improvement in the precision of the dark matter density prediction with respect to ${\Delta m_{\tilde{t}_1}}= 0.42 {\,\, \mathrm{GeV}}$. Discovery of the Light Stop Quark {#S:discovery} --------------------------------- The main focus of this paper is the measurement of the stop quark mass. It is interesting, nonetheless, to consider the utility of these selections for discovering the light stop quark at the ILC. The IDA-based selection, in particular, achieves a very low background and a high efficiency – see Table \[tab:ida2\] in Subsection \[sec:ida\]. We examined this issue assuming that the ILC collects data at ${\sqrt{s_{\mathrm{pk}}}}= 500$ GeV, with [*unpolarized*]{} beams, as one might expect at start-up. The signal cross-section for this scenario is $\sigma_{\tilde{t}} = 118$ fb. The nominal IDA selection efficiency is $\epsilon = 0.416$ and the background cross-section for unpolarized beams is $\sigma_b = 10.3$ fb. Tightening the selection to reduce the background improves the sensitivity of the analysis only very slightly. This information allows a calculation of the expected tail probability or $p$-value as a function of integrated luminosity, ${{\cal{L}}}$. Specifically, we computed the $p$-value setting the hypothetical number of observed events equal to the mean of the corresponding Poisson distributions (signal and background), as a function of ${{\cal{L}}}$. The result is shown in Fig. \[f:pvalues\] by the thick red line. The black dots on the line show hypothetical integral numbers of observed events, starting at $N = 1$ for ${{\cal{L}}}= 0.02~{{\mathrm{fb}^{-1}}}$. The plot clearly indicates that a luminosity of only ${{\cal{L}}}\approx 240~{{\mathrm{pb}^{-1}}}$ would produce eleven observed events, on average, and the significance of ten signal events over the expected background would be more than $5\sigma$. The uncertainty on the background estimate and the signal efficiency have a negligible impact on this result. This example applies only to our given scenario, with ${m_{\tilde{t}_1}}= 122.5{\,\, \mathrm{GeV}}$, ${m_{\tilde{\chi}^0_{1}}} = 107.2{\,\, \mathrm{GeV}}$ and $\cos\theta_{\tilde{t}} = 0.01$. Further investigations would be needed in order to understand how well this IDA selection would perform for other mass and mixing combinations. Summary {#sec:concl} ======= A new method for a precise measurement of the stop quark mass has been described, based on the ratio of yields at the peak stop quark pair production cross section, and near threshold. This ratio is far less sensitive to experimental uncertainties than other methods, leading to a very low estimated uncertainty, still dominated by the statistical uncertainty and the theoretical uncertainty (which is also present for other methods based on a cross-section measurement). We studied a specific scenario in detail, with an emphasis on analysis techniques and systematic uncertainties. We placed special emphasis on the modeling of the stop quark and charm fragmentation uncertainties, and suggest how fragmentation models could be constrained with data taken at the ILC. Previous studies had not considered this source of uncertainty. This method is general, and could be applied to other species, provided an accurate prediction for the excitation curve is or can be available. For weakly interacting particles, such as staus, the theoretical uncertainty is much smaller and the advantage of the new method would be even more impressive. An important part of our studies is the use of multi-variate methods to isolate a very clean stop quark signal. For this we utilize the Iterative Discriminant Analysis (IDA) used previously at LEP. It is interesting that a carefully-tuned set of sequential cuts achieves a much smaller systematic uncertainty, and hence a better overall result for the stop quark mass measurement in this method. The superior background rejection of the IDA, however, is extremely useful when searching for a stop signal, and we give an illustration for $\sqrt{s} = 500{\,\, \mathrm{GeV}}$, which shows that a five-sigma significance could be obtained by the IDA selection with only $240~{{\mathrm{pb}^{-1}}}$. The reduction of the uncertainty on the stop mass from about $\Delta{m_{\tilde{t}_1}}= 1.2{\,\, \mathrm{GeV}}$ in Ref. [@stop; @heavyq] to $\Delta{m_{\tilde{t}_1}}= 0.42{\,\, \mathrm{GeV}}$ in this analysis is crucial for testing theoretical explanations of the dark matter relic density in the light-stop co-annihilation scenario. With these new results, the theoretical calculation has an accuracy equal to the two-sigma uncertainty of the WMAP measurements. The remaining uncertainty is no longer dominated by $\Delta{m_{\tilde{t}_1}}$. 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[^2]: Note that the error of 1.2 GeV is slightly larger than reported in Eq. (17) in Ref. [@stop], since we are using the scenario from Ref. [@heavyq] with large slepton masses, leading to a larger neutralino mass error, which in turn increases the stop mass uncertainty.

--- author: - 'J. Klein' - 'A. Hötger' - 'M. Florian' - 'A. Steinhoff' - 'A. Delhomme' - 'T. Taniguchi' - 'K. Watanabe' - 'F. Jahnke' - 'A. W. Holleitner' - 'M. Potemski' - 'C. Faugeras' - 'J. J. Finley' - 'A. V. Stier' bibliography: - 'full.bib' title: 'Supplemental Material - Quantized many-body spin-valley textures in charge tunable monolayer MoS$_2$' --- Field-effect device for carrier density control in monolayer MoS$_{2}$ ====================================================================== We make use of field-effect devices to control the carrier density in monolayer MoS$_{2}$. [@CastellanosGomez.2014] SI Figure \[SIfig1\] shows the two device geometries used in this manuscript. In both devices, monolayer MoS$_{2}$ is encapsulated in hBN. We use the hBN for two main reasons: (i) Encapsulation reduces inhomogeneous linewidth broadening of excitons [@Wierzbowski.2017] and (ii) as gate dielectric that withstands high breakdown fields, [@Dean.2010] thus preventing leakage currents. We use few-layer graphite as the gate-electrode and to directly contact the MoS$_2$. Sample A is a dual-gate device with top and bottom gates as shown in SI Fig. \[SIfig1\](a) while device $B$ is a single-gated. In device $A$, equal gate voltages with the same polarity are applied to the gates $V_{bg}=V_{tg}$ with respect to the monolayer MoS$_{2}$, thus controlling the carrier density. In the dual-gated device $A$, we are also able to tune from the n- into the p-doped regime, while in device $B$, we can tune from the intrinsic regime to the n-doped regime only. The observation that only the n-doped regime is accessible with a single-gate is common in the literature for monolayer MoS$_{2}$. [@Robert.2018; @Roch.2019] It is likely that the dual-gate device allows to access the p-doped regime since it overcomes Fermi-level pinning effects by the symmetric device geometry. Furthermore, dual gates allow larger applied effective fields for tuning the carrier density. We determine the carrier density by using a simple plate capacitor model where the device capacitance is $C = \epsilon_0 \epsilon_{hBN}/d$ with the dielectric constant of multilayer hBN $\epsilon_{hBN} = 2.5$ [@Dean.2010; @Kim.2012; @Hunt.2013; @Laturia.2018] and the hBN layer thickness $d$ which is determined by atomic force microscopy. The carrier density is given by $n = C V_{bg}/e$ for the single-gate device. Since top and bottom hBN thickness are very similar for the dual-gate device we relate the carrier density with the gate voltage through $n = C (V_{tg} + V_{bg}) /e = 2CV /e$. Transfer characteristics in high magnetic fields ================================================ For the gate-dependent magneto-photoluminescence measurements on device $A$ and B, we apply a magnetic field and vary the gate voltage in steps of $\SI{100}{\milli\volt}$ while collecting PL spectra for every gate voltage step. We perform the same gate biasing sequence for every magnetic field, thus ensuring that the voltage sweeps at different magnetic fields are directly comparable. We first apply a static magnetic field and then we tune the bias voltage from max. $V_{+}$ to max. $V_{-}$. Typical current voltage characteristics of device $A$ and $B$ are presented in SI Fig. \[SIfig2\](a) and (b). For device $A$, we apply equal voltage to top- and bottom-gate ($V=V_{bg}=V_{tg}$). From the transfer characteristics of both devices, we find that our biasing scheme is highly reproducible for all magnetic fields applied in the experiment. The reproducibility is due to the graphite contacts to the MoS$_{2}$ which is known for low contact resistance and small Schottky barrier heights. [@Cui.2015; @Allain.2015] The leakage currents are in the noise floor for most of the range and negligible leakage currents of $<\SI{1.5}{\nano\ampere}$, that correspond to current densities of $< 10^{-2}\SI{}{\ampere\per\centi\meter\squared}$, at the highest bias voltages. The data are collected for a laser excitation power of $\SI{30}{\micro\watt}$ with a laser excitation energy of $\SI{2.41}{\electronvolt}$. Trion fine structure in monolayer MoS$_{2}$ from theory ======================================================= The linear absorption spectrum of MoS$_2$ is calculated by solving the semiconductor Bloch equations (SBE) for the microscopic interband polarizations $\psi^{he}_{\mathbf k} = \braket{a^h_{\mathbf k}a^{e}_{\mathbf k}}$, where the operators $a^{\lambda}_{\mathbf k}$ annihilate a carrier in band $\lambda$ with momentum $\mathbf k$. In the limit of vanishing excitation/doping density, the SBE become formally equivalent to the Bethe-Salpeter equation [@rohlfing_electron-hole_2000] and read in Fourier space $$\begin{aligned} \begin{split} &(\varepsilon^e_{\mathbf k} + \varepsilon^h_{\mathbf k} - \hbar\omega)\psi^{he}_{\mathbf k}(\omega) - \frac{1}{A}\sum_{\mathbf{k'}}\sum_{h'e'} \left( V^{eh'he'}_{\mathbf k,\mathbf k',\mathbf k,\mathbf k'} - U^{eh'e'h}_{\mathbf k,\mathbf k',\mathbf k',\mathbf k}\right)\psi^{h'e'}_{\mathbf k'}(\omega) = (d^{he}_{\mathbf k})^*E(\omega)\, , \end{split} \label{eq:SBE}\end{aligned}$$ with bandstructures $\varepsilon^{e/h}_{\mathbf k}$, Coulomb matrix elements $V^{\lambda_1\lambda_2\lambda_3\lambda_4}_{\mathbf k+\mathbf q,\mathbf k'-\mathbf q,\mathbf k',\mathbf k}$ and dipole matrix elements $d^{he}_{\mathbf k}$ that are projected in the polarization direction of the electric field $E(t)$, which is assumed to propagate vertical to the single-layer plane. $A$ denotes the crystal area. Note that the electron-hole exchange interaction in Eq.  is described by unscreened Coulomb matrix elements $U$ [@sham_many-particle_1966; @denisov_longitudinal_1973; @Qiu.2015], while $V$ denotes Coulomb matrix elements that are screened by carriers in occupied bands and the dielectric environment of the TMDC layer. The linear response of the material is given by the macroscopic susceptibility $\chi(\omega) = \frac{1}{A}\sum_{\mathbf k}\sum_{he}\left( \mathrm d^{he}_{\mathbf{k}}\psi^{he}_{\mathbf k} + c.c.\right)/E(\omega)$, which contains excitons as discrete resonances below a continuum of optical interband transitions. At moderate charge carrier densities, signatures of tightly bound trions additionally appear in experimental spectra. To capture this effect, we extend the semiconductor Bloch equations presented in Eq.  and explicitly include correlation functions that describe three-particle states. For this purpose, we examine the equation of motion for the microscopic polarization for finite electron density, which reads $$\begin{aligned} \begin{split} &(\varepsilon^e_{\mathbf k} + \varepsilon^h_{\mathbf k} - \hbar\omega)\Psi^{he}_{\mathbf k}(\omega) - \frac{1}{A}\sum_{\mathbf{k'}}\sum_{h'e'} \left(V^{eh'he'}_{\mathbf k,\mathbf k',\mathbf k,\mathbf k'} - U^{eh'e'h}_{\mathbf k,\mathbf k',\mathbf k',\mathbf k}\right)\Psi^{h'e'}_{\mathbf k'}(\omega) \\ &+\frac{1}{A^2}\sum_{\mathbf{Q}\mathbf q}\sum_{e_2h_3e_4}\left( V^{e_2h_3he_4}_{\mathbf Q,\mathbf k-\mathbf q,\mathbf k,\mathbf Q-\mathbf q} - U^{e_2h_3e_4h}_{\mathbf Q,\mathbf k-\mathbf q,\mathbf Q-\mathbf q,\mathbf k} \right) \mathrm T_{ee_4h_3e_2}(\mathbf k, \mathbf Q-\mathbf q, \mathbf Q) \\ &-\frac{1}{A^2}\sum_{\mathbf{Q}\mathbf q}\sum_{e_2e_3e_4}\left(V^{e_2ee_4e_3}_{\mathbf Q,\mathbf k,\mathbf k+\mathbf q,\mathbf Q-\mathbf q} - V^{e_2ee_3e_4}_{\mathbf Q,\mathbf k,\mathbf Q-\mathbf q,\mathbf k+\mathbf q}\right)\mathrm T_{e_4e_3he_2}(\mathbf k + \mathbf q, \mathbf Q-\mathbf q, \mathbf Q) \\ &=(1 - f^e_{\mathbf{k}})(\mathrm d^{he}_{\mathbf k})^*E(\omega)\,. \end{split} \label{eq:eom_psi_trion}\end{aligned}$$ Here, $f^\lambda_{\mathbf{k}}$ denotes the carrier population in the band $\lambda$. We assume the system to be in thermal quasi-equilibrium described by Fermi functions with given temperature and carrier density. The Coulomb interaction couples the polarization $\Psi^{he}$ to the trion amplitude $$\begin{aligned} T_{e_1e_2h_3e_4}(\mathbf k_1, \mathbf k_2, \mathbf Q)=\braket{{a_{\mathbf{Q}}^{e_4}}^\dagger a_{-(\mathbf{k}_1+\mathbf{k}_2-\mathbf{Q})}^{h_3} a_{\mathbf{k}_2}^{e_2} a_{\mathbf{k}_1}^{e_1}}\,,\end{aligned}$$ which is a four-operator expectation value and describes the correlated process of annihilating two electrons and one hole, leaving behind an electron with momentum $\mathbf Q$ in the conduction band and is linked to the optical response of an electron trion $X^-$. For moderate carrier density and linear optics a closed expression for the trion amplitudes can be obtain [@esser_theory_2001; @Florian.2018] $$\begin{aligned} \begin{split} &(\varepsilon^{e_1}_{\mathbf k_1} + \varepsilon^{e_2}_{\mathbf k_2} + \varepsilon^{h_3}_{\mathbf k_3} - \varepsilon^{e_4}_{\mathbf Q} - \hbar\omega) T_{e_1e_2h_3e_4}(\mathbf k_1, \mathbf k_2, \mathbf Q) \\ &-\frac{1}{A}\sum_{\mathbf q}\sum_{h_5,e_6}\left(V^{e_2h_5h_3e_6}_{\mathbf k_2,\mathbf k_3-\mathbf q,\mathbf k_3,\mathbf k_2-\mathbf q} - U^{e_2h_5e_6h_3}_{\mathbf k_2,\mathbf k_3-\mathbf q,\mathbf k_2-\mathbf q,\mathbf k_3}\right) T_{e_1e_6h_5e_4}(\mathbf k_1, \mathbf k_2- \mathbf q, \mathbf Q) \\ &-\frac{1}{A}\sum_{\mathbf q}\sum_{h_5,e_6}\left( V^{e_1h_5h_3e_6}_{\mathbf k_1,\mathbf k_3-\mathbf q,\mathbf k_3,\mathbf k_1-\mathbf q} - U^{e_1h_5e_6h_3}_{\mathbf k_1,\mathbf k_3-\mathbf q,\mathbf k_1-\mathbf q,\mathbf k_3}\right) T_{e_6e_2h_5e_4}(\mathbf k_1- \mathbf q , \mathbf k_2, \mathbf Q) \\ &+\frac{1}{A}\sum_{\mathbf q}\sum_{e_5,e_6}\left(V^{e_1e_2e_5e_6}_{\mathbf k_1,\mathbf k_2,\mathbf k_2+\mathbf q,\mathbf k_1-\mathbf q} - V^{e_1e_2e_6e_5}_{\mathbf k_1,\mathbf k_2,\mathbf k_1-\mathbf q,\mathbf k_2+\mathbf q}\right) T_{e_6e_5h_3e_4}(\mathbf k_1- \mathbf q , \mathbf k_2+\mathbf q, \mathbf Q) \\ &=f^{e_1}_{\mathbf{Q}}\left( \mathrm d^{he}_{\mathbf{k_2}} \delta_{\mathbf k_1,\mathbf Q} \delta_{e,e_1} - \mathrm d^{he}_{\mathbf{k_1}}\delta_{\mathbf k_2,\mathbf Q} \right)E(\omega)\,, \end{split} \label{eq:eom_trion}\end{aligned}$$ where the homogeneous part of these equations represents a generalized three-particle Schrödinger equation in reciprocal space. The oscillator strength of a transition between a trion state with total momentum $\mathbf Q$ and an electron state are directly proportional to the carrier population $f^{\lambda}_{\mathbf{Q}}$, which is apparent from the inhomogenous part of Eq. . Thus, trions visible in optical spectra are hosted in the $K$ and $K^{\prime}$ valleys of the two lowest conduction (highest valence) bands to which we limit our single-particle basis. We combine the above many-body theory of trions with ab-initio methods providing material-realistic band structures and bare as well as screened Coulomb matrix elements on a G$_0$W$_0$ level as input for the equations of motion (Eq. and ). More details on the G$_0$W$_0$ calculations are given in Ref. [@Steinhoff.2014]. First- and second-order Rashba spin-orbit coupling are added subsequently to account for the spin-orbit splitting in the conduction- and the valence-band $K$ valleys, see Ref. [@Liu.2013; @Steinhoff.2014]. Direct dipole transition matrix elements are calculated using a Peierls approximation [@Steinhoff.2014]. Coulomb matrix elements for hBN encapsulated MoS$_2$ are obtained in two steps: Bare Coulomb matrix elements for a freestanding monolayer and the RPA dielectric function are calculated in a localized Wannier basis $\ket{\alpha}$ with dominant W-d-orbital character. Substrate effects are then included using the *Wannier function continuum electrostatics* (WFCE) approach as described in Refs. [@rosner_wannier_2015; @Florian.2018]. The approach combines a continuum electrostatic model for the screening by the dielectric environment with a localized description of the Coulomb interaction. The actual parametrization is provided in Ref. [@Florian.2018] and has been shown to provide reasonable agreement with experiments. Subsequently, the Coulomb matrix elements are transformed in Bloch-state representation $$\begin{split} & U^{\lambda_1\lambda_2\lambda_3\lambda_4}_{{\mathbf{k}}+{\mathbf{q}},{\mathbf{k}'}-{\mathbf{q}},{\mathbf{k}'},{\mathbf{k}}} = \sum_{\alpha\beta\gamma\delta} \big[c_{\alpha}^{\lambda_1}({\mathbf{k}}+{\mathbf{q}})\big]^*\big[c_{\beta}^{\lambda_2}({\mathbf{k}'}-{\mathbf{q}})\big]^* \, c_{\gamma}^{\lambda_3}({\mathbf{k}'}) c_{\delta}^{\lambda_4}({\mathbf{k}}) \,\, U_{{\mathbf{q}}}^{\alpha\beta\gamma\delta}\,. \label{eq:unitary_transform} \end{split}$$ where the coefficients $c_{\alpha}^{\lambda}({\mathbf{k}})$ connect the localized and the Bloch basis, see Eq. . As the Coulomb interaction is spin-diagonal, only the electron-hole exchange terms couple trions with different spin combinations and are known to give rise to a fine structure splitting of the trion resonances [@Yu.2014]. A proper description of electron-hole exchange requires matrix elements with density-density-like $U_{{\mathbf{q}}}^{\alpha\beta\beta\alpha}$ and exchange-like contributions $U^{\alpha\beta\alpha\beta}$ in the local representation [@Qiu.2015; @steinhoff_biexciton_2018]. While the density-density-like matrix elements are momentum-dependent and thus nonlocal in real space, the exchange-like matrix elements are practically momentum-independent and correspond to local interaction processes between carriers within the same unit cell. For monolayer MoS$_2$ we find that the dominant exchange-like matrix elements amount to 0.378 eV (0.20 eV) per unit-cell area for interaction between $d_{m=0}$ and $d_{m=\pm2}$ (among $d_{m=\pm2}$) orbitals. To compute optical absorption spectra numerically, the coupled equations of motion (Eq.  and ) are solved in frequency space. Due to translational symmetry of the crystal, trions with different total momentum $\mathbf Q$ do not couple to each other, thus, we restrict the calculation to the main contribution at the $K$/$K^{\prime}$ point. Note that contributions from finite center-of-mass momentum give rise to an asymmetric line shape caused by electron recoil [@esser_theory_2001]. The first Brillouin zone is sampled by a 84 $\times$ 84 Monkhorst-Pack mesh and a region around the $K$/$K’$ point defined by a radius of 3.5nm$^{-1}$. Due to the numerical discretization of momentum space, there is an uncertainty of binding energies of $\pm 0.5$ meV. For the matrix inversion problem we utilize an iterative Krylov-space method as contained in the PETSc toolkit [@balay_petsc_2016]. A qualitative understanding of the trion fine structure can be obtained from a configuration model. The homogenous part of Eq. (\[eq:eom\_trion\]) constitutes a three-particle Hamiltonian whose eigenstates describe trions with total momentum $\mathbf Q$. The Hamiltonian can be split into a part $H_0$ without electron-hole exchange and an exchange part $H_U$ according to the Coulomb matrix element. Microscopically, trions can be interpreted as a superposition of an exciton and an electron state with different momenta. Configurations are eigenstates of the three-particle Hamiltonian $H_0$ that contains the kinetic energies of two electrons and a hole as well as the direct Coulomb interaction. There are six optically bright configurations in the subspace of zero-momentum trions with a hole located in the highest valance band [@Courtade.2017]. Due to time-reversal symmetry the configurations are pairwise degenerate and are connected by changing $K$ into $K^{\prime}$ and flipping all spins. In Fig. 3a of the main text the configurations are shown for $\mathbf Q=\mathbf K$. By adding electron-hole exchange to this picture, interaction between the configurations is introduced and leads to new eigenstates and -energies. It is the new eigenstates including interaction which are observed in experiment instead of the configurations. It turns out that the non-local exchange, which mixes excitons in the $K$ and $K^{\prime}$ valleys with finite total momentum, leads to a singlet-triplet splitting of two trion configurations and gives rise to the threefold fine structure observed in the experiment. Carrier density dependent magneto-photoluminescence of monolayer MoS$_{2}$ ========================================================================== The $\sigma^-$ circularly polarized, charge carrier density dependent magneto-photoluminescence of device $A$ is shown in SI Fig. \[SIfig5\]. The above described biasing sequence is used to maintain sample stability throughout the individual voltage sweeps for static magnetic fields ranging from $B = \SI{-28}{\tesla}$ to $B = \SI{28}{\tesla}$. Landau levels in the negatively charged *intra*valley trion =========================================================== SI Figure \[SIfig7\] shows density dependent false color plots of the low-temperature ($T = \SI{5}{\kelvin}$) magneto-PL from device $B$. In particular, the intravalley trion $X^-$ is shown for magnetic fields ranging from $\SI{-22}{\tesla}$ to $\SI{22}{\tesla}$. Similar to the *intra*valley trion in device $A$, Landau levels with filling factors $\nu = +0$ and $\nu = +1$ which period increases with the applied magnetic field. The quantum oscillations are analogous to Shubnikov-de-Haas oscillations in transport studies. [@Pisoni.2018] Here, the position of the LLs is derived from the kinks of the quantum oscillations which are a direct consequence of the Fermi level being tuned in resonance with the individual LL, thus, enhancing trion oscillator strength mediated by carrier scattering. Oppositely, when the Fermi level is situated in between the LLs, no scattering occurs resulting in a reduction of the trion formation rate. This oscillation is clearly visible in the *intra*valley trion $X^-$ in both, device $A$ and $B$. SI Figure \[SIfig8\] shows the integrated PL intensity of the $X^-$ as a function of the applied carrier density for static applied magnetic fields. We find the same LL structure in both devices with filling factors of $\nu = +0$ and $\nu = +1$. The LLs vanish in both devices around the same carrier density $n \sim 4 \cdot 10^{12} \SI{}{\per\centi\meter\squared}$ which is defined by the situation the Fermi level touches the upper conduction band. Extracting the electron mass from Landau levels observed in the trion PL ======================================================================== For an electron in a given valley, the energy difference between subsequent LLs in a magnetic field is given by the cyclotron energy $$E_C = \hbar \omega_C \label{eq:cyclotron}$$ with the cyclotron frequency $\omega_C = e B / m_e$. All contributions from spin and Berry curvature related Zeeman energies cancel out. [@Stier.2018] From our magneto-PL measurements in SI Figure \[SIfig7\], we observe LLs with filling factors $\nu = +0$ and $\nu = +1$. We can calibrate the applied gate voltage into an absolute energy scale by relating the carrier density for which the Fermi level tunes into the upper conduction band $c_2$ to the dark-bright exciton splitting. In our experiment, the Fermi energy enters $c_2$ for a density $n = 4 \cdot 10^{12} \SI{}{\per\centi\meter\squared}$. At this density, the $X^{\prime-}$ PL feature emerges and this emergence has unambiguously been related to $E_F$ in the $c_2$ band. [@Pisoni.2018] The dark-bright splitting of the conduction bands has recently been determined to be $\SI{14}{\milli\electronvolt}$. [@Pisoni.2018; @Jadczak.2020] SI Figure \[landau\_fan\] shows the shift of the LL difference as a function of the applied magnetic field for device $A$ and $B$. In order to obtain the electron mass, we apply a linear fit to the charge carrier difference between the 0$^{th}$ and 1$^{st}$ LL as a function of magnetic field where we express $\Delta n$ as an energy by normalizing to the carrier density at which the upper conduction band is reached using the dark-bright splitting $\Delta_{db}=\SI{14}{\milli\electronvolt}$. Utilizing Eq. \[eq:cyclotron\] yields a value of $m_e = 0.44 \pm 0.15 m_0$ ($m_e = 0.36 \pm 0.15 m_0$) for device $A$ ($B$). Calculating the density dependent hole mass =========================================== We determined the electron mass in device $A$ to be $m_e = 0.44 \pm 0.15 m_0$. Here, we determine the effective hole mass $m_h$ based on the assumption that the electron mass stays constant up to carrier densities of $5 \cdot 10^{12}\SI{}{\per\centi\meter\squared}$. [@Pisoni.2018] The Zeeman energy of the trion is given by [@Lyons.2019] $$H_Z = \frac{1}{2} \bigg[\tau_1 g_{ve}^{vb} + s_{1}g_{s}^{vb}+\sum_{i=2}^{N=3}\bigg(\tau_ig_{ve}^{cb}+s_ig_s^{cb}\bigg)\bigg]\mu_B B \label{}$$ where the index $i = 1$ is assigned to the bound hole and indices $i = 2, 3$ are assigned to the bound and excess electron of the trion. Here, $g_{ve}$ and $g_s$ are the valley and spin $g$-factors in the conduction and valence band, respectively. The valley and spin indices are $\tau_i = \pm 1$ ($K=+1$ and $K^{\prime}=-1$) and $s_i = \pm 1$ ($+1$ spin-$\uparrow$ and $-1$ spin-$\downarrow$). The change in the emitted photon energy due to radiative recombination of the negative trion in a magnetic field is [@Lyons.2019] $$\Delta E_{\hbar \omega} = \Delta E_Z - \Delta E_R + \Delta E_D \label{}$$ with $\Delta E_Z$ the Zeeman shift of the electron-hole pair within the trion, $\Delta E_R$ the excess electron recoil shift and $\Delta E_D$ the diamagnetic shift. Since in the applied magnetic field range here, the diamagnetic shift is negligible, [@Stier.2018] we omit this term in the following discussions. Unlike neutral exciton recombination, trion recombination cannot be a zero momentum process as the same momentum is imprinted onto the excess electron as recoil, detracting from the absorbed photon energy. The excess electron recoil depends on the temperature and the Fermi energy. The recoil cost is the energy difference in the trion and the free electron in the first LL. The change in energy of the excess electron after trion recombination in a magnetic field is [@Lyons.2019] $$\Delta E_{e}^{\tau} = \frac{1}{2} \hbar \omega_c \bigg(1- \frac{m_3}{m_T}\bigg) + \frac{1}{2} \tau_e g_e \mu_B B \label{}$$ with the cyclotron energy $\hbar \omega_c = \frac{e B}{m_3}$, the valley index $\tau_e$, the trion effective mass $m_T = \sum_{i = 1}^{N = 3} m_i$ and the bare electron $g$-factor $g_e$. Importantly, this recoil energy cost is equal in both valleys and simplifies to $$\Delta E_{e}^{\tau} = \frac{1}{2} \hbar \omega_c - \frac{1}{2} \hbar \omega_T + \frac{1}{2} \tau_e g_e \mu_B B \label{}$$ where the first term describes the final state $\ket{f}$ of the trion, which is an electron in a LL, and the second term is the initial trion state $\ket{i}$. The difference is the recoil energy. The third term is the electron spin Zeeman energy that has no valley contribution. From these considerations, we can write the energy Zeeman shift for positive and negative magnetic field as measured in our experiment. For positive magnetic field we obtain $$\Delta E_{\hbar \omega}^+ = E_{Z} - \bigg[\frac{1}{2} (\hbar \omega_c - \hbar \omega_T) + \frac{1}{2} g_e \mu_B B\bigg]~, \label{}$$ while for negative field $$\Delta E_{\hbar \omega}^- = - E_{Z} - \bigg[\frac{1}{2} (\hbar \omega_c - \hbar \omega_T) - \frac{1}{2} g_e \mu_B B\bigg]~. \label{}$$ The sum of the above Zeeman shifts for positive and negative field is $$\Delta E_{\hbar \omega}^+ + \Delta E_{\hbar \omega}^- = - (\hbar \omega_c - \hbar \omega_T)~,$$ which is proportional to the difference between the cyclotron energy of an electron in a LL and the trion LL. This can be expressed as $$\Delta E_{\hbar \omega}^+ + \Delta E_{\hbar \omega}^- = - \hbar e B\bigg(\frac{1}{m_3}-\frac{1}{m_T}\bigg)~,$$ with $m_3 = 0.44 m_0$ and $m_T = 2 \cdot 0.44 m_0 + m_h m_0$. From our experiment we directly determine the Zeeman shift and corresponding $g$-factor of the negatively charged trion as a function of the carrier concentration. Hence, we can relate the measured $g$-factor for positive and negative magnetic field $g(s^+)$ and $g(s^-)$ to the energy difference of initial and final state of the trion, and therefore to the hole mass. We obtain $$\frac{1}{2} g(s^+) \mu_B B + \frac{1}{2} g(s^-) \mu_B B = - \hbar e B \cdot \bigg(\frac{1}{m_3}-\frac{1}{m_T}\bigg)~,$$ which we can express as a function of the hole mass $$m_h(n) = \Bigg| \frac{-m_e (1 + m_e |g_{ave}(n)|)}{1/2 \cdot m_e |g_{ave}(n)| + 1} \Bigg|~,$$ with $|g_{ave}| = \frac{1}{2} (g(s^+) + g(s^-))$ and by using $\frac{\mu_B m_0}{\hbar e} \sim 0.5$. Quantized spin-valley texture ============================= The Zeeman shift of electrons in the spin-orbit split conduction band valleys in monolayer MoS$_2$ manifests from spin, the Berry phase and due to quantization of electrons in Landau levels in each valley. The shift for the lower conduction band $c_1$ is given by $$E_{c_1} = \tau_s s_z 2 \mu_B B + \tau_i m_e \mu_B B + \nu \frac{ \hbar e B}{m_e}~, \label{Eq:Ec1}$$ while the shift of the upper conduction band $c_2$ is $$E_{c_2} = \Delta_{db} + \tau_s s_z 2 \mu_B B + \tau_i m_e \mu_B B + \nu \frac{ \hbar e B}{m_e}~. \label{Eq:Ec2}$$ Here, the valley and spin indices are $\tau_s = \pm 1$ ($K=+1$ and $K^{\prime}=-1$) and $\tau_i = \pm 1$ ($+1$ spin-$\uparrow$ and $-1$ spin-$\downarrow$). Moreover, $\nu$ is the filling factor of the LL and $\Delta_{db}$ the energy splitting between $c_1$ and $c_2$ for no magnetic field applied. We model two LLs for each valley with $\nu = +0$ and $\nu = +1$ in $K^{\prime}$. [@Rose.2013] After quantifying the energy shift of each LL of each spin in every valley, we can further calculate the density of states (DOS) as a function of the Fermi level $E_F$ (applied gate voltage). From this quantity, we can then infer the number of electrons populating the spin-$\downarrow$ and spin-$\uparrow$ LLs to deduce the degree of spin polarization for a given $E_F$. We model each LL by using a Gaussian function $$DOS_{LL} = \frac{e |B|}{h}\frac{1}{\sigma \sqrt{2 \pi}} \exp{-\frac{(E-E_{LL})^2}{2\sigma^2}}~,$$ with $eB/h$ as the degeneracy per unit area and the energetic position of the LL $E_{LL}$ as defined in Eq. \[Eq:Ec1\] and \[Eq:Ec2\] and a FWHM of each LL of $\Gamma = 2 \sqrt{2 ln(2)} \sigma = \SI{4}{\milli\electronvolt}$ accounting for the experimentally observed inhomogeneous broadening in device $A$. The total DOS for all spin-$\uparrow$ electrons is given through $$DOS^{\uparrow}_{LL} = \sum_{i=0}^{\nu=1} DOS^{\uparrow, K^{\prime}}_{LL} + \sum_{i=1}^{\nu=2} DOS^{\uparrow, K}_{LL}~,$$ and the DOS for all spin-$\downarrow$ electrons are given by $$DOS^{\downarrow}_{LL} = \sum_{i=1}^{\nu=2} DOS^{\downarrow, K}_{LL} + \sum_{i=0}^{\nu=1} DOS^{\downarrow, K^{\prime}}_{LL}~.$$ By integrating the DOS to $E_F$ we obtain the number of electrons populating each LL with the total number of electrons with spin-$\uparrow$ $$N_{\uparrow} = \int_0^{E_F} DOS^{\uparrow}_{LL} dE$$ and spin-$\downarrow$ $$N_{\downarrow} = \int_0^{E_F} DOS^{\downarrow}_{LL} dE~.$$ We can now compute the global degree of spin polarization all magnetic fields and $E_F$ $$\eta_s(B,E_F) = \frac{N_{\downarrow} - N_{\uparrow}}{N_{\downarrow} + N_{\uparrow}}~.$$ Valley Zeeman shift of $X^{-}$, $X^{+}$ and $X^{\prime-}$ ========================================================= Magnetic field and density dependent valley dichroism ===================================================== Magnetic moment of Bloch electrons: lattice Hamiltonian vs. atomic contributions ================================================================================ The magnetic moment is comprised of a contribution due to the orbital motion of a Bloch electron and a contribution due to the electron spin. The z-component of the orbital magnetic moment is given by: $$\mu^{\textrm{orb},n}_{z\boldsymbol{k}}=-\frac{e}{2 m_e}\langle\Phi^{n}_{\boldsymbol{k}}|\hat{l}_z|\Phi^{n}_{\boldsymbol{k}}\rangle =-\frac{e}{2 m_e}\langle\Phi^{n}_{\boldsymbol{k}}|\hat{x}\hat{p}_y-\hat{y}\hat{p}_x|\Phi^{n}_{\boldsymbol{k}}\rangle. \label{m_z_start}$$ Consider the general expression $\langle\Phi^{n}_{\boldsymbol{k}}|\hat{x}_i\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle$ and insert a complete set of Bloch states: $$\langle\Phi^{n}_{\boldsymbol{k}}|\hat{x}_i\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle= \sum_{n'\boldsymbol{k}'}\langle\Phi^{n}_{\boldsymbol{k}}|\hat{x}_i|\Phi^{n'}_{\boldsymbol{k}'}\rangle\langle\Phi^{n'}_{\boldsymbol{k}'}|\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle. \label{xipj_completeness}$$ The momentum matrix elements are diagonal in $\boldsymbol{k}$ due to translational invariance: $$\langle\Phi^{n'}_{\boldsymbol{k}'}|\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle=\langle\Phi^{n'}_{\boldsymbol{k}}|\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle\delta_{\boldsymbol{k},\boldsymbol{k}'}. \label{momentum_diag}$$ The position matrix element can be transformed using the Schrödinger equation of Bloch states, $$H\big|\Phi^{n}_{\boldsymbol{k}}\big>=\varepsilon_{\boldsymbol{k}}^{n}\big|\Phi^{n}_{\boldsymbol{k}}\big>, \label{sgl_bloch}$$ and the commutator relation [@Gajdo.2006] $$\frac{1}{m_e}\boldsymbol{\hat{p}}=\frac{i}{\hbar}\big[{H},{\boldsymbol{\hat{r}}}\big], \label{H_r_comm}$$ which holds in case of a local one-electron potential. It is still valid in the presence of spin-orbit interaction, as long as the latter can be approximately treated as an on-site potential. Using Eqs. (\[momentum\_diag\]), (\[sgl\_bloch\]) and (\[H\_r\_comm\]), we find: $$\langle\Phi^{n}_{\boldsymbol{k}}|\hat{x}_i\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle= \sum_{n'}\frac{\hbar}{i m_e}\langle\Phi^{n}_{\boldsymbol{k}}|\hat{p}_i|\Phi^{n'}_{\boldsymbol{k}}\rangle\frac{1}{\varepsilon_{\boldsymbol{k}}^{n}-\varepsilon_{\boldsymbol{k}}^{n'}}\langle\Phi^{n'}_{\boldsymbol{k}}|\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle. \label{xipj_transform}$$ The crystal wave functions can be constructed as a linear combination of localized orbitals in the following way such that they fulfill Bloch’s theorem: $$\big|\Phi^{n}_{\boldsymbol{k}}\big\rangle=\sum_{\alpha}c^{n}_{\alpha}(\boldsymbol{k})\big|\boldsymbol{k}\alpha\big\rangle,\quad\big|\boldsymbol{k}\alpha\big\rangle=\frac{1}{\sqrt{N}}\sum_{\boldsymbol{R}}e^{i\boldsymbol{k}\cdot\boldsymbol{R}}\big|\boldsymbol{R}\alpha\big\rangle \label{TB_ansatz}$$ with the orthonormality relations $$\big\langle \boldsymbol{R}\alpha \big|\boldsymbol{R}'\alpha'\big\rangle=\delta_{\boldsymbol{R}\boldsymbol{R}'}\delta_{\alpha\alpha'} \label{orth_wann}$$ and $$\big\langle\boldsymbol{k}\alpha\big|\boldsymbol{k}'\alpha'\big\rangle=\delta_{\boldsymbol{k}\boldsymbol{k}'}\delta_{\alpha\alpha'}, \label{orth_bloch}$$ where $N$ is the number of lattice sites. We can formulate the crystal (lattice) Hamiltonian in terms of the localized orbitals: $$H=\sum_{\boldsymbol{R}\boldsymbol{R}'\alpha\alpha'}t^{\alpha\alpha'}_{\boldsymbol{R}\boldsymbol{R}'}\big|\boldsymbol{R}\alpha\big\rangle\big\langle \boldsymbol{R}'\alpha' \big|. \label{H_lat}$$ Inserting the ansatz (\[TB\_ansatz\]) into Eq. (\[xipj\_transform\]), we obtain: $$\langle\Phi^{n}_{\boldsymbol{k}}|\hat{x}_i\hat{p}_j|\Phi^{n}_{\boldsymbol{k}}\rangle= \sum_{n'}\frac{\hbar}{i m_e}\frac{1}{\varepsilon_{\boldsymbol{k}}^{n}-\varepsilon_{\boldsymbol{k}}^{n'}} \sum_{\alpha\alpha'}(c^{n}_{\alpha}(\boldsymbol{k}))^*c^{n'}_{\alpha'}(\boldsymbol{k}) \langle\boldsymbol{k}\alpha|\hat{p}_i|\boldsymbol{k}\alpha'\rangle \sum_{\alpha\alpha'}(c^{n'}_{\alpha'}(\boldsymbol{k}))^*c^{n}_{\alpha}(\boldsymbol{k}) \langle\boldsymbol{k}\alpha'|\hat{p}_j|\boldsymbol{k}\alpha\rangle. \label{xipj_final}$$ Following [@Tomczak.2009], we analyze the momentum matrix element by using the commutator relation (\[H\_r\_comm\]) again, transforming the momentum states according to (\[TB\_ansatz\]) and inserting a complete set of position states: $$\langle\boldsymbol{k}\alpha'|\hat{p}_j|\boldsymbol{k}\alpha\rangle=-\frac{i m_e}{\hbar}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')} \int \boldsymbol{dr}\big[ \langle\boldsymbol{R}'\alpha'|\hat{r}_j|\boldsymbol{r}\rangle\langle\boldsymbol{r}|H|\boldsymbol{R}\alpha\rangle -\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{r}\rangle\langle\boldsymbol{r}|\hat{r}_j|\boldsymbol{R}\alpha\rangle\big]. \label{p_transform}$$ One has to distinguish between the continuous space variable $\boldsymbol{r}$ and the discrete unit-cell label $\boldsymbol{R}$. The position operator acts as $\hat{r}_j|\boldsymbol{r}\rangle=r_j|\boldsymbol{r}\rangle$. The momentum matrix element contains contributions that can be directly related to the discrete lattice as well as contributions that arise due to the spatial extension of orbitals. We separate these contributions by shifting $\boldsymbol{r}\rightarrow\boldsymbol{r}+\boldsymbol{R}'$ in the first term and $\boldsymbol{r}\rightarrow\boldsymbol{r}+\boldsymbol{R} $ in the second term: $$\begin{split} \langle\boldsymbol{k}\alpha'|\hat{p}_j|\boldsymbol{k}\alpha\rangle=&-\frac{i m_e}{\hbar}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')} \int\boldsymbol{dr} \\ &\big[\langle\boldsymbol{R}'\alpha'|r_j+R'_j|\boldsymbol{r}+\boldsymbol{R}'\rangle\langle\boldsymbol{r}+\boldsymbol{R}'|H|\boldsymbol{R}\alpha\rangle \\ &-\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{r}+\boldsymbol{R}\rangle\langle\boldsymbol{r}+\boldsymbol{R}|r_j+R_j|\boldsymbol{R}\alpha\rangle\big] \\ =&-\frac{i m_e}{\hbar}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')}(R'_j-R_j)\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{R}\alpha\rangle \\ &-\frac{i m_e}{\hbar}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')} \int\boldsymbol{dr}r_j \\ &\big[\langle\boldsymbol{R}'\alpha'|\boldsymbol{r}+\boldsymbol{R}'\rangle\langle\boldsymbol{r}+\boldsymbol{R}'|H|\boldsymbol{R}\alpha\rangle \\ &-\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{r}+\boldsymbol{R}\rangle\langle\boldsymbol{r}+\boldsymbol{R}|\boldsymbol{R}\alpha\rangle\big], \end{split} \label{p_shift}$$ where we made use of the completeness of position states again to derive the first term of the second line. This so-called Peierls contribution given by *inter*-site hopping can be written as a generalized Fermi velocity: $$\begin{split} -\frac{i m_e}{\hbar}&\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')}(R'_j-R_j)\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{R}\alpha\rangle \\ =\frac{m_e}{\hbar}&\frac{\partial}{\partial k_j}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')}\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{R}\alpha\rangle \\ =\frac{m_e}{\hbar}&\frac{\partial}{\partial k_j}\tilde{H}^{\alpha'\alpha}_{\boldsymbol{k}}. \end{split} \label{Fermi_velocity}$$ It follows from the Schrödinger equation (\[sgl\_bloch\]) that the Hamiltonian $\tilde{H}^{\alpha\alpha'}_{\boldsymbol{k}}$ defines the tight-binding-like eigenvalue problem $$\sum_{\alpha'}\tilde{H}^{\alpha\alpha'}_{\boldsymbol{k}}c^{n}_{\alpha'}(\boldsymbol{k})=\varepsilon_{\boldsymbol{k}}^{n}c^{n}_{\alpha}(\boldsymbol{k}). \label{TB_eig}$$ The second term in Eq. (\[p\_shift\]) contains continuum contributions due to the finite extension of the electron wave functions. It can be written as $$\begin{split}-\frac{i m_e}{\hbar}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')} \int\boldsymbol{dr}r_j\sum_{\boldsymbol{R}'',\alpha''}&\big[\langle\boldsymbol{R}'\alpha'|\boldsymbol{r}+\boldsymbol{R}'\rangle\langle\boldsymbol{r}+\boldsymbol{R}'|\boldsymbol{R}''\alpha''\rangle\langle\boldsymbol{R}''\alpha'' |H|\boldsymbol{R}\alpha\rangle \\ &-\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{R}''\alpha''\rangle\langle\boldsymbol{R}''\alpha''|\boldsymbol{r}+\boldsymbol{R}\rangle\langle\boldsymbol{r}+\boldsymbol{R}|\boldsymbol{R}\alpha\rangle\big] \\ =-\frac{i m_e}{\hbar}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')} \int\boldsymbol{dr}r_j\sum_{\boldsymbol{R}'',\alpha''}&\big[(\chi_{\boldsymbol{R}'\alpha'}(\boldsymbol{r}+\boldsymbol{R}'))^*\chi_{\boldsymbol{R}''\alpha''}(\boldsymbol{r}+\boldsymbol{R}')\langle\boldsymbol{R}''\alpha'' |H|\boldsymbol{R}\alpha\rangle \\ &-\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{R}''\alpha''\rangle(\chi_{\boldsymbol{R}''\alpha''}(\boldsymbol{r}+\boldsymbol{R}))^*\chi_{\boldsymbol{R}\alpha}(\boldsymbol{r}+\boldsymbol{R})\big] \end{split} \label{p_second_term}$$ with the wave functions $\chi_{\boldsymbol{R}\alpha}(\boldsymbol{r})=\langle \boldsymbol{r} | \boldsymbol{R}\alpha \rangle $. This term accounts on the one hand for all atomic or *intra*-site $(\boldsymbol{R}=\boldsymbol{R}')$ processes and on the other hand for corrections to the *inter*-site processes contained in the Peierls term. Hence, we can split the momentum matrix element into three contributions: $$\langle\boldsymbol{k}\alpha'|\hat{p}_j|\boldsymbol{k}\alpha\rangle=\frac{m_e}{\hbar}\frac{\partial}{\partial k_j}\tilde{H}^{\alpha'\alpha}_{\boldsymbol{k}}+p^{\alpha'\alpha}_{\boldsymbol{k},j}\big|_{\textrm{inter-site corr.}}+p^{\alpha'\alpha}_{\boldsymbol{k},j}\big|_{\textrm{intra-site corr.}}\,. \label{momentum_contrib}$$ In the limit of well-localized orbitals, the dominant correction is given by the *intra*-site term [@Tomczak.2009]. It is obtained by using the lattice periodicity, $\chi_{\boldsymbol{R}\alpha}(\boldsymbol{r}+\boldsymbol{R})=\chi_{\boldsymbol{0}\alpha}(\boldsymbol{r}) $, to shift the origins of all wave functions to the same unit cell, which is labeled $\boldsymbol{0}$. We then identify those terms where the wave function arguments also lie within the same unit cell ($\boldsymbol{R}''=\boldsymbol{R}' $ in the first term, $\boldsymbol{R}''=\boldsymbol{R} $ in the second term): $$\begin{split} p^{\alpha'\alpha}_{\boldsymbol{k},j}\big|_{\textrm{intra-site corr.}}&=-\frac{i m_e}{\hbar}\frac{1}{N}\sum_{\boldsymbol{R},\boldsymbol{R}'}e^{i\boldsymbol{k}\cdot(\boldsymbol{R}-\boldsymbol{R}')} \int\boldsymbol{dr}r_j\sum_{\alpha''} \\ &\big[(\chi_{\boldsymbol{0}\alpha'}(\boldsymbol{r}))^*\chi_{\boldsymbol{0}\alpha''}(\boldsymbol{r})\langle\boldsymbol{R}'\alpha'' |H|\boldsymbol{R}\alpha\rangle -\langle\boldsymbol{R}'\alpha'|H|\boldsymbol{R}\alpha''\rangle(\chi_{\boldsymbol{0}\alpha''}(\boldsymbol{r}))^*\chi_{\boldsymbol{0}\alpha}(\boldsymbol{r})\big]\\ &=-\frac{i m_e}{\hbar}\int \boldsymbol{dr}r_j\sum_{\alpha''} \big((\chi_{\boldsymbol{0}\alpha'}(\boldsymbol{r}))^* \chi_{\boldsymbol{0}\alpha''}(\boldsymbol{r}) \tilde{H}^{\alpha''\alpha}_{\boldsymbol{k}} -\tilde{H}^{\alpha'\alpha''}_{\boldsymbol{k}}(\chi_{\boldsymbol{0}\alpha''}(\boldsymbol{r}))^* \chi_{\boldsymbol{0}\alpha}(\boldsymbol{r}) \big)\\ &=-\frac{i m_e}{\hbar}\sum_{\alpha''}\big(r_j^{\alpha'\alpha''}\tilde{H}^{\alpha''\alpha}_{\boldsymbol{k}}-\tilde{H}^{\alpha'\alpha''}_{\boldsymbol{k}}r_j^{\alpha''\alpha} \big) \end{split} \label{intra_site}$$ with the atom-like dipole matrix elements $$\begin{split} r_j^{\alpha\alpha'}=\int \boldsymbol{dr}r_j (\chi_{\boldsymbol{0}\alpha}(\boldsymbol{r}))^* \chi_{\boldsymbol{0}\alpha'}(\boldsymbol{r}) \end{split} \label{atomic_dipole}$$ forcing the usual optical selection rules $\Delta l = \pm 1,\Delta m =0,\pm 1 $. Note that the *intra*-site term (\[intra\_site\]) needs to take into account the same set of localized orbitals that is used to set up the lattice Hamiltonian (\[H\_lat\]). The fact that the coupling to the magnetic field via the vector potential has to be two-fold in a tight-binding or lattice approach has also been discussed in Ref. [@Wang.2015]. There are always contributions that can not be captured by the so-called Peierls substitution [@Tomczak.2009] in the lattice Hamiltonian leading to the first term in (\[momentum\_contrib\]). If we nevertheless use the Peierls contribution alone, we end up with the following lattice formulation of the orbital magnetic moment: $$\begin{split} \mu^{\textrm{orb},n}_{z\boldsymbol{k}}\big|_{\textrm{lat}}&=-\frac{e}{2 m_e}\frac{\hbar}{i m_e}\big(\frac{m_e}{\hbar}\big)^2\sum_{n'}\frac{1}{\varepsilon_{\boldsymbol{k}}^{n}-\varepsilon_{\boldsymbol{k}}^{n'}} \sum_{\alpha\alpha'}(c^{n}_{\alpha}(\boldsymbol{k}))^*c^{n'}_{\alpha'}(\boldsymbol{k}) \frac{\partial}{\partial k_x}\tilde{H}^{\alpha\alpha'}_{\boldsymbol{k}} \sum_{\alpha\alpha'}(c^{n'}_{\alpha'}(\boldsymbol{k}))^*c^{n}_{\alpha}(\boldsymbol{k}) \frac{\partial}{\partial k_y}\tilde{H}^{\alpha'\alpha}_{\boldsymbol{k}} \\ &-(x\leftrightarrow y) \\ &=\frac{ie}{2\hbar}\sum_{n'}\frac{1}{\varepsilon_{\boldsymbol{k}}^{n}-\varepsilon_{\boldsymbol{k}}^{n'}} \Big\{\sum_{\alpha\alpha'}(c^{n}_{\alpha}(\boldsymbol{k}))^*c^{n'}_{\alpha'}(\boldsymbol{k}) \frac{\partial}{\partial k_x}\tilde{H}^{\alpha\alpha'}_{\boldsymbol{k}} \sum_{\alpha\alpha'}(c^{n'}_{\alpha'}(\boldsymbol{k}))^*c^{n}_{\alpha}(\boldsymbol{k}) \frac{\partial}{\partial k_y}\tilde{H}^{\alpha'\alpha}_{\boldsymbol{k}}-c.c.\Big\} \\ &=-\frac{e}{\hbar}\sum_{n'}\frac{1}{\varepsilon_{\boldsymbol{k}}^{n}-\varepsilon_{\boldsymbol{k}}^{n'}} \textrm{Im}\,\Big\{\sum_{\alpha\alpha'}(c^{n}_{\alpha}(\boldsymbol{k}))^*c^{n'}_{\alpha'}(\boldsymbol{k}) \frac{\partial}{\partial k_x}\tilde{H}^{\alpha\alpha'}_{\boldsymbol{k}} \sum_{\alpha\alpha'}(c^{n'}_{\alpha'}(\boldsymbol{k}))^*c^{n}_{\alpha}(\boldsymbol{k}) \frac{\partial}{\partial k_y}\tilde{H}^{\alpha'\alpha}_{\boldsymbol{k}}\Big\}. \end{split} \label{m_z_lat}$$ In a simple two-band model, the structure of this expression leads to equal orbital magnetic moments for conduction and valence electrons. Corrections are expected due to transitions from each band, respectively, to energetically higher and lower bands. Moreover, if the two fundamental bands are composed of localized orbitals that do not allow for atom-like dipole transitions (e.g. $d_{z^2}$- and $d_{x^2-y^2}$/$d_{xy}$-orbitals in transition metal dichalcogenide monolayers), the *intra*-site correction will vanish. The exciton g-factor, calculated directly from the magnetic moments of conduction and valence bands at the $\boldsymbol{K}$-point as $g_X=2(\mu^{c}_{z\boldsymbol{K}}-\mu^{v}_{z\boldsymbol{K}})/\mu_{\textrm{B}}$, will therefore be zero since the net spin of the exciton is zero. This is consistent with the $k\cdot p$-picture discussed in Ref. [@Wang.2015].

--- abstract: 'Faraday Rotation Measure (RM) Synthesis, as a method for analyzing multi-channel observations of polarized radio emission to investigate galactic magnetic fields structures, requires the definition of complex polarized intensity in the wavelength range $-\infty <\lambda^2 < \infty$. The problem is that the measurements at negative $\lambda^2$ are not possible. We introduce a simple method for continuation of the observed complex polarized intensity $P(\lambda^2)$ into the domain $\lambda^2<0$ using symmetry arguments. The method is suggested in context of magnetic field recognition in galactic disks where the magnetic field is supposed to have a maximum in the equatorial plane. The method is quite simple when applied to a single Faraday-rotating structure on the line of sight. Recognition of several structures on the same line of sight requires a more sophisticated technique. We also introduce a wavelet-based algorithm which allows us to consider a set of isolated structures in the ($\phi,\lambda^2$) plane (where $\phi$ is the Faraday depth). The method essentially improves the possibilities for reconstruction of complicated Faraday structures using the capabilities of modern radio telescopes.' bibliography: - 'ref.bib' date: 'Accepted 2009 .... Received 2009 ....; in original form 2009' title: 'Wavelet-based Faraday Rotation Measure Synthesis' --- \[firstpage\] Methods: polarization – methods: data analysis – galaxies: magnetic fields – RM Synthesis – wavelets Introduction ============ Observations of polarized radio emission are the main sources of information on magnetic fields of galaxies. The basic idea of magnetic field analysis from polarized radio emission data originates in the classical paper of [@Burn1966MNRAS.133...67B] (for a later development see [@1998MNRAS.299..189S]). In particular, [@Burn1966MNRAS.133...67B] noted that the complex polarized intensity $P$ obtained from a radio source is related to the Faraday dispersion function $F(\phi)$ as $$\label{p_to_f} P(\lambda^2) = \int_{-\infty}^{\infty} F(\phi) e^{2i\phi \lambda^2} d \phi.$$ $F(\phi)$ is the fraction of radiation with the Faraday depth $\phi$ multiplied by intrinsic complex polarization and it is an important emission characteristic of interest. Here the Faraday depth $\phi$ is defined by $$\phi(z) = -0.81\int_{z}^{0} {\,B_{\parallel}}n_e dz', \label{fardep}$$ where ${\,B_{\parallel}}$ is the line-of-sight magnetic field component measured in $\mu$G, $n_e$ is the thermal electron density measured in cm$^{-3}$ and the integral is taken from the observer at $z=0$ over the region which contains both, magnetic fields and free electrons, and $z$ is measured in parsecs. Following Eq. (\[p\_to\_f\]) $P$ is the inverse Fourier transform of $F$. Correspondingly, the Faraday dispersion function $F$ is the Fourier transform of the complex polarized intensity: $$\label{f_to_p1} F(\phi) = {{1} \over{\pi}} \hat P(k), \label{Burn}$$ where $k=2\phi$, and the Fourier transform is defined as $$\label{four} f\left( x \right) = {{1}\over{2\pi}} \int_{-\infty}^{\infty} {\hat {f}\left( k \right)e^{i k x}d k}, \quad \hat {f}\left( k \right) = \int_{-\infty}^{\infty} {f\left( x \right)e^{ - i k x} dx}.$$ Implementation of multichannel spectro-polarimetry on modern radio telescopes provided observations of $P$ over a wide range of $\lambda$ which made the use of Eq. (\[Burn\]) possible. This is the idea of Faraday Rotation Measure Synthesis (RM Synthesis) which opened new perspectives in investigations of magnetic field of galaxies and clusters of galaxies . A key problem of RM Synthesis application is that $P$ is defined only for $\lambda^2>0$ and in practice can be observed only in a finite spectral band. Moreover, the maximum of $P$ in practice can be located outside the available spectral band [ (see e.g. Fig. \[classex\]b)]{}. Development of robust methods for the reconstruction of $F$ from $P$ in a given spectral range becomes crucial for the practical implementation of RM Synthesis. Fig. \[classex\] shows results of RM Synthesis applied to a standard test as exploited by . Panel (a) shows the function $F$, which includes three [*real-valued*]{} box-like structures, panel (b) - the corresponding polarized intensity $P$ (the dashed horizontal line shows the spectral window $ 0.6 < \lambda < 0.78$m). We used a channel spacing of $\delta\lambda=0.4$cm. Hereafter, $F$ and $P$ are numerically evaluated in arbitrary but mutually consistent units. Note that $F$ is in general a complex-valued function. Its modulus defines the emission and its phase defines the intrinsic position angle. Panel (c) shows the result of the straightforward application of the RM Synthesis algorithm to the physical range $\lambda^2 >0$, while $P(\lambda^2)$ is set to zero for all negative $\lambda^2$. We see that the real part of the reconstructed signal is the same as the initial one (except that it has a twice lower amplitude), however, the reconstructed signal obtains a substantial imaginary part with a shape which is quite remote from the real part. This leads to a change of the emission distribution and a loss of any information concerning the position angle (apart from the central point of the emission region, where the position angle correctly is zero). In the context of chaotic magnetic fields in galaxy clusters this loss is less important , but in galactic magnetic field studies it becomes crucial because the intrinsic position angle determines the orientation of the regular magnetic field component perpendicular to the line of sight. Fig. \[classex\]d shows that the reconstruction becomes much more difficult if we restrict the data to a relatively narrow spectral band $ 0.6 < \lambda < 0.78$m. We see that even the sign of the reconstructed real part can be wrong. In that case the algorithm for finite spectral band introduced by was used. A general message obtained from Fig. \[classex\] is that in order to envisage possible ways to get a practical implementation of RM Synthesis one has to include some additional information based on the nature of the physical phenomena which provide the Faraday rotation. Here we concentrate our efforts on the problems associated with missing $P(\lambda^2)$ for $\lambda^2<0$. Improving the RM Synthesis algorithm ==================================== The complex-valued intensity of polarized radio emission for a given wavelength $$\label{p-def} P(\lambda^2) = \int_{0}^{\infty} \varepsilon(z)e^{2i\chi(z)} e^{2i\phi(z) \lambda^2} d z,$$ is defined by the emissivity $\varepsilon$ and the intrinsic position angle $\chi$ along the line of sight. Here $z$ is the distance from observer to a point in the emitting region; the integral is taken over the whole emitting region. If the Faraday depth $\phi$ is a monotonic function of $z$ (which means that $z$ is a single-valued function of $\phi$), we can define the Faraday dispersion function as a function of Faraday depth $$\label{F-to-z} F(\phi)=\varepsilon(\phi)e^{2i\chi(\phi)}\left({d\phi}\over{dz}\right)^{-1}.$$ In the ideal case, reconstructing the Faraday dispersion function $F$ from (\[Burn\]) and knowing the Faraday depth $\phi$ for any $z$, one can derive the characteristics of radio emission ($\varepsilon$ and $\chi$) along the line of sight. They can be used [ as a tomography]{} in order to derive some characteristics of the magnetic field distribution from $F$. The task of RM Synthesis is much more modest and concerns the reconstruction of the Faraday function from the observed polarized emission which itself is already a complicated problem. Let us consider a physically motivated simple example, i.e. $P$ produced by a two-layer system , to isolate and overcome the shortcomings of the RM Synthesis technique. Each layer contains a homogeneous magnetic field which has non-vanishing line-of-sight and perpendicular components. Both layers are thought to be emitting and rotating polarized radio waves. The corresponding $F(\phi)$ is shown in Fig. 2a. It is important for the discussion below that the analyzed signal has non-vanishing real and imaginary parts. The absolute value of $F(\phi)$ indicates how much polarized emission comes from a region with Faraday depth $\phi$ and its phase gives the intrinsic position angle (about $13^\circ$ and $31^\circ$) of the emission. Just to illustrate the variety of possible situations, we choose two different shapes of the slabs, i.e. one slab with sharp boundaries and one with a Gaussian shape. The result of the straightforward application of RM Synthesis where the integral is taken over the physically admissible region $\lambda^2>0$ is shown in Fig. \[classex1\]b. RM Synthesis reproduces to some extent the absolute value of the signal, but fails to reproduce its phase. A naive interpretation of this result could be that field reversals occur in each layer, but is obviously incorrect. In the same figure we show the result of $F(\phi)$ reconstruction within the spectral band $ 0.6 < \lambda < 0.78$m (panel c). Then both structures become diffuse with a more or less arbitrary phase. The last panel illustrates what happens if the upper wavelength boundary will be extended up to $\lambda =2.5$m (as expected for the Low Frequency Array (LOFAR) and the Square Kilometre Array (SKA) telescopes). This extension essentially improves the recognition of the sharp structure (the right one in the figure) but almost does not affect the reconstruction of the left (Gaussian) structure. To avoid the non-uniqueness in the Faraday dispersion function reconstruction, some additional information (or hypothesis) is required. We suggest to improve the above reconstruction by some constraint concerning the possible symmetry of an isolated object. Suppose that the expected objects are mainly galactic disks with magnetic fields believed to be symmetric with respect to the galactic equator. Then the desired $F$ should be even with respect to the center of the given object. Therefore, we consider each maximum of the reconstructed $F(\phi)$ separately and prescribe that the continuation of $P(\lambda^2)$ to the region of $\lambda^2 <0$ has to be chosen in a way which makes $F(\phi)$ symmetric with respect to the point $\phi = \phi_0$, where $\phi_0$ is the position of the maximum under consideration. This means that $F(2\phi_0-\phi)=F(\phi)$ and using the shift theorem one gets $$P(-\lambda^2) = \exp (-4 i \phi_0 \lambda^2) P(\lambda^2). \label{cont}$$ The antisymmetric case can be considered as well with slight change in the algorithm: Eq. (\[cont\]) changes to $P(-\lambda^2) = -\exp (-4 i \phi_0 \lambda^2) P(\lambda^2)$. Fig. \[classex2\] shows the results of reconstruction of the same test but following the suggested continuation. The test function includes two objects, while the algorithm includes only one parameter $\phi_0$. Firstly, we performed the continuation adjusting $\phi_0$ to the position of the left object (panel a). Then the method gives realistic result for this object. The reconstructed structure has no apparent internal field reversal and the ratio of real and imaginary parts of $F(\phi)$, i.e. the phase, is correctly reproduced. Position angles are restored with the accuracy of $3^\circ$. Of course, the result for the other layer, i.e. the second maximum of $|F(\phi)|$ in Fig. \[classex2\] remains false. Panel (b) shows what happens if the range of $\lambda$ covered by the observation is reduced to $ 0.6 < \lambda < 0.78$m. Instead of one peak one gets a sequence of peaks, which is a usual result for a Fourier reconstruction using a narrow spectral window. The suggested procedure does not suppress the sidelobes in the standard Rotation Measure Spread Function (RMSF) [@2009arXiv0905.3995H] but corrects the phase within the main central peak. Of course, the amplitude of each peak is much less than the amplitude of the peak in panel (a), however, the ratio of real and imaginary parts of $F(\phi)$ in the central peak remains realistic. If the parameter $\phi_0$ is chosen following the position of the second object the method gives a correct reconstruction for the right layer and fails to reproduce the left one. An obvious shortcoming of the method exploited is its local nature: We obtain a realistic shape of a chosen maximum and ignore what happens with the other one. A natural extension is to apply the recommendation of Eq. (\[cont\]) locally to each maximum. This extension brings the idea of wavelets into consideration. RM Synthesis and wavelets ========================= Wavelet transform presents a kind of “local” Fourier transform, allowing us to isolate a given structure in physical space and the Fourier space. Let us define the wavelet transform of the Faraday dispersion function $F(\phi)$ as $$\label{wF_d} w_F(a,b) = {{1}\over {|a|}} \int\limits_{ - \infty }^\infty {F(\phi)\psi ^\ast \left( {\frac{\phi - b}{a}} \right)d\phi} ,$$ where $\psi(\phi)$ is the analyzing wavelet, $a$ defines the scale and $b$ defines the position of the wavelet. Then the coefficient $w_F$ gives the contribution of corresponding structure into the function $F$. The function $F$ can be reconstructed using the inverse transform (see, e.g. [@1992ApMat..61.....D]) $$\label{wF_inv} F(\phi) = \frac{1}{C_\psi }\int\limits_{-\infty}^\infty {\int\limits_{ - \infty }^\infty {\psi \left( {\frac{\phi - b}{a}} \right)w_F\left( {a,b} \right)\frac{d a \, d b}{a^2}} }.$$ The reconstruction formula (\[wF\_inv\]) exists under condition that $$\label{adm} C_\psi = \frac1{2}\int\limits_{ - \infty }^\infty {\frac{\vert \hat {\psi }(k )\vert ^2}{ |k| }d k < \infty } .$$ Here $\hat{\psi }(k ) = \int {\psi (\phi)e^{ - ik\phi}d\phi} $ is the Fourier transform of the analyzing wavelet $\psi(\phi)$. Let us emphasize that the inverse formula (\[wF\_inv\]) is usually written for real signals. Then the scale parameter $a$ is positively defined and the integral is taken for $0< a<\infty$. In the case of a complex-valued function, the range of $a$ can be limited by positive values $a>0$ by taking a [*real*]{} analyzing wavelet $\psi(x)$. In general case of a complex-valued function and a complex wavelet, the scale parameter $a$ should be extended into the domain of negative values (like wave numbers in Fourier space). For the sake of definiteness, we use as the analyzing wavelet the so-called Mexican hat $\psi (\phi) = (1-\phi^2) \exp (-\phi^2/2)$. The wavelet is real, however, the function $P$ is complex, so that the wavelet coefficients $w_F$ are complex as well. For the chosen wavelet $w_F(-a,b)=w_F(a,b)$ and $C_\psi = 1$. Using the definition of the wavelet transform (\[wF\_d\]) and relation (\[Burn\]) we can directly define the wavelet decomposition of the Faraday dispersion function from the polarized intensity $P(\lambda^2)$ $$\label{wF_P} w_F(a,b) = {{1}\over {\pi}} \int\limits_{ - \infty}^\infty {P(\lambda^2)e^{-2ib \lambda^2} \hat{\psi}^\ast \left( -2a \lambda^2 \right)d \lambda^2} .$$ Note that in the case of real $F$ the problem of negative $\lambda^2$ can be solved using progressive wavelets, whose Fourier image is localized in the domain of positive wave numbers. Thus using this kind of wavelets one avoids the problem of the $P(\lambda^2)$ continuation in the domain $\lambda^2 <0$. For the general case, we divide Eq. (\[wF\_P\]) in two parts $w_F(a,b) =w_-(a,b)+w_+(a,b)$, where $$\begin{aligned} \label{wF_P2} w_-(a,b)&=& {{1}\over {\pi}} \int\limits_{ - \infty }^0 {P(\lambda^2)e^{-2ib \lambda^2} \hat{\psi}^\ast \left( -2a \lambda^2 \right)d\lambda^2}, \\ w_+(a,b)&=&{{1}\over {\pi}} \int\limits_0^\infty {P(\lambda^2)e^{-2ib \lambda^2} \hat{\psi}^\ast \left( -2a \lambda^2 \right)d\lambda^2}.\end{aligned}$$ We propose the following algorithm: Firstly, knowing $P(\lambda^2)$ for $\lambda^2>0$ we calculate the coefficients $w_+(a,b)$ and we recognize the dominating structures in the map $|w_+(a,b)|$. The coordinate $b$ of the corresponding maximum gives us the value of $\phi_0^i$, where upper index $i$ indicates the number of the structure. Then we reconstruct the coefficients $w_-(a,b)$ following the idea of Eq. (\[cont\]), but reformulated for the local domain in wavelet space $(a,b)$. Namely, we define $$\label{w-w+} w_-(a,b)=w_+\left(a,2\phi_0^i(a,b)-b\right),$$ where the parameter $\phi_0^i(a,b)$ for the given point $(a,b)$ is chosen according to the structure $i$ which dominates in its vicinity. Now we apply the suggested algorithm to the test function from Fig. \[classex1\]. The map $|w_+(a,b)|$ presented in Fig. \[classex3\]a demonstrates two well-defined structures. The $b$-coordinates of the maxima are taken as $\phi_0^i$. The result of the reconstruction (see Fig.  \[classex3\]b) shows that the method reproduces the amplitude and phase of $F(\phi)$ for both layers. The reconstruction here is performed using $P(\lambda^2)$ for the whole range $\lambda^2>0$. [ The comparison of the reconstructed position angle using standard and wavelet-base RM Synthesis is shown in Fig. \[comp\]. The suggested algorithm gives correct value for $\chi$ within both emission regions.]{} Panels (c,d) show what happens for the reconstruction using the spectral window $ 0.6 < \lambda < 0.78$m. One can see the wavelet map is empty in its substantial part $a>2$, however, the structures remain well-recognizable (panel c). The reconstructed $F$ contains several oscillations in domains related to both layers. The amplitude of each oscillation becomes much lower than that in panel (b), however, the ratio of the real and imaginary parts in the central maxima remain correct. The third couple of panels shows the reconstruction within the extended window $ 0.6 < \lambda < 2.5$m. This extension allows one to keep the horn-like structures in the bottom of the wavelet plane (panel e) which provide the reconstruction of sharp boundaries of the box-like structure (panel f). Conclusions =========== The development of multi-channel observations of polarized radio emission opens promising perspectives in the understanding of cosmic magnetic fields on galactic and intergalactic scales. The first fruitful applications of RM Synthesis suggested in this context include the recognition of local structures in the Milky Way , clusters of galaxies and spiral galaxies [@2009arXiv0905.3995H]. However, in general the RM Synthesis algorithm contains a fundamental problem emerging from the fact that the reconstruction formula requires the definition of complex polarized intensity in the range $-\infty <\lambda^2 < \infty$. In this paper we introduce a simple method for continuation of observed complex polarized intensity $P(\lambda^2)$ into the domain of negative $\lambda^2<0$. The method is suggested in context of magnetic field recognition in galactic disks, for which the magnetic field strength is supposed to have a maximum in the equatorial plane. The suggested method is quite simple when applied to a single structure on the line of sight. Recognition of several structures on the same line of sight requires a more sophisticated technique. The problem of structure separation is resolved using the wavelet decomposition. A simple test example demonstrates the applicability of this method. [ The polarization angle reconstruction is significantly improved over the standard technique.]{} The wavelets can be useful to also overcome some other problems of RM Synthesis, related to the multi-band structure of the observational domain in $\lambda$-space, noise filtration, etc [e.g. @1997ApJ...483..426F; @2001MNRAS.327.1145F]. The method essentially improves the possibilities for reconstruction of complicated Faraday structures using the capabilities of modern radio telescopes. Finally note that our simple examples illustrate that the extension of the observational band into the long-wavelength domain is helpful for the recognition of structures with sharp boundaries, while the short-wavelength domain is crucial for the reconstruction of smooth structures. \[lastpage\] Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the DFG-RFBR grant 08-02-92881.

--- abstract: 'Scaling laws that describe the structure of river networks are shown to follow from three simple assumptions. These assumptions are: (1) river networks are structurally self-similar, (2) single channels are self-affine, and (3) overland flow into channels occurs over a characteristic distance (drainage density is uniform). We obtain a complete set of scaling relations connecting the exponents of these scaling laws and find that only two of these exponents are independent. We further demonstrate that the two predominant descriptions of network structure (Tokunaga’s law and Horton’s laws) are equivalent in the case of landscapes with uniform drainage density. The results are tested with data from both real landscapes and a special class of random networks.' author: - Peter Sheridan Dodds - 'Daniel H. Rothman' title: 'Unified View of Scaling Laws for River Networks\' --- [^1] Introduction ============ If it is true that scaling laws abound in nature [@mandelbrot83], then river networks stand as a superb epitome of this phenomenon. For over half a century, researchers have uncovered numerous power laws and scaling behaviors in the mathematical description of river networks [@horton45; @langbein47; @strahler52; @hack57; @tarboton88; @labarbera89; @tarboton90; @maritan96a]. These scaling laws, which are usually parameterized by exponents or ratios of fundamental quantities, have been used to validate scores of numerical and theoretical models of landscape evolution [@leopold62; @shreve66; @howard71b; @stark91; @meakin91; @willgoose91a; @willgoose91c; @willgoose91; @kramer92; @leheny93; @sun94; @sun94b; @somfai97; @rodriguez-iturbe97] and have even been invoked as evidence of self-organized criticality [@rodriguez-iturbe97; @bak97]. However, despite this widespread usage, there is as yet no fundamental understanding of the origin of scaling laws in river networks. It is the principal aim of this paper to bring together a large family of these scaling laws within a simple, logical framework. In particular, we demonstrate that from a base of three assumptions regarding network geometry, all scaling laws involving planform quantities may be obtained. The worth of these consequent scaling laws is then seen to rest squarely upon the shoulders of the structural assumptions themselves. We also simplify the relations between the derived laws, demonstrating that only two scaling exponents are independent. The paper is composed in the following manner. We first present preliminary definitions of network quantities and a list of empirically observed scaling laws. Our assumptions will next be fully stated along with evidence for their validity. Several sections will then detail the derivations of the various scaling laws, being a combination of both new insights of our own as well as previous results. Progressing in a systematic way from our assumptions, we will also be required to amend several inconsistencies persistent in other analyses. The theory will be tested with comparisons to data taken from real landscapes and [Scheidegger]{}’s random network model [@scheidegger67b; @scheidegger90]. The ordering of streams ======================= A basic tool used in the analysis of river networks is the device of stream ordering. A stream ordering is any scheme that attaches levels of significance to streams throughout a basin. Most orderings identify the smallest tributaries as lowest order streams and the main or ‘trunk’ stream as being of highest order with the intermediary ‘stream segments’ spanning this range in some systematic fashion. Stream orderings allow for logical comparisons between different parts of a network and provide a basic language for the description of network structure. Here, we build our theory using the most common ordering scheme, one that was first introduced by [Horton]{} in his seminal work on erosion [@horton45]. [Strahler]{} later improved this method [@strahler57] and the resulting technique is commonly referred to as Horton-Strahler stream ordering [@rodriguez-iturbe97]. The most natural description of this stream ordering, due to [Melton]{} [@melton59], is based on an iterative pruning of a tree representing a network as shown in Figure \[fig:scaling.order\]. All source (or external) streams are pared away from the tree, these being defined as the network’s first order ‘stream segments’. A new tree is thus created along with a new collection of source streams and these are precisely the second order stream segments of the original network. The pruning and order identification continues in like fashion until only the trunk stream segment of the river network is left. The overall order of the basin itself is identified with the highest stream order present. The usual and equivalent description details how stream orders change at junctions [@rodriguez-iturbe97]. When a stream segment of order ${\omega}_1$ merges with a stream segment of order ${\omega}_2$, the outgoing stream will have an order of ${\omega}$ given by $${\omega}= \max({\omega}_1,{\omega}_2) + \delta_{{\omega}_1,{\omega}_2} \label{eq:scaling.streamordering}$$ where $\delta$ is the Kronecker delta. In other words, stream order only increases when two stream segments of the same order come together and, otherwise, the highest order is maintained by the outflowing stream. Planform network quantities and scaling laws {#sec:scaling.defs} ============================================ The results of this paper pertain to networks as viewed in planform. As such, any effects involving relief, the vertical dimension, are ignored. Nevertheless, we show that a coherent theory of planform quantities may still be obtained. This section defines the relevant quantities and their various permutations along with scaling laws observed to hold between them. The descriptions of these laws will be short and more detail will be provided in later sections. The two essential features in river networks are basins and the streams that drain them. The two basic planform quantities associated with these are drainage area and stream length. An understanding of the distribution of these quantities is of fundamental importance in geomorphology. Drainage area, for example, serves as a measure of average discharge of a basin while its relationship with the length of the main stream gives a sense of how basins are shaped. General network quantities {#sec:scaling.genquants} -------------------------- Figure \[fig:scaling.planformq\] shows a typical drainage basin. The basin features are $a$, the area, $l$, the length of the main stream, and $L_\parallel$ and $L_\perp$, the overall dimensions. The main (or trunk) stream is the dominant stream of the network—it is traced out by moving all the way upstream from the outlet to the start of a source stream by choosing at each junction (or fork) the incoming stream with the largest drainage area. This is not to be confused with stream segment length which only makes sense in the context of stream ordering. We will usually write $L$ for $L_\parallel$. Note that any point on a network has its own basin and associated main stream. The sub-basin in Figure \[fig:scaling.planformq\] illustrates this and has its own primed versions of $a$, $l$, $L_\parallel$ and $L_\perp$. The scaling laws usually involve comparisons between basins of varying size. These basins must be from the same landscape and may or may not be contained within each other. Several scaling laws connect these quantities. One of the most well known is Hack’s law [@hack57]. Hack’s law states that $l$ scales with $a$ as $$l \sim a^{h} \label{eq:scaling.hack}$$ where $h$ is often referred to as Hack’s exponent. The important feature of Hack’s law is that $h \neq 1/2$. In particular, it has been observed that for a reasonable span of basin sizes that $0.57 < h < 0.60$ [@hack57; @maritan96a; @gray61; @rigon96]. The actual range of this scaling is an unresolved issue with some studies demonstrating that very large basins exhibit the more expected scaling of $h=1/2$ [@mueller72; @mosley73; @mueller73]. We simply show later that while the assumptions of this paper hold so too does Hack’s law. Further comparisons of drainage basins of different sizes yield scaling in terms of $L (= L_\parallel)$, the overall basin length. Area, main stream length, and basin width are all observed to scale with $L$ [@maritan96a; @tarboton88; @labarbera89; @tarboton90; @labarbera90], $$a \sim L^D, \qquad l \sim L^{{d}}, \qquad L_\perp \sim L^{H}. \label{eq:scaling.scalingwithL}$$ Turning our attention to the entire landscape, it is also observed that histograms of stream lengths and basin areas reveal power law distributions [@maritan96a; @rodriguez-iturbe97]: $$P(a) \sim a^{-\tau} \qquad \mbox{and} \qquad P(l) \sim l^{-\gamma}. \label{eq:scaling.powerlawdist}$$ There are any number of other definable quantities and we will limit ourselves to a few that are closely related to each other. We write $\lambda$ for the average distance from a point on the network to the outlet of a basin (along streams) and $\Lambda$ for the unnormalized total of these distances. A minor variation of these are $\tilde{\lambda}$ and $\tilde{\Lambda}$, where only distances from junctions in the network to the outlet are included in the averages. The scaling law involving these particular quantities is Langbein’s law [@langbein47] which states that $$\Lambda \sim a^\beta. \label{eq:scaling.langbeinslaw}$$ Similarly, we have $\lambda \sim L^\varphi$, $\tilde{\Lambda} \sim a^{\tilde{\beta}}$ and $\tilde{\lambda} \sim L^{\tilde{\varphi}}$, [@maritan96a]. Network quantities associated with stream ordering -------------------------------------------------- With the introduction of stream ordering, a whole new collection of network quantities appear. Here, we present the most important ones and discuss them in the context of what we identify as the principal structural laws of river networks: Tokunaga’s law and Horton’s laws. ### Tokunaga’s law Tokunaga’s law concerns the set of ratios, $\{T_{{\omega},{\omega}'}\}$, first introduced by [Tokunaga]{} [@tokunaga66; @tokunaga78; @tokunaga84; @peckham95; @newman97]. These ‘Tokunaga ratios’ represent the average number of streams of order ${\omega}'$ flowing into a stream of order ${\omega}$ as side tributaries. In the case of what we will call a ‘structurally self-similar network’, we have that $T_{{\omega},{\omega}'} = T_{{\omega}-{\omega}'} = T_\nu$ where $\nu = {\omega}-{\omega}'$ since quantities involving comparisons between features at different scales should only depend on the relative separation of those scales. These $T_\nu$, in turn, are observed to be dependent such that [@tokunaga66], $$T_{\nu+1}/T_{\nu} = R_T \label{eq:scaling.tokdef}$$ where $R_T$ is a fixed constant for a given network. Thus, all of Tokunaga’s ratios may be specified by two fundamental parameters $T_1$ and $R_T$: $$T_\nu = T_1 (R_T)^{\nu-1}. \label{eq:scaling.toklaw}$$ We refer to this last identity as Tokunaga’s law. The network parameter $T_1$ is the average number of major side tributaries per stream segment. So for a collection of stream segments of order ${\omega}$, there will be on average $T_1$ side tributaries of order ${\omega}-1$ for each stream segment. The second network parameter $R_T$ describes how numbers of side tributaries of successively lower orders increase, again, on average. As an example, consider that the network in Figure \[fig:scaling.order\] is part of a much larger network for which $T_1 = 2$ and $R_T = 4$. Figure \[fig:scaling.order\] (b) shows that the third order stream segment has two major side tributaries of second order which fits exactly with $T_1 = 2$ (Note that the two second order stream segments that come together to create the third order stream segment are not side tributaries). Figure \[fig:scaling.order\] (a) further shows nine first order tributaries, slightly above the average eight suggested by $T_2 = T_1 R_T^{1} = 8$. Finally, again referring to Figure \[fig:scaling.order\] (a), there are $9/4 = 2.25$ first order tributaries for each second order stream segment, not far from the expected number $T_1 = 2$. ### Horton’s laws [Horton]{} introduced several important measurements for networks in conjunction with his stream ordering [@horton45]. The first is the bifurcation ratio, $R_n$. This is the ratio of the number $n_\omega$ of streams of order $\omega$ to the number $n_{\omega+1}$ of streams of order $\omega + 1$ and is, moreover, observed to be independent of $\omega$ over a large range. There is next the stream length ratio, $R_\ell = \bar{\ell}_{{\omega}+1}/\bar{\ell}_{{\omega}}$, where $\bar{\ell}_{{\omega}}$ is the average length of stream segments of order ${\omega}$. These lengths only exist within the context of stream ordering. In contrast to these are the main stream lengths, which we have denoted by $l$ and described in section \[sec:scaling.genquants\]. Main stream lengths are defined regardless of stream ordering and, as such, are a more natural quantity. Note that stream ordering gives rise to a discrete set of basins, one for each junction in the network. We therefore also have a set of basin areas and main stream lengths defined at each junction. Taking averages over basins of the same order we have $\bar{a}_{\omega}$ and $\bar{l}_{\omega}$ to add to the previously defined $\bar{\ell}_{\omega}$ and $n_{\omega}$. The connection between the two measures of stream length is an important, if simple, exercise [@scheidegger68c]. Assuming $\bar{\ell}_{{\omega}+1} = R_{\ell} \bar{\ell}_{{\omega}}$ holds for all $\omega$, one has $$\bar{l}_{{\omega}} = \sum_{i=1}^{{\omega}} \bar{\ell}_{i} = \sum_{i=1}^{{\omega}} (R_{\ell})^{i-1}\bar{\ell}_{1} = \bar{l}_{1}\frac{(R_{\ell})^{{\omega}}-1}{R_{\ell}-1} \label{eq:scaling.lconnect}$$ where $\bar{l}_1 = \bar{\ell}_1$ has been used. Since typically $R_\ell > 2$ [@kirchner93], $\bar{l}_{{\omega}+1}/\bar{l}_{\omega}\rightarrow R_{\ell}$ rapidly. For ${\omega}=4$ and $R_\ell = 2$, the error is only three per cent. On the other hand, starting with the assumption that main stream lengths satisfy Horton’s law of stream lengths for all $\omega$ implies that the same is true for stream segments. Thus, for most calculations, Horton’s law of stream lengths may involve either stream segments or main streams and, for convenience, we will assume that the law is fully satisfied by the former. Furthermore, this small calculation suggests that studies involving only third- or fourth-order networks cannot be presumed to have reached asymptotic regimes of scaling laws. We will return to this point throughout the paper. Schumm [@schumm56a] is attributed with the concrete introduction of a third and final law that was also suggested by Horton. This last ratio is for drainage areas and states that $R_a = \bar{a}_{{\omega}+1}/\bar{a}_{{\omega}}$. We will later show in section \[sec:scaling.Ra=Rn\] that our assumptions lead to the result that $R_a \equiv R_n$. At this stage, however, we write Horton’s laws as the three statements $$\frac{n_{{\omega}}}{n_{{\omega}+1}} = R_n, \ \frac{\bar{\ell}_{{\omega}+1}}{\bar{\ell}_{{\omega}}} = R_{\ell}, \ \mbox{and} \ \frac{\bar{a}_{{\omega}+1}}{\bar{a}_{{\omega}}} = R_a. \label{eq:scaling.hortslaws}$$ A summary of all of the scaling laws presented in this section is provided in Table \[tab:scaling.scalinglaws\]. Empirically observed values for the relevant exponents and ratios are presented in Table \[tab:scaling.values\]. [Law:]{} [Name or description:]{} ------------------------------------------------------------ ---------------------------------------- $T_{\nu} = T_1 (R_T)^{\nu-1}$ [Tokunaga’s law]{} $l \sim L^{{d}}$ [self-affinity of single channels]{} $n_{{\omega}+1}/n_{{\omega}} = R_n$ Horton’s law of stream numbers $\bar{\ell}_{{\omega}+1}/\bar{\ell}_{{\omega}} = R_{\ell}$ Horton’s law of stream segment lengths $\bar{l}_{{\omega}+1}/\bar{l}_{{\omega}} = R_{\ell}$ Horton’s law of main stream lengths $\bar{a}_{{\omega}+1}/\bar{a}_{{\omega}} = R_a$ Horton’s law of stream areas $l \sim a^h$ Hack’s law $a \sim L^D$ scaling of basin areas $L_\perp \sim L^H$ scaling of basin widths $P(a) \sim a^{-\tau}$ probability of basin areas $P(l)\sim l^{-\gamma}$ probability of stream lengths $\Lambda \sim a^\beta$ Langbein’s law $\lambda \sim L^\varphi$ variation of Langbein’s law $\tilde{\Lambda} \sim a^{\tilde{\beta}}$ as above $\tilde{\lambda} \sim L^{\tilde{\varphi}}$ as above : A general list of scaling laws for river networks. All laws and quantities are defined in section \[sec:scaling.defs\]. The principal finding of this paper is that these scaling laws follow from the first two relations, Tokunaga’s law (structural self-similarity) and the self-affinity of single channels, and the assumption of uniform drainage density (defined in section \[sec:scaling.assumptionsdd\]).[]{data-label="tab:scaling.scalinglaws"} Scheidegger’s random networks ----------------------------- To end this introductory section, we detail some of the features of the random network model of Scheidegger [@scheidegger67b; @scheidegger90]. Although originally defined without reference to a real surface, Scheidegger networks may be obtained from a completely uncorrelated landscape as follows. Assign a random height between 0 and 1 at every point on a triangular lattice and then tilt the lattice so that no local minima (lakes) remain. Scheidegger networks are then traced out by following paths of steepest descent. Surprisingly, these networks still exhibit all of the scaling laws observed in real networks. It thus provides an important point in ‘network space’ and accordingly, also provides an elementary test for any theory of scaling laws. Exact analytical results for various exponents are known due to the work of [Takayasu]{} [et al.]{} on the aggregation of particles with injection [@takayasu88; @takayasu89a; @takayasu89b; @takayasu90; @takayasu91; @huber91]. While there are no analytic results for the Tokunaga ratio $T_1$ or the Horton ratios $R_n$ and $R_{\ell}$, our own simulations show that these stream order laws are strictly obeyed. Table \[tab:scaling.values\] lists the relevant exponents and their values for the [[Scheidegger]{}]{} model along with those found in real networks. [Quantity:]{} [Scheidegger:]{} [Real networks:]{} --------------- ------------------ ------------------------------- $R_n$ $5.20 \pm .05$ 3.0–5.0 [@abrahams84] $R_a$ $5.20 \pm .05$ 3.0–6.0 [@abrahams84] $R_\ell$ $3.00 \pm .05$ 1.5–3.0 [@abrahams84] $T_1$ $1.30 \pm .05$ 1.0–1.5 [@tokunaga78] ${d}$ $1$ $1.1 \pm 0.01$ [@maritan96a] $D$ $3/2$ $1.8 \pm 0.1$ [@maritan96a] $h$ $2/3$ 0.57–0.60 [@maritan96a] $\tau$ $4/3$ $1.43 \pm 0.02$ [@maritan96a] $\gamma$ $3/2$ $1.8 \pm 0.1$ [@rigon96] $\varphi$ $1$ $1.05 \pm 0.01$ [@maritan96a] $H$ $1/2$ 0.75–0.80 [@maritan96a] $\beta$ $5/3$ 1.56 [@langbein47] $\varphi$ $1$ $1.05 \pm 0.01$ [@maritan96a] : Ratios and scaling exponents for Scheidegger’s random network model and real networks. For Scheidegger’s model, exact values are known due to the work of [Takayasu]{} [et al.]{} [@takayasu88; @takayasu89a; @takayasu89b; @takayasu90; @takayasu91; @huber91] and approximate results are taken from our own simulations. For real networks, the references given are generally the most recent and further appropriate references may be found within them and also in section \[sec:scaling.defs\].[]{data-label="tab:scaling.values"} Assumptions {#sec:scaling.assumptions} =========== We start from three basic assumptions about the structure of river networks: structural self-similarity, self-affinity of individual streams and uniformity of drainage density. We define these assumptions and their relevant parameters and then discuss their mutual consistency. We end with a discussion of the correspondence between the laws of Tokunaga and Horton. It should be stressed that while we make a case for each assumption there is also considerable proof to ponder in the pudding that these ingredients create. Structural self-similarity {#sec:scaling.assumptionsss} -------------------------- Our first assumption is that networks are structurally self-similar. It has been observed that river networks exhibit self-similarity over a large range of scales [@mandelbrot83; @tarboton88; @rodriguez-iturbe97]. Naturally, the physical range of this self-similarity is restricted to lie between two scales. The large scale cutoff is the overall size of the landscape and the small scale cutoff is of the order of the characteristic separation of channels [@montgomery92]. In order to quantify this phenomenon, we look to laws of network structure such as Tokunaga’s law and Horton’s laws of stream number and length. We demonstrate in the following section that these descriptions are mutually consistent within the context of our third assumption, uniformity of drainage density. Thus, we may assume a network where both Tokunaga’s and Horton’s laws hold. For convenience, we write these laws as if they hold for all orders down to the first order. Any actual deviations from these laws for low orders will not affect the results since we are interested in how laws behave for increasing stream order. Self-affinity of individual streams {#sec:scaling.assumptionsdl} ----------------------------------- Our second assumption is that individual streams are self-affine curves possessing a dimension ${d}>1$, as introduced in equation [(\[eq:scaling.scalingwithL\])]{}. Empirical support for this premise is to be found in [@tarboton88; @labarbera89; @tarboton90; @maritan96a; @rodriguez-iturbe97; @tarboton89]. In reality, this is at best a weak fractality with measurements generally finding ${d}$ to be around $1.1$ [@maritan96a]. We assume ${d}$ to be constant throughout a given network, true for each stream independent of order. In general, it is most reasonable to consider this in the sense of a growing fractal: stream length $l$ will grow like $L^{{d}}$ where $L$ is the overall length of a box containing a portion of a stream. So, rather than examine one fixed section of a stream, we take larger and larger pieces of it. Moreover, this is the most reasonable method for actually measuring ${d}$ for a real network. Uniform drainage density {#sec:scaling.assumptionsdd} ------------------------ Our third and final assumption is that drainage density is uniform throughout a network. For a given basin, the drainage density, $\rho$, is a measure of the average area drained per unit length of stream by overland flow (i.e., excluding contributions from tributary streams). Its usual form is that given by Horton [@horton45]: $$\rho = \frac{\sum\ell}{a} \label{eq:scaling.ddensity}$$ where, for a given basin, $\sum\ell$ represents the summed total length of all stream segments of all orders and $a$ is the drainage area. More generally, one can in the same way measure a local drainage density for any connected sections of a network within a landscape. Such sections should cover a region at least $\ell_1$ in diameter, the typical length of a first order stream. Drainage density being uniform means that the variation of this local drainage density is negligible. There is good support in the literature for the uniformity of drainage density in real networks [@hack57; @shreve67; @haggett69; @gardiner73; @morisawa62; @devries94] while there are some suggestions that it may vary slightly with order [@hack57; @tokunaga78]. Uniform drainage density may also be interpreted as the observation that the average distance between channels is roughly constant throughout a landscape [@horton45; @rodriguez-iturbe97], an estimate of this distance being simply $1/\rho$. This is due to the fact that there is a finite limit to the channelization of a landscape determined by a combination of soil properties, climate and so on. Implicit in this assumption is that the channel network has reached its maximum extension into a landscape [@shreve67; @glock31]. Indeed, In the bold words of [Glock]{} [@glock31], we are considering river networks at the “time of completed territorial conquest.” Furthermore, [Shreve]{} [@shreve67] notes that drainage density would be uniform in a “mature topography developed in a homogeneous environment.” Importantly, our third assumption connects the planform description to the surface within which the network lies. Computationally, the uniformity of drainage density allows for the use of the length of a stream as a proxy for drainage area [@devries94]. Further, the average distance between streams being roughly constant implies that, on average, tributaries are spaced evenly along a stream. Tokunaga’s law and Horton’s laws are equivalent =============================================== This section demonstrates an equivalence between Tokunaga’s law and Horton’s two laws of stream number and stream length in the case of a landscape with uniform drainage density. From Tokunaga’s law to Horton’s laws ------------------------------------ [Tokunaga]{} has shown that Horton’s law for stream numbers follows from Tokunaga’s law (given in equation [(\[eq:scaling.toklaw\])]{}) [@tokunaga78; @peckham95]. This follows from the observation that $n_{\omega}$, the number of streams of order ${\omega}$, in a basin of order ${\Omega}$ may be expressed as $$n_{\omega}= 2n_{{\omega}+1} + \sum_{\nu=1}^{{\Omega}-{\omega}} T_\nu n_{{\omega}+\nu}. \label{eq:scaling.nwtok}$$ The $2n_{{\omega}+1}$ accounts for the fact that each order ${\omega}+1$ stream is initiated by the confluence of two streams of order ${\omega}$. Presuming Tokunaga’s law, a simple analysis of equation [(\[eq:scaling.nwtok\])]{} shows that in the limit of large ${\Omega}$, the ratio $n_{\omega}/n_{{\omega}+1}$ does indeed approach a constant. This leads to an expression for the Horton ratio $R_n$ in terms of the two Tokunaga parameters $T_1$ and $R_T$ (first obtained by [Tokunaga]{} in [@tokunaga78]): $$2 R_n = (2 + R_T + T_1) + \left[(2 + R_T + T_1)^2 - 8R_T\right]^{1/2}. \label{eq:scaling.tokhortlink1}$$ [Tokunaga]{}’s work has been recently generalized by [Peckham]{} who deduces links to the other Horton ratios $R_{\ell}$ and $R_a$ [@peckham95]. In contrast to the purely algebraic calculation of $R_n$, these results require the step of equating topological properties to metric basin quantities. In determining $R_{\ell}$, Peckham uses the number of side tributaries to a stream as an estimate of stream segment length. This is based on the assumption that tributaries are evenly spaced. As discussed in section \[sec:scaling.assumptionsdd\], this even spacing of tributaries follows for networks with uniform drainage density. Therefore, we may write, after [Peckham]{}, that $$\bar{\ell}_{\omega}\propto 1+\sum_{\nu=1}^{{\omega}-1}{T_\nu} \label{eq:scaling.tokhortlink2a}$$ where the dimension of length absent on the right-hand side is carried by an appropriate constant of proportionality. This sum is simply the total number of tributaries that, on average, enter a stream of order ${\omega}$. The number of lengths of stream between tributaries is then simply one more in number. Using Tokunaga’s law (equation [(\[eq:scaling.toklaw\])]{}) we find that $$\bar{\ell}_{{\omega}+1}/\bar{\ell}_{{\omega}} = R_T \left( 1 + O (R_T)^{-\omega} \right), \label{eq:scaling.tokhortlink2b}$$ obtaining [Horton]{}’s stream length ratio with the simple identification: $$R_{\ell} = R_T \label{eq:scaling.tokhortlink2}$$ and we will use $R_{\ell}$ in place of $R_T$ throughout the rest of the paper. As already noted we will see that $R_a \equiv R_n$ for landscapes where drainage density is uniform. This redundancy means that there are only two independent Horton ratios, $R_{\ell}$ and $R_n$, which sits well with the two independent quantities required for Tokunaga’s law, $T_1$ and $R_T$. Presupposing this result, we can invert equations [(\[eq:scaling.tokhortlink1\])]{} and [(\[eq:scaling.tokhortlink2\])]{} to obtain Tokunaga’s parameters from the two independent Horton ratios: $$\begin{aligned} R_T & = & R_{\ell} \\ T_1 & = & R_n - R_{\ell} - 2 + 2R_{\ell}/R_n. \label{eq:scaling.tokhortinv}\end{aligned}$$ From Horton’s laws to Tokunaga’s law ------------------------------------ We now provide an heuristic argument to show that Tokunaga’s law in the form of equation [(\[eq:scaling.toklaw\])]{} follows from Horton’s laws of stream number and length and uniform drainage density. Note that even though we have shown in equations [(\[eq:scaling.tokhortlink1\])]{}, [(\[eq:scaling.tokhortlink2\])]{}, and [(\[eq:scaling.tokhortinv\])]{} that the parameters of Tokunaga’s law and those of Horton’s laws may be obtained from each other, it is not a priori clear that this result would be true. Indeed, Tokunaga’s law contains more direct information about network structure than Horton’s laws and it is the additional constraint of uniform drainage density that provides the key. Consider a stream of order ${\omega}$ along with its side tributaries of order ${\omega}' = 1$ through ${\omega}' ={\omega}-1$, the numbers of which are given by the usual $T_{\nu}$ where $\nu = {\omega}-{\omega}'$ (see Figure \[fig:scaling.hort2tok\]). Since the presumed adherence to Horton’s laws implies that a network is self-similar we need only consider the form of the $T_\nu$ and not the more general $T_{{\omega}',{\omega}}$. Now, again since networks are self-similar, a typical stream of order ${\omega}+ 1$ can be obtained by scaling up the picture of this order ${\omega}$ stream. As per Horton’s law of stream lengths, this is done by increasing the length of each stream by a factor of $R_{\ell}$ (Figure \[fig:scaling.hort2tok\] (a) becomes Figure \[fig:scaling.hort2tok\] (b)). However, since order ${\omega}'$ streams become ${\omega}'+1$ streams in this rescaling, the picture in Figure \[fig:scaling.hort2tok\] (b) is missing first order streams. Also, the average distance between tributaries has grown by a factor of $R_{\ell}$. Therefore, to retain the same drainage density, an extra $(R_{\ell}-1)$ first order streams must be added for each link (one more than the number of tributaries) along this new order ${\omega}+ 1$ stream (Figure \[fig:scaling.hort2tok\] (c)). Since the number of first order streams is now given by $T_{{\omega}+1}$ we have $$T_{{\omega}+1} = (R_{\ell}-1)\left(\sum_{\nu=1}^{{\omega}} T_\nu + 1 \right). \label{eq:scaling.hort2tok1}$$ It may be simply checked that this equation is satisfied, for large ${\omega}$, by Tokunaga ratios given by equation [(\[eq:scaling.toklaw\])]{}. Thus, Horton’s laws of stream number and stream length and the uniform drainage density are seen to imply Tokunaga’s law. In general, Horton’s ratios rather than the parameters of Tokunaga’s law will be the most useful parameters in what follows. In particular, we will see that the two independent quantities $R_n$ and $R_{\ell}$ will be needed only in the form $\ln R_n / \ln R_{\ell}$. All other exponents will be expressible as algebraic combinations of $\ln R_n / \ln R_{\ell}$ and ${d}$, the fractal dimension of an individual stream. Furthermore, example (or modal) values for the parameters of Horton and Tokunaga are [@tokunaga78; @kirchner93] $$T_1 = 1, \ R_T = R_\ell = 2, \ \mbox{and} \ R_n = 4. \label{eq:scaling.typical}$$ The parameters have been chosen so as to satisfy the inversion relations of equation [(\[eq:scaling.tokhortinv\])]{}. As shown in Table \[tab:scaling.values\], real networks provide some variation around these modal values. These will be used as rough checks of accuracy throughout the rest of the paper. Hack’s law ========== One of the most intriguing scalings found in river networks is Hack’s law [@hack57] which relates main stream length to basin area as $l \sim a^h$. This equation has been empirically shown to hold true for a large range of drainage basin sizes on many field sites [@rodriguez-iturbe97]. The salient feature is that for smaller basins [@rigon96], $h$ is typically found to be in the range $(0.56,0.60)$, whereas $0.5$ would be expected from simple dimensional analysis [@rodriguez-iturbe97]. It should be emphasized that Hack’s law is only true on average as are, for that matter, Tokunaga’s law and Horton’s laws. An extension of Hack’s law to a more natural statistical description of the connection between stream lengths and drainage areas was suggested by [Maritan]{} [et al.]{} [@maritan96a] with some further developments to be found in [@dodds98d]. Horton’s other law of stream numbers ------------------------------------ In order to obtain Hack’s law, we will use the uniformity of drainage density to estimate the area of an order ${\Omega}$ basin by calculating the total length of streams within the same basin. So we simply need the typical length and number of each stream order present. Taking the length of a source stream, $\bar{\ell}_1$, to be the finest resolution of the network and the basic unit of length, the length of a stream segment of order ${\omega}$ is $\bar{\ell}_{{\omega}}=(R_{\ell})^{{\omega}-1}\bar{\ell}_1$. However, in finding the frequency of such streams we find that some care must be taken for the following reasons. Horton’s law of stream numbers is potentially misleading in that it suggests, at first glance, that within a basin of order ${\omega}$ there should be one stream of order ${\omega}$, $R_n$ streams of order ${\omega}-1$, $R_n^2$ streams of order ${\omega}-2$ and so on. Indeed, many calculations involving Horton’s laws use this assumption [@labarbera89; @rodriguez-iturbe97; @devries94; @rosso91]. But Horton’s $R_n$ actually provides the ratio of the number of streams of consecutive orders as totalled for a *whole basin*. To illustrate this fact, consider streams of order ${\omega}$ and ${\omega}+1$ within a basin of order ${\Omega}\gg {\omega}$. As Tokunaga’s law makes clear, streams of order ${\omega}$ are not all found within sub-basins of order ${\omega}+1$. Indeed, a certain number of order ${\omega}$ streams will be tributaries to streams of order greater than ${\omega}+1$ (see the example network of Figure \[fig:scaling.order\] (a)). Tokunaga’s law shows that we should in fact expect $T_1+2$ rather than than $R_n$ streams of order ${\omega}$ entering into a stream of order ${\omega}+1$. For the typical values $T_1=1$ and $R_n=4$ in [(\[eq:scaling.typical\])]{} this is a substantial error. We proceed then to find a corrected version of Horton’s law of stream numbers. Returning to equation [(\[eq:scaling.nwtok\])]{}, we see that it is only valid in the limit ${\Omega}\rightarrow \infty$. Defining $n'({\omega},{\Omega})$ as the actual number of streams of order ${\omega}$ *within* a basin of order ${\Omega}$, we have $$n'({\omega},{\Omega}) = 2n'({\omega}+1,{\Omega}) + \sum_{\nu=1}^{{\Omega}-{\omega}} T_\nu n'({\omega}+\nu,{\Omega}). \label{eq:scaling.nwtokbetter}$$ This equation may be exactly solved. Considering the above expression for $n'({\omega},{\Omega})$ and the corresponding one for $n'({\omega}+1,{\Omega})$ we can reduce this to a simple difference equation, $$n'({\omega},{\Omega}) = (2+R_{\ell}+T_1)n'({\omega}+1,{\Omega}) -2R_{\ell} n'({\omega}+2,{\Omega}) \label{eq:scaling.nwtokdiffeq}$$ which has solutions of the form $\mu^k$. Applying the constraints that $n'({\Omega},{\Omega})=1$ and $n'({\Omega}-1,{\Omega})=T_1+2$, we obtain $$n'({\omega},{\Omega}) = c (\mu_{+})^{{\Omega}-{\omega}} + (1-c) (\mu_{-})^{{\Omega}-{\omega}} \label{eq:scaling.nwtokdiffeqsoln}$$ where $$2\mu_{\pm} = (2+R_{\ell}+T_1) \pm \left[(2 + R_\ell + T_1)^2 - 8R_\ell \right]^{1/2} \label{eq:scaling.mudef}$$ and $$c=R_n(R_n-R_{\ell})/(R_n^2-2R_{\ell}). \label{eq:scaling.cdef}$$ Note that $R_n = \mu_+$ and we will use the notation $R_n^\ast$ in place of $\mu_-$. This observation regarding Horton’s law of stream numbers was first made by [[Tokunaga]{}]{} [@tokunaga66] and later by [Smart]{} [@smart67]. In particular, Tokunaga noted that this would explain the deviation of Horton’s law for the highest orders of a basin, a strong motivation for his work. We can now define an effective Horton ratio, ${R_n}'({\omega},{\Omega})$ as follows: $$\begin{aligned} {R_n}'({\omega},{\Omega}) & = & n'({\omega}-1,{\Omega})/n'({\omega},{\Omega}) \nonumber\\ & = & R_n \left( 1 + O(R_n^\ast/R_n)^{({\Omega}-{\omega})} \right)\end{aligned}$$ The typical values of Horton’s ratios in [(\[eq:scaling.typical\])]{} give $R_n^\ast = 1$. In this case, ${R_n}'({\omega},{\Omega})$ converges rapidly to $R_n$ with an error of around one per cent for ${\omega}={\Omega}-3$. Hack’s law {#sec:scaling.hackslaw} ---------- As discussed in section \[sec:scaling.assumptionsdd\], an estimate of total drainage area of a basin is given by the total length of all streams within the basin. Summing over all stream orders and using the numbers $n'({\omega},{\Omega})$ given by equations [(\[eq:scaling.nwtokdiffeqsoln\])]{} and [(\[eq:scaling.mudef\])]{} we have that $$\begin{aligned} \bar{a}_{\Omega}& \propto & \sum_{{\omega}=1}^{{\Omega}} n'({\omega},{\Omega}) (R_{\ell})^{{\omega}-1} \nonumber \\ & = & c_1(R_n)^{{\Omega}} + c_2(R_{\ell})^{{\Omega}} - c_3(R^\ast_n)^{{\Omega}} \label{eq:scaling.areawRBRL}\end{aligned}$$ where $c_1=c/(R_n-R_\ell)$, $c_3=(1-c)/(R_\ell-R^\ast_n)$ and $c_2=c_3-c_1$ with $c$ being given in equation [(\[eq:scaling.cdef\])]{}. Slightly more complicated is the estimate of $\bar{a}({\omega},{\Omega})$, the drainage area of a basin of order ${\omega}$ within a basin of order ${\Omega}$: $$\begin{aligned} \bar{a}({\omega},{\Omega}) & \propto & 1/n'({\omega},{\Omega}) \sum_{{\omega}'=1}^{{\omega}} n'({\omega}',{\Omega}) (R_{\ell})^{{\omega}'-1} \nonumber \\ & = & 1/n'({\omega},{\Omega}) \left[ c_1 (R_n)^{\Omega}(1-(R_\ell/R_n)^{\omega}) \right. \nonumber \\ & & + \left. c_3 (R_\ell)^{\omega}(R_n^\ast)^{{\Omega}-{\omega}}(1-(R_n^\ast/R_\ell)^{\omega}) \right]. \label{eq:scaling.areawRBRLsub}\end{aligned}$$ Now, for $1 \ll {\omega}\ll {\Omega}$ (typically, $3<{\omega}<{\Omega}-2$ is sufficient), this expression is well approximated as $$\bar{a}({\omega},{\Omega}) \sim (R_n)^{\omega}. \label{eq:scaling.areaapprox}$$ since $R_n > R_{\ell} > R_n^\ast$. Thus, we have also shown here that $R_a \equiv R_n$. While it is true that we would have obtained the same with a naive use of Horton’s laws, we have both made the derivation thorough and established the correction terms found in equation [(\[eq:scaling.areawRBRLsub\])]{}. This will be investigated further in the next section. Finally, using this result and the estimate $\bar{l}_{{\omega}} \propto (R_{\ell})^{{\omega}}$ from equation [(\[eq:scaling.lconnect\])]{}, it follows that $$\bar{l}_{{\omega}} \propto (R_{\ell})^{{\omega}} = (R_n)^{{\omega}\ln R_{\ell}/\ln R_n} \sim (\bar{a}_{{\omega}})^{\ln R_{\ell}/\ln R_n} \label{eq:scaling.RBRLhack}$$ which is precisely Hack’s law. Comparing equations [(\[eq:scaling.RBRLhack\])]{} and [(\[eq:scaling.hack\])]{}, Hack’s exponent is found in terms of the Horton ratios $R_n$ and $R_{\ell}$ as $$h = \frac{\ln R_{\ell}}{\ln R_n}. \label{eq:scaling.hackRBRL}$$ There is one minor caveat to the derivation in [(\[eq:scaling.RBRLhack\])]{} and, for that matter, to most other derivations in this paper. Equation [(\[eq:scaling.RBRLhack\])]{} only holds for the characteristic areas and lengths $\bar{a}_{\omega}$ and $\bar{l}_{\omega}$. Since these quantities grow exponentially with ${\omega}$, the derivation gives evenly spaced points on a log-log plot lying on a straight line. Clearly, this would indicate that the actual relationship is continuous and linear on a log-log plot. Indeed, there is no obvious reason that a network would prefer certain lengths and areas. The averaging of stream lengths and areas brought about by the imposition of stream ordering necessarily removes all information contained in higher order statistics. Motivated by this observation, generalizations of the laws of Tokunaga, Horton and Hack to laws of distributions rather than averages is in progress [@dodds98d]. There are only two Horton ratios {#sec:scaling.Ra=Rn} ================================ In deriving Hack’s law in the previous section we obtained from equation [(\[eq:scaling.areaapprox\])]{} that $R_a \equiv R_n$. This redundancy in Horton’s laws is implicit in, amongst others, the works of Horton [@horton45] and Hack [@hack57] but has never been stated outright. As noted previously, [Peckham]{} also obtains a similar result for a topological quantity, the number of source streams in a basin, that is used as an estimate of area. Thus, we see that for a landscape with uniform drainage density, Horton’s laws are fully specified by only two parameters $R_n$ and $R_\ell$. This further supports our claim that Tokunaga’s law and Horton’s laws are equivalent since we have shown that there is an invertible transformation between $(T_1,R_T)$, the parameters of Tokunaga’s law, and $(R_n,R_\ell)$ (equations [(\[eq:scaling.tokhortlink1\])]{}, [(\[eq:scaling.tokhortlink2\])]{} and [(\[eq:scaling.tokhortinv\])]{}). In this section, we present data from real networks that support the finding $R_n = R_a$. We also address reported cases that do not conform to this result and consider a possible explanation in light of the correction terms established in equation [(\[eq:scaling.areawRBRL\])]{}. Excellent agreement for the result $R_n = R_a$ in real networks is to be found in the data of [Peckham]{} [@peckham95]. The data is taken from an analysis of digital elevation models (DEM’s) for the Kentucky River, Kentucky and the Powder River, Wyoming. Figure \[fig:scaling.kentucky\] shows average area and stream number plotted as a function of order for the Kentucky River while Figure \[fig:scaling.powder\] shows the same for the Powder river. Note that stream number has been plotted against decreasing stream order to make the comparison clear. The exponents $R_a$ and $R_n$ are indistinguishable in both cases. For the Kentucky river, $R_n \approx R_a = 4.65 \pm 0.05$ and for the Powder river, $R_n \approx R_a = 4.55 \pm 0.05$. Also of note here is that the same equality is well satisfied by [[Scheidegger]{}]{}’s model where numerical simulations yield values of $R_a = 5.20 \pm 0.05$ and $R_n = 5.20 \pm 0.05$. Note the slight deviation from a linear form for stream numbers for large ${\omega}$ in both cases. This upwards concavity is as predicted by the modified version of Horton’s law of stream numbers for a single basin, equation [(\[eq:scaling.nwtokdiffeqsoln\])]{}. At the other extreme, the fit for both stream areas and stream numbers extends to ${\omega}= 1$. While this may seem remarkable, it is conceivable that at the resolution of the DEM’s used, some orders of smaller streams may have been removed by coarse-graining. Thus, ${\omega}=1$ may actually be, for example, a third order stream. Note that such a translation in the value of ${\omega}$ does not affect the determination of the ratios as it merely results in the change of an unimportant multiplicative constant. If ${\omega}_r$ is the true order and ${\omega}= {\omega}_r - m$, where $m$ is some integer, then, for example, $$n_{\omega}\propto (R_n)^{{\omega}} \sim (R_n)^{{\omega}_r - m} = \mbox{const} \times (R_n)^{{\omega}_r}. \label{eq:scaling.omok}$$ This is only a rough argument as coarse-graining does not necessarily remove all streams of low orders. At odds with the result that $R_n \equiv R_a$ are past measurements that uniformly find $R_a > R_n$ at a number of sites. For example, Rosso [et al.]{} in [@rosso91] examine eight river networks and find $R_a$ to be on average 40 % greater than $R_n$. Clearly, this may be solely due to one or more of the our assumptions not being satisfied. The most likely would be that drainage density is not uniform. However, the limited size of the data sets points to a stronger possibility which we now discuss. In the case of [@rosso91], the networks considered are all third or fourth order basins with one exception of a fifth order basin. As shown by equation [(\[eq:scaling.areawRBRL\])]{}, if Horton’s laws of stream number and length are exactly followed for all orders, Horton’s law of area is not obeyed for lower orders. Moreover, the former are most likely asymptotic relations themselves. It is thus unsatisfactory to make estimates of Horton’s ratios from only three or four data points taken from the lowest order basins. Note that the Kentucky and Powder rivers are both eighth order networks and thus provide a sufficient range of data. We consider more precisely how the corrections to the scaling of area given in equation [(\[eq:scaling.areawRBRLsub\])]{} would affect the measurement of the Horton ratios. Figure \[fig:scaling.RAfit\] shows an example of how stream number, length and area might vary with ${\omega}$. It is assumed, for the sake of argument, that stream number and length scale exactly as per Horton’s laws and that area behaves as in equation [(\[eq:scaling.areawRBRLsub\])]{}, satisfying Horton’s law of area only for higher values of ${\omega}$. The plot is made for the example values $R_n=4$ and $R_{\ell}=2$. The prefactors are chosen arbitrarily so the ordinate is of no real significance. A measurement of $R_a$ from a few data points in the low ${\omega}$ range will overestimate its asymptotic value as will a similar measurement of $R_n$ underestimate its true value. Estimates of $R_n$ and $R_a$ from a simple least squares fit for various ranges of data are provided in Table \[tab:scaling.RaRnmess\]. ${\omega}$ range $1,2,3$ $1,2,3,4$ $1,2,3,4,5$ $4,5,6,7,8$ ------------------ --------- ----------- ------------- ------------- $R_n$ 2.92 3.21 3.41 3.99 $R_a$ 5.29 4.90 4.67 4.00 : Values of Horton ratios obtained from least squares estimates of slopes for data represented in Figure \[fig:scaling.RAfit\]. The range indicates the data points used in the estimate of the slopes. The ratios obtained from the low order data demonstrate substantial error whereas those obtained from the middle data essentially give the true values of $R_n = R_a = 4$.[]{data-label="tab:scaling.RaRnmess"} Thus, the validity of the methods and results from past work are cast in some doubt. A reexamination of data which has yielded $R_a \gg R_n$ appears warranted with an added focus on drainage density. Moreover, it is clear that networks of a much higher order must be studied to produce any reasonable results. Fractal dimensions of networks: a revision ========================================== A number of papers and works over the past decade have analyzed the relationships that exist between Horton’s laws and two fractal dimensions used to describe river networks [@tarboton88; @labarbera89; @tarboton90; @rosso91; @feder88; @labarbera94; @stark97]. These are $D$, the dimension which describes the scaling of the total mass of a network, and ${d}$, the fractal dimension of individual streams that comprises one of our assumptions. In this section, we briefly review these results and point out several inconsistencies. We then provide a revision that fits within the context of our assumptions. Our starting point is the work of [La Barbera]{} and [Rosso]{} [@labarbera89] which was improved by [Tarboton]{} [et al.]{} to give [@tarboton90] $$D = {d}\frac{\ln R_n}{\ln R_{\ell}}. \label{eq:scaling.DdRBRL}$$ We find this relation to be correct but that the assumptions and derivations involved need to be redressed. To see this, note that equation [(\[eq:scaling.DdRBRL\])]{} was shown to follow from two observations. The first was the estimation of $N(\bar{\ell}_1)$, the number of boxes of size $\bar{\ell_1} \times \bar{\ell_1}$ required to cover the network [@labarbera89]: $$N(\bar{\ell}_1) \sim (\bar{\ell}_1)^{-\ln R_n/\ln R_{\ell}} \label{eq:scaling.Nl1}$$ where $\bar{\ell_1}$ is the mean length of first order stream segments. Note that Horton’s laws were directly used in this derivation rather than the correctly modified law of stream numbers for single basins (equation [(\[eq:scaling.nwtokdiffeqsoln\])]{}). Nevertheless, the results are the same asymptotically. The next was the inclusion of our second assumption, that single channels are self-affine [@tarboton90]. Thus, it was claimed, $\bar{\ell}_1 \sim \delta^{-{d}}$ where $\delta$ is now the length of the measuring stick. Substitution of this into equation [(\[eq:scaling.Nl1\])]{} gave $$N(\delta) \sim \delta^{-{d} \ln R_n/\ln R_{\ell}}, \label{eq:scaling.Ndelta}$$ yielding the stated expression for $D$, equation [(\[eq:scaling.DdRBRL\])]{}. However, there is one major assumption in this work that needs to be more carefully examined. The network is assumed to be of infinite order, i.e., one can keep finding smaller and smaller streams. As we have stated, there is a finite limit to the extension of any real network. The possible practical effects of this are pictorially represented in Figure \[fig:scaling.crossover\]. Consider that the network in question is of actual order ${\Omega}$. Then there are three possible scaling regimes. Firstly, for a ruler of length $\delta \gg \bar{\ell}_1$, only the network structure may be detected, given that individual streams are almost one-dimensional. Here, the scaling exponent will be $\ln R_n/\ln R_{\ell}$. Next, as $\delta$ decreases, the fractal structure of individual streams may come into play and the exponent would approach that of equation [(\[eq:scaling.Ndelta\])]{}. Depending on the given network, this middle section may not even be present or, if so, perhaps only as a small deviation as depicted. Finally, the contribution due to the overall network structure must vanish by the time $\delta$ falls below $\bar{\ell}_1$. From this point on, the measurement can only detect the fractal nature of individual streams and so the exponent must fall back to ${d}$. We therefore must rework this derivation of equation [(\[eq:scaling.DdRBRL\])]{}. As suggested in the definition of ${d}$ in section \[sec:scaling.assumptionsdl\], it is more reasonable to treat networks as growing fractals. Indeed, since there is a finite limit to the extent of channelization of a landscape, there is a lower cut-off length scale beyond which most network quantities have no meaning. The only reasonable way to examine scaling behavior is to consider how these quantities change with increasing basin size. This in turn can only be done by comparing different basins of increasing order as opposed to examining one particular basin alone. With this in mind, the claim that equation [(\[eq:scaling.DdRBRL\])]{} is the correct scaling can be argued as follows. Within some basin of order ${\Omega}$, take a sub-basin of order ${\omega}$. Consider $N({\omega})$, the number of boxes of side length $\bar{\ell}_1$ required to cover the sub-network. This is essentially given by the total length of all the streams in the network. This is given by the approximation of equation [(\[eq:scaling.areaapprox\])]{} and so we have that $N({\omega}) \propto (R_n)^{\omega}$. Using the fact that $\bar{\ell}_{\omega}= (R_{\ell})^{{\omega}-1}\bar{\ell}_1$ we then have that $N({\omega}) \propto (\ell_{\omega}/\bar{\ell}_1)^{\ln R_n/ \ln R_{\ell}}$. The difference here is that $\bar{\ell}_1$ is fixed and pertains to the actual first order streams of the network. By assumption, we have that $\ell_{\omega}\propto L^{{d}}$ and thus $$N(L) \propto L^{{d} \ln R_n/\ln R_{\ell}}, \label{eq:scaling.Dcorrect}$$ which gives the same value for $D$ as equation [(\[eq:scaling.DdRBRL\])]{}. There are two other relations involving fractal dimensions that also need to be reexamined. Firstly [Rosso]{} [et al.]{} [@rosso91] found that $${d} = 2\frac{\ln R_{\ell}}{\ln R_a}. \label{eq:scaling.dRLRA}$$ Combining equations [(\[eq:scaling.DdRBRL\])]{} and [(\[eq:scaling.dRLRA\])]{}, they then obtained $$D = 2\frac{\ln R_n}{\ln R_a}. \label{eq:scaling.dRBRA}$$ However, equation [(\[eq:scaling.dRLRA\])]{} and hence equation [(\[eq:scaling.dRBRA\])]{} are both incorrect. There is a simple explanation for this discrepancy. In deriving equation [(\[eq:scaling.dRLRA\])]{}, [Rosso]{} [et al.]{} make the assumption that $h={d}/2$, a hypothesis first suggested by Mandelbrot [@mandelbrot83]. In arriving at the relation $h={d}/2$, Mandelbrot states in [@mandelbrot83] that “$(\mbox{basin area})^{1/2}$ should be proportional to (distance from source to mouth as the crow flies).” In other words, $a \propto L^{1/2}$. However, as noted in equation [(\[eq:scaling.scalingwithL\])]{}, observations of real networks show that $a \propto L^D$ where $D < 2$ [@maritan96a]. Furthermore, on examining the result $h = \ln R_\ell / \ln R_n$ with the expression for $D$ in equation [(\[eq:scaling.DdRBRL\])]{} we see that $$h = \frac{{d}}{D}, \label{eq:scaling.hackdD}$$ which suggests that this hypothesis is valid only when $D=2$. Consider also the test case of the [[Scheidegger]{}]{} model where $h = 2/3$, $D = 3/2$ and ${d} = 1$ (see Table \[tab:scaling.values\]). Using these values, we see that equation  [(\[eq:scaling.hackdD\])]{} is exactly satisfied while the relation $h={d}/2$ gives $h=1/2 \neq 2/3$. Now, if $h={d}/D$ is used in place of $h={d}/2$ in deriving equation [(\[eq:scaling.dRLRA\])]{} then equation [(\[eq:scaling.DdRBRL\])]{} is recovered. It also follows that equation [(\[eq:scaling.dRBRA\])]{} simplifies to the statement $R_a = R_n$, further demonstrating the consistency of our derivations. Thus, the two equations [(\[eq:scaling.dRLRA\])]{} and [(\[eq:scaling.dRBRA\])]{} become redundant and the only connection between Horton’s ratios and network dimensions is given by equation [(\[eq:scaling.DdRBRL\])]{}. An important point is that $D < 2$ *does not imply* that drainage basins are not space filling. This exponent shows how basin area changes when comparing different basins with different values of $L$, i.e., $a \propto L^D$. Any given single basin has of course a fractal dimension of 2. The equating of the way basin sizes change with the actual dimension of any one particular basin is a confusion evident in the literature (see, for example, [@tarboton88]). Incorporating the effects of measuring basin area with boxes of side length $\delta$ in the relation $a \propto L^D$ would lead to the form $$a_L(\delta) \propto \delta^{-2} L^{D}, \label{eq:scaling.adeltaLH}$$ where the subscript $L$ has been used to emphasize that different values of $L$ correspond to different basins. Thus, for any given basin (i.e., for fixed $L$), the area scales with $\delta$ while for a fixed $\delta$, areas of different basins scale as per equation [(\[eq:scaling.scalingwithL\])]{}. It should also be emphasized that the relationship found here between Hack’s exponent and the fractal dimensions ${d}$ and $D$ is one that is explicitly derived from the assumptions made. The observation that basin areas scale non-trivially with $L$ follows from these starting points and thus there is no need to assume it here. Other scaling laws ================== We now address three remaining sets of scaling laws. These are probability distributions for areas and stream lengths, scaling of basin shape and Langbein’s law. As introduced in equation [(\[eq:scaling.powerlawdist\])]{}, probability distributions for $a$ and $l$ are observed to be power law with exponents $\tau$ and $\gamma$ [@rodriguez-iturbe97]. Both of these laws have previously been derived from Horton’s laws. [De Vries]{} [et al.]{} [@devries94] found a relationship between $\tau$, $R_n$ and $R_{\ell}$ but did not include ${d}$ in their calculations while [Tarboton]{} [et al.]{} [@tarboton88] obtained a result for $\gamma$ that did incorporate ${d}$. Again, both of these derivations use Horton’s laws directly rather than the modified version of equation [(\[eq:scaling.nwtokdiffeqsoln\])]{}. Asymptotically, the same results are obtained from both approaches, $$\tau = 2 - \frac{\ln R_{\ell}}{\ln R_n} \quad \mbox{and} \quad \gamma = \frac{\ln R_n}{\ln R_{\ell}}. \label{eq:scaling.tau}$$ Using the form of the Hack exponent found in equation [(\[eq:scaling.hackdD\])]{} and equation [(\[eq:scaling.DdRBRL\])]{}, further connections between these exponents are found: $$\tau = 2 - h \quad \mbox{and} \quad \gamma = \frac{1}{h}. \label{eq:scaling.tauh}$$ One important outcome concerns the fact that only one of the exponents of the triplet $(h,\tau,\gamma)$ is independent. Previously, for the particular case of directed networks, this has been shown by [Meakin]{} [et al.]{} [@meakin91] and further developed by [Colaiori]{} [et al.]{} [@colaiori97]. Directed networks are those networks in which all flow has a non-zero positive component in a given direction. In a different setting, [Cieplak]{} [et al.]{} also arrive at this same conclusion for what they deem to be the separate cases of self-similar and self-affine networks although their assumptions are that $d < 1$ and $D < 2$ are mutually exclusive contrary to empirical observations [@cieplak98a]. In the case of non-directed networks, [Maritan]{} [et al.]{} have found one scaling relation for these three exponents, $\gamma = 1 + (\tau - 1)/h$ and, therefore, that two of these three exponents are independent. They further noted that $\tau = 2 - h$ is an “intriguing result” suggested by real data [@maritan96a]. In the present context, we have obtained this reduction of description in a very general way with, in particular, no assumption regarding the directedness of the networks. The scaling of basin shapes has been addressed already but it remains to show how it simply follows from our assumptions and how the relevant exponents are related. It is enough to show that this scaling follows from Hack’s law. Now, the area of a basin is related to the longitudinal length $L$ and the width $L_\perp$ by $a = L_\perp L$, while the main stream length scales by assumption like $l \sim L^{{d}}$. Hence, $$\begin{aligned} l \sim a^h & \Rightarrow & L^{{d}} \sim (L_\perp L)^h \nonumber \\ & \Rightarrow & L_\perp \sim L^{{d}/h - 1} = L^{D - 1} \label{eq:scaling.self-aff}\end{aligned}$$ where the fact that $h = {d}/D$ has been used. Comparing this to equation [(\[eq:scaling.scalingwithL\])]{} we obtain the scaling relation $$H = D - 1. \label{eq:scaling.HD}$$ The last set of exponents we discuss are those relating to Langbein’s law [@langbein47]. Langbein found that $\tilde{\Lambda}$, the sum of the distances (along streams) from stream junctions to the outlet of a basin, scales with the area of the basin. Recently, Maritan [et al.]{} [@maritan96a] introduced the quantity $\lambda$, which is an average of Langbein’s $\tilde{\Lambda}$ except now the sum is taken over all points of the network. Citing the case of self-organized critical networks, they made the claim that $$\lambda \propto L^\varphi. \label{eq:scaling.elldef}$$ Further, they assumed that $\varphi = {d}$ although it was noted that there is no clear reason why this may be so since there are evident differences in definition ($\lambda$ involves distances downstream while ${d}$ involves distances upstream). We find this scaling relation to hold in the present framework. We further consider the two related quantities $\Lambda$ and $\tilde{\lambda}$, respectively the sum over all points and the average over all junctions of distances along streams to the basin outlet. The calculations are straightforward and follow the manner of previous sections. We first calculate $\lambda({\omega},{\Omega})$, the typical distance to the outlet from a stream of order ${\omega}$ in an order ${\Omega}$ basin. Langbein’s $\tilde{\Lambda}$, for example, is then obtained as $\sum_{{\omega}=1}^{\Omega}n({\omega},{\Omega}) \lambda({\omega},{\Omega})$. We find the same scaling behavior regardless of whether sums are taken over all points or all junctions. Specifically we find $$\Lambda \sim \tilde{\Lambda} \sim a^{1+\ln R_\ell/\ln R_n} \quad \mbox{and} \quad \lambda \sim \tilde{\lambda} \sim L^d \label{eq:scaling.lambascaling}$$ yielding the scaling relations $$\beta = \tilde{\beta} = {1+\ln R_\ell/\ln R_n} \quad \mbox{and} \quad \tilde{\varphi} = \varphi = {d}. \label{eq:scaling.varphid}$$ Note that the second pair of scaling relations admit other methods of measuring ${d}$. The large amount of averaging inherent in the definition of the quantity $\lambda$ would suggest that it is a more robust method for measuring ${d}$ than one based on measurements of the sole main stream of the basins. [Maritan]{} [et al.]{} [@maritan96a] provide a list of real world measurements for various exponents upon which several comments should be made. Of particular note is the relationship between $\tau = 2 - h$. This is well met by the cited values $1.41 < \tau < 1.45$ and $0.57 < h < 0.60$. Also reasonable is the estimate of $h$ given by ${d}/D$ ($D=\phi$ in their notation) which is $0.58<h<0.65$. The values of $\gamma$ and $\varphi$, however, do not work quite so well. The latter does not match ${d}$ within error bars, although they are close in absolute value with $\varphi = 1.05\pm0.01$ and ${d}=1.10\pm0.01$. The length distribution exponent $\gamma$ may be found via 3 separate routes: $\gamma = 1/h = D/{d} = 1/(2-\tau)$. The second and third equalities have been noted to be well satisfied and so any one of the 3 estimates of $\gamma$ may be used. Take, for example, the range $0.58 < h < 0.59$, which falls within that given by $h=2-\tau$, $h = {d}/D$ and the range given for $h$ itself. This points to the possibility that the measured range $1.8 < \gamma < 1.9$ is too high, since using $\gamma = 1/h$ yields $\gamma = 1.74 \pm .02$. Also of note is that Maritan [et al.]{}’s own scaling relation $\gamma = 1 + (\tau-1)/h$ would suggest $\gamma = 1.74 \pm .05$. Better general agreement with the scaling relations is to be found in [@rigon96] in which [Rigon]{} [et al.]{} detail specific values of $h$, $\tau$ and $\gamma$ for some thirteen river networks. Here, the relations $\tau = 2-h$ and $\gamma = 1/h$ are both well satisfied. Comparisons for this set of data show that, on average and given the cited values of $h$, both $\tau$ and $\gamma$ are overestimated by only 2 per cent. Concluding remarks ================== We have demonstrated that the various laws, exponents and parameters found in the description of river networks follow from a few simple assumptions. Further, all quantities are expressible in terms of two fundamental numbers. These are a ratio of logarithms of Horton’s ratios, $\ln R_n/\ln R_{\ell}$, and the fractal dimension of individual streams, ${d}$. There are *only two* independent parameters in network scaling laws. These Horton ratios were shown to be equivalent to Tokunaga’s law in informational content with the attendant assumption of uniform drainage density. Further support for this observation is that both the Horton and Tokunaga descriptions depend on two parameters each and an invertible transformation between them exists (see equations [(\[eq:scaling.tokhortlink1\])]{}, [(\[eq:scaling.tokhortlink2\])]{} and [(\[eq:scaling.tokhortinv\])]{}). A summary of the connections found between the various exponents is presented in Table \[tab:scaling.laws\]. It should be emphasized that the importance of laws like that of Tokunaga and Horton in the description of networks is that they provide explicit structural information. Other measurements such as the power law probability distributions for length and area provide little information about how a network fits together. Indeed, information is lost in the derivations as the Horton ratios cannot be recovered from knowledge of $\ln R_n/ \ln R_{\ell}$ and ${d}$ only. The basic assumptions of this work need to be critically examined. Determining how often they hold and why they hold will follow through to a greater understanding of all river network laws. One vital part of any river network theory that is lacking here is the inclusion of the effects of relief, the third dimension. Another is the dynamics of network growth: why do mature river networks exhibit a self-similarity that gives rise to these scaling laws with these particular values of exponents? Also, extensive studies of variations in drainage density are required. The assumption of its uniformity plays a critical role in the derivations and needs to be reexamined. Lastly, in those cases where these assumptions are valid, the scaling relations gathered here provide a powerful method of cross-checking measurements. Finally, we note that work of a similar nature has recently been applied to biological networks [@west97]. The assumption analogous to network self-similarity used in the biological setting is considerably weaker as it requires only that the network is a hierarchy. A principle of minimal work is then claimed to constrain this hierarchy to be self-similar. It is conceivable that a similar approach may be found in river networks. However, a generalization of the concept of a hierarchy and perhaps stream ordering needs to be developed since a ‘Tokunagic network’ is not itself a simple hierarchy. [law:]{} [parameter in terms of $R_n$, $R_{\ell}$ and ${d}$:]{} ------------------------------------------------------------ -------------------------------------------------------- $T_{\nu} = T_1 (R_T)^{\nu-1}$ $T_1 = R_n -R_\ell - 2 + 2R_\ell/R_n$ $R_T = R_\ell$ $l \sim L^{{d}}$ — $n_{{\omega}+1}/n_{{\omega}} = R_n$ — $\bar{\ell}_{{\omega}+1}/\bar{\ell}_{{\omega}} = R_{\ell}$ — $\bar{l}_{{\omega}+1}/\bar{l}_{{\omega}} = R_{\ell}$ — $\bar{a}_{{\omega}+1}/\bar{a}_{{\omega}} \sim R_a$ $R_a = R_n$ $l \sim a^h$ $h = \ln R_{\ell}/ \ln R_n$ $a \sim L^D$ $D = {d} \ln R_n/\ln R_{\ell}$ $L_\perp \sim L^H$ $H = {d} \ln R_n/\ln R_{\ell} - 1$ $P(a) \sim a^{-\tau}$ $\tau = 2 - \ln R_{\ell}/ \ln R_n$ $P(l)\sim l^{-\gamma}$ $\gamma = \ln R_n/ \ln R_{\ell}$ $\Lambda \sim a^\beta$ $\beta = 1 + \ln R_{\ell}/\ln R_n$ $\lambda \sim L^\varphi$ $\varphi = {d}$ $\tilde{\Lambda} \sim a^{\tilde{\beta}}$ $\tilde{\beta} = 1 + \ln R_{\ell}/\ln R_n$ $\tilde{\lambda} \sim L^{\tilde{\varphi}}$ $\tilde{\varphi} = {d}$ : Summary of scaling laws and the scaling relations found between the various exponents. Compare with table \[tab:scaling.scalinglaws\]. []{data-label="tab:scaling.laws"} Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful to R. Pastor-Satorras, J. Pelletier, G. West, J. Weitz and K. Whipple for useful discussions. The work was supported in part by NSF grant EAR-9706220. 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[^1]: Author to whom correspondence should be addressed

--- abstract: 'Font design is now still considered as an exclusive privilege of professional designers, whose creativity is not possessed by existing software systems. Nevertheless, we also notice that most commercial font products are in fact manually designed by following specific requirements on some attributes of glyphs, such as italic, serif, cursive, width, angularity, etc. Inspired by this fact, we propose a novel model, Attribute2Font, to automatically create fonts by synthesizing visually pleasing glyph images according to user-specified attributes and their corresponding values. To the best of our knowledge, our model is the first one in the literature which is capable of generating glyph images in new font styles, instead of retrieving existing fonts, according to given values of specified font attributes. Specifically, Attribute2Font is trained to perform font style transfer between any two fonts conditioned on their attribute values. After training, our model can generate glyph images in accordance with an arbitrary set of font attribute values. Furthermore, a novel unit named Attribute Attention Module is designed to make those generated glyph images better embody the prominent font attributes. Considering that the annotations of font attribute values are extremely expensive to obtain, a semi-supervised learning scheme is also introduced to exploit a large number of unlabeled fonts. Experimental results demonstrate that our model achieves impressive performance on many tasks, such as creating glyph images in new font styles, editing existing fonts, interpolation among different fonts, etc.' author: - Yizhi Wang - Yue Gao - Zhouhui Lian bibliography: - 'sample-bibliography.bib' title: 'Attribute2Font: Creating Fonts You Want From Attributes' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010178.10010224&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Computer vision&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010371.10010382&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Image manipulation&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; {width="\textwidth"} \[fig:teaser\] Introduction ============ Traditional font design workflow sets a high barrier for common users, which requires creativity and expertise in this field. Automatic font designing remains a challenging and ongoing problem in areas of Computer Graphics (CG), Computer Vision (CV) and Artificial Intelligence (AI). In this paper, we aim to handle the task of generating glyph images according to the user-specified font attributes (such as italic, serif, cursive, angularity, etc.) and their values. Our model significantly lowers the barrier and provides various and customizable fonts for common users. We believe that our model can also inspire professional designers and assist them in creating new fonts. To the best of our knowledge, our work is the first to automatically generate glyph images from the attributes of font styles. Existing works on attribute-controllable image synthesis are unsuitable for this task due to the following three reasons: (1) Existing works aim to generate images such as faces and fashion, whose appearances (such as color and texture) vary with the attributes but shapes generally remain unchanged. By contrast, the shapes of glyphs vary dramatically with the font attributes. (2) Existing works treat different attributes equally, whose effects are stacked into the generated images. However, for each kind of font, some font attributes have much more impact on the glyph shapes than the other attributes. Thereby, the correlation of different attributes should be explored and more attention should be paid to important attributes. (3) The attribute values in these works are either binary (with or without) or discrete, which cannot be arbitrarily manipulated by users. Up to now, large numbers of font generation methods have been reported intending to simplify the workflow of font designers. There exist some works which attempt to “create” new fonts by manifold learning and interpolating between different fonts. Nevertheless, these works fail to handle the situation when users have specific demands for the font style. For example, one may seek for one kind of font which is attractive, sharp, with a little serif but not thin. An alternative is employing font retrieval systems when they have such requirements. However, these searching engines can only retrieve the most similar fonts from a font database that may not meet the users’ needs (see Fig. \[fig:teaser\]). In addition, the majority of them cannot accept attributes with values as keywords. To address the above-mentioned problems, this paper proposes a novel model which is capable of generating glyph images according to user-specified font attributes, named **Attribute2Font** [^1]. In the first place, we assume that each font corresponds to a set of font attribute values. On the basis of this assumption, we train a deep generative network to transfer glyph images in a font style to another according to their font attribute values. In the inference stage, we choose an appropriate font as source and transform it into any fonts the users want from given attribute values. Technically, we propose a semi-supervised learning scheme and the Attribute Attention Module to boost our model’s performance. The semi-supervised learning scheme is introduced to deal with the shortage of annotated training data. The Attribute Attention Module assists our model to concentrate on the most important attributes and better portray the prominent characteristics of glyphs in the generation stage. Experimental results on publicly available datasets demonstrate the effectiveness of our model in many applications, such as creating glyph images in new font styles, editing existing fonts, interpolation among different fonts, etc. Last but not least, our experiments verify that our model can also be applied to deal with fonts in any other writing systems (e.g., Chinese) which might contain hundreds of thousands of different characters. To sum up, major contributions of this paper are as follows: - For the first time, we establish a mapping relation from descriptive font attributes to the glyph image space. Users are allowed to arbitrarily set values of the predefined font attributes and create any fonts they want. - The devised semi-supervised learning scheme and attribute attention module significantly promote the quality of generated glyph images. Besides, they are not only limited to our task but also applicable to other image synthesis tasks. - Our model is capable of synthesizing highly varied and visually pleasing glyph images. Therefore, our model has a high practical value in font generation for both ordinary users and font designers. Related Work ============ Attribute-controllable Image Synthesis -------------------------------------- Image synthesizing methods essentially fall into two categories: parametric and non-parametric. The non-parametric methods generate target images by copying patches from training images. In recent years, the parametric models based on Generative Adversarial Networks (GANs) [@goodfellow2014generative] have been popular and achieved impressive results. Image-to-image translation is a specific problem in the area of image synthesis. Most recent approaches utilize CNNs (Convolutional Neural Networks) to learn a parametric translation function by training a dataset of input-output examples. Inspired by the success of GANs in generative tasks, the “pix2pix” framework [@isola2017image] uses a conditional generative adversarial network to learn a mapping from input to output images. Many recent works utilize GAN-based models to synthesize attribute-controllable images, such as face images with controllable hair colors, ages, genders, etc. Xu et al.  directly mapped the descriptive texts into images by combining the techniques of natural language processing and computer vision. Choi et al. , He et al. , Liu et al.  and Wu et al.  achieved this goal by implementing image-to-image translation conditioned on image attributes. Compared to the direct mapping from attributes to images, image-to-image translation usually generates more realistic images on the basis of the strong prior of source images. However, there is no existing work attempting to synthesize glyph images based on font style attributes. Besides, the attribute values in existing works are discrete rather than continuous, which constrains the variety and diversity of synthesized images. {width="\textwidth"} Font Generation --------------- Existing works of font generation have two main practical applications. The first one is simplifying the workflow of font designers. They typically take a few (or part) glyphs of a font as reference and then generate the rest glyphs in this font. They fully exploit the shared stylistic elements within the same font and the similar structure that character glyphs present across different fonts. The second one is “creating" new fonts by integrating and mixing up existing font styles via interpolation or/and manifold learning methods. This benefits the designers and users who desire to freely customize their favourite font styles. In terms of the representation of generated glyphs, these works can be classified into two categories: vector font generation and glyph image synthesis. ### Vector Font Generation Generating vector fonts is a typical problem in the area of CG. Campbell and Kautz  built a font manifold and generated new fonts by interpolation in a high dimensional space. Lian et al.  proposed a system to automatically generate large-scale Chinese handwriting fonts by learning styles of stroke shape and layout separately. Balashova et al.  proposed to learn a stroke-based font representation from a collection of existing typefaces. Lopes et al.  attempted to model the drawing process of fonts by building sequential generative models of vector graphics. However, vector font generation suffers from the contradiction between various and flexible font styles and structured representations of visual concepts. For example, the model proposed by Campbell and Kautz  is limited by the need of having all glyphs of a certain class be composed of equivalent shapes; When trained on a large-scale font dataset, the model presented by Lopes et al.  generates vector glyphs whose quality is far from satisfactory. ### Glyph Image Synthesis Glyph image synthesis loses the benefit of providing a scale-invariant representation but is able to generate glyphs in complex shapes and novel styles. For example, Lyu et al.  devised a model to synthesize Chinese calligraphy. Azadi et al.  proposed an end-to-end stacked conditional GAN model to generate a set of multi-content glyph images following a consistent style from very few examples. Guo et al.  proposed a method to synthesize Chinese glyph images in new font styles by building font manifolds learned from existing font libraries. Jiang et al.  and Gao et al.  viewed font style transfer as an image-to-image translation problem and mapped glyph images from one font style to another style. Given a few glyph image pairs of the source and target fonts, the deep generative networks they utilized learn a mapping function and then transfer the other glyph images in the source font style to the target font style. However, the mapping function is abstract and users are not allowed to get their desired fonts by specifying their expected attributes. To remedy this issue, we design an attribute-controllable font style transfer model where users can customize the attribute categories and arbitrarily assign the attribute values. Font Selection and Retrieval ---------------------------- Font selection and retrieval is about how to select fonts from a font database according to the conditions provided by users. Wang et al.  designed the DeepFont system which is queried by a real-world text image and retrieves the most similar fonts from a font library. O’Donovan et al.  first proposed to use high-level descriptive attributes for font selection, such as “dramatic" or “legible." Choi et al.  and Chen et al.  addressed the task of large-scale tag-based font retrieval, employing deep neural networks to build the bridge between descriptive tags and glyph images in various fonts. However, these works are merely aimed at searching existing fonts for users, which limits the creativity and flexibility of their models and may not satisfy the needs of users. In this paper, we propose to directly generate glyph images from given descriptive attributes. Our model receives continuous attribute values as input instead of binary tags in [@chen2019large]. Users are allowed to control the value of each attribute and get the glyph images they want. Method Description ================== Overview {#sec:method_des_ov} -------- We first briefly describe the pipeline of Attribute2Font shown in Fig. \[fig:Pipeline\]. Generally, our model aims to transfer the glyph images in one font style to another according to their font attributes. Let $x(a,k)$ and $x(b,k)$ be the glyph image pair with the same $k$-th character category but in the $a$-th and $b$-th fonts, respectively, where $1 \leq k \leq N_{c}$ and $1 \leq a,b \leq N_{f}$, $N_{c}$ and $N_{f}$ are the total numbers of character categories and fonts in a database, respectively. $x(a,k)$ and $x(b,k)$ denote the source and target in the transfer, respectively, and are marked as “source content” and “ground truth” in Fig. \[fig:Pipeline\]. The glyphs in the same font share the same font attribute. Let $\alpha(a)$ and $\alpha(b)$ be the attribute values of $x(a,k)$ and $x(b,k)$, respectively. Let $N_{\alpha}$ be the category number of predefined attributes and thus $\alpha(a),\alpha(b) \in \mathbb{R}^{N_{\alpha}}$. Note that the categories of font attributes can be freely customized but they remain fixed when the model is deployed. There is no requirement of these attributes being independent of each other. In principle the attribute values can be arbitrarily assigned but need to be normalized into the range of $[0,1]$ before sent into the model (‘0’ denotes minimum and ‘1’ denotes maximum). The Visual Style Transformer (VST) aims to estimate the style feature of the $b$-th font on the basis of glyph images of the $a$-th font and the attribute difference between these two fonts. A few glyph images in the font $a$ but different character categories are taken as input, denoted as $\{x(a,k_{1}), ...,x(a,k_{m})\}$ and presented as “source styles” in Fig. \[fig:Pipeline\], where $k_{1}, ..., k_{m}$ are randomly sampled from $\{1, ..., N_{c}\}$ and $m$ is a hyperparameter denoting the number of these glyph images. The estimated style feature of font $b$ is formulated as: $$\label{equ:GlyGen} \hat{s}(b) = F_{T}(x(a,k_{1}), ...,x(a,k_{m}),\alpha(b)-\alpha(a)),$$ where $F_{T}$ denotes the function of VST. The Attribute Attention Module (AAM) is devised to further refine the attributes so that they can serve better in the stage of glyph image generation: $$\label{equ:GlyGen} \alpha^{*}(a,b) = F_{A}(\alpha(b) - \alpha(a)),$$ where $\alpha^{*}$ denotes the refined attribute difference by applying the attention mechanism and $F_{A}$ denotes the function of AAM. The Generator takes the source image $x(a,k)$, the estimated style feature $\hat{s}(b)$, and the refined attribute difference $\alpha^{*}(a,b)$ as input to reconstruct the glyph image ${x}(b,k)$: $$\label{equ:GlyGen} \hat{x}(b,k) = G(x(a,k), \hat{s}(b), \alpha^{*}(a,b)),$$ where $G$ denotes the function of glyph generator. Following the adversarial training scheme of GANs, we employ a glyph discriminator to discriminate between the generated image and the ground-truth image, i.e., $\hat{x}(b,k)$ and $x(b,k)$. The glyph discriminator takes $x(b,k)$ or $\hat{x}(b,k)$ as input and predicts the probability of the input image being real (denoted as $p(y_{d} = 1)$) and its corresponding attributes $\hat{\alpha}$: $$\label{equ:GlyGen} p(y_{d} = 1 | x(b,k)), \hat{\alpha}(x(b,k)) = D( x(b,k)),$$ $$\label{equ:GlyGen} p(y_{d} = 1 | \hat{x}(b,k)), \hat{\alpha}(\hat{x}(b,k)) = D(\hat{x}(b,k)).$$ Through the adversarial game between the generator and the discriminator, the quality of generated glyphs can be continuously improved. More details of above-mentioned modules utilized in our method are presented in the following sections. Visual Style Transformer ------------------------ We employ a CNN style encoder to transform the selected glyph images of the $a$-th font into its style feature which is denoted as $s(a)$: $$s(a) = F_{S}([x(a,k_{1}); ...;x(a,k_{m})]),$$ where the square bracket denotes concatenation, in other words, all images are concatenated along the depth channel and then fed into the style encoder (see Fig. \[fig:Pipeline\]). $F_{S}$ denotes the function of style encoder. Empirically, a single glyph is unable to sufficiently embody its belonging font style. Therefore, we set $m \textgreater 1$ for more accurate estimation of $s(a)$. In Section \[sec:experiments\], we will show experimental results that demonstrate how the setting of $m$ affects our model’s performance. Next, the encoded feature $s(a)$ is concatenated with the attribute difference between the attribute values of $x(a,k)$ and $x(b,k)$, i.e., $\alpha(b) - \alpha(a)$. Afterwards we send them into several residual blocks [@he2016deep] and finally have the estimated target style features: $$\hat{s}(b) = F_{R}([s(a);\alpha(b) - \alpha(a)]),$$ where $F_{R}$ denotes the function of residual blocks. The number of residual blocks is denoted as $N_{rb}$ and its effect on our method will also be investigated in Section \[sec:experiments\]. Attribute Attention Module -------------------------- In this section, we give a detailed description of the Attribute Attention Module (AAM) which is illustrated in Fig. \[fig:channel\_attention\]. We introduce attribute embeddings $e \in \mathbb{R}^{N_{\alpha} \times N_{e}}$ as more concrete representations of the features of each attribute, where $N_{e}$ is interpreted as the dimension of attribute embeddings. The attribute values $\alpha$ could be viewed as the weight coefficients of attribute embeddings. The attribute feature difference between the $b$-th and $a$-th fonts is represented as: $$\beta(a,b)= (\alpha^{t}(b) - \alpha^{t}(a)) \odot e,$$ where $\odot$ denotes the element-wise multiplication; $\alpha^{t}(b)$, $\alpha^{t}(a) \in \mathbb{R}^{N_{\alpha} \times N_{e}}$ are tiled from $\alpha(b)$ and $\alpha(a)$, respectively; $\beta(a,b) \in \mathbb{R}^{N_{\alpha} \times N_{e}}$. We map the feature vector of each attribute in $\beta(a,b)$ into a two-dimensional feature map by: $$\gamma(a,b) = \beta_{0}(a,b) \otimes \beta'_{0}(a,b),$$ where $\beta_{0}(a,b) \in \mathbb{R}^{N_{\alpha} \times N_{e} \times 1}$ is unsqueezed from $\beta(a,b) \in \mathbb{R}^{N_{\alpha} \times N_{e}} $; $\beta'_{0}(a,b)\in \mathbb{R}^{N_{\alpha} \times 1 \times N_{e}}$ is the transpose of $\beta_{0}(a,b)$; $\otimes$ denotes the matrix multiplication over the last two dimensions and thus we have $\gamma(a,b) \in \mathbb{R}^{N_{\alpha} \times N_{e} \times N_{e}}$. The feature map for each attribute in $\gamma(a,b)$ is a symmetric matrix and is in favour of convolution operations in the following stage. Afterwards, we perform channel attention on $\gamma(a,b)$: $$\alpha^{*}(a,b) = F_{CA}(\gamma(a,b)),$$ where $F_{CA}$ denotes the channel attention block. The channel attention operation was first proposed in [@woo2018cbam] and [@zhang2018image], which produces a channel attention map by exploiting the inter-channel relationship of features. Specifically, the feature maps in $\gamma(a,b)$ are first aggregated by an average pooling layer and then sent into two convolution layers with channel squeeze and stretch to output the channel attention map $M(a,b) \in \mathbb{R}^{N_{\alpha} \times 1 \times 1}$. $\alpha^{*}(a,b)$ is computed by the channel-wise multiplication between $M(a,b)$ and $\gamma(a,b)$. We utilize the channel attention to explore the correlation of different attributes and help our model concentrate on more important attributes. The attribute attention is performed on the different stages of glyph generation, and the output will be re-scaled into the desired scale of each stage (details will be revealed in the next section). ![The architecture of our Attribute Attention Module and how it is incorporated into the feature maps in the decoder.[]{data-label="fig:channel_attention"}](channel_attention){width="8cm"} Glyph Image Generation ---------------------- A hierarchical encoder-decoder framework is deployed to generate the target glyph image (i.e., $x(b,k)$) on the basis of $\hat{s}(b)$, $\alpha^{*}(a,b)$ and $x(a,k)$. The encoder, named as the content encoder, is a CNN which progressively maps $x(a,k)$ into its content feature $c(a,k)$. Multi-scale features from the content encoder are utilized to more accurately reconstruct the glyph’s shape, which is shown in the skip-connection in Fig. \[fig:Pipeline\]. Note that only one example of skip-connection is demonstrated for brevity. We denote them as $c_{1}(a,k)$, $c_{2}(a,k)$, ..., $c_{L}(a,k)$ and $L$ is the number of feature scales. A softmax classifier is attached to $c_{L}(a,k)$ to predict the character category of $x(a,k)$. The probability of $x(a,k)$ belonging to the ${k}$-th category is denoted as $p(y_{c} =k|x(a,k))$. We also have $L$ up-sampling layers in the decoding step. The output of the $i$-th up-sampling layer in the decoder is formulated as: $$g_{i}=\begin{cases}F_{u}([F_{CA}([g_{i-1};\alpha^{*}_{i-1}]);h_{i-1}]), &2 \leq i \leq L \cr F_{u}(h_{i-1}), &i = 1\end{cases},$$ where $F_{u}$ is the function of up-sampling, including deconvolution, instance normalization [@ulyanov2016instance] and activation operations; $\alpha^{*}_{i-1}$ denotes the re-scaled output of AAM in the $(i-1)$-th decoding stage; $h_{i}$ is the fusion of ${c}_{L-i}(a,k)$ and $\hat{s}(b)$. Specifically, $\hat{s}(b)$ is first tiled according to the shape of ${c}_{L-i}(a,k)$ and then concatenated with ${c}_{L-i}(a,k)$: $$h_{i}=\begin{cases}F_{c}([\hat{s}_{i}^{t}(b);{c}_{L-i}(a,k)]), &1 \leq i \leq L-1 \cr [\hat{s}(b);{c}_{L-i}(a,k)], &i = 0\end{cases},$$ where $F_{c}$ denotes the function of convolution and $\hat{s}_{i}^{t}(b)$ is tiled from $\hat{s}(b)$. We have $\hat{x}(b,k) = g_{L}$ and the last up-sampling layer uses $tanh$ as the activation function. Loss Functions -------------- We define five losses in the step of generation. The pixel-level loss $l_{pixel}$ measures the dissimilarity between $\hat{x}(b,k)$ and ${x}(b,k)$ in the level of pixel values using the L1 Norm: $$l_{pixel} = \Vert \hat{x}(b,k) - {x}(b,k)\Vert.$$ The content loss $l_{char}$ is the cross-entropy loss for character recognition: $$l_{char} = -\log p( y_{c} = k | x(a,k)).$$ $l_{attr}$ is the prediction loss of the discriminator predicting the attribute values of generated images: $$l_{attr} = smooth_{L_{1}}(\hat{\alpha}(\hat{x}(b,k))- \alpha(b)),$$ and the $smooth_{L_{1}}$ function is defined as: $$smooth_{L_{1}}(x)=\begin{cases}0.5x^{2}, &\left |x \right |\leq 1 \cr \left |x \right|-0.5, &otherwise\end{cases}.$$ The contextual loss was recently proposed in [@mechrez2018contextual]. It is a new and effective way to measure the similarity between two images, requiring no spatial alignment. As the spatial alignment is required for the L1 loss, if the synthesized image is not exactly spatially aligned to the ground truth image (e.g., a small displacement or rotation), the L1 loss will be high but the synthesis result is often visually acceptable. The contextual loss leads the model to pay more attention to style features at a high level, not just differences in pixel values. Therefore, we regard the contextual loss as a complementary to the L1 loss: $$l_{CX} = CX(\hat{x}(b,k), {x}(b,k)).$$ The vanilla generation loss $l_{G}$ is defined as: $$l_{G} = -\log p(y_{d} = 1 | \hat{x}(b,k)).$$ The total loss of the generation step is formulated as: $$L_G = \lambda_{1} l_{G} + \lambda_{2} l_{pixel} + \lambda_{3} l_{char} + \lambda_{4} l_{CX} + \lambda_{5} l_{attr},$$ where $\lambda_{1},\lambda_{2},...,\lambda_{5}$ are hyperparameters denoting the weights of loss functions. In the step of discrimination, we define two losses. The first one $l'_{attr}$ is the prediction loss of the discriminator that predicts the attribute values of ground-truth images: $$l'_{attr} = smooth_{L_{1}}(\hat{\alpha}(x(b,k))- \alpha(b)).$$ The other one, the vanilla discrimination loss, $l_{D}$ is formulated as: $$l_{D} = -\log p(y_{d} = 1 | x(b,k)) -\log p(y_{d} = 0 | \hat{x}(b,k)).$$ The total loss of the discrimination step is formulated as: $$L_D = l_{D} + l'_{attr}.$$ We optimize these two objective functions ($L_{G}$ and $L_{D}$) alternately. Semi-supervised Learning ------------------------  [@o2014exploratory] released a font dataset consisting of 1,116 fonts where the attribute values of 148 fonts are annotated. Such a small quantity of training data is far from enough to train a deep neural network with satisfactory performance. Nevertheless, they paid a huge effort to obtain these annotations by sending questionnaires to the Mechanical Turk and sum them up by machine learning models. To remedy this issue, we propose a semi-supervised learning scheme to exhaustively exploit unlabeled fonts. Our main idea is to incorporate the annotating of unlabeled fonts’ attribute values into the font style transfer process. The glyph images in a training image pair could be either selected from the labeled fonts or the unlabeled fonts. We first assign pseudo attribute values for the unlabeled fonts and they will be fine-tuned by the gradient descent algorithm. In this manner, our model learns to annotate the attribute values of unlabeled fonts by referring to the distribution of human-annotated attributes. Details of our semi-supervised learning scheme are described as follows. Assume that we have $N_{sf}$ fonts whose attribute values are labeled and $N_{uf}$ fonts whose attribute values are unlabeled ($N_{uf} + N_{sf} = N_{f}$ and $N_{uf} \gg N_{sf}$, typically). Let $\Phi{s}$ consist of the indexes of labeled fonts while $\Phi{u}$ consist of the indexes of unlabeled fonts. The font of the source image is randomly selected, i.e., the probability of the source image being in a labeled font $p(a \in \Phi{s}) = \frac{ N_{sf}}{N_{sf} + N_{uf}}$ and being in a unlabeled font $p(a \in \Phi{u}) = \frac{ N_{uf}}{N_{sf} + N_{uf}}$. For the target image, selecting from the labeled fonts and unlabeled fonts are equally likely, i.e., $p(b \in \Phi{s}) = 0.5$ and $p(b \in \Phi{u}) = 0.5$. We increase the proportion of labeled fonts in target images compared to source images because the strongest supervision (such as $l_{pixel}$, $l_{CX}$) comes from the target image. The details of this combination strategy are illustrated in Fig \[fig:training\_strategy\]. We first randomly initialize the attribute values of unlabeled fonts according to the standard Gaussian distribution $\mathcal N(0,1)$ attached by the sigmoid function: $$\alpha(i) = sigmoid(z), i \in \Phi{u},$$ where $z \sim \mathcal N(0,1)$; $sigmoid$ is the sigmoid function which maps $z$ into the range of $(0,1)$. In the training phase, the attribute values of labeled fonts are fixed but the ones of unlabeled fonts are fine-tuned via the gradient descent algorithm. Namely, if $a$ or $b \in \Phi{s}$, $\alpha(a)$ or $\alpha(b)$ remains fixed; If $a$ or $b$ $\in \Phi{u}$, $\alpha(a)$ or $\alpha(b)$ is fine-tuned by using the gradient descent algorithm. The scheme is proved to be very effective through our experiments, which will be further discussed in Section \[sec:experiments\]. ![How we select from the labeled and unlabeled fonts to make image pairs for training.[]{data-label="fig:training_strategy"}](training_strategy){width="8cm"} The Choice of Source Font in Inference Stage -------------------------------------------- In the training stage, our model performs font style transfer between any two fonts in the training dataset. But in the inference stage, our model is only given a set of attribute values. This raises an issue of how to select an appropriate source font so that it can be easily transferred into the desired font. Since we estimate the style feature $\hat{s}(b)$ from the glyph images of the source font, $\hat{s}(b)$ is fused with the source font’s style more or less. Intuitively, selecting a source font which is similar with the desired font results in good performance; selecting a source font which differs a lot against the desired font results in bad performance. The similarity is measured by comparing the attribute values of the user’s input and the candidate source font. However, our experiments described in Section \[sec:experiments\] demonstrate that the generated glyph images are nearly the same for most source fonts. Eventually, we draw the conclusion that there is no strict restriction on the source font in our model. Experiments {#sec:experiments} =========== Dataset ------- We adopt the font dataset released by [@o2014exploratory] which consists of 148 attribute-labeled fonts and 968 attribute-unlabeled fonts. For convenience, we refer this dataset as “AttrFont-ENG”. There are 37 kinds of font attributes ($N_{\alpha} = 37$), containing both concrete attributes such as “thin” and “angular,” and more nebulous concepts like “friendly” and “sloppy”. The annotation task was done by asking Mechanical Turk workers to compare different fonts according to their attributes and estimating relative scalar values for each font. Among these labeled fonts, we set the first 120 fonts as our supervised training data, the last 28 fonts as our validation data. Namely, $N_{sf} = 120$ and $N_{uf} = 968$. In this dataset, each font is represented by 52 glyph images (a-b and A-B), i.e., $N_{c} = 52$. Implementation Details ---------------------- The proposed model is implemented in PyTorch and trained on a NVIDIA 2080ti GPU. The whole network is trained in an end-to-end manner using the ADAM optimizer [@kingma2015adam]. The learning rate is set to $0.0002$ and the batch size $N_{bs}$ is set as 16. The image of each glyph is rendered with the font size 128 and resized into the resolution of 64 $\times$ 64. The dimension of attribute embeddings $N_{e}$ is set as 64. As we mentioned before, the category number of attributes $N_{\alpha}$ is set as 37. The original attribute values in [@o2014exploratory] vary from 0 to 100 and we re-scale them into \[0,1\] by dividing them by 100. One training epoch consists of $\frac{N_{f} \cdot N_{c} }{N_{bs}} $ steps. Empirically, we set $\lambda_{1} = 5$, $\lambda_{2} = 50$, $\lambda_{3} = 5$, $\lambda_{4} = 5$ and $\lambda_{5} = 20$ in loss functions. Effect of the Choice of Source Font ----------------------------------- Intuitively, the font style of generated glyphs would be affected by the choice of source font in our model. Surprisingly, through our experiments, we find that the influence is negligibly small when we select regular fonts as source. In Fig. \[fig:source\_input\], we randomly select a font from the validation dataset as the target font and four different fonts from the training dataset as the source fonts. The validation font is shown in the “ground truth” row and the four training fonts are shown in the “source input 1” - “source input 4” rows. The generated glyph images from four different source fonts are shown in the “output 1” - “output 4” rows, respectively, with nearly the same shapes and subtle differences. In the training process, the source font $a$ and the target font $b$ are both randomly selected and different fonts could match the same font as target. This phenomenon suggests that our model manages to map different font styles to a fixed font style based on their attribute values. We can also observe that the letter ‘g’ transferred from the fourth font preserves the original artistic shape while the others are more regular. In general, there is no strict restriction on the choice of source font but we highly recommend to choose a more regular font as the source. ![Generating glyph images with the same target attribute values from different source fonts. The pixel value of each grayscale grid represents each attribute’s value. A darker grid indicates a higher attribute value. In the following figures we use the same way to display attribute values.[]{data-label="fig:source_input"}](source_input){width="\columnwidth"} {width="\textwidth"} Evaluation Metrics ------------------ We adopt five commonly-used metrics for image generation to evaluate our model: Inception Score (IS) [@salimans2016improved], Fréchet Inception Distance (FID) [@heusel2017gans], Learned Perceptual Image Patch Similarity (LPIPS) [@zhang2018unreasonable], structural similarity index (SSIM) and pixel-level accuracy (pix-acc). Specifically, IS is used to measure the realism and diversity of generated images. FID is employed to measure the distance between two distributions of synthesized glyph images and ground-truth images, while SSIM aims to measure the structural similarity between them. LPIPS evaluates the distance between image patches. Since FID and IS cannot directly reflect the quality of synthesized glyph images, we also adopt the pixel-level accuracy (pix-acc) to evaluate performance. Higher values of IS, SSIM and pix-acc are better, whereas for FID and LPIPS, the lower the better. We also adopt two metrics which are widely used for shape matching, including the Hausdorff distance and Chamfer distance. We first extract the contours of glyph images using OpenCV and then utilize the points on the contours for calculating these two kinds of distances. Ablation Study -------------- For the purpose of analyzing the impacts of different modules, we conduct a series of ablation studies by removing or changing the proposed modules in our model. “Full Attribute2Font” denotes the full proposed model. “w/o skip-connection”, “w/o VST” and “w/o AAM” denote the proposed models without the skip-connection, Visual Style Transformer and Attribute Attention Module, respectively. “w/o unsupervised data” denotes that we train our model only with the attribute-labeled fonts. ### Quantitative Experiments The loss curves of $l_{pixel}$, which can be found in the supplementary material, demonstrate how different modules promote the model’s performance of reconstructing the target glyphs. When finishing training, the quantitative results for ablation studies are shown in Table \[tab:ablation\]. We witness a notable improvement brought by the proposed semi-supervised learning scheme via the exploitation of unlabeled fonts. AAM also boosts our model’s performance in a significant degree. It is worth noting that higher IS score does not always guarantee higher quality of images. The IS score is calculated from the classification predictions of a CNN trained with the ImageNet dataset. The domain mismatch between the glyph images and natural images makes the CNN consider the glyph images to be “unrealistic”. IS FID LPIPS SSIM pix-acc -------------------- ------------ ------------- ------------- ------------ ------------ w/o SL **3.2654** 77.7443 0.11503 0.7098 0.7703 w/o SC 3.0067 38.5884 0.10981 0.7181 0.7919 w/o VST 3.1082 35.9498 0.09503 0.7366 0.8049 w/o AAM 3.0908 46.5920 0.08790 0.7502 0.7948 **Full Attr2Font** 3.0740 **26.8779** **0.08742** **0.7580** **0.8153** : Quantitative results for ablation studies, w/o denotes without. “SC” denotes skip-connection. “SL” denotes semi-supervised learning. \[tab:ablation\] ### Qualitative Experiments Fig. \[fig:ablation\] shows some examples of synthesized glyph images of our model under different configurations. We select a representative font named as “Simontta-Black” from the validation dataset, as the target font for our model to generate. Note that only the attribute values of “Simontta-Black” are sent into our model. We expect our generated glyph images to share the most prominent styles with “Simontta-Black” instead of a perfect replica of “Simontta-Black”. The glyphs rendered with this font is shown in the “ground truth" row. The glyphs synthesized by our models with different configurations are shown in the second to sixth rows. Without the semi-supervised learning or skip-connection or VST, our model tends to bring more artifacts on the synthesis results. Without AMM, our model tends to miss some important characteristics such as “delicate” and “warm”. As we can see from Fig. \[fig:ablation\], the effectiveness of each proposed module is vividly verified. Parameter Studies ----------------- We conduct experiments to investigate how the number of glyph images sent into the style encoder (i.e., $m$) and the number of residual blocks in the Visual Style Transformer (i.e., $N_{rb}$), affect our model’s performance. The results are shown in Table \[tab:parameter\_m\] and \[tab:parameter\_nrb\], from which we can observe that generally larger $m$ and $N_{rb}$ result in better performance. However, larger values of $m$ and $N_{rb}$ increase the computational cost of our model. To achieve a balance between the model size and performance, we choose $m$ = 4 and $N_{rb}$ = 16 as the default settings of our model unless otherwise specified. Settings IS FID LPIPS SSIM pix-acc ---------- ------------ ------------- ------------- ------------ ------------ $m$ = 1 **3.1513** 56.8889 0.10355 0.7224 0.7836 $m$ = 2 3.1012 35.8145 0.09333 0.7423 0.8090 $m$ = 4 3.0740 26.8779 0.08742 0.7580 0.8153 $m$ = 8 3.0403 26.1309 0.08580 0.7592 0.8201 $m$ = 16 3.0907 **25.5825** **0.07922** **0.7645** **0.8236** : How the setting of $m$ affects our model’s performance when $N_{rb}$ is fixed to 16. \[tab:parameter\_m\] Settings IS FID LPIPS SSIM pix-acc --------------- ------------ ------------- ------------- ------------ ------------ $N_{rb}$ = 4 3.1720 29.0150 **0.07494** 0.7338 0.7742 $N_{rb}$ = 16 3.0740 26.8779 0.08742 0.7580 0.8153 $N_{rb}$ = 32 **3.1957** **24.0153** 0.07935 **0.7862** **0.8308** : How the setting of $N_{rb}$ affects our model’s performance when $m$ is fixed to 4. \[tab:parameter\_nrb\] ![Generating glyph images by interpolation between the attribute values of two different fonts. Three interpolation processes (Font 1 to Font 2, Font 2 to Font 3, Font 3 to Font 4) are presented in succession.[]{data-label="fig:interp"}](interp){width="\columnwidth" height="\textheight"} {width="\textwidth"} ![Generating glyphs from random attribute values.[]{data-label="fig:random_target"}](random_target){width="\columnwidth"} Attribute-controllable Interpolation ------------------------------------ Existing methods such as  [@campbell2014learning] and  [@guo2018creating] search for new fonts in an abstract manifold space. Our model makes it feasible for interpolation between different fonts by explicitly modifying the font attribute values. The interpolated attribute of two fonts is formulated as: $$\alpha^{ip} = (1 -\lambda) \cdot \alpha(a) + \lambda \cdot \alpha(b),$$ where $\lambda \in [0,1]$ denotes the interpolation coefficient. Fig. \[fig:interp\] shows that our model achieves smooth interpolation between different fonts and generates visually pleasing glyph images. Compared to [@campbell2014learning] and  [@guo2018creating], the synthesized glyph images from interpolated attribute values are more interpretable. Editing Fonts by Modifying Attribute Values {#sec:edit_font} ------------------------------------------- Font designers often have the requirement of editing an existing font into their desired one by modifying just a few or single attributes. In this section, we show our model’s ability to delicately edit a font by manipulating the value of a single attribute. We first set the font to be edited as the source font with attribute $\alpha(a)$. Let $i^{*}$ be the index of the attribute category that we want to modify with the value of $v$, then we set the target attribute $\alpha(b)$ as: $$\alpha_{i}(b)=\begin{cases}\alpha_{i}(a), & i \ne i^{*} \cr v, &i = i^{*}\end{cases},$$ where $i$ is the index of attribute category and $\alpha_{i}$ is the value of the $i$-th attribute ($ 1 \le i \le N_{\alpha}$) . Fig. \[fig:editing\] shows some examples of our synthesized glyph images by modifying the value of a single attribute, such as serif, cursive, display, italic, strong, thin and wide. We set $v$ as 6 different values (0.0, 0.2, 0.4, 0.6, 0.8, 1.0) to illustrate the generated glyph images with 11 different levels of attribute values. We can see that a smooth and gradual transition can be achieved by our model when the attribute value varies from 0.0 to 1.0. Generating From Random Attribute Values --------------------------------------- Our model is capable of generating glyphs from random attribute values, regardless of the relationships among different attributes. Fig. \[fig:random\_target\] demonstrates the generated glyph images of our model from several random sets of attribute values. The source font presented in the first row is fixed in this experiment and we randomly assign the attribute values of each character in the following rows (i.e., each character in the same row has different attribute values). As we can see, the font styles of synthesized glyph images are highly varied and most of them are visually pleasing. Our model can generate infinite typefaces from randomly assigned attribute values. It is easy to find glyphs with high qualities and their corresponding sets of attribute values. We believe this is inspiring for ordinary users and font designers to find or create their desired fonts. We also demonstrate the glyph images of a whole char-set generated from random attribute values in the supplemental material. Attribute Study --------------- {width="\textwidth"} ### Distribution Analysis In this section, we illustrate how our semi-supervised learning scheme works by demonstrating the distribution of attribute values of different fonts (including both labeled fonts and unlabeled fonts) in Fig. \[fig:attribute\_embedding\_distribution\]. The attribute values of each font are reduced into two dimensions by PCA (Principal Component Analysis). The green points correspond to unlabeled fonts and the blue points correspond to fonts with labels. As we can see from this figure, the distribution changes significantly in the early stage to fit the actual distribution of all font styles. Afterwards, it remains relatively stable in the training phase. In the beginning (Epoch 10), the green points are almost completely separated from the blue points. In the later period, the green points are mixed into the blue points according to a specific distribution. The attribute values of unlabeled fonts evolve from a stochastic state to a meaningful state, which verifies the effectiveness of our semi-supervised learning scheme. We attach the glyph images (‘A’ or ‘a’) of some fonts to their corresponding points in Epoch 400. We can observe that similar fonts are located closer while dissimilar fonts are located farther away. Thereby, the predicted attributes of unlabeled fonts are reasonable. ### Impact of Different Attributes {#sec:attr_impact} In this section, we investigate the impact of all kinds of attributes in font editing. Quantitative statistics of the glyph changes are conducted after modifying the value of each target attribute. The font editing is performed on all 28 fonts in the validation dataset. When investigating the impact of the $i$-th attribute, the values of other attributes in $\alpha(b)$ are set to the same as $\alpha(a)$. Let $\alpha_{i}(b)$ be the value of the $i$-th attribute of target font. In the beginning we set $\alpha_{i}(b)$ to 0.0, and then increase $\alpha_{i}(b)$ by 0.2 each time until $\alpha_{i}(b)$ reaches 1.0. We measure the shape difference of generated glyphs between the current step and the previous step by utilizing three metrics (pix-diff, SSIM, LPIPS), where “pix-diff” denotes the difference of pixel values. We calculate the mean value of the five steps for three metrics respectively. As we can see from Fig. \[fig:influence\_of\_different\_attributes\], nebulous attributes such as “complex, cursive and friendly” and concrete attributes such as “thin and wide” bring significant changes on the glyph shapes. ![The impact of different attributes on the glyph shape. The horizontal axis represents all kinds of attributes in our model. The vertical axis represents the value change of three metrics (pix-diff, SSIM, LPIPS) after modifying each attribute’s value. Please refer to Section \[sec:attr\_impact\] for details.[]{data-label="fig:influence_of_different_attributes"}](influence_of_different_attributes){width="8cm"} ### Correlation between Attributes In section \[sec:method\_des\_ov\], we mention that it is unnecessary for our model to have attributes being independent of each other, making it very convenient for users to define their own attribute categories. The only important thing is to accurately annotate the value of each attribute, instead of considering the correlation among different attributes. To prove this statement, we investigate the correlation of different attributes in the font dataset employed in this paper. The attributes are mutually related in a certain degree, which can be observed from Fig. \[fig:atributes\_correlation\]. This figure reveals some attribute pairs with strong correlations, such as “sloppy” and “italic”, “dramatic” and “attractive”, “disorderly” and “clumsy”. The correlation matrix also provides users with the guidance of manipulating attribute values into a meaningful set. ![The correlation matrix of different attributes in the font dataset released by [@o2014exploratory].[]{data-label="fig:atributes_correlation"}](atributes_correlation){width="\columnwidth"} Comparison with Other Models ---------------------------- ### Comparison with Attribute-controllable Image Synthesis Models In this section, we compare our method with existing methods of attribute-controllable image synthesis, including AttGAN [@he2019attgan], StarGAN [@choi2018stargan], RelGAN [@wu2019relgan] and STGAN [@liu2019stgan]. AttGAN and StarGAN are among the earliest works that address the problem of image attribute editing. They tackle arbitrary attribute editing by taking the target attribute as input to the transform model. STGAN and RelGAN utilize the difference between the target and source attributes as the transfer condition. In Fig. \[fig:comparison\], we select four fonts from the validation dataset whose attribute values are set as the target attributes in our model. Because these existing GANs can only accept binary attribute values, the attribute values they receive are set to 0 or 1 if they are less or higher than 0.5. Glyph images rendered from these fonts are shown in the “ground truth" rows as reference. We can see from Fig. \[fig:comparison\] that AttGAN and RelGAN tend to generate very blurry and low-quality glyphs. STGAN generates glyphs with higher quality than AttGAN and RelGAN, which proves the effectiveness of the attribute difference. Although the Selective Transfer Units (STU) were introduced to improve the skip-connection, STGAN still tends to bring many fake textures in the glyph images. The training strategy of RelGAN makes it very unstable in our task and result in very bad results. The quantitative results on the whole validation dataset are presented in Table \[tab:comparison\] which demonstrates that our model significantly outperforms the others. We also compare the model size of different methods in Table \[tab:comparison-para-prefer\], showing that our model possesses less parameters than STGAN but achieves much better performance than STGAN. \[tab:comparison\] ![Comparison with existing methods of attribute-controllable image synthesis.[]{data-label="fig:comparison"}](comparison){width="8cm"} Model \# Parameters User prefer. --------------- --------------- -------------- AttGAN 63.32M 0.0242 StarGAN **53.33**M 0.1020 RelGAN 61.74M 0.0305 STGAN 94.78M 0.1314 **Attr2Font** 69.85M **0.7119** : Comparison of model size and user preference for different image synthesizing methods. \[tab:comparison-para-prefer\] ### Comparison with Font Retrieval Models {width="\textwidth"} Model Hausdorff Chamfer attr-acc User prefer. ------------------ ------------ ------------- ------------ -------------- O’Donovan et al. 10.2917 327.747 0.7735 0.0980 Chen et al. 7.9193 273.221 0.8213 0.2902 **Attr2Font** **7.1954** **241.670** **0.9372** **0.6118** : Comparison of our model against two font retrieval methods. \[tab:comparison-attr-acc\] Font retrieval models search in a font database and then return fonts which are most relevant to the attributes (or tags) in a query list. However, choosing from existing fonts limits the model’s flexibility and the variety of available fonts. The lack of flexibility leads the retrieved fonts to miss some important attribute characteristics. For example, our model achieves smooth and continuous interpolation between different fonts. But when we set the interpolated attributes of two different fonts as query, the font retrieval models cannot provide such diverse and precise results on account of the limited number of existing fonts. Increasing the scale and diversity of font database may solve the problem to some extent but the computational cost will also increase dramatically. Another problem with the font retrieval models is the ignorance of some remarkable attributes in pursuit of global similarities. Fig. \[fig:comparison\_retrieval\] compares the synthesis results of our model with the retrieval results of [@o2014exploratory] and [@chen2019large] when given two sets of attribute values in the validation dataset. Our model obtains the most precise result compared to the other two methods. In the first example, [@o2014exploratory] ignores the weak “italic” attribute and [@chen2019large] ignores the strong “wide” attribute. In the second example, [@o2014exploratory] ignores the strong “formal” attribute and [@chen2019large] ignores the weak “strong” attribute. We train an attribute predictor and implement it on glyph images outputted from our model and these two font retrieval methods, respectively. As shown in Table \[tab:comparison-attr-acc\], our model achieves the highest accuracy, which gives a solid evidence of our method’s superiority to the state of the art. ### User Study We conduct a user study among ordinary users to compare our model with other existing methods. The first part of this study investigates the users’ preference among AttGAN, StarGAN, RelGAN, STGAN and our model. Specifically, for each font in the validation dataset, we send the attribute values of this font into the five models respectively, and get five sets of generated glyph images. For each set of glyph images, a participant is asked to choose the one that possesses the best quality and has the most similar style as the glyph images rendered by the validation font. The second part of this study investigates the users’ preference among two above-mentioned font retrieval methods and our model. For each set of provided glyph images, a participant is asked to choose the one that corresponds best to the given attribute values. 50 participants have taken part in this user study. Participants in this user study consist of students and faculties from different colleges. Most of them have no expertise in font design. Statistical results are shown in the “User prefer.” column of Table \[tab:comparison-para-prefer\] and \[tab:comparison-attr-acc\], respectively, from which we can see that our model outperforms the others by a large margin. Our model is initially designed for ordinary users, but it can also inspire/help professional designers to create new vector fonts based on the synthesized glyph images. There exists a widely-used font designing procedure that is to first obtain characters’ raster images and then automatically/manually convert them to vector glyphs. To further verify whether our system is helpful in practical use, we conduct another user study among professional font designers in Founder Group, one of the world’s largest font producers. 10 professional font designers have taken part in this user study. We develop a user interface for them and they are allowed to arbitrarily manipulate the attribute values and observe our generated glyph images. After a deep experience, they are asked to answer the following questions: (1) Is the system useful for font designers? (2) Does the system provide glyph images which are visually pleasing and embody the input attributes? (3) Can some of those machine-generated fonts inspire their creations? We conclude the 4 different perspectives as usefulness, quality, attribute-reflection and creativity, respectively. For each perspective, they give a rating among 1 to 5, where 1 denotes the most negative and 5 denotes the most positive. The results are shown in Fig. \[fig:user\_study\_designers\], from which we can see that most of the designers affirm the practical usefulness of our system and agree that it can generate satisfactory glyph images. Many of them have been deeply inspired by some creative fonts synthesized by our system and are willing to convert them into vector fonts. For instance, a senior font designer spoke highly of our system by commenting : “it is very useful for assisting and inspiring font designers and will significantly improve the efficiency of font design and production in the future." ![A user study among professional font designers. 10 participants have evaluated our model from 4 different perspectives and give ratings which are among 1-5 (higher is better).[]{data-label="fig:user_study_designers"}](user_study_designers){width="8cm"} Application on Chinese Fonts ---------------------------- ![Generating Chinese fonts from attributes.[]{data-label="fig:cn_synthesis"}](cn_synthesis){width="8cm"} ![Editing Chinese fonts by changing a single attribute’s value.[]{data-label="fig:cn_editing"}](cn_editing){width="8cm"} ![Interpolation among Chinese fonts.[]{data-label="fig:cn_interp"}](cn_interp){width="8cm"} To verify the generality of our model, we apply our model on Chinese fonts in addition to English fonts. We collected 245 Chinese fonts to form a new dataset named as AttrFont-CN. These fonts contain both Chinese and English glyphs and in general the Chinese and English glyphs in the same font share the same style. 78 fonts from AttrFont-ENG and 78 fonts from AttrFont-CN are matched in terms of their font style similarity on English glyphs. Then we annotate the latter with the attribute values of the former. The selected 78 fonts in AttrFont-CN are divided into 50 training fonts and 28 validation fonts. There are altogether 217 training fonts, including 50 labeled fonts and 167 unlabeled fonts, and 28 validation fonts. We train our model with Chinese glyphs from a character set with the size of 100. In Fig. \[fig:cn\_synthesis\], \[fig:cn\_editing\] and \[fig:cn\_interp\] we demonstrate our model’s performance on synthesizing glyph images from attributes, editing fonts and interpolation among fonts, respectively. Our model still achieves good performance on creating Chinese fonts from attributes although Chinese glyphs are much more complicated than English glyphs. Discussion ========== Generating Fonts in Strange Styles ---------------------------------- Fig. \[fig:difficult\_target\] shows some synthesis results of our model in strange font styles. The two target fonts are selected from the validation dataset and possess very unique styles. Our generated results share some remarkable styles with the ground truth although there exist some differences. ![Generating fonts in strange styles.[]{data-label="fig:difficult_target"}](difficult_target){width="8cm"} Limitations ----------- ![Some failure cases of our model.[]{data-label="fig:limitation"}](limitation){width="8cm"} Fig. \[fig:limitation\] reveals two limitations of our model: (1) The encoder-decoder architecture usually requires that the source image and the target image should share a similar geometric structure. The upper and lower cases of some characters have very different shapes. Thereby, the proposed model cannot handle capital fonts well, producing lowercase glyph images instead of capital ones even though the attribute “capital” is specified. Besides, ‘a’ and ‘g’ both have two different typologies. If we transfer one typology to another, the generated glyph tends to preserve the structure of the source typology to some extend (see the last row in Fig. \[fig:source\_input\]). (2) The assumption of each font corresponds to a set of font attribute values is not completely accurate. A novel font may have some other characteristics beyond the pre-defined attributes. If a desired font style is so unique that the pre-defined attributes cannot sufficiently describe it, our model will certainly fail to portray the font style, such as the second to fourth cases in Fig. \[fig:limitation\]. Application in Other Image Synthesis Tasks {#sec:app_on_other_tasks} ------------------------------------------ To further verify the generality of our Attribute Attention Module (AAM), we apply it on the task of face image synthesis. We integrate AAM into StarGAN (denoted as StarGAN + AAM) and compare it with the original StarGAN. Specifically, AAM is imposed on the different layers of the decoder in StarGAN, which is the same as our model. We show some cases in Fig. \[fig:stargan\_aam\] to compare the performance of these two models, where the input images are selected from a publicly-available database [@liu2015faceattributes] whose training set is adopted to train the models. More results are presented in the supplemental material. Generally, our model aims to handle the task of image-to-image translation conditioned on attributes. Thereby, potentially our model can also be applied in many other tasks, such as scene translation according to timing (from day to night), climate (from spring to winter), etc. Theoretically, the semi-supervised learning scheme is applicable to any scenarios when there is a shortage of attribute annotations. ![Comparison of StarGAN+AAM and StarGAN. The red rectangles emphasize some cases where StarGAN+AAM significantly differs with StarGAN. Note that StarGAN suffers from the problem of identity inconsistency. Thereby, synthesized faces may look quite different against input faces in this figure.[]{data-label="fig:stargan_aam"}](stargan_aam){width="\columnwidth"} Exploring the Variations of Our Work ------------------------------------ To sum up, our work learns a style mapping function based on the attribute difference. Previous to this paper, style similarity metrics have been widely used in design systems, such as [@o2014exploratory; @garces2014similarity; @lun2015elements; @serrano2018intuitive]. The goals of both the style similarity metrics and our mapping function are to build relationships between different styles. However, our mapping function concentrates on mapping one style to another according to the attribute difference while those existing style similarity metrics aim to calculate affinity scores between different styles. As a result, our system performs well for the task of synthesizing novel styles by arbitrarily assigning attribute values while those metric-learning systems are good at matching existing styles for specific needs. These two kinds of systems can benefit from each other: (1) The synthesized styles and existing styles can be combined to increase the diversity of styles and better meet the design needs. (2) The feature representations of these two kinds of systems might be complementary. In metric-learning systems, features of different styles are typically extracted/learnt individually. In our system, features of different styles are mutually learnt from the mappings between them. (3) Our semi-supervised learning scheme can also be applied to measure the style similarity when the manually annotated data are insufficient for supervised learning. We also come up with some other potential variations of our work. A simple variation of our work is to synthesize some other kinds of objects (other 2D images and 3D shapes, etc.), which has been partially discussed in Section \[sec:app\_on\_other\_tasks\]. Previous works, such as StarGAN, STGAN, etc., are limited to synthesizing faces or fashions from discrete attribute values, but our methods go one step further. Another variation of our work is to incorporate controllable attributes into the task of style transfer, which aims to change the style of a content image according to a reference style image. In previous works, such as [@gatys2016image] and [@ma2014analogy], the styles of the reference image are fully blended into the content image. With the help of our methods, the reference image’s styles can be selectively transferred through the control of attributes. Conclusion ========== In this paper, we presented a novel model to create new fonts by generating glyph images in accordance with user-specified attributes and their corresponding values. The Attribute Attention Module and semi-supervised learning scheme were introduced to deal with the difficulties in this task. Extensive experiments were conducted to demonstrate the effectiveness and superiority of our model compared to existing approaches. Last but not least, our model is not only limited to generating English glyphs, but also applicable to synthesizing fonts in any other writing systems (e.g., Chinese). We would like to investigate the following issues in the future: (1) extending our method to directly handle the task of vector font generation; (2) constructing a 2D font manifold for users to easily explore the fonts generated by our system; (3) improving the architecture of deep learners to promote the quality of generated fonts. This work was supported by National Natural Science Foundation of China (Grant No.: 61672043 and 61672056), Beijing Nova Program of Science and Technology (Grant No.: Z191100001119077), Center For Chinese Font Design and Research, and Key Laboratory of Science, Technology and Standard in Press Industry (Key Laboratory of Intelligent Press Media Technology). [^1]: Source code is available at <https://hologerry.github.io/Attr2Font/>

--- abstract: 'Jamming and avalanche statistics are studied in a simulation of the discharge of a polydisperse ensemble of disks from a 2-D silo. Exponential distributions are found for the avalanche sizes for all sizes of the exit opening, in agreement with reported experiments. The average avalanche size grows quite fast with the size of the exit opening. Data for this growth agree better with a critical divergence with a large critical exponent, as reported for 3-D experiments, than with the exponential growth reported for 2-D experiments.' author: - 'Gabriel Pérez$^*$' title: 'Statistics of Jamming in the discharge of a 2-D Silo' --- introduction ============ The discharge of granular matter from a container is one of the most common phenomena of everyday life, but it is not yet completely understood. Be it the pouring of salt from a salt-shaker, of corn form a hopper, or of gravel from a truck, we count only with a few empirical rules to explain the process [@beverloo1; @nedderman1]. Due to both its intrinsic interest and its obvious practical applications this phenomenon has received the attention of many researchers in recent times. It also provides a paradigmatic example of *jamming* [@liu1], a recently proposed transition to a particular state of matter (also called *fragile matter*) that can support some stresses, called *compatible*, but flow under *incompatible* ones. Jammed systems are then non-solids that, under the influence of external forcing, develop some particular structures that block their flow [@zuriguel1]. A particularly simple and common example of granular matter discharge is given in hoppers and silos [@hopper-silo]. Three states of flow have been identified in these systems: dilute (gas-like), dense (liquid-like) and jammed (static, solid-like) [@hou1; @zhong1]. The transition from dilute to dense, as a function of the exit opening, is discontinuous, and shows hysteresis [@hou1]. In these containers jamming is known to appear as soon as the size of the exit hole is reduced to a few times the average diameter of the particles inside. Experimental work in both 2-D and 3-D hoppers [@to1; @to2] and silos [@to2; @zuriguel1; @zuriguel2; @hou1; @zhong1] have shown that jamming depends only on the ratio between particle and exit hole sizes, as long as the diameter and height of the silo are large enough, that is, the silo is in its “thermodynamic limit”. Among the many questions still remaining in the dynamics of silos, one of the simplest and more fundamental is is the possible existence of a critical hole size such that for larger holes the flow cannot jam. The existence of such a size —actually, of a critical value for the ratio $R = \mbox{exit-hole-size/grain-size}$— has been shown in experiments for several types of granular media, including smooth and rough spheres, rice grains and lentils [@zuriguel2]. This is a somewhat puzzling result, since the same work shows that the distribution of avalanche sizes can be very well fitted to an exponential, and this in turn is consistent with a simple model where each grain —maybe cluster of grains— has a given probability of exiting the silo, uncorrelated to the behavior of other grains (clusters). Accepting this model, it becomes difficult to understand how a fully uncorrelated process can give rise to the long distance correlations one usually associates to criticality. It should also be mentioned that the experiments carried in [@to1; @to2] also support the hypothesis of a fixed probability of exiting the silo for each grain, but point towards a probability of jamming that is exponentially decaying on $R$; however, Ref. [@to2] also argues that it may be possible to fit its experimental data to either one of the two behaviors. In this work we have carried out a simulation of a 2-D silo with variable hole size, with the intention of getting some information on the statistics of its discharge, and on the possible presence of criticality in this process. Even though simulational approaches cannot reproduce completely the dynamics of real experiments, they do give very good approximations to real flows, and should be able to find signatures of critical behavior, if present. They have the added advantage of allowing for continuous and unconstrained adjustments in the main parameters of the flow. There are some obvious limitations to this particular simulation that puts it at a disadvantage with respect to actual experiments, chiefly the fact that tracking the very long avalanches that appear for large exit hole sizes consume an inordinate amount of computer time. Still, a systematic study of the avalanches for different exit hole sizes allows the identification of a well defined trend. Simulation ========== The simulations reported here were done over an ensemble of $N$ polydisperse disks, with diameter given by $d_i = d_{ave} + x \Delta d$, where $d_{ave}$ is the average diameter, $\Delta d$ is its maximum fluctuation, and $x$ is randomly chosen from the $[-1,1]$ uniform distribution. The silo has a bottom size $D$ and indefinite height. At the center of the bottom there is a hole of size $d_H$. The disks have a 2-D mass density $\sigma$, and the gravitational acceleration $g$ acts in the negative $z$ direction. Upon contact, the disks interact with the (perfectly rigid) walls of the silo, and with each other, via a linear spring with a constant that on loading has the value $\kappa$, and on unloading is reduced by a restitution factor $\epsilon$. This is an implementation of the *linear spring-dashpot model* [@cundall1; @schaefer1], using the two-couplings approach given in [@walton1]. This approach is commonly used because of its robustness and simplicity. The interaction is complemented by dynamic and static friction, using for both the same coefficient $\mu$. For the static part the the tangential spring model given in [@cundall1] is used, following the specific formulation of [@schaefer1], with the corrections given in [@brendel1]. Equations of motion were integrated using a velocity-Verlet algorithm [@allen1]. The only fixed numerical input in the problem is given by the gravitational constant; all quantities can be scaled, say to natural units where the disks’ average diameter and mass are set to one. In this work we have preferred to implement a numerical experiment with standard units, and have used $N = 2000$ disks with $d_{ave} = 0.5$, $\Delta d = 0.05$, $D = 15$, $\sigma = 0.8$ and $\kappa = 4\times10^6$, with all quantities given in the cgs system. This value for $\kappa$ is not as large as could be expected for some hard real systems (steel or glass spheres, for instance), but it allows for a more efficient use of computer time. It can be realistic enough for softer grains, like the rice or lentils used in [@zuriguel2]. It has also been reported that changes of up one or two orders of magnitude in the stiffness of the grains have little influence on the results of these types of simulations [@silbert1]. For the adimensional quantities $\mu$ and $\epsilon$ we have set $0.5$ and $0.9$ respectively. Gravity is fixed as $g = 981$. The ratio between the diameters of the disks and the silo gives $D/d_{ave} = 30$, and for the number of disks used the silo gets filled up to a height of around 2.5 times $D$. These two values are large enough to put the silo in the thermodynamic limit [@zuriguel1; @zuriguel2; @hirshfeld1], at least for the 3-D case. A time-step of 0.01 times the disk-disk collision time $t_{coll}$, which for the linear interaction used here is given by $$t_{coll} = \pi \sqrt{\frac{m}{\kappa}},$$ neglecting a weak dependence on $\epsilon$. For later convenience, we also define the time and speed scales $t_0 = \sqrt{d_{ave}/g}$ and $v_0 = g\, t_0$. The simulation proceeds as follows: first the silo is closed from below and the disks are placed in a regular grid with random initial velocities. The system is then allowed to relax, under the influence of gravity, up to the moment where the maximum speed detected is some small fraction of $v_0$. At his point a hole of length $d_h$ appears at the center of the bottom line, and disks start pouring out of the silo. These falling disks are followed until their centers are a distance $\approx 1.3\, d_{ave}$ below the bottom, and at this moment they are eliminated from the exiting flow and re-injected on top of the system, at a distance of $5.5\, d_{ave}$ from the surface of the dense aggregate of disks. In re-injection the disks preserve their $z$ velocity, but their horizontal velocity is set to zero. The re-injection point is chosen so as to keep the top of the material roughly flat. With these conditions the observed flow is of a mixed type, neither massic nor funnel-like; the disks close to the center of the silo fall faster than those close to the walls, but the speed difference is not too large. Occasionally, an arch is formed above the exiting hole and the flow stops. Given that we are including static friction, these arcs are not always convex. This jammed state is detected by checking that both (1) the *maximum* speed in the system is less than $v_0/c$, and (2) no disk has exited the silo in a time longer than $c\, t_0$. It has been found that for the purposes of this simulation $c = 8$ is adequate. Once these two conditions are fulfilled, the silo receives a tap given by vertical displacement $z_{tap} = A \sin( 2 \pi \nu t)$, applied for half a period. Here we have used $A = 0.6$ and $\nu = 8.0$. In most cases this tap is enough to break the arch or arches that are blocking the flow; however, given that the tap moves the whole material in parallel, it does occasionally happen that there is not enough rearrangement of the disks as to break the blockage. It is possible therefore with this unjamming protocol to get *null avalanches*, which are time intervals between two taps where no material flows out of the silo. These null avalanches are highly correlated among themselves, in the sense that, for small openings, they tend to appear next to each other in the time record. This type of events have also appeared in the experiments reported by Zuriguel [@zuriguel3]. Null avalanches are common for very small hole sizes, less so for larger openings. results ======= The simulation has been carried on for hole sizes from 1.70 to 2.25, in steps of 0.05, corresponding to hole/particle ratios $R$ from 3.4 to 4.5 in steps of 0.1. In all cases we have performed several runs starting from different grain configurations. For each size we have obtained at least 1000 avalanches. For all hole/particle ratios the distributions of avalanches $n(s)$ show basically an exponential form, except for a spike at $s = 0$ (null avalanches), and a weak dip for small $s$ (see Fig. (\[small-s\])). These two characteristics are probably a peculiarity of the method used here to unjam the silo. It should be noticed that the decrease in $n(s)$ found for small $s$ is not as pronounced as the one reported from the experiments [@zuriguel2]. As for the probability of finding null avalanches, it goes from a maximum of 0.20 at $R = 3.4$ to a minimum of 0.019 at $R = 4.4$, but there is not enough statistics as to be able to predict their presence or absence for larger values of $R$. To avoid having to fix a bin size in the histograms, we have used the normalized cumulative distribution $$w(s) = {\sum_{s^\prime = s}^{\infty} n(s^\prime)}/ {\sum_{s^\prime = 1}^{\infty} n(s^\prime)},$$ that is, we count the number of avalanches with $s$ or more disks. Notice that in this measure we are leaving out the null avalanches. For a properly normalized exponential distribution the normalized cumulative happens to be identical to the distribution itself. Fig. (\[cumulative\]) shows the cumulative avalanche distribution for $R = 3.6$. Given the claim that the distribution is exponential for all cases, it should be possible to scale $s$ to obtain a collapse of all cumulatives. This is shown in Fig. (\[colapso\]). Even so, it should be remembered that the distribution is not a perfect exponential, due to the smaller probabilities found for very small avalanches. This effect is almost imperceptible in the cumulatives. The main question that remains to elucidate is the behavior of the average avalanche $\langle s \rangle$ with respect to the hole/particle ratio. As intuitively expected, one finds a rapidly growing curve. This growth can be interpreted as evidence of an exponential divergence of the form $\langle s(R) \rangle \approx s_0 \exp(R/R_0)$, meaning that there is a non-zero probability of jamming for any size of the exit hole, even if for large openings the typical avalanche becomes astronomically large. However, as pointed out in [@to2], it is also possible to obtain a good fit to a power law of the form $\langle s(R) \rangle \approx s_0/(R_c - R)^\gamma$. Trying both types of fit for the results of this simulation, we find that the the avalanche averages have a better fit to the power-law expression, with $R_c = 6.7 \pm 0.4$ and $\gamma = 8.16 \pm 1.10$. For this fit the $\chi^2/\mbox{dof}$ is $0.41$. For the exponential fit we get $R_0 = 0.768$ and a $\chi^2/\mbox{dof}$ of $1.61$. The two different fits are shown in the semi-log graph given in Fig. (\[powerlaw\]). It is then clear that a power-law divergence is favored over an exponential behavior; it is is also clear, however, that with the available data the difference between the two fits is not really large enough as to allow for a definite conclusion. Moreover, some other types of divergence have been hinted at, like an essential singularity given by $\langle s \rangle \approx s_0 \exp[1/(R_c - R)]$. Still, the results agree with the most extensive experiments performed at this time, and therefore adds support the existence of criticality in the jamming of silos. This leaves open the more fundamental question about the origin of the correlations that may lead to critical behavior in this type of phenomena. Electronic address: [email protected] W. A. Beverloo, H. A. Leniger and J. van de Velde, Chem. Eng. Sci. [**15**]{}, 260 (1961). R. M. Nedderman, U. Tüzün, S. B. Savage and G. T. Houlsby, Chem. Eng. Sci. [**37**]{}, 1597 (1982). A. J. Liu and S. R. Nagel, Nature (London) [**396**]{}, 21 (1998). I. Zuriguel, L. A. Pugnaloni, A. Garcimartín and D. Maza, Phys. Rev. E [**68**]{}, 030301(R) (2003). Here a silo is any hopper with a flat bottom. J. Zhong, M. Hou, Q. Shi and K. Lu, J. Phys.: Condens. Matter [**18**]{}, 2789 (2006). M. Hou *et al.*, Phys. Rev. Lett. [**91**]{}, 204301 (2003). K. To, P. Y. Lai and H. K. Pak, Phys. Rev. Lett. [**86**]{}, 71 (2001). K. To, Phys. Rev. E [**71**]{}, 060301(R) (2005). I. Zuriguel [*et al.*]{}, Phys. Rev. E [**71**]{}, 051303 (2005). P. A. Cundall and O. D. L. Strack, Géotechnique [**29**]{}, 47 (1979). J. Schäfer, S. Dippel and D. E. Wolf, J. Phys. I France [**6**]{}, 5 (1996). O. R. Walton and R. L. Brown, J. Rheol. [**30**]{}, 949 (1986). L. Brendel and S. Dippel, *Physics of dry granular matterials*, NATO ASI Series [**E 350**]{}, 313 (1998). M. P. Allen and D. J. Tildesley, *Computer simulation of liquids*, Oxford U. P., Oxforf (1987). L. E. Silbert *et al.*, Phys. Rev. E [**64**]{}, 051302 (2001). D. Hirschfeld and D. C. Rapaport, Eur. Phys. J. E [**4**]{} 193 (2001). I. Zuriguel, Ph. D. Thesis, Universidad de Navarra (2005) (in spanish). ![Histogram of avalanches for $R = 3.6$. Bin size has been set to $4$, and null avalanches have not been included.[]{data-label="small-s"}](small_s_new.ps){width="14.0cm"} ![Cumulative curve $w(s)$ for $R = 3.6$. Null avalanches are excluded.[]{data-label="cumulative"}](acumulado_1.800_new.ps){width="14.0cm"} ![Collapse of the cumulatives $w(s)$ for all values of $R$ considered. The regions of non-exponential behavior found for small $s$ have some incidence over the observed dispersion, since in all cases the normalization is $w(1) = 1$.[]{data-label="colapso"}](colapso_new.ps){width="14.0cm"} ![Scaling of the average avalanche $\langle s \rangle$ against the exit-hole/disk ratio $R$. The continuous line give the power-law fit, with $R_c = 6.7$ and $\gamma = 8.16$. The dotted line shows the best linear fit, which corresponds to an exponential growth for the averages.[]{data-label="powerlaw"}](semilog_new.ps){width="14.0cm"}

--- abstract: 'In this note we give an explicit geometric description of some of the Nakajima’s quiver varieties. More precisely, we show that the $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ and the Hilbert scheme $X_\Gamma^{[n]}$ (where $X=\C^2$, $\Gamma\subset SL(\C^2)$ is a finite subgroup, and $X_\Gamma$ is a minimal resolution of $X/\Gamma$) are quiver varieties for the affine Dynkin graph, corresponding to $\Gamma$ via the McKay correspondence, the same dimension vectors, but different parameters $\zeta$ (for earlier results in this direction see [@H; @VV; @W]). In particular, it follows that the varieties $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $\Gamma=\Z/d\Z$) via the fixed points of $(\C^*\times\C^*)$-action we deduce the following combinatorial identity: the number $UCY(n,d)$ of uniformly coloured in $d$ colours Young diagrams consisting of $nd$ boxes coincides with the number $CY(n,d)$ of collections of $d$ Young diagrams with the total number of boxes equal to $n$.' address: ' Institute for Problems of Information Transmission, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow 101447, Russia ' author: - Alexander Kuznetsov title: Quiver varieties and Hilbert schemes --- addtoreset[equation]{}[section]{} \[equation\][Theorem]{} \[equation\][Proposition]{} \[equation\][Lemma]{} \[equation\][Corollary]{} \[equation\][Definition]{} \[equation\][Example]{} \[equation\][Remark]{} [[[M]{}od]{}]{} [[|[K]{}]{}]{} [[\_+]{}]{} [[\_-]{}]{} [[\_\*]{}]{} [[|[Q]{}]{}]{} [[|[h]{}]{}]{} \[1\] [^1] Introduction ============ The quiver varieties defined by Nakajima in [@Na] are of fundamental importance in algebraic and differential geometry, theory of representations and other branches of mathematics. They provide a rich source of examples of hyperkähler manifolds with very interesting geometry. For example, the quiver variety $\FM_\zeta(\bv^0,0)$, corresponding to an affine Dynkin graph, the dimension vector $\bv^0$ given by the minimal positive imaginary root of the corresponding affine root lattice, and any generic parameter $\zeta$ is diffeomorphic to the minimal resolution $X_\Gamma$ of the simple singularity $X/\Gamma$, where $X=\C^2$ and $\Gamma$ is a finite subgroup in $SL(\C^2)$, corresponding to the graph via the McKay correspondence (see [@Kr]). Furthermore, Nakajima proved that for some dimension vectors $\bv$, $\bw$ and for generic $\zeta$ the regular locus $\FM^{\text{reg}}_\zeta(\bv,\bw)$ of the quiver variety $\FM^{\text{reg}}_\zeta(\bv,\bw)$ is diffeomorphic to the framed moduli space of instantons on the 1-point compactification of $\FM_{-\zeta}(\bv^0,0)$. So, it would be natural to hope that one can give an algebraic description of $\FM_\zeta(\bv,\bw)$ as some moduli space of coherent sheaves on $\FM_{-\zeta}(\bv^0,0)$. The main result of this note is a description of this kind in the case $\bv=n\bv^0$, $\bw=\bw^0$ — the simple root of the extending vertex $0$ of the graph, and $\zeta=(0,\zeta_\R)$ with either $\zeta_\R\in\cplus$, or $\zeta_\R\in\cminus(n)$, where $$\cplus = \{\zeta_\R\in\R^d\ |\ \zeta_\R^k>0\quad 0 \le k \le d-1\}$$ $$\cminus(n)=\left\{\zeta_\R\in\R^d\ \left|\ \frac1n\zeta_\R^k > \sum_{i=0}^{d-1}\zeta_\R^i\bv^0_i > 0 \quad 1\le k\le d-1\right.\right\}$$ We prove that for $\zeta_\R\in\cplus$ the quiver variety $\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0)$ is isomorphic to the $\Gamma$-equivariant Hilbert scheme of points on the plane $X$. This fact is well known, see e.g. [@W], [@VV]; we include a proof only for the sake of completeness. We also prove that for $\zeta_\R\in\cminus(n)$ the quiver variety $\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0)$ is isomorphic to the Hilbert scheme of points on $X_\Gamma$. After this paper was written H. Nakaijma kindly informed me that this fact was known to him (see [@W] and [@H]), but his arguments are different (see Remark \[pn\]). Our proof is based on an interpretation of quiver varieties as moduli spaces of representations of the corresponding double quivers suggested by Crawley-Boevey in [@CB]. The paper is organized as follows, In section 2 we recollect the necessary background: the definition of quiver varieties, representations of quivers and the construction of Crawley-Boevey. In section 3 we reproduce in a short form a geometric version of the McKay correspondence, based on investigation of $X_\Gamma$. We also prove here a generalization of the result of Kapranov and Vasserot [@KV]. More precisely, the authors of [*loc. cit.*]{} have constructed equivalences of (bounded) derived categories $$\xymatrix{ \D^b(\Coh_\Gamma(X)) \ar[rr]<.5ex>^{\Phi} && \D^b(\Coh(X_\Gamma)) \ar[ll]<.5ex>^{\Psi} },$$ where $\Coh$ stands for the category of coherent sheaves, and $\Coh_\Gamma$ stands for the category of $\Gamma$-equivariant coherent sheaves. We show that there is a whole family of equivalences $$\xymatrix{ \D^b(\Coh_\Gamma(X)) \ar[rr]<.5ex>^{\Phi_{\zeta_\R}} && \D^b(\Coh(X_\Gamma)) \ar[ll]<.5ex>^{\Psi_{\zeta_\R}} },$$ which differ when parameters $\zeta_\R$ lie in distinct chambers of $\R^n$ with respect to roots hyperplanes. In section 4 we prove the main results of this paper, isomorphisms $$\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0)\cong \begin{cases} X^{\Gamma[n]}, & \text{for $\zeta_\R\in\cplus$}\\ X_\Gamma^{[n]}, & \text{for $\zeta_\R\in\cminus(n)$} \end{cases}$$ where $X^{\Gamma[n]}$ is the $\Gamma$-equivariant Hilbert scheme on $X$, and $X_\Gamma^{[n]}$ is the Hilbert scheme on $X_\Gamma$. It follows immediately from the general properties of quiver varieties that $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are diffeomorphic. We also prove a generalization of the first isomorphism in the Calogero-Moser context: we check that for arbitrary $\tau\ne0$ and $\zeta_\R\in\cplus$ we have $$\FM_{((\tau,\dots,\tau),\zeta_\R)}(n\bv^0,\bw^0) \cong (\CM_{nN})^\Gamma_{reg},$$ where $N=|\Gamma|$, $\CM_{nN}$ is the Calogero-Moser space and $(\CM_{nN})^\Gamma_{reg}$ is the connected component of the set of $\Gamma$-fixed points, where the tautological bundle is a multiplicity of the regular representation. Finally, in section 5 we consider $(\C^*\times\C^*)$-actions on $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ for cyclic $\Gamma\cong\Z/d\Z$. We show that these actions have only finitely many number of fixed points and that the number of fixed points equal to the dimension of the cohomology. Furthermore, we check that the fixed points on $X^{\Gamma[n]}$ are in a bijection with the set of uniformly coloured in $d$ colours Young diagrams with $dn$ boxes, and that the fixed points on $X_\Gamma^{[n]}$ are in a bijection with the set of collections of $d$ Young diagrams with total number of boxes equal $n$. Recalling that $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are diffeomorphic we obtain the following combinatorial identity (see [@JK] for a combinatorial proof) $$UCY(n,d) = CY(n,d),$$ where $UCY$ and $CY$ denote the number of uniformly coloured diagrams and the number of collections of diagrams respectfully. Acknowledgements {#acknowledgements .unnumbered} ---------------- I am very grateful to V. Ginzburg for bringing quiver varieties into the area of my interests and to H. Nakajima for helpful comments. Also I would like to thank M. Finkelberg and D. Kaledin for fruitful discussions. Quiver varieties ================ In this section we recollect the definition and the basic properties of quiver varieties. We will follow the notation of [@Na]. Definition {#defqv .unnumbered} ---------- Choose an arbitrary finite graph and let $H$ be the set of pairs, consisting of an edge together with an orientation on it. Let $in(h)$ (resp. $out(h)$) denote the incoming (resp. outgoing) vertex of $h\in H$. Let $\bh$ denote the edge $h$ with the reverse orientation. Further, we choose an orientation of the graph, that is a subset $\Omega\subset H$ such that $\Omega\cup\bOmega=H$, $\Omega\cap\bOmega=\emptyset$. We define $\eps(h)=1$ for $h\in\Omega$ and $\eps(h)=-1$ for $h\in\bOmega$. Finally, we identify the set of vertices of our graph with the set $\{0,1,\dots,d-1\}$. Choose a pair of hermitian vector spaces $V_k$, $W_k$ for each vertex of the graph and let $$\bv=(\dim_\C V_0,\dots,\dim_\C V_{d-1}),\ \bw=(\dim_\C W_0,\dots,\dim_\C W_{d-1})\in\Z^d$$ be their dimension vectors. Then the complex vector space $$\begin{gathered} \BM = \BM(\bv,\bw) = \\ \left(\bigoplus_{h\in H}\Hom(V_{out(h)},V_{in(h)})\right)\bigoplus \left(\bigoplus_{k=0}^{d-1}\big[\Hom(W_k,V_k)\oplus\Hom(V_k,W_k)\big]\right) \end{gathered}$$ can be identified with the cotangent bundle of the hermitian vector space $$\BM_\Omega(\bv,\bw)= \left(\bigoplus_{h\in\Omega}\Hom(V_{out(h)},V_{in(h)})\right)\oplus \left(\bigoplus_{k=0}^{d-1}\Hom(W_k,V_k)\right).$$ In particular, $\BM$ can be considered as a flat hyper-Kähler manifold. Note that the group $G_\bv=\prod_{k=0}^{d-1} U(V_k)$ acts on $\BM$: $$g = (g_k)_{k=0}^{n-1}: (B_h,i_k,j_k) \mapsto (g_{in(h)}B_hg_{out(h)}^{-1},g_ki_k,j_kg_k^{-1}),$$ where $B_h\in\Hom(V_{out(h)},V_{in(h)})$, $i_k\in\Hom(W_k,V_k)$, $j_k\in\Hom(V_k,W_k)$. This action evidently preserves the hyper-Kähler structure. The corresponding moment map $\mu=(\mu_\R,\mu_\C)$ is given by the following explicit formulas $$\mu_\C(B,i,j) = \left(\sum_{in(h)=k}\eps(h)B_hB_\bh+i_kj_k\right) \in \bigoplus_{k=0}^{d-1}\fgl(V_k) = \fg_\bv\otimes\C,$$ $$\mu_\R(B,i,j) = \frac i2\left(\sum_{in(h)=k} B_hB_h^\dagger-B_\bh^\dagger B_\bh+i_ki_k^\dagger-j_k^\dagger j_k \right)\!\!\!\! \in \bigoplus_{k=0}^{d-1}\fu(V_k) = \fg_\bv,$$ where $\fg_\bv$ is the Lie algebra of $G_\bv$ which is identified with its dual space $\fg_\bv^*$ via the canonical hermitian inner product. Let $Z_\bv\subset\fg_\bv$ denote the center of the Lie algebra $\fg_\bv$. For any element $\zeta=(\zeta_\C,\zeta_\C)\in (Z_\bv\otimes\C)\oplus Z_\bv$ the corresponding quiver variety $\FM_\zeta$ is defined as a hyper-Kähler quotient $$\FM_\zeta = \FM_\zeta(\bv,\bw) = \{(B,i,j)\ |\ \mu(B,i,j)=-\zeta\}/G_\bv.$$ In general, $\FM_\zeta$ has singularities, however its open subset $$\FM_\zeta^\reg = \left.\left\{(B,i,j)\in\mu^{-1}(-\zeta)\ \left|\ \parbox{0.34\textwidth}{the stabilizer of $(B,i,j)$ in $G_\bv$ is trivial} \right.\right\}\right/G_\bv.$$ is a smooth hyper-Kähler manifold (but maybe empty). Roots and genericity {#roots-and-genericity .unnumbered} -------------------- The center $Z_\bv$ of $\fg_\bv=\bigoplus_{k=0}^{d-1}\fu(V_k)$ is the product of the set of scalar matrices on $V_k$, thus it can be considered as a subspace of $\R^d$. Let $A$ denote the adjacency matrix of the graph and let $C=2I-A$ be the generalized Cartan matrix. Then we consider the set of positive roots $$R_+ = \{\theta = (\theta_k)\in(Z_{\ge0})^d\ |\ {}^t\theta C\theta\le 2\}\setminus\{0\}.$$ Further we denote $$R_+(\bv) = \{\theta\in R_+\ |\ \theta_k\le\bv_k\ \text{for all $k=0,\dots,d-1$}\},$$ and for any positive root $\theta$ we consider a hyperplane $$D_\theta=\{x=(x_k)\in\R^d\ |\ \sum x_k\theta_k=0\}\subset \R^d.$$ The element $\zeta$ is called generic (with respect to $\bv$) if for any $\theta\in R_+(\bv)$ we have $$\zeta\not\in\R^3\otimes D_\theta\subset\R^3\otimes\R^d \supset \R^3\otimes Z_\bv = (Z_\bv\otimes\C)\oplus Z_\bv.$$ The importance of generic parameters $\zeta$ is explained by the following Theorem of Nakajima (see [@Na] and [@CB] for connectedness). \[na\] For any $\zeta$, $\bv$, and $\bw$ the quiver variety $\FM_\zeta(\bv,\bw)$ is either empty, or nonemty and connected. Further, if $\zeta$ is generic then $\FM_\zeta$ is smooth and there is a canonical map $\pi_0:\FM_{(\zeta_\C,\zeta_\R)}\to\FM_{(\zeta_\C,0)}$ which is a resolution of singularities, provided $\FM_{(\zeta_\C,0)}^\reg$ is nonempty. Finally, if both $\zeta$ and $\zeta'$ are generic then the varieties $\FM_\zeta(\bv,\bw)$ and $\FM_{\zeta'}(\bv,\bw)$ are diffeomorphic. Representations of quivers {#representations-of-quivers .unnumbered} -------------------------- Recall that a quiver $Q$ is a finite oriented graph. Let $I(Q)$ denote the set of vertices and $A(Q)$ denote the set of arrows of $Q$. For any arrow of the quiver $\alpha\in A(Q)$ we denote by $in(\alpha)$ and $out(\alpha)$ the incoming and outgoing vertex of $\alpha$. A representation $\rho$ of a quiver $Q$ is the following datum: $$\rho = (V,B),$$ where $V = (V_i)_{i\in I(Q)}$ is a collection of vector spaces for each vertex of the quiver, and $B = (B_\alpha)_{\alpha\in A(Q)}$ is a collection of linear maps $B_\alpha:V_{out(\alpha)}\to V_{in(\alpha)}$ for each arrow of the quiver. A morphism of representations $$\phi:\rho=(V,B)\to\rho'=(V',B')$$ is a collection of linear maps $\phi_i:V_i\to V'_i$ for each vertex of the quiver, such that for any arrow $\alpha$ we have $$\phi_{in(\alpha)}B_\alpha = B'_\alpha\phi_{out(\alpha)}.$$ The dimension of a representation $\rho=(V,B)$ is the collection of dimensions of the vector space $V_i$: $$\dim\rho = (\dim V_i)_{i\in I(Q)}.$$ For any dimension vector $\bv=(\bv_i)_{i\in I(Q)}$ let $$\Rep_Q(\bv) = \mathop{\bigoplus}_{\alpha\in A(Q)} \Hom(\C^{\bv_{out(\alpha)}},\C^{\bv_{in(\alpha)}})$$ the space of $\bv$-dimensional representations of $Q$. The group $$GL(\bv,\C) = \prod_{i\in I(Q)} GL(\bv_i,\C)$$ acts on $\Rep_Q(\bv)$ by conjugation: $$g = (g_i)_{i\in I(Q)}:(B_\alpha)_{\alpha\in A(Q)} \mapsto (g_{in(\alpha)}B_\alpha g_{out(\alpha)}^{-1}).$$ This action evidently factors through the quotient group $$PGL(\bv,\C) = GL(\bv,\C)/\C^*,\qquad$$ where the embedding $\C^*\to GL(\bv,\C)$ takes $\lambda\in\C^*$ to the element $(\mathop{\rm diag}(\lambda))_{i\in I(Q)}$. Moreover, it is clear that the set of $PGL(\bv,\C)$ orbits in $\Rep_Q(\bv)$ is the set of isomorphism classes of $\bv$-dimensional representations of $Q$. Let $\chi$ be a map $I(Q)\to\R$, $\chi(i)=\chi^i$ (so-called polarization). For any dimension vector $\bv$ we define $$\chi(\bv) = \sum_{i\in I(Q)}\chi^i\bv_i.$$ A representation $\rho$ of a quiver $Q$ is called $\chi$-stable (resp. $\chi$-semistable) if $\chi(\dim\rho)=0$ and for any subrepresentation $\rho'\subset\rho$ such that $0\ne\rho'\ne\rho$ we have $\chi(\dim\rho') > 0$ (resp. $\chi(\dim\rho')\ge 0$). Representations $\rho$ and $\rho'$ are called $S$-equivalent with respect to a polarization $\chi$ if both $\rho$ and $\rho'$ are $\chi$-semistable and admit filtrations $$0 = \rho_0 \subset \rho_1 \subset \dots \subset \rho_n = \rho \quad\text{and}\quad 0 = \rho'_0 \subset \rho'_1 \subset \dots \subset \rho'_n = \rho'$$ such that $\chi(\dim\rho_i)=\chi(\dim\rho'_i)=0$ for all $i$, and $$\bigoplus_{i=1}^n\rho_i/\rho_{i-1} \cong \bigoplus_{i=1}^n\rho'_i/\rho'_{i-1}.$$ The following Theorem has been proved in [@Ki] and [@Na2]. \[mq\] For any quiver $Q$, dimension vector $\bv$ and polarization $\chi$, such that $\chi(\bv)=0$, there exists a coarse moduli space $\Mod_Q(\bv,\chi)$ of $\bv$-dimensional $\chi$-semistable representations of $Q$. Furthermore, if every $\chi$-semistable $\bv$-dimensional representation is $\chi$-stable and the dimension vector $\bv$ is indivisible, then $\Mod_Q(\bv,\chi)$ is a fine moduli space. Quivers with relations {#quivers-with-relations .unnumbered} ---------------------- Recall that the path algebra $\C[Q]$ of a quiver $Q$ is an algebra with a basis given by (oriented) paths in quiver and with multiplication given by concatenation of paths. It is clear that a representation of a quiver $\rho=(V,B)$ is the same as the structure of a right $\C[Q]$-module on the vector space $\bigoplus_{i\in I(Q)}V_i$. A quiver with relations is a pair $(Q,J)$, where $Q$ is a quiver and $J$ is a two-sided ideal $J\subset\C[Q]$ in its path algebra. A representation of a quiver with relations $(Q,J)$ is a representation $\rho$ of its underlying quiver $Q$, such that the ideal $J$ acts by zero in the corresponding right $\C[Q]$-module. In the other words, it is just a right $\C[Q]/J$-module. We denote by $\Rep_{Q,J}(\bv)$ the space of $\bv$-dimensional representations of the quiver with relations $(Q,J)$. It is clear that $\Rep_{Q,J}(\bv)$ is a closed algebraic subset in the vector space $\Rep_Q(\bv)$. It is easy to see that an analogue of Theorem \[mq\] is true for quivers with relations. \[mqr\] For any quiver with relations $(Q,J)$, dimension vector $\bv$ and polarization $\chi$, such that $\chi(\bv)=0$, there exists a coarse moduli space $\Mod_{Q,J}(\bv,\chi)$ of $\bv$-dimensional $\chi$-semistable representations of $(Q,J)$. Furthermore, if every $\chi$-semistable $\bv$-dimensional representation is $\chi$-stable and the dimension vector $\bv$ is indivisible, then $\Mod_{Q,J}(\bv,\chi)$ is a fine moduli space. \[qcham\] Let us fix the quiver $(Q,J)$ and the dimension vector $\bv$ and let $\chi$ vary in the space $D_\bv = \{\chi\ |\ \chi(\bv) = 0\}$. Then it is easy to see that the notion of $\chi$-(semi)stability can change only when the polarization $\chi$ crosses a hyperplane $D_{\bv'}\cap D_{\bv} \subset D_{\bv}$ for some . Thus the space $D_\bv$ decomposes by walls $D_{\bv'}\cap D_{\bv}$ into a union of chambers. \[explconstr\] One of constructions of the moduli space $\Mod_{Q,J}(\bv,\chi)$ is given in the terms of symplectic reduction. Namely, we can consider the representations space $\Rep_{Q,J}(\bv)$ as a symplectic variety with respect to the Kähler form (corresponding to the flat Kähler metric on the vector space $\Rep_Q(\bv)$). Then compact algebraic group $$PU(\bv) = U(\bv)/U(1) = \left(\prod_{i\in I(Q)}U(\bv_i)\right)/U(1)$$ (a real form of $PGL(\bv,\C)$) acts on $\Rep_{Q,J}(\bv)$ preserving the symplectic structure. Let $\mu:\Rep_{Q,J}(\bv)\to\fpu(\bv)^*$ denote the corresponding moment map. The polarization $\chi$ can be thought of as an element of the center of dual Lie algebra $\fpu(\bv)^*$. Then one can check (see e.g. [@Ki]) that $$\Mod_{Q,J}(\bv,\chi) = \mu^{-1}(-\chi)/PU(\bv).$$ Modular description of quiver varieties {#modular-description-of-quiver-varieties .unnumbered} --------------------------------------- The following construction, relating quiver varieties to the moduli spaces of representation of quivers was suggested in [@CB]. Assume that we are interested in a quiver variety $\FM_\zeta(\bv,\bw)$ corresponding to a graph with the set of vertices $\{0,\dots,d-1\}$ and with the set of oriented edges $H$. We define a quiver $Q = Q(\bw)$ (depending on the dimension vector $\bw$) with the set of vertices $$I(Q) = \{*\}\cup\{0,\dots,d-1\}$$ and with the following set of arrows. Firstly, we consider every oriented edge $h\in H$ as an arrow in the quiver with the same outgoing and incoming vertices. Secondly, for every $0\le i\le d-1$ we draw $\bw_i$ arrows from the vertex $*$ to the vertex $i$ and backwards. Then it is easy to see that a choice of a basis in spaces $W_0$, …, $W_{d-1}$ gives an identification $$\BM(\bv,\bw) = \Rep_{Q(\bw)}((1,\bv)).$$ Further, the expression $\mu_\C + \zeta_\C$ can be considered as an element of the path algebra $\C[Q(\bw)]$ of the quiver $Q(\bw)$. We denote by $J(\zeta_\C)$ the two-sided ideal in $\C[Q(\bw)]$, generated by $\mu_\C+\zeta_\C$. Then the algebraic subvariety $\mu_\C^{-1}(-\zeta_\C)\subset\BM(\bv,\bw)$ gets identified with the algebraic subvariety $\Rep_{Q(\bw),J(\zeta_\C)}((1,\bv)) \subset \Rep_{Q(\bw)}((1,\bv))$: $$\label{mci} \mu_\C^{-1}(-\zeta_\C) = \Rep_{Q(\bw),J(\zeta_\C)}((1,\bv)).$$ Further, note that the quiver variety $$\FM_\zeta(\bv,\bw) = \mu_\C^{-1}(-\zeta_\C)\cap\mu_\R^{-1}(-\zeta_\R)/G_\bv$$ is nothing but the symplectic reduction of the variety $\mu_\C^{-1}(-\zeta_\C) = \Rep_{Q(\bw),J(\zeta_\C)}((1,\bv))$ with respect to its canonical Kähler form. Finally, the group $G_\bv$ acting on $\mu_\C^{-1}(-\zeta_\C)$ is canonically isomorphic to the group $PU((1,\bv))$ acting on $\Rep_{Q(\bw),J(\zeta_\C)}((1,\bv))$: $$G_{\bv} = \prod_{i=0}^{d-1}U(\bv_i) \cong \left(U(1)\times\prod_{i=0}^{d-1}U(\bv_i)\right)/U(1),$$ and under this identificationthe element $\zeta_\R\in\fg_\bv$ goes to the polarization $$\label{chiz} \chi_\bv(\zeta_\R) = (-\zeta_\R(\bv),\zeta_\R)\in \fpu^*((1,\bv)). $$ Summing up and taking into account Remark \[explconstr\] we obtain the following Proposition. \[md\] For any $(\zeta_\C,\zeta_\R)$ we have an isomorphism of algebraic varieties $$\label{id} \FM_{(\zeta_\C,\zeta_\R)}(\bv,\bw) = \Mod_{Q(\bw),J(\zeta_\C)}((1,\bv),\chi_\bv(\zeta_\R)).$$ \[qvcham\] The arguments of Remark \[qcham\] show that $\FM_\zeta(\bv,\bw)$ depend on $\zeta_\R$ as follows. The space $\R^n$ of all $\zeta_\R$ is decomposed by root hyperplanes (walls) into a union of chambers. Whenever $\zeta_\R$ varies within a chamber the complex structure of $\FM_{(\zeta_\C,\zeta_\R)}(\bv,\bw)$ doesn’t change at all (but the hyper-Kähler metric does), while when $\zeta_\R$ crosses a wall $\FM_{(\zeta_\C,\zeta_\R)}(\bv,\bw)$ endures a (usually birational) transformation. We will finish this section with the following useful Lemma. \[fine\] If $\zeta$ is generic then $\Mod_{Q(\bw),J(\zeta_\C)}((1,\bv),\chi_\bv(\zeta_\R))$ is a fine moduli space. [*Proof:*]{} Let $\chi = \chi_\bv(\zeta_\R)$. Since the dimension vector $(1,\bv)$ is evidently indivisible it suffices to check that every $\chi$-semistable $(1,\bv)$-dimensional representation $\rho$ is $\chi$-stable. So assume that $\rho'\subset\rho$ is a $\chi$-stable subrepresentation, such that $0\ne\rho'\ne\rho$ and $\chi(\dim\rho')=0$. We have two possibilities: either $\dim\rho'=(0,\bv')$ or $\dim\rho'=(1,\bv')$ for some $\bv'$. Assume for example that $\dim\rho'=(0,\bv')$. Then we have $$0 = \chi(\dim\rho') = (-\zeta_\R(\bv),\zeta_\R)(0,\bv') = \zeta_\R(\bv').$$ Without loss of generality we can assume that $\rho'$ is $\chi$-stable. It follows from Theorem 0.2 of [@CB] that $(0,\bv')$ is a positive root for the quiver $Q(\bw)$, hence $\bv'\in R_+(\bv)$, hence $\zeta_\R\in D_{\bv'}$. On the other hand, by Proposition \[md\] the representation $\rho'$ corresponds to a point $(B',0,0)$ of the quiver variety $\FM_\zeta(\bv',\bw)$, hence $$\begin{gathered} \zeta_\C(\bv') = \sum_{k=0}^{n-1} Tr(\zeta_\C^k) = \sum_{k=0}^{n-1} Tr(-\mu_\C(B',0,0)) = \\ = - Tr\left(\sum_{h\in H}\eps(h)B'_hB'_\bh\right) = - Tr\left(\sum_{h\in \Omega}\big[B'_h,B'_\bh\big]\right) = 0.\end{gathered}$$ Thus $\zeta\in\R^3\otimes D_{\bv'}$, a contradiction with the assumtion that $\zeta$ is generic. Similarly, assume that $\dim\rho'=(1,\bv')$. Consider the quotient representation $\rho'' = \rho/\rho'$. Then $\dim\rho'' = (0,\bv'')$ with $\bv'' = \bv-\bv'$ and without loss of generality we can assume that $\rho''$ is $\chi$-stable. Then repeating the above arguments for $\rho''$ and $\bv''$ instead of $\rho'$ and $\bv'$ we again deduce that $\zeta$ is not generic. Thus we have proved that every $\chi$-semistable representation is $\chi$-stable, hence the moduli space is fine. The McKay correspondence ======================== The McKay correspondence associates to every (conjugacy class of a) finite subgroup of the group $SL(\C^2)$ an affine Dynkin graph of type $ADE$. The graph corresponding to a subgroup $\Gamma\subset SL(\C^2)$ can be constructed as follows. Let $R_0$, …, $R_{d-1}$ be the set of all (isomorphism classes of) irreducible representations of the group $\Gamma$, and assume that $R_0$ is the trivial representation. Let $L=\C^2$ be the tautological representation. Further, let $a_{k,l}$ be the multiplicities in the decomposition of the tensor product $R_k\otimes L$ into the sum of irreducible representations: $$\label{rl} R_k\otimes L \cong \mathop{\bigoplus}_{l=0}^{d-1} R_l^{\oplus a_{k,l}}.$$ Then the graph with the set of vertices $\{0,\dots,d-1\}$ and with $a_{k,l}$ edges between the vertices $k$ and $l$ is the corresponding affine Dynkin graph. Quiver varieties for affine Dynkin graphs {#quiver-varieties-for-affine-dynkin-graphs .unnumbered} ----------------------------------------- From now on we will consider quiver varieties corresponding to an affine Dynkin graph. Let $\Gamma$ be the corresponding subgroup of $SL(\C^2)$. Let $d$ denote the number of irreducible representations of $\Gamma$ (that is equal to the number of vertices in the graph), and let $N$ be the order of the group $\Gamma$. Let $V$ and $W$ be representations of $\Gamma$. Then $$\label{uudecomp} V = \bigoplus_{k=0}^{d-1}V_k\otimes R_k,\qquad W = \bigoplus_{k=0}^{d-1}W_k\otimes R_k.$$ Then the dimension vectors $$\bv = (\dim V_0,\dots,\dim V_{d-1}),\quad \bw = (\dim W_0,\dots,\dim W_{d-1}).$$ can be thought of as the classes of $V$ and $W$ in the Grothendieck ring $K_0(\Gamma)$. Consider a triple $$(B,i,j) \in \Hom_\Gamma(V\otimes L,V) \oplus \Hom_\Gamma(W,V) \oplus \Hom_\Gamma(V,W)$$ where $\Hom_\Gamma$ denotes the space of all $\Gamma$-equivariant linear maps. Then from (\[uudecomp\]) and from $$V\otimes L = \bigoplus_{k=0}^{d-1}V_k\otimes R_k\otimes L = \bigoplus_{k,l=0}^{d-1}V_k\otimes R_l^{\oplus a_{k,l}} = \bigoplus_{k,l=0}^{d-1}V_k^{\oplus a_{k,l}}\otimes R_l$$ it follows that a choice of $(B,i,j)$ is equivalent to a choice of a collection $(B_h,i_k,j_k)\in\BM(\bv,\bw)$. Thus $$\BM(\bv,\bw) = \Hom_\Gamma(V\otimes L,V) \oplus \Hom_\Gamma(W,V) \oplus \Hom_\Gamma(V,W).$$ Further, it is easy to check that $$\mu_\C(B,i,j) = [B,B] + ij,\qquad \mu_\R(B,i,j) = B B^\dagger - B^\dagger B + i i^\dagger - j^\dagger j,$$ where $[B,B]$, $B B^\dagger$, and $B^\dagger B$ are defined as the following compositions $$\begin{array}{rcl} [B,B] &:& V @>{\lambda_L^{-1}}>> V\otimes L\otimes L @>{B\otimes1}>> V\otimes L @>{B}>> V,\\ B B^\dagger &:& V @>{h_V}>> V^* @>{B^*}>> V^*\otimes L^* @>{h_V^{-1}\otimes h_L^{-1}}>> V\otimes L @>{B}>> V,\\ B^\dagger B &:& V @>{B}>> V\otimes L^* @>{h_V\otimes h_L^{-1}}>> V^*\otimes L @>{B^*}>> V^* @>{h_V^{-1}}>> V, \end{array}$$ and $\lambda_L$, $h_L$ and $h_V$ stand for $\Gamma$-invariant symplectic form and hermitian inner products on $L$ and $V$ respectively. Further, we can consider the parameters $\zeta_\C$ and $\zeta_\R$ as elements of $Z(\C[\gamma])$ and $\Z(\R[\Gamma])$, the center of the group algebra of $\Gamma$ over $\C$ and $\R$ respectively (element $\zeta_\C$ acts in $R_k$ as $\zeta_\C^k$-multiplication). Finally, it is easy to see that $G_\bv = U_\Gamma(V)$ and we obtain the following Lemma. \[gqv\] We have $$\FM_{(\zeta_\C,\zeta_\R)}(\bv,\bw) = \left.\left\{(B,i,j)\ \left|\ \begin{array}{rcl} {}[B,B]+ij &=& - \zeta_\C\\ {}[B,B^\dagger] + i i^\dagger - j^\dagger j &=& -\zeta_\R \end{array} \right.\right\}\right/U_\Gamma(V).$$ Let $X=\C^2$ with tautological action of $\Gamma$ and identify $L$ with the space of linear functions on $X$. \[vvk\] For any point $(B,i,j)\in\FM_{(0,\zeta_\R)}(\bv,\bw)$ such that $j=0$, the map $B$ induces the structure of a $\Gamma$-equivariant $\C[X]$-module on $V$ [(]{}where the linear functions on $X$ act via $B$[)]{}. [*Proof:*]{} We have $$[B,B] = \mu_\C(B,i,j) - ij = 0 - ij = 0,$$ hence the actions of linear functions on $X$ commute. Therefore, they induce an action of $\C[X]$. The obtainde $\C[X]$-module structure on $V$ is evidently $\Gamma$-equivariant. Combining this with (\[mci\]) we obtain the following. \[vvk1\] We have an isomorphism $$\begin{gathered} \{(V_\bullet,B,i,j)\in\Rep_{Q(\bw),J(0)}((1,\bv))\ |\ j=0 \} \cong \\ \left\{\parbox{0.85\textwidth}{\raggedright$\Gamma$-equivariant $\C[X]$-module structures on $V = \bigoplus\limits_{k=0}^{d-1}V_k\otimes R_k$\\ with a framing $i\in\Hom_\Gamma(W,V)$}\right\}\end{gathered}$$ Abusing the notation we will denote by $V_\bullet$ the representation of the quiver $(Q(\bw),J(0))$ corresponding to a $\Gamma$-equivariant $\C[X]$-module $V$ and $\Gamma$-morphism $i:W\to V$. Vice versa, the $\Gamma$-equivariant $\C[X]$-module, corresponding to a representation $\rho$ of the quiver $(Q(\bw),J(0))$ will be denoted by $\Tot(\rho)$. \[morita\] In fact, one can reformulate Corollary \[vvk1\] in terms of Morita equivalence. Let $e_*\in\C[Q(\bw)]$ denote the idempotent of the vertex $*$ (the path of length $0$ with $*$ being the outgoing (and incoming) vertex). Then the algebra $\C[Q(\bw)]/\langle J(0),e_*\rangle$ is Morita-equivalent to the smash product algebra $\C[X]\#\Gamma$. Resolutions of simple singularities {#resolutions-of-simple-singularities .unnumbered} ----------------------------------- We fix a pair of dimension vectors: $$\bv^0 = (\dim R_0,\dim R_1,\dots,\dim R_{d-1}),\quad\text{and}\quad \bw^0 = (1,0,\dots,0).$$ Note that $$\label{n} N = \sum_{k=0}^{d-1}(\bv^0_i)^2 = |\Gamma|.$$ We fix also the complex parameter $\zeta_\C=0$ but let the real parameter $\zeta_\R$ vary. The quiver varieties $\FM_{(0,\zeta_\R)}(\bv^0,\bw^0)$ are described by the following Theorem. \[kr\] $(i)$ The quiver variety $\FM_{(0,0)}(\bv^0,\bw^0)$ is isomorphic to the quotient variety $X/\Gamma$. $(ii)$ For any generic $\zeta_\R$ the quiver variety $X_\Gamma(\zeta_\R)=\FM_{(0,\zeta_\R)}(\bv^0,\bw^0)$ is a minimal resolution of the quotient variety $X/\Gamma$ via the canonical map $\pi_0:\FM_{(0,\zeta_\R)}(\bv^0,\bw^0)\to\FM_{(0,0)}(\bv^0,\bw^0)$. Actually, results of [@Kr] concern varieties $\FM_\zeta(\bv^0,0)$ rather than $\FM_\zeta(\bv^0,\bw^0)$. However, it is easy to see that these varieties are canonically isomorphic. Indeed, arguing like in the proof of Lemma \[li0j0\] below, one can check that for any point of $\FM_\zeta(\bv^0,\bw^0)$ we have $$j_0i_0 = \sum\zeta_\C^k\bv^0_k,\qquad i_0i_0^\dagger - j_0^\dagger j_0 = -2\sqrt{-1}\sum\zeta_\R^k\bv^0_k.$$ It follows that the map $\BM(\bv^0,\bw^0)\supset\mu^{-1}(\zeta)\to\mu^{-1}(\zeta)\subset\BM(\bv^0,0)$, induced by the canonical projection $\BM(\bv^0,\bw^0)\to\BM(\bv^0,0)$ (forgeting of $i_0$ and $j_0$) is a principal $U(1)$-bundle. Moreover, this map is $G_{\bv^0}$-equivariant and the action of $U(1)$ on fibers is nothing but the action of a subgroup $U(1)\subset G_{\bv^0}$. Hence the quotients modulo $G_{\bv^0}$ are canonically isomorphic. In dimension 2 any two minimal resolutions necessarily coincide. Thus, we have a canonical identification $X_\Gamma(\zeta_\R)\cong X_\Gamma(\zeta'_\R)$ for any generic $\zeta_\R$ and $\zeta'_\R$, compatible with the projection to $X/\Gamma$. So, we can (and will) write $X_\Gamma$ instead of $X_\Gamma(\zeta_\R)$ without risk of misunderstanding. Theorem \[md\] implies that for any $\zeta_\R$ we have an isomorphism $$X_\Gamma = \Mod_{Q(\bw),J(0)}((1,\bv^0),\chi_{\bv^0}(\zeta_\R)),$$ while Lemma \[fine\] implies that $X_\Gamma$ is a fine moduli space whenever $\zeta_\R$ is generic. This means that for any generic $\zeta_\R$ we have a universal representation $\urho^{\zeta_\R}$ of the quiver $(Q(\bw^0),J(0))$ over $X_\Gamma$, that is a collection of vector bundles $\CV_k^{\zeta_\R}$ over $X_\Gamma$, $k\in\{*\}\cup\{0,\dots,d-1\}$, which ranks are given by $$\label{rvk} \rk(\CV_*^{\zeta_\R})=1,\qquad \rk(\CV_k^{\zeta_\R})=\bv^0_k\quad\text{for $k\in\{0,\dots,d-1\}$,}$$ and morphisms $$\uB_h^{\zeta_\R}:\CV_{out(h)}^{\zeta_\R}\to\CV_{in(h)}^{\zeta_\R},\quad \ui_0^{\zeta_\R}:\CV_*^{\zeta_\R}\to\CV_0^{\zeta_\R},\quad \uj_0^{\zeta_\R}:\CV_0^{\zeta_\R}\to\CV_*^{\zeta_\R}$$ (recall the choice of $\bw^0$), satisfying the equations $$\label{mmc} \begin{array}{rl} \sum_{in(h)=k} \eps(h)\uB_h^{\zeta_\R}\uB_\bh^{\zeta_\R} = 0 & \text{for $k=1,\dots,d-1$} \smallskip\\ \ui_0^{\zeta_\R}\uj_0^{\zeta_\R} + \sum_{in(h)=0} \eps(h)\uB_h^{\zeta_\R}\uB_\bh^{\zeta_\R} = 0, \end{array}$$ and such that for any $x\in X_\Gamma$ the corresponding representation $$\urho(x)^{\zeta_\R} = ({\CV}_{|x}^{\zeta_\R},\uB^{\zeta_\R}(x), \ui_0^{\zeta_\R}(x),\uj_0^{\zeta_\R}(x))$$ of the quiver $Q(\bw^0)$ is $\chi_{\bv^0}(\zeta_\R)$-stable. \[rcham\] In fact, the family $\rho^{\zeta_\R}$ doesn’t change whenever $\zeta_\R$ varies within a chamber. Note that universal representation is defined up to a twist by a line bundle, so without loss of generality we may (and will) assume that $\CV_*^{\zeta_\R}=\CO$. \[li0j0\] For any $(1,\bv)$-dimensional representation $(V,B_h,i_0,j_0)$ of the quiver $(Q(\bw^0),J(0))$ we have $j_0i_0 = 0$. [*Proof:*]{} Since $\dim V_* = 1$ it follows that $$j_0i_0 = \mathop{Tr} j_0i_0 = \mathop{Tr} i_0j_0.$$ On the other hand, summing up equations (\[mmc\]) and taking the trace we see that $$\mathop{Tr}i_0j_0 = \sum_{h\in\Omega} \mathop{Tr}[B_h,B_\bh] + \mathop{Tr}i_0j_0 = \mathop{Tr}\mu_\C(B_h,i_0,j_0) = 0.$$ Since $\rk(\CV^{\zeta_\R}_*) = \rk(\CV^{\zeta_\R}_0) = 1$ by (\[rvk\]), it follows from Lemma \[li0j0\] that for any $x\in X_\Gamma$ either we have $\uj_0^{\zeta_\R}(x)=0$, or $\ui_0^{\zeta_\R}(x)=0$. Assume that $$\label{pos} \zeta_\R(\bv) > 0.$$ Then $\ui_0^{\zeta_\R}(x)=0$ would violate the $\chi_{\bv^0}(\zeta_\R)$-stability, because then $\urho^{\zeta_\R}(x)$ would admit a $(1,0,\dots,0)$-dimensional subrepresentation, but $$\chi_{\bv^0}(\zeta_\R)(1,0,\dots,0) = (-\zeta_\R(\bv^0),\zeta_\R)(1,0,\dots,0) = -\zeta_\R(\bv^0)<0.$$ So it follows that when (\[pos\]) holds we have $\uj_0^{\zeta_\R}=0$, and that $\ui_0^{\zeta_\R}$ is an embedding of line bundles, and hence an isomorphism. Thus we have proved the following Lemma. \[norm\] For any generic $\zeta_\R$ satisfying $(\ref{pos})$ there exists a universal $\chi_{\bv^0}(\zeta_\R)$-stable family on $X_\Gamma$ such that $$\CV_*^{\zeta_\R}\cong\CV_0^{\zeta_\R}\cong\CO_{X_\Gamma},\qquad \uj_0^{\zeta_\R}=0,\quad\text{and}\quad \ui_0^{\zeta_\R}=\id.$$ Applying Corollary \[vvk1\] we deduce. For any generic $\zeta_\R$ such that $(\ref{pos})$ holds, $\Tot(\rho^{\zeta_\R})$ is a family of $\Gamma$-equivariant $\C[X]$-modules over $X_\Gamma$. From now on we will denote by $\urho^{\zeta_\R}$ the universal representation normalized as in Lemma \[norm\]. Moreover, to unburden the notation we will omit the superscript $\zeta_\R$ when it won’t lead us to a confusion. Let $\rho_0$ denote the unique $(1,0,\dots,0)$-dimensional representation of the quiver $(Q(\bw^0),J(\zeta_\C))$. It follows from Lemma \[norm\] that for any $x$ we have canonical surjective homomorphism $\urho(x)\to\rho_0$. Let us denote its kernel by $\hrho(x)$. Derived equivalences {#derived-equivalences .unnumbered} -------------------- For each generic $\zeta_\R$ we define functors $$\xymatrix{ \D^b(\Coh_\Gamma(X)) \ar[rr]<.5ex>^{\Phi_{\zeta_\R}} && \D^b(\Coh(X_\Gamma)) \ar[ll]<.5ex>^{\Psi_{\zeta_\R}} },$$ where $\Coh_\Gamma(X)$ denotes the category of $\Gamma$-equivariant coherent sheaves on $X$, $\Coh(X_\Gamma)$ denotes the category of coherenet sheaves on $X_\Gamma$, and $\D^b$ stands for the bounded derived category. To this end we consider the family $\Tot(\urho^{\zeta_\R})$ of $\Gamma$-equivariant $\C[X]$-modules as a sheaf on the product $X_\Gamma\times X$. Denoting the projections to $X_\Gamma$ and to $X$ by $p_1$ and $p_2$ respectfully, we define $$\begin{array}{l} \Phi_{\zeta_\R}(F) = (Rp_{1*}R\CHom(\Tot(\urho^{\zeta_\R}),p_2^*F))^G, \quad\text{and}\\ \Psi_{\zeta_\R}(F) = Rp_{2*}(p_1^*F\otimes^L\Tot(\urho^{\zeta_\R})) \end{array}$$ For any generic $\zeta_\R$ the functors $\Phi_{\zeta_\R}$ and $\Psi_{\zeta_\R}$ are mutually inverse equivalences of categories. This Theorem is nothing but a slight generalization of Theorem 1.4 of Kapranov and Vasserot (see [@KV]). In fact, the equivalences $\Phi$ and $\Psi$ from [*loc. cit.*]{} are isomorphic to the functors $\Phi_{\zeta_\R}$ and $\Psi_{\zeta_\R}$ with $$\zeta_\R\in \{ \zeta_\R\in \R^n\ |\ \zeta_\R^i<0 \text{ for all $0\le i\le n-1$}\}$$ The proof of this Theorem can be done by the same arguments as in [*loc. cit.*]{} In fact, the equivalences $\Phi_{\zeta_\R}$ and $\Psi_{\zeta_\R}$ don’t change whenever $\zeta_\R$ varies within a chamber (see Remark \[rcham\]). For any pair of generic $\zeta_\R$ and $\zeta'_\R$ the compositions of functors $\Phi_{\zeta_\R}\cdot\Psi_{\zeta'_\R}$ and $\Psi_{\zeta'_\R}\cdot\Phi_{\zeta_\R}$ are autoequivalences of the derived categorie $\D^b(\Coh(X_\Gamma))$ and $\D^b(\Coh_\Gamma(X))$ respectively. One can check that they generate the action of the affine braid group described in [@ST]. Interpretation of quiver varieties ================================== Symmetric power of $X/\Gamma$ {#symmetric-power-of-xgamma .unnumbered} ----------------------------- From now on we will be interesting in an explicit geometric description of quiver varieties $\FM_\zeta(n\bv^0,\bw^0)$ for $\zeta_\C=0$, $\bv^0$ and $\bw^0$ as above and various $n$ and $\zeta_\R$. We begin with the simplest case, $\zeta_\R=0$. In fact, this is well known, but we put here a proof for the sake of completeness. \[psi0\] We have $\FM_{(0,0)}(n\bv^0,\bw^0) = S^n(X/\Gamma)$, where $S^n$ stands for the symmetric power. [*Proof:*]{} The case $n=1$ follows from the first assertion of Theorem \[kr\]. So assume that $n>1$. It follows from Proposition \[md\] that it suffices to check that $$\Mod_{Q(\bw),J(0)}((1,n\bv^0),0) = S^n(X/\Gamma).$$ Fix some generic $\zeta_\R$. For a collection of points $(x_1,\dots,x_n)$, $x_i\in X/\Gamma$ we put $$g_0(x_1,\dots,x_n) = \rho_0 \oplus\hrho(\tx_1)\oplus\dots\hrho(\tx_n),$$ where $\tx_i\in\pi_0^{-1}(x_i)\in X_\Gamma(\zeta_\R)$ are arbitrary lifts of the points $x_i$. Since every representation is semistable with respect to the trivial polarization $\chi=0$, it follows that $g_0$ induces a map $$g_0:S^n(X/\Gamma) \to \Mod_{Q(\bw^0),J(0)}((1,n\bv^0),0).$$ To construct the inverse we need the following Lemma. \[s0\] Any $(1,\bv)$-dimensional representation of $(Q(\bw^0),J(0))$ is $S$-equivalent [(]{}with respect to $\chi=0$[)]{} to a representation with $j=0$. [*Proof:*]{} Let $\rho=(V_*,V_\bullet)$ be a $(1,\bv)$-dimensional representation of $(Q(\bw^0),J(0))$. Let $B:V\otimes L\to V$, $i:W\to V$, and $j:V\to W$ be the corresponding $\Gamma$-equivariant morphisms. Let $U$ be the minimal subspace of $V$ such that $i(W)\subset U$ and $B(U\otimes L)\subset U$. Then $U$ is invariant under the action of $\Gamma$, hence $U=\oplus U_l\otimes R_k$. Moreover, $(V_*,U_\bullet)$ is a subrepresentation of $(V_*,V_\bullet)$, thus $$(V_*,V_\bullet) \seq0 (V_*,U_\bullet) \oplus (0,V_\bullet/U_\bullet),$$ where $\seq0$ denotes $S$-equivalence with respect to $\chi=0$. On the other hand, the arguments of [@Na2], Lemma 2.8 show that $j=0$ in the first summand in the RHS, and in the second summand $j=0$ for trivial reasons. Now we need to construct a map inverse to $g_0$. Take arbitrary $\rho\in\Mod_{Q(\bw^0),J(0)}((1,n\bv^0),0)$. By Lemma \[s0\] we can assume that $j=0$ in $\rho$. Then $\Tot(\rho)$ is a $\Gamma$-equivariant $\C[X]$-module. Regarding it as a $\Gamma$-equivariant coherent sheaf on $X$ we define $$f_0(\rho) = \supp(\Tot(\rho)) \in (S^{nN}X)^\Gamma = S^n(X/\Gamma).$$ Note that the definition of the map $f_0$ is correct. Indeed, assume that $\rho'\seq0\rho$ and $j=0$ in $\rho'$. Then $\rho$ and $\rho'$ admit filtrations with isomorphic associated factors (up to a permutation). These filtrations induce filtrations of sheaves $\Tot(\rho)$ and $\Tot(\rho')$ with isomorphic associated factors. But then $\supp(\Tot(\rho)) = \supp(\Tot(\rho'))$. Now we have to check that the maps $g_0$ and $f_0$ are mutually inverse. More precisely, one have to check that $$\label{todo} g_0(f_0(\rho)) \seq0 \rho\quad\text{and}\quad f_0(g_0(x_1,\dots,x_n)) = (x_1,\dots,x_n).$$ Before we begin the proof let us note that from the explicit construction of isomorphisms in Theorem \[kr\] it follows that for any point $\tx\in X_\Gamma$ we have $$\supp\Tot(\hrho(\tx)) = \pi_0(\tx).$$ Now, take $\rho\in\Mod_{Q(\bw^0),J(0)}((1,n\bv^0),0)$. By Lemma \[s0\] we can assume that $j=0$ in $\rho$, hence $\rho\seq0\rho_0\oplus\hrho$. Now consider the sheaf $\Tot(\hrho)$. It is a $\Gamma$-equivariant sheaf of finite length on $X$, hence it admits a filtration with associated factors being either the structure sheaves of a $\Gamma$-orbit corresponding to a point $0\ne x_i\in X/\Gamma$, or a length 1 skyscraper sheaf supported at zero. This filtration induces a filtration on $\hrho$ with associated factors being either $\hrho(\tx_i)$, with $\tx_i = \pi_0^{-1}(x_i)$, or $\rho_k$, where $\rho_k$ is the unique representation with $\dim(\rho_k)=\bw^k = (0,\dots,0,1,0,\dots,0)$ ($1$ stands on the $k$-th position). Thus $$\label{seq0} \rho \seq0 \rho_0 \oplus \left(\bigoplus_{i=1}^m\hrho(\tx_i)\right) \oplus \left(\bigoplus_{k=0}^{d-1}(\rho_k)\right)^{\oplus(n-m)\bv^0_k}$$ Then $$\begin{gathered} g_0(f_0(\rho)) = g_0(\supp\Tot(\rho)) = g_0(x_1,\dots,x_m,0,\dots,0) = \\ \rho_0 \oplus \left(\bigoplus_{i=1}^m\hrho(\tx_i)\right) \oplus \hrho(\tx_0)^{\oplus(n-m)\bv^0_k},\end{gathered}$$ where $\tx_0\in\pi_0^{-1}(0)$. But decomposition (\[seq0\]) for the representation $\rho_0 \oplus \hrho(\tx_0)^{\oplus(n-m)\bv^0_k}$ gives $$\hrho(\tx_0)^{\oplus(n-m)\bv^0_k} \seq0 \left(\bigoplus_{k=0}^{d-1}(\rho_k)\right)^{\oplus(n-m)\bv^0_k},$$ hence $g_0(f_0(\rho))\seq0\rho$, the first equality of (\[todo\]) is proved. Now, take $x_1,\dots,x_n\in X/\Gamma$. Then $$\begin{gathered} f_0(g_0(x_1,\dots,x_n)) = f_0(\rho_0\oplus\hrho(x_1)\oplus\dots\oplus\hrho(x_n)) = \\ \supp(\Tot(\hrho(x_1))\oplus\dots\oplus\Tot(\hrho(x_n))) = \{x_1,\dots,x_n\}.\end{gathered}$$ This completes the proof of the Proposition $\Gamma$-equivariant Hilbert scheme {#gamma-equivariant-hilbert-scheme .unnumbered} ----------------------------------- The action of the group $\Gamma$ on $X$ induces an action of $\Gamma$ on the Hilbert scheme $X^{[nN]}$ of $nN$-tuples of points on $X$. Let $\left(X^{[nN]}\right)^\Gamma$ denote the $\Gamma$-invariant locus. Since the group $\Gamma$ acts on the tautological bundle $\CV$ on $X^{[nN]}$, the restriction of $\CV$ to $\left(X^{[nN]}\right)^\Gamma$ decomposes into the direct sum of locally free sheaves indexed by the set of irreducible representations of $\Gamma$: $$\label{udecomp} \CV_{|\left(X^{[nN]}\right)^\Gamma} = \bigoplus_{k=0}^{d-1}\CV_k\otimes R_k.$$ Let $X^{\Gamma[n]}$ denote the locus of $\left(X^{[nN]}\right)^\Gamma$, where the rang of every $\CV_k$ equals $n\bv^0_k$ (or, in the other words, where $\CV$ is a multiplicity of the regular representation). It is clear that $X^{\Gamma[n]}$ is a union of some of the connected components of $\left(X^{[nN]}\right)^\Gamma$. We will reffer to $X^{\Gamma[n]}$ as the $\Gamma$-equivariant Hilbert scheme of $X$. Denote by $\cplus$ the positive octant of $\R^d$, that is $$\cplus = \{\zeta_\R\in\R^d\ |\ \zeta_\R^k>0\ \text{for all $0\le k\le d-1$}\}$$ Note that any $\zeta$ with $\zeta_\R\in\cplus$ is generic (because its real component $\zeta_\R$ is stricktly positive) and any $\zeta_\R\in\cplus$ satisfies the restriction (\[pos\]). The following Theorem is well known (see [@W], [@VV]). \[mplus\] For any integer $n\ge0$ and any $\zeta_\R\in\cplus$ the quiver variety $\FM_{(0,\zeta_R)}(n\cdot\bv^0,\bw^0)$ is isomorphic to the $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$. We begin with the following evident Lemma. \[st\] Let $U$ be a $\Gamma$-equivariant finite dimensional $\C[X]$-module with a $\Gamma$-invariant vector $u\in U$ and let $\rho$ be the corresponding $(1,\bu)$-dimensional representation of the quiver $(Q(\bw^0),J(0))$. Then $\rho$ is $\chi_{\bu}(\zeta_\R)$-stable with $\zeta_\R\in\cplus$ iff $U$ has no proper $\Gamma$-equivariant $\C[X]$-submodule containing $u$. [*Proof:*]{} Assume that $u\in U'\subset U$ is a $\Gamma$-equivariant proper $\C[X]$-submodule. Let $\rho'$ be the corresponding $(1,\bu')$-dimensional representation of the quiver. Then $\rho'$ is a subrepresentation in $\rho$ and $$\chi_\bu(\zeta_\R)(\rho') = (-\zeta_\R(\bu),\zeta_\R)(1,\bu') = \zeta_R(\bu') - \zeta_\R(\bu) = - \zeta_\R(\bu-\bu') < 0,$$ since $\bu>\bu'$ and $\zeta_\R$ is positive. Hence $\rho$ is unstable. Similarly, assume that $\rho$ is unstable. Then it contains a subrepresentation $\rho'$ such that $$\chi_\bu(\zeta_\R)(\dim\rho') = (-\zeta_\R(\bu),\zeta_\R)(\bu'_*,\bu') = \zeta_R(\bu') - \bu'_*\zeta_\R(\bu) < 0,$$ where $(\bu'_*,\bu') = \dim\rho'$. Since $\zeta_\R$ is positive the case $\bu'_*=0$ is impossible. Hence $\bu'_*=1$. This means that the subspace $\Tot(\rho')\subset\Tot(\rho) = U$ contains the vector $u$. On the other hand, it is a proper $\Gamma$-equivariant $\C[X]$-submodule in $U$. Now we can prove the Theorem. [*Proof:*]{} Recall that $\CV$ carries the structure of a family of quotient algebras of the algebra of functions $\C[X]$, and in particular, of $\Gamma$-equivariant $\C[X]$-module. In particular, the unit of the algebra $\C[X]$ induces a $\Gamma$-equivariant morphism $i:\CO\to\CV$. Applying relative analog of Corollary \[vvk1\] we see that for $\CV_*=\CO$ we obtain on $(\CV_*,\CV_k)$ the natural structure of a family of $(1,n\bv^0)$-dimensional representations of the quiver $(Q(\bw^0),J(0))$. Since $\CV$ is a family of quotient algebras of $\C[X]$ it follows that it is pointwise generated by the image of $1\in\C[X]$ as a family of $\C[X]$-modules. In other words, it has no proper $\Gamma$-invariant $\C[X]$-submodules containig the image of $1$. Hence Lemma \[st\] implies $\chi_{n\bv^0}(\zeta_\R)$-stability of the family $(\CV_*,\CV_k)$. Thus, we obtain a map $$f_+:X^{\Gamma[n]} \to \Mod_{Q(\bw^0),J(0)}((1,n\bv^0),\chi_{n\bv^0}(\zeta_\R)) = \FM_{(0,\zeta_\R)}(n\bv^0,\bw^0).$$ Similarly, let $\urho = (\CV_*,\CV_k)$ be a universal family on the moduli space $\Mod_{Q(\bw^0),J(0)}((1,n\bv^0),\chi_{n\bv^0}(\zeta_\R))$. Twisting it by the line bundle $\CV^*_*$ we can assume that $\CV_*=\CO$. Then Corollary \[vvk1\] implies that $\Tot(\urho)$ is a family of $\Gamma$-equivariant $\C[X]$-modules, endowed with a morphism $i:\CO\to\Tot(\urho)$. Furthermore, the $\chi_{n\bv^0}(\zeta_\R)$-stability of the family $\urho$ with Lemma \[st\] imply that $\Tot(\urho)$ is in fact a family of quotient algebras of $\C[X]$, when $1\in\C[X]$ goes to $i(1)\in\Tot(\urho)$. Since $$\rk(\Tot(\urho)) = \sum_{k=0}^{d-1}\rk(\CV_k)\dim R_k = \sum_{k=0}^{d-1}(n\bv^0_k)\bv^0_k = n\sum_{k=0}^{d-1}(\bv^0_k)^2 = nN$$ we obtain a map $$g_+:\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0) = \Mod_{Q(\bw^0),J(0)}((1,n\bv^0),\chi_{n\bv^0}(\zeta_\R)) \to X^{[nN]}.$$ The $\Gamma$-equivariance of $\CV$ implies that $f_+$ goes in fact to the $\Gamma$-invariant part of $X^{[nN]}$ and moreover to $X^{\Gamma[n]}$. It remains to notice that the maps $f_+$ and $g_+$ are evidently mutually inverse. The Hilbert scheme of $X_\Gamma$ {#the-hilbert-scheme-of-x_gamma .unnumbered} -------------------------------- Let $K_n,\BK_n\subset\Z^d$ denote the following finite sets of dimension vectors $$\begin{array}{c} K_n = \{ \bv\in\Z_{\ge0}^d\ |\ \bv_0\bv^0 < \bv < n\bv^0 \}\smallskip\\ \BK_n = \{ \bv\in\Z_{\ge0}^d\ |\ \bv_0\bv^0 \le \bv \le n\bv^0 \} \end{array}$$ Then $$\cminus(n) = \left\{\zeta_\R\in\R^d\ \left|\ \begin{cases} \zeta_\R(\bv^0)>0,\\\zeta_\R(\bv) > n\zeta_\R(\bv^0),&\quad \text{for all $\bv\in K_n$} \end{cases}\right.\right\}$$ is a convex polyhedral cone. \[cm\] We have $$\cminus(n) = \{\zeta_\R\in\R^d\ |\ \frac1n\min_{k=1}^{d-1}\zeta_\R^k-\sum_{k=1}^{d-1}\zeta_\R^k\bv^0_k > \zeta_\R^0 > - \sum_{k=1}^{d-1}\zeta_\R^k\bv^0_k \}$$ In particular, the cone $\cminus(n)$ is nonempty and for any $\zeta_\R\in\cminus(n)$ we have $\zeta_\R^k>0$ for all $1\le k\le d-1$. [*Proof:*]{} Easy. \[mminus\] For any $n\ge0$ and any $\zeta_\R\in\cminus(n)$ the quiver variety $\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0)$ is isomorphic to the Hilbert scheme $X_\Gamma^{[n]}$. [*Proof:*]{} Let $\urho=\urho^{\zeta_\R}=(\CV_*,\CV_k,\uB_h,\ui_0,\uj_0)$ be the universal family on $X_\Gamma$ normalized as in Lemma \[norm\]. Let $Z\subset X_\Gamma$ be a length $n$ subscheme in $X_\Gamma$ and let $\CO_Z$ be its structure sheaf. We associate to $Z$ the following representation $\rho(Z)=(V_*(Z),V_k(Z),B(Z),i_0(Z),j_0(Z))$ of the quiver $Q(\bw^0)$. We put $$\begin{array}{l} V_k(Z) = \Gamma(X_\Gamma,\CV_k\otimes\CO_Z)\qquad k=0,\dots,d-1,\\ V_*(Z)= \Gamma(X_\Gamma,\CO_{X_\Gamma}) \cong \C \end{array}$$ with morphisms $B_h(Z):V_{out(h)}(Z)\to V_{in(h)}(Z)$ induced by the maps $\uB_h\otimes\id_{\CO_Z}:\CV_{out(h)}\otimes\CO_Z\to\CV_{in(h)}\otimes\CO_Z$, while $j_0(Z)=0$ and $i_0(Z)$ is induced by the composition $$\CO_{X_\Gamma} \cong \CV_0 \to \CV_0\otimes\CO_Z,$$ where the first is the isomorphism of Lemma \[norm\], while the second morphism is obtained from the canonical projection $\CO_{X_\Gamma}\to\CO_Z$ by tensoring with $\CV_0$. Since $Z$ is a length $n$ subscheme of $X_\Gamma$ and $\rk\CV_k=\bv^0_k$ it follows that $V_k(Z)$ is an $n\bv^0_k$-dimensional vector space for any $0\le k\le d-1$. On the other hand, $V_*(Z)$ is $1$-dimensional by definition. Hence $\rho(Z)$ is $(1,n\bv^0)$-dimensional reresentation of the quiver $Q(\bw^0)$. Further, it follows from (\[mmc\]) and Lemma \[li0j0\] that $\rho(Z)$ satisfies the relations $J(0)$. Now we want to check that $\rho(Z)$ is $\chi_{n\bv^0}(\zeta_\R)$-stable for any $\zeta_\R\in\cminus(n)$. We begin with the following Proposition. \[subrep\] If $\rho'\subset\rho(Z)$ is a subrepresentation and $\rho'\ne\rho(Z)$ then we have $$\dim\rho' = (0,\bv)\text{ with $\bv\in\BK_n$, or } \dim\rho' = (1,\bv)\text{ with $\bv\in K_n$.}$$ [*Proof:*]{} Choose a chain of subschemes $$Z = Z_n\supset Z_{n-1}\supset \dots \supset Z_1\supset Z_0=\emptyset,$$ such that $Z_i$ is a length $i$ subscheme. This chain of subschemes induces a chain of surjections of structure sheaves $$\CO_Z = \CO_{Z_n}\to \CO_{Z_{n-1}}\to \dots \to \CO_{Z_1}\to \CO_{Z_0} = 0,$$ and of representations of quivers $$\label{rhosur} \rho(Z) = \rho(Z_n) \to \rho(Z_{n-1}) \to \dots \to \rho(Z_1) \to \rho(Z_0) = \rho_0.$$ Note that for any $i=1,\dots,n$ we have exact sequence of sheaves $$0 \to \CO_{x_i} \to \CO_{Z_i} \to \CO_{Z_{i-1}} \to 0,$$ for some $x_i\in X_\Gamma$, which induces exact sequence of representations $$0 \to \hrho(x_i) \to \rho(Z_i) \to \rho(Z_{i-1}) \to 0.$$ Now assume that $\rho'$ is a subrepresentation in $\rho(Z)$. Let $\rho'_i$ denote the image of $\rho'$ in $\rho(Z_i)$ with respect to the surjection (\[rhosur\]). Then (\[rhosur\]) induces a chain of surjections $$\rho' = \rho'_n \to \rho'_{n-1} \to \dots \to \rho'_1 \to \rho'_0 = \rho_0.$$ Let $\rho''_i$ denote the kernel of the map $\rho'_i\to\rho'_{i-1}$. Then $\rho''_i$ is a subrepresentation in $\hrho(x_i)$, hence $\dim\rho''_i=(0,\bu^i)$ for some dimension vector $\bu^i\le\bv^0$. Since $\bv^0_0=1$, it follows that either $\bu^i_0=0$ or $\bu^i_0=1$. Assume that $\bu^i_0=1$. Then it is easy to see that we can extend the subrepresentation $\rho''_i\subset\hrho(x_i)$ to $(1,\bu^i)$-dimensional subrepresentation $\widetilde{\rho''_i}\subset\rho(x_i)$. Now, note that if $\bu^i<\bv^0$ then since $\bu^i_0=\bv^0_0$ and since $\zeta_\R^k>0$ for all $1\le k\le d-1$ by Lemma \[cm\], it follows that $$\chi_{\bv^0}(\zeta_\R)(1,\bu^i) = (-\zeta_\R(\bv^0),\zeta_\R)(1,\bu^i) < (-\zeta_\R(\bv^0),\zeta_\R)(1,\bv^0) = 0,$$ which contradicts the $\chi_{\bv^0}(\zeta_\R)$-stability of $\rho(x_i)$. Thus we have proved that if $\bu^i_0=1$ then $\bu^i=\bv^0$. On the other hand, if $\bu^i_0=0$ then certainly $\bu^i\ge 0$. Thus in both cases we have $$\bu^i \ge \bu^i_0\bv^0.$$ Summing up these inequalities from $i=1$ to $n$ we see that $$\bv = \sum_{i=1}^n\bu^i \ge \sum_{i=1}^n\bu^i_0\bv^0 = \bv_0\bv^0,$$ hence $\bv\in\BK_n$. It remains to check that if $\dim\rho'=(1,\bv)$ then $\bv\in K_n$. So, assume that it is not true. Then $\bv=m\bv^0$ for some $0\le m\le n$. Let us check that the case $m<n$ is impossible. For this we will use induction in $n$. The base of induction, $n=0$, is trivial. So assume that $n>0$, $Z\subset X_\Gamma$ is a length $n$ subscheme, and that $\rho'\subset\rho(Z)$ is a $(1,m\bv^0)$-dimensional subrepresentation with $m<n$. Consider the following commutative diagram with exact rows $$\begin{CD} 0 @>>> \rho''_n @>>> \rho' @>>> \rho'_{n-1} @>>> 0 \\ @. @VVV @VVV @VVV \\ 0 @>>> \hrho(x_n) @>>> \rho(Z) @>>> \rho(Z_{n-1}) @>>> 0 \end{CD}$$ where $\rho''_n$ and $\rho'_{n-1}$ was defined above. Denote $$\dim\rho''_n = (0,\bu),\qquad \dim\rho'_{n-1} = (1,\bv').$$ Then the above arguments show that $$\bu \ge \bu_0\bv^0,\qquad \bv' \ge \bv'_0\bv^0.$$ But since $\bu + \bv' = m\bv^0 = (\bu_0+\bv'_0)\bv^0$ it follows that $$\bu = m''\bv^0,\qquad \bv'= m' \bv^0,\qquad m'+m''=m.$$ In particular, $\rho(Z_{n-1})$ contains a $(1,m'\bv^0)$-dimensional subrepresentation. The induction hypothesis for $Z_{n-1}$ then implies that $m'=n-1$. On the other hand, we have $m''=1$ or $m''=0$. In the first case we have $m=n$, a contradiction. In the second case, it follows that $\rho'=\rho(Z_{n-1})$, hence the exact sequence $$0 \to \hrho(x_n) \to \rho(Z) \to \rho(Z_{n-1}) \to 0$$ splits. Comparing the definition of $\rho(Z)$ with the definition of the equivalence $\Psi_{\zeta_\R}$, we see that the splitting of the above sequence implies that the sequence $$0 \to \CO_{x_n} \to \CO_{Z} \to \CO_{Z_{n-1}} \to 0,$$ also splits, and that this splitting is compatible with the projections $\CO_{X_\Gamma}\to\CO_Z$ and $\CO_{X_\Gamma}\to\CO_{Z_{n-1}}$. But this means that $$J_{Z_{n-1}} \cong J_Z \oplus \CO_{x_n},$$ which is false. Thus we again come to a contradiction, and the Lemma is proved. Now we can finish the proof of the Theorem. The $\chi_{n\bv^0}(\zeta_\R)$-stability of the representation $\rho(Z)$ follows immediately form the definition of the cone $\cminus(n)$ and from Proposition \[subrep\]. Thus $\rho(Z)$ form a family of $(1,n\bv^0)$-dimensional $\chi_{n\bv^0}(\zeta_\R)$-stable representations of the quiver $(Q(\bw^0),J(0))$ over the Hilbert scheme $X_\Gamma^{[n]}$. This family induces a map $$g_-:X_\Gamma^{[n]} \to \Mod_{Q(\bw^0),J(0)}((1,n\bv^0),\chi_{n\bv^0}(\zeta_\R)) = \FM_{(0,\zeta_\R)}(n\bv^0,\bw^0).$$ Moreover, it is easy to see that the map fits into the following commutative diagram: $$\begin{CD} X_\Gamma^{[n]} @>{g_-}>> \FM_{(0,\zeta_\R)}(n\bv^0,\bw^0) \\ @VVV @VV{\pi_0}V \\ S^n(X/\Gamma) @>{g_0}>> \FM_{(0,0)}(n\bv^0,\bw^0) \end{CD}$$ where the left vertical arrow is the composition of the Hilbert-Chow morphism $X_\Gamma^{[n]}\to S^n(X_\Gamma)$ with the map induced by the projection $\pi_0:X_\Gamma\to X/\Gamma$. Since both vertical maps are birational, and $g_0$ is an isomorphism by Proposition \[psi0\] it follows that $g_-$ is a regular birational map. On the other hand, both $X_\Gamma^{[n]}$ and $\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0)$ are holomorphically symplectic varieties, hence any regular birational map between them is an isomorphism. \[pn\] H. Nakaijma indicated to me the direction of his arguments. He constructs a map $\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0)\to X_\Gamma^{[n]}$ (note that the direction is opposite to that of the map $g_-$) using certain complex and utilizing the stability condition to ensure that certain cohomology groups vanish. Combining Theorems \[mplus\] and \[mminus\] with Theorem \[na\] we get the following Corollary. \[cor\] The $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ is diffeomorphic to the Hilbert scheme $X_\Gamma^{[n]}$. In particular, we have an isomorphism of cohomology groups $$H^\bullet(X^{\Gamma[n]},\Z) \cong H^\bullet(X_\Gamma^{[n]},\Z).$$ It is also easy to deduce that for any $$\zeta_\R \in \{\zeta_\R\ |\ \zeta_\R(\bv^0) = 0,\quad\text{and}\quad \zeta_\R^k > 0\quad\text{for all $1\le k\le d-1$}\}$$ we have $$\FM_{(0,\zeta_\R)}(n\bv^0,\bw^0) \cong S^n(X_\Gamma).$$ Generalizations: the Calogero-Moser space {#generalizations-the-calogero-moser-space .unnumbered} ----------------------------------------- Let $\Gamma$ be a finite subgroup in $SL(2,\C)$ and $\Gamma'\subset\Gamma$ its central subgroup. Let $\FM^\Gamma$ and $\FM^{\Gamma'}$ denote the quiver varieties corresponding to the affine Dynkin graphs of $\Gamma$ and $\Gamma'$ respectively. Since $\Gamma'$ is central in $\Gamma$ the embedding $\Gamma'\to\Gamma$ induces embeddings $$\sigma_\C^*:Z(\C[\Gamma']) \to Z(\C[\Gamma]),\qquad \sigma_\R^*:Z(\R[\Gamma']) \to Z(\R[\Gamma]).$$ In terms of the Dynkin graphs the embedding $\sigma^*$ can be described as follows. Let $I$ and $I'$ be the sets (of isomorphism classes) of irreducible representations of $\Gamma$ and $\Gamma'$ respectively. In other words, $I$ and $I'$ are the sets of vertices of the corresponding affine Dynkin graphs. Consider an irreducible $\Gamma$-module $R_i^\Gamma$, $i\in I$. Since $\Gamma'$ is central it follows from the Schur Lemma that the restriction $(R_i^\Gamma)_{|\Gamma'}$ is a multiplicity of an irreducible representation of the group $\Gamma'$, say $R^{\Gamma'}_{i'}$, $i'\in I'$. So, associating this way to arbitrary vertex $i\in I$ the vertex $i'\in I'$ we obtain a map $\sigma:I\to I'$. The map $\sigma^*$ is induced by $\sigma$. Let $N=|\Gamma/\Gamma'|$ denote the index of $\Gamma'$ in $\Gamma$. Choose arbitrary generic $\zeta\in Z(\C[\Gamma'])\oplus Z(\R[\Gamma'])$. Let also $W$ be an arbitrary representation of $\Gamma$, let $W'$ be the restriction of $W$ to $\Gamma'$ and let $V'$ be an arbitrary representation of $\Gamma'$. Let $\bv'$, $\bw'$ and $\bw$ denote the classes of $V'$, $W'$ and $W$ in the Grothendieck ring of $\Gamma'$ and $\Gamma$ respectively (i.e. their dimension vectors). Recall that by Lemma \[gqv\] the quiver variety $\FM=\FM^{\Gamma'}_\zeta(\bv',\bw')$ coincides with the set of all triples $$(B,i,j)\in\Hom_{\Gamma'}(V'\otimes L,V')\oplus \Hom_{\Gamma'}(W',V')\oplus\Hom_{\Gamma'}(V',W')$$ such that $$[B,B]+ij=-\zeta_\C, \qquad [B,B^\dagger]+ii^\dagger-j^\dagger j=-\zeta_\R$$ and modulo action of $U_{\Gamma'}(V')$. We define for every $\gamma\in\Gamma$ another triple $$B^\gamma = B \cdot (1\otimes\gamma) : V\otimes L @>{1\otimes\gamma}>> V\otimes L @>{B}>> V,\quad i^\gamma = i\cdot\gamma,\quad j^\gamma = \gamma^{-1}\cdot j.$$ Since $\Gamma$ commutes with $\Gamma'$, it follows that $(B^\gamma,i^\gamma,j^\gamma)$ gives another point of $\FM$, hence the correspondence $B\mapsto B^\gamma$ defines an action of the group $\Gamma$ on the quiver variety $\FM$. Let $\FM^\Gamma$ denote the fixed points set of $\Gamma$ on $\FM$. Take arbitrary $(B,i,j)\in\FM^\Gamma$. Then $(B,i,j)$ and $(B^\gamma,i^\gamma,j^\gamma)$ should be conjugated under the action of $U_{\Gamma'}(V')$. Hence there exists $g_\gamma\in U_{\Gamma'}(V')$ such that $$\label{gg} g_\gamma B g_\gamma^{-1} = B^\gamma,\quad g_\gamma i = i^\gamma,\quad j g_\gamma^{-1} = j^\gamma.$$ Moreover, when $\zeta$ is generic such $g_\gamma$ is unique (because the action of $U_{\Gamma'}(V')$ is free in this case). It follows that $\gamma\mapsto g_\gamma$ defines an action of $\Gamma$ on $V'$ extending the action of $\Gamma'$. Now let $V$ be an arbitrary representation of $\Gamma$, such that its restriction to $\Gamma'$ is isomorphic to $V'$. Let $\bv$ be its class in the Grothendieck ring of $\Gamma$. Let $\FM^\Gamma_\bv$ denote the locus of the set $\FM^\Gamma$, where the defined above structure of a representation of $\Gamma$ on $V'$ is isomorphic to $V$. Then we have. \[gen\] For any generic $\zeta\in Z(\C[\Gamma'])\oplus Z(\R[\Gamma'])$, $\bv,\bw\in K_0(\Gamma)$ let $\bv'=\bv_{|\Gamma'}$, $\bw'=\bw_{|\Gamma'}$. Then we have $$\left(\FM^{\Gamma'}_\zeta(\bv',\bw')\right)^\Gamma_\bv = \FM^\Gamma_{\sigma^*\zeta}(\bv,\bw).$$ [*Proof:*]{} Note that the equations (\[gg\]) mean that the triple $(B,i,j)$ is $\Gamma$-equivariant with respect to the action $\gamma\to g_\gamma$ of the group $\Gamma$ on $V$ and its canonical actions on $W$ and $L$. Choose an arbitrary $\Gamma$-equivariant isomorphism $V'\to V$. Then $$(B,i,j)\in\Hom_\Gamma(V\otimes L,V)\oplus \Hom_\Gamma(W,V)\oplus\Hom_\Gamma(V,W).$$ Furthermore, it follows from the definition of $\sigma^*$ that $$[B,B]+ij=-\sigma_\C^*\zeta_\C, \qquad [B,B^\dagger]+ii^\dagger-j^\dagger j=-\sigma_\R^*\zeta_\R.$$ Thus we obtain a map $$\left(\FM^{\Gamma'}_\zeta(\bv',\bw')\right)^\Gamma_\bv \to \FM^\Gamma_{\sigma^*\zeta}(\bv,\bw).$$ Similarly, for any $(B,i,j)\in\FM^\Gamma_{\sigma^*\zeta}(\bv,\bw)$ forgetting the $\Gamma$-structure on $V$ and $W$ we can consider it as a point of $\FM^{\Gamma'}_\zeta(\bv',\bw')$. Thus we obtain the inverse map. Consider the case $\Gamma'=\{1\}$, $W=R_0$, $W'=\C$, $V'=\C^{nN}$. Then the quiver variety $\FM^{\Gamma'}_\zeta(\bv',\bw')$ coincides with the Hilbert scheme $X^{[nN]}$ when $\zeta_\C=0$ and with the so-called Calogero-Moser space $\CM_{nN}$ when $\zeta_\C\ne0$. Thus in the case $\zeta_\C=0$, Theorem \[gen\] specializes to Theorem \[mplus\], and in the case $\zeta_\C=\tau\ne0$ we obtain the following. For any $\tau\ne0$ and $n\ge0$ we have $$(\CM_{nN})^\Gamma_{n\bv^0} = \FM^\Gamma_\zeta(n\bv^0,\bw^0),$$ where $\zeta_\C=(\tau,\dots,\tau)$ and $\zeta_\R$ is arbitrary. Combinatorial applications ========================== From now on assume that $\Gamma\cong\Z/d\Z$ (the $\widetilde{A}_{d-1}$-case). Then both $X$ and $X_\Gamma$ admit a $\Gamma$-equivariant action of the torus $\C^*\times\C^*$. The first action is a coordinatewise dilation, and the second is induced by the first one. These actions induce actions on the Hilbert schemes $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$. In both cases, there are only finite number of fixed points. Now we will give their combinatorial description. Let us consider a Young diagram as a domain in the top right-hand octant of the coordinate plane. We associate to a box of a diagram the coordinates of its bottom left-hand corner. Thus the coordinates of any box are nonnegative integers. We denote by $(p,q)$ the box with the coordinates $(p,q)$. A Young diagram $\Delta$ is called [*uniformly coloured in $d$ colours*]{} if the integer $$n_i(\Delta) = \#\{(p,q)\in\Delta\ |\ p-q\equiv i\pmod{d}\}, \qquad0\le i\le d-1$$ (the number of boxes of the colour $i$) doesn’t depend on $i$. One can say that we colour the diagonals of the digram $\Delta$ periodically in $d$ colours and call the diagram uniformly coloured if it has equal number of boxes of each colour. Let $UCY(n,d)$ denote the number of uniformly coloured in $d$ colours Young diagrams with $n$ boxes of each colour. The $\C^*\times\C^*$-fixed points on the $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ are in a $1-1$ correspondence with uniformely coloured in $d$ colours Young diagrams with $n$ boxes of each colour. [*Proof:*]{} Note that by definition $X^{\Gamma[n]}$ is a connected component of $(X^{[dn]})^\Gamma$. The fixed points on $X^{[dn]}$ are numbered by Young diagrams with $dn$ boxes. All these points are $\Gamma$-invariant (because the action of $\Gamma$ factors through the torus action), so it remains to understand, which of these points lie in the component. Note that the fiber of the tautological bundle over $X^{[dn]}$ at the $\C^*\times\C^*$-fixed point corresponding to a Young diagram $\Delta$ is isomorphic as a $\Gamma$-module to the representation $\oplus_{(p,q)\in\Delta} R_{p-q}$. In particular, it is a multiplicity of the regular representation iff $\Delta$ is uniformly coloured in $d$ colours. Let $CY(n,d)$ be the number of all ordered collections $(\Delta_1,\dots,\Delta_d)$ of Young diagrams such that $$\sum_{k=1}^d |\Delta_k|=n,$$ where $|\Delta|$ is the number of boxes in $\Delta$. The $\C^*\times\C^*$-fixed points on the Hilbert scheme $X_\Gamma^{[n]}$ are in a $1-1$ correspondence with ordered collections $(\Delta_1,\dots,\Delta_d)$ of Young diagrams, such that $\sum_{k=1}^d|\Delta_k|=n$. [*Proof:*]{} First note that the number of $\C^*\times\C^*$ fixed points on $X_\Gamma$ equals $d$. It follows for example, from the evident fact, that $X_\Gamma=X^{\Gamma,1}$, and $UCY(1,d)=d$. Further, denoting these points by $x_1,\dots, x_d$ it is easy to see that $Z$ is a fixed point on $X_\Gamma^{[n]}$ iff it splits as a union $Z=Z_1\cup\dots\cup Z_d$, where $Z_i$ is a $\C^*\times\C^*$-invariant length $n_i$-subscheme in $X_\Gamma$ with support at $x_i$. Linearizing the action of the torus in a neighbourhood of $x_i\in X_\Gamma$ we see that such $Z_i$ are numbered by Young diagrams $\Delta_i$ with $|\Delta_i|=n_i$. Finally, the condition $Z\in X_\Gamma^{[n]}$ is just $\sum_{i=1}^dn_i=n$, hence the Lemma. For any $n,d > 0$ we have $UCY(n,d) = CY(n,d)$. See [@JK] for the combinatorial proof of this identity. [*Proof:*]{} Using the arguments of Nakajima (see [@Na2], Chapter 5) one can check that the dimension of the cohomolgy groups of the Hilbert schemes $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are equal to the number of the fixed points with respect to the torus action. Thus $$H^\bullet(X^{\Gamma[n]},\Z) \cong \Z^{UCY(n,d)},\qquad H^\bullet(X_\Gamma^{[n]},\Z) \cong \Z^{CY(n,d)}$$ and the Theorem follows from Corollary \[cor\]. [XXX]{} V. Baranovsky, V. Ginzburg, A. Kuznetsov, [*Quiver varieties and a noncommutative ${\mathbb P}^2$*]{}, to appear in Comp. Math. W. Crawley-Boevey, [*Geometry of the moment map for representations of quivers*]{}. W. Crawley-Boevey, M. Holland, [*Noncommutative defformations of Kleinian singularities*]{}, Duke Math. J. [**92**]{} (1998), 605–635. M. Haiman, [*Hilbert schemes as quiver varieties*]{}, http://www.math.berkeley.edu/ hausel/quiver/seminar10.html G. James, A. Kerber, [*The representation theory of the symmetric group*]{}, Encyclopedia of Mathematics and its applications, [**16**]{} (1981). M. Kapranov, E. Vasserot, [*Kleinian singularities, derived categories and Hall algebras*]{}, Math. Ann. [**316**]{} (2000), 565–576. A.D. King, [*Moduli of representations of finite dimensional algebras*]{}, Q. J. Math., Oxf. II. Ser. 45, [**180**]{} (1994), 515–530. P. Kronheimer, [*The construction of ALE spaces as a hyper-Kähler quotients*]{}, J. Diff. Geom. [**29**]{} (1989), 665–683. H. Nakajima, [*Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras*]{}, Duke Math. J. [**76**]{} (1994), 365–416. H. Nakajima, [*Lectures on Hilbert schemes of points on surfaces*]{}. P. Seidel, R. Thomas, [*Braid group actions on derived categories of coherent sheaves*]{}, math.AG/0001043. M. Varagnolo, E. Vasserot, [*On the K-theory of the cyclic quiver variety*]{}, Internat. Math. Res. Notices, [**18**]{} (1999), 1005–1028. W. Wang, [*Algebraic structures behind Hilbert schemes and wreath products*]{}, math.QA/0011103. [^1]: I was partially supported by RFFI grants 99-01-01144 and 99-01-01204.

--- abstract: | Using multi-wavelength imaging observations, in EUV, white light and radio, and radio spectral data over a large frequency range, we analyzed the triggering and development of a complex eruptive event. This one includes two components, an eruptive jet and a CME which interact during more than 30 min, and can be considered as physically linked. This was an unusual event. The jet is generated above a typical complex magnetic configuration which has been investigated in many former studies related to the build-up of eruptive jets; this configuration includes fan-field lines originating from a corona null point above a parasitic polarity, which is embedded in one polarity region of large Active Region (AR). The initiation and development of the CME, observed first in EUV, does not show usual signatures. In this case, the eruptive jet is the main actor of this event. The CME appears first as a simple loop system which becomes destabilized by magnetic reconnection between the outer part of the jet and the ambient medium. The progression of the CME is closely associated with the occurrence of two successive types II bursts from distinct origin. An important part of this study is the first radio type II burst for which the joint spectral and imaging observations allowed: i) to follow, step by step, the evolution of the spectrum and of the trajectory of the radio burst, in relationship with the CME evolution; ii) to obtain, without introducing an electronic density model, the B-field and the Alfvén speed. author: - 'P. Zucca $^{1, 2}$, M. Pick $^{2}$, P. D[é]{}moulin$^{2}$, A. Kerdraon$^{2}$, A. Lecacheux $^{2}$,P.T. Gallagher $^{1}$' title: Understanding CME and associated shocks in the solar corona by merging multi wavelength observations --- Introduction {#Sect:Introduction} ============ Coronal mass ejections (CME) are large-scale energetic events associated with various manifestations of solar activity (e.g., flares, eruptive prominences, shocks). The correlation between the kinematics of the CMEs with these different forms of solar activity has been, for several decades, a major tool to shed light into the physical mechanisms of CME development. CMEs have been frequently observed in white light coronagraph images as having a so called three-part structure, consisting of a bright rim surrounding a dark void which contains a bright core [@Illing1985]. The SOHO/LASCO and more recently STEREO observations showed that CMEs are consistent with a two-dimensional projection of a three-dimensional magnetic flux rope [@Chen1997; @Chen2000; @Thernisien2006; @Thernisien2009]. The authors concluded that the cavity, seen in white light, can be interpreted as the cross section of an expanded flux rope. @Vourlidas2013 gave arguments implying that at least 40% of the observed CMEs have flux-rope structures. In recent years, new prominent results on CME initiation mechanisms and their early development in the low corona have arisen from EUV observations with the EUV Imager of the STEREO/SECCHI telescope [EUVI; @Wuelser2004] and from the Atmospheric Imaging Assembly on board the *Solar Dynamic Observatory* [$SDO$/AIA; @Lemen2012]. @Patsourakos2010 showed that the CME formation is first characterized by slow, self-similar, expansion of slowly-rising loops, possibly triggered by a rising filament, that leads to the formation of a bubble-shaped structure within about 2 minutes. This is consistent with the transformation, by magnetic reconnection, of loops into a flux rope structure as predicted by several models [e.g., @Lynch2008]. The AIA multi-temperature observations have given access to detailed description of a CME namely: i) the ejection of a plasma blob transforming rapidly into a growing hot flux rope that stretches the upper field lines; ii) the appearance of a Y-type magnetic configuration at the bottom of the flux-rope, in which a bright thin line (i.e., a Current Sheet, CS) extends downward; and iii) the shrinkage of magnetic field lines observed underneath the CME [@Cheng2011; @Cheng2013]. All these above observations are consistent with the CME eruption model proposed by @Lin2004. This model is based on a flux rope magnetic configuration overlying a photospheric polarity inversion line. This flux rope becomes unstable and erupts building up behind a CS, which convert the surrounding B-field in a new poloidal around the flux rope. In radio, the formation and development of reconnecting CS behind an erupting flux rope was imaged by the Nançay radio-heliograph [@Pick2005; @Huang2011; @Demoulin2012]. CMEs are frequently associated with type II radio bursts which are a signature of a shock formation and propagation in the corona at speeds higher than those of the Alfvén speed. These bursts are generated by shocks exciting Langmuir waves which decay into radio waves at the local plasma frequency and/or its harmonics [see e.g., @Melrose1980]. A long debate on the physical mechanisms which generate these shocks is still ongoing [see, e.g., @Vrsnak2008; @Vasanth2011]. @Nindos2011 has led to the conclusion that coronal shocks may be generated by two different mechanisms: blast-waves initiated by the plasma pressure of a flare, and piston driven shocks due to CMEs. Several statistical studies on the association of CMEs with type II radio bursts can be found in the literature [see, e.g., @Gopalswamy2009]. @Ramesh2012 have found that 92% of the type II bursts observed at 109 MHz are associated with CMEs and are located near their leading edge. However, the sources of the coronal type II were often found to be located not in front but on the flanks of CMEs, [see, e.g., @Classen2002; @Cho2007; @Demoulin2012; @Zucca2014]. Coronal type II bursts were also often observed jointly with the occurrence of EUV waves, which are large-scale, bright, wave-like disturbances visible in EUV. Several authors have recently taken advantage of the high cadence observations and of the simultaneous dual (or sometimes triple) view-points obtained with STEREO/EUVI, $SDO$/AIA and PROBAB2/SWAP [@Berghmans2006] instruments to study the association between CMEs and EUV waves [see, e.g., @Wang00; @Gallagher2011]. For instance, @Veronig2010 found that the development of an EUV wave exhibits two phases: a first phase consistent with a wave driven by the expanding flanks of the CME [e.g., @Carley2013], and a second one where the wave propagates freely. However, the physical nature of these waves and their association with type II bursts is still unclear and no single model can account for the large variety of EUV waves observed [@Warmuth2010; @Zhukov2011]. To understand the nature of the shock and its association with CMEs, EUV waves, and flares, detailed studies of the complex morphology present in radio burst spectra are required. This complex morphology shows up, for instance, under the form of a splitting of the emission bands into two lanes [@Smerd1974; @Vrsnak2001], or a fragmentation and an abrupt change of their drift rates [see, e.g., @Pohjolainen2008; @Kong2012]. These various morphologies are related to the characteristics of the eruption and to the properties of the surrounding corona in which the shock is propagating. In particular, the electron density and the magnetic field characterize the ambient medium, which then determine the Alfvén speed. This characteristic speed is important for the formation of a shock and for the conditions under which the radio burst can be initiated. Furthermore, both the coronal density and the magnetic field configuration are crucial to determine the radio burst frequency drift and its duration. While numerous studies have been realized on the origin of the shocks and their association with CMEs, there are in fact a very small number of cases for which it has been possible to study such events simultaneously through radio spectra obtained on a large frequency scale, and through radio and EUV images obtained with a high enough time resolution to follow their evolution in detail. To contribute to the understanding of when and where the CMEs and coronal shocks are produced and how they relate to type II bursts and EUV waves, we study here the complex morphological spectral features of a radio event observed on 06 November 2013 together with imaging EUV and radio observations. The November 06 event includes two components, an eruptive jet and a CME, which interact during more than 30 min, and can be considered as physically linked. The magnetic configuration, in which eruptive jets are produced, has already been studied with several magneto-hydrodynamic (MHD) numerical simulations [e.g. @Pariat2009; @Pariat2010]. Conversely, eruptive events, such as the 06 November one which was accompanied by a CME, are not frequently described in the literature. The results of the data analysis presented here take advantage from particularly favorable conditions: i) same field of view on EUV and radio instruments; ii) joint radio spectral (0.5-1000 MHz) and multi-frequency imaging observations (150-450 MHz) at high cadence (better than 1s) and with an high sensitivity; iii) a broad-band frequency spectrogram obtained by the combination of different spectrographs. We identify step by step the causes of the type II spectral fragmentation in relationship with the CME evolution and the ambient medium. We obtain for each step, without introducing an electronic density model or a MHD simulation, the upstream plasma density, the Alfvén Mach number for the shock and the magnetic strength. The end of this type II burst is followed, several minutes later, by a second type II burst of shorter duration. In the absence of imaging observations, the spectral versus time evolution would have led us to conclude to a reactivation of the original type II burst. We will show that this is not the case. The paper is organized as follows: Section \[Sect:Joint\] provides first a description of the observations, and then we present the data analysis which mainly includes: i) a brief overview of the radio event properties; ii) the magnetic configuration of the active region and of its environment; iii a detailed joint EUV and radio analysis of the CME and of the associated type II bursts. Section \[Sect:Shock\] presents the method through which the observations of the first radio type II burst leads to an estimation of physical parameters such as the Alfvén velocity, the density and the magnetic field of the ambient medium. In Section \[Sect:Discussion\], we discuss what we learnt on: i) the rôle of the eruptive jet and of the ambient medium for setting up the CME; ii) the nature of the two shocks, associated with two radio type II bursts, which occur during the CME progression. The main findings are summarized in Section \[Sect:Conclusions\]. Joint evolution of the EUV, white light, and radio emissions {#Sect:Joint} ============================================================ Observations {#Sect:Observations} ------------ A GOES M3.8 class flare started on 2013 November 06 at 13:39 UT in the active region (AR) NOAA 11890 (S12 E35). The flare maximum occurred at 13:46 UT and an associated coronal mass ejection (CME) was observed at low altitude by $SDO$/AIA starting to rise at $\sim$13:44:00 UT. It was also later observed at $\sim$14:36 UT with the Large Angle and Spectrometric Coronagraph on board the *Solar and Heliospheric Observatory* [$SOHO$/LASCO; @Brueckner1995]. An $SDO$/HMI magnetogram and a $SDO$/AIA EUV image at 94 Å are displayed in Figure \[Figure1\]; NOAA 11890 was located on the eastern side of an extended coronal hole. Spectral radio observations were obtained with different instruments (Figure \[Figure2\]): the radio spectrograph ORFEES (Observation Radio Frequence pour l’Etude des Eruptions Solaires) which is a new radio-spectrograph located in Nançay and observing between 140 and 1000 MHz, the e-Callisto spectrograph at the Rosse Solar-Terrestrial observatory [RSTO; @Zucca2012], the Decametric Array in Nançay [DAM; @Lecacheux2000] observing between 70 and 30 MHz, and the Wind WAVES spectrograph [@Bougeret1995] observing between 13.825 and 1.075 MHz. Radio Imaging was obtained with the Nançay radio-heliograph [NRH; @Kerdraon1997] which observed at 9 different frequencies between 445 and 150 MHz on November 06, 2013. Overview of the radio event {#Sect:Overview} --------------------------- During the hours preceding the onset of the event, the main activity, in the eastern hemisphere, consisted in a noise storm [@Elgaroy1976] which was observed in the whole frequency range of the NRH. This noise storm is located south east of AR 11890, and has a negative circular polarization. We will name hereafter negative, respectively positive polarization, the polarization of the ordinary mode in a negative, entering in the photosphere (respectively positive, going out the photosphere) magnetic field. For the present noise storm, which is supposed to be emitted in the ordinary mode, and is located in large scale negative magnetic fields, the negative polarization is what is expected [@Elgaroy1976] . We shall also notice that type II and type III radio bursts are expected to have the polarization of the ordinary mode. An overview of the development of the radio event is shown in Figure \[Figure2\], which displays a synthetic spectrum of the event obtained by combining the data from the different spectrographs. This first radio emission is a group of decimetric (dm) type III bursts starting at 13:42:58 UT. These bursts are observed only at frequencies higher than 100 MHz and they end at 13:43:57 UT. They are followed by interplanetary (IP) type III bursts, starting around 70 MHz, approximately at the time when the CME observed by $SDO$ starts to rise. The red dashed line in Figure \[Figure2\] marks the transition time between the dm and IP type III bursts. Two other groups of dm type III bursts are recorded later. The first group of IP type III bursts is followed at 13:45:59 UT by the onset of a type II radio burst. This burst shows both the fundamental (F) and harmonic (H) emission and also a band splitting particularly visible in the harmonic emission. The F and H emissions fade in the spectrum at respectively 13:49:00 UT and 13:51:00 UT and are observed to start again at $\sim$13:55:30 UT, respectively at $\sim$45 MHz (F) and $\sim$80 MHz (H). They end at $\sim$14:02 UT, in coincidence with the occurrence of a second group of IP type III bursts (see Figure \[Figure2\]). A last group of IP type III bursts is observed at $\sim$14:10:30 UT. Event Movie time number --------------- ------------------- ---------------- -------- SDO/AIA 171 Å Direct 13:30-14:20 UT 1 SDO/AIA 171 Å Run-Diff. 13:40-14:20 UT 2 SDO/AIA 193 Å Run-Diff. and NRH 13:45-13:52 UT 3 SDO/AIA 193 Å Run-Diff. 13:30-14:30 UT 4 SDO/AIA 131 Å Run-Diff. 13:30-14:30 UT 5 : List of available movies[]{data-label="Table:movies"} Magnetic configuration of the active region and of its environment ------------------------------------------------------------------ To understand the successive phases of the eruption, it is necessary to determine first the magnetic configuration of this AR and of its environment; we deduce it from the $SDO$/HMI and $SDO$/AIA 171 Å. In Figure \[Figure3\], the $SDO$/HMI magnetogram is shown (Panel a) together with the $SDO$/AIA EUV image at 171 Å (Panel b). Inside the eastern trailing region of negative polarity, we note the presence of an embedded small positive and parasitic polarity. Using a potential field extrapolation and a three-dimensional MHD numerical simulation, @Masson2009 showed, in a similar case, that the active region includes fan-field lines originating from a coronal null point. This structure has the shape of a dome with the null point at its top. @Masson2009 linked the circular shape of the observed flare ribbons with the photospheric mapping of the fan-field lines. The ribbon brightening would be due to the chromospheric impact of the particles accelerated near the null point by reconnection between the field lines located just below and above the fan. By analogy with the Masson et al. event, in our case, the coronal structure is evidenced in Figure \[Figure3\]a and Figure \[Figure3\]b by, a blue dashed circle and by the red short field lines. Figure \[Figure3\]c displays the $SDO$/HMI magnetogram together with the potential-field source-surface (PFSS) extrapolation [@Schatten1969] using the software of @Schrijver2003[^1]. We also note the presence of more than one embedded (parasitic) polarity observed in the trailing spots; the structure might then be more complex than the one described by @Masson2009. A coronal loop system is also plotted in Figure \[Figure3\]c and Figure \[Figure3\]d. It is anchored in two regions of opposite polarity A and B, south-east of the active region. EUV eruptive jet and type III bursts {#Sect:EUV_type_III} ------------------------------------ The eruption took place above the active region (AR) NOAA 11890 which was classified as a $\beta\gamma\delta$ region. The eruptive jet is first detected at $\sim$ 13:41:48 UT, in the 171 Å  channel of SDO/AIA, as a thin ascending structure. A sudden brightening appears at its basis at 13:42:12 UT, resulting in an increased lateral size. The jet then shows a more complex shape with different branches visible at 13:44:37 UT, 13:47:02 UT and later (see Movie 1). A brightening, visible on running differences (see Movie 2) appears above the main body of the jet at 13:45:36. The jet lasts until 14:10 UT, beginning to turn and move downward after 14:00 UT, pointing toward a belt of positive magnetic polarities located south of AR 11890. ### Decimeter (dm) type III burst groups The sources of the three dm type III burst groups were imaged every second at different frequencies by the NRH. All of them have *positive polarization*. Such polarization means that they are emitted along magnetic field lines emerging from the photosphere. Their locations, measured respectively at 13:43:06 UT, 13:48:37 UT and 13:51:12 UT, are reported, for each group, in the three direct and running difference images of SDO/AIA at 171 Å  displayed in Figure \[Figure4\]a. For the two last groups, because of the progression of a type II burst, simultaneously detected by the NRH (Figure \[Figure2\]), the source locations could be determined only at frequencies higher than 150 MHz. These locations, which are nearly identical for the three groups, trace the path followed by the electron beams responsible for the type III radio emission (see also Figure \[Figure3\]c). The three groups have starting frequencies higher than 1 GHz, which implies an acceleration region located low in the corona. The first group occurs soon after the sudden broadening near the base of the jet (see Figure \[Figure4\]a), and at the time when a EUV bright and compact EUV source starts to be observed above the region of positive polarity near B (see the arrow). Some elongated thin brightenings are also observed in the same region (see in particular Movie 1 at 171 Å  and Movie 5 at 131Å). Soon after, a bright “like-bridge shape” appears between this bright source and the western part of the jet. It is particularly clear in the run-difference images at 131 Å. This observation suggests that this bridge and the EUV bright source result from the reconnection between the western side of the eruptive jet and field lines anchored in this region of positive polarity. A few weaker bright points also appear near this main EUV source, which suggests that other field lines undergo a similar reconnection process. This process will also accelerate the electrons responsible for the radio type III bursts. This is consistent with the trajectory of the electron beams (revealed by the positions of the radio sources) which lies precisely above the EUV bright source. Furthermore the sign of polarization of these type III implies that they originate above a region of positive polarity which is the case. However, we note that no field lines above this region are observed in Figure \[Figure3\]c on the PFSS map. ### The first interplanetary type III bursts Two successive interplanetary groups of IP type III bursts started respectively at 13:43:57 UT and 13:48:20 UT. The first group took place shortly after the dm-type III bursts. The spectrum displayed in Figure \[Figure2\] shows that both IP groups are detected at frequencies below $\sim$70 MHz with the DAM spectrograph. The first one was detected by the NRH at 173 MHz and 150 MHz (but the second one was not). Its emission measured at 150 MHz was not polarized. The two upper Panels (a) and (b) of Figure \[Figure5\] show that the respective locations, at 150 MHz, of the dm and IP type III bursts are different. The IP burst location, which is superposed on an HMI magnetogram in Panel (c) is quite close to the noise storm position, above the A-B coronal loop system (defined in Figure \[Figure3\]c and d). Moreover, the running-difference image at 193 Å (see Figure \[Figure5\]d) shows that these bursts are also concomitant with a sudden brightening, identified in the $SDO$ image at 13:45 UT, which appears on the east side of the eruptive jet, near the B foot of the A-B coronal region and persists until at least 13:48 UT. Open field lines, identified in the magnetic field line extrapolation are present nearby this brightening. Most of them originate in A. This set of observations leads us to propose that the electron beams, which produce the IP type III emission, result from the reconnection of these open field lines with the magnetic field structure of jet. CME and type II radio burst {#Sect:CME_type_II} --------------------------- The onset of the CME rising loops at 13:44 UT (see Movie 2) was followed, soon after, by a type II burst starting at 13:45:59 UT. In this section, we investigate the relationship between the type II progression and the CME evolution. An expanded view of the type II burst is presented in Figure \[Figure6\]. This burst exhibits the (F) fundamental and (H) harmonic emission bands. The H band is clearly splitted in two parallel lanes. Following the interpretation proposed originally by @Smerd1974, these two lanes are a consequence of the plasma emission of the upstream and downstream shock regions. The H component is fragmented in four main segments, highlighted by red dashed lines in Figure \[Figure6\]. The positions of the upstream type II source, measured by the NRH at different times and frequencies, are superposed in Figure \[Figure7\] on $SDO$/AIA running difference images at 193 Å. These images display the progression of the CME. The diamond symbol indicates the position of the upstream radio source, while the location of the CME leading edge (LE) is indicated by a dashed line (see the figure caption for the color code of frequencies). We note the following sequences in the radio spectrum of the type II burst: a\) The initial position of the upstream type II burst measured at 13:45:59 UT is located above the LE of the CME (Figure \[Figure7\]a). Both the CME LE and the upstream radio sources propagate in the south direction. The distance between the type II source and the CME LE is increasing; that indicates that the shock source is moving faster than the LE. b\) At 13:47:06 UT, the dynamic spectra (Figure \[Figure6\]) shows an abrupt change in the drift rate of the type II burst, passing from 1.25 to 0.55 MHz/s (*end of the first segment*). Between $\sim$ 13:47:00 and $\sim$ 13:48:36 UT, the type II source stops its southward progression, being westward deviated (Figure \[Figure7\]b,c and e). During the same period of time, the shape and the orientation of the CME LE are also modified (Figure \[Figure7\]c, Movie 4). The CME LE is now southward elongated and slightly westward oriented. We thus conclude that the change in the frequency drift and in the trajectory of the type II burst coincides with the change in the orientation of the CME. Moreover, these modifications occur during the same period as the bright “like-bridge” which connects the eruptive jet with the B region. We thus conclude that the change in the type II burst trajectory and in the orientation of the CME result from their approach from the eruptive jet (see Figure \[Figure3\]d) which strongly affects their development. (*end of the second segment*). c\) After 13:48:20 UT, the type II burst returns again at a drift rate of 1.1 MHz/s (*start of the third segment*). There is another change of its drift rate at $\sim$ 13:49 UT (*start of the fourth segment*). It becomes $\sim$ 0.2 MHz/s until $\sim$ 13:51:00 UT. At this time, the type II emission fades and disappears from the spectrogram, marking the end of *the fourth segment* (Figure \[Figure7\]d). d\) While the CME continues its southward progression, its lateral expansion is limited on one side by the eruptive jet and, on the other side by the neighboring CH (Figure \[Figure8\]). Its western edge becomes slightly westward deviated after 13:49 UT; however, the base-difference images displayed in Figure \[Figure8\] show that its lateral expansion seems to be limited by the pressure generated by the neighboring CH and stops its westward progression at $\sim$ 13:48:50 UT. *The second type II burst* Type II burst emission reappears at 13:55:30 UT at a frequency below 90 MHz (Figure \[Figure2\]), at the same time as the development of a dark feature near the western edge of the CME (see Figure \[Figure8\] and Movie 2). We suggest that the dark feature and the type II burst have a common origin attributed to the pressure exercised by the CH on the CME edge; it was indeed shown that, the build-up of such a compression region can be accompanied by compression waves, or shocks detected in EUV and white light images [@Vourlidas2003; @Yan2006]. This last assumption is consistent with the western compressed shape of the CME edge observed later, at 16:48:06 UT, in LASCO-C2 coronagraph (Figure \[FigureCME\]b). e\) The m-type II burst fades abruptly around 14:02 UT, which is the time when the shock reaches the boundary of the southern coronal hole. The spectrogram in Figure \[Figure2\] shows the onset of an IP type III burst, followed a few minutes later by another group of IP type III bursts. The latter were observed by the NRH at 150 MHz; their locations at this frequency are indicated by two crosses reported in Figure \[Figure8\]d. These bursts are probably due to the interaction between the CME LE (or the jet) and the open magnetic field lines in the polar region. The progression of the SDO CME is later observed by LASCO C2-C3. A distance-time plot of the erupting CME is shown in Figure \[FigureCME\]a. This plot was obtained using the running difference images of the SDO/AIA 193 Å for the range 1-1.6 R$_{\odot}$ and of SOHO/LASCO C2 and C3 for the range 3-8 R$_{\odot}$. A composite running difference image of the CME observed with SDO/AIA at 13:48:23 UT, and at its later expansion stage with LASCO C2 at 16:48:06 UT is shown in Figure \[FigureCME\]b; the western edge of the CME appears to be compressed by the interaction of the CH at the western side of the AR. The initial velocity of the CME, as measured with LASCO C2 at 3 R$_{\odot}$, is of $\sim$400 km s$^{-1}$ and its final velocity at 8 R$_{\odot}$, observed by LASCO C3, is $\sim$280 km s$^{-1}$, with an acceleration of -13.8 m s$^{-2}$. Shock and ambient medium characteristics {#Sect:Shock} ======================================== In this section, we describe the shock properties, relating its kinematics to the kinematics of its driver, and to the ambient medium characteristics. For that we use the well observed splitting of the type II harmonic emission in two parallel lanes (Figure \[Figure6\]) which correspond to the plasma emission of the upstream and downstream shock regions. The upstream and downstream shock regions {#Sect:up_downstream} ----------------------------------------- The NRH observations obtained at the 9 frequencies quoted above allow us to determine the respective location of the upstream and downstream regions. Figure \[FigureUpDownSources\] shows, at three different times, the respective positions of these two components and of the CME LE, which are superposed on $SDO$/AIA running difference images at 193 Å. The CME LE is highlighted with a blue dashed line. Each NRH source position is indicated by a contour (at 90% of the peak flux), the color referring to the selected frequency (see the figure caption). For the three chosen times, 13:46:18 UT, 13:47:06 UT and 13:49:06 UT, it was possible to locate simultaneously the upstream and downstream regions. The three couples of selected points are reported in the spectrum in Figure \[Figure6\], using the same color code as in Figure \[FigureUpDownSources\]. We note that the orientation of the two sources in Panel (b) is different from those of Panels a and c. This observation is consistent with the sudden change of the type II orientation described in Section \[Sect:CME\_type\_II\] (during the segment 2, Figure \[Figure7\]). The timing of the events is shown in the top panel of Figure \[FigureDistance-Time\] with the radio dynamic spectrum. The red dashed lines correspond, successively, to the starting time of the flare, 13:39 UT, the starting and end time of the first group of dm-type III, and to the time period of the second segment (when the type II drift rate is lower). In the bottom panel of Figure \[FigureDistance-Time\] we compare the *projected* distance-time plots of the CME LE and of the NRH sources for the H-low band of the type II burst (blue squares). \[\] This distance is measured along the direction indicated in yellow in the inset. As already seen in Figure \[FigureUpDownSources\], the upstream position is located in front of the CME LE and increases its separation from the LE, as the former travels faster. Speed comparison {#Sect:Speed} ---------------- Figure \[FigureVA\_MA\_B\]a shows a comparison between the projected speeds of the CME-LE (black line) and of the type II NRH sources. A 2D Gaussian fit was applied to obtain the location of each radio source at a given time frequency, i.e. at a given density. The uncertainty on the NRH speed estimation depends exclusively on the uncertainty on the source location; the error bars are plotted at 2$\sigma$. Speeds are here projected speeds. As we do not know the angle between the CME or the type II burst and the plan of the sky, the CME, shock and Alfvén speeds given below may be underestimated by $ \sim$ 30 $\%$. As a consequence, the magnetic fields may be underestimated by the same amount. The projection effect does not affect comparison between the CME and Alfvén speeds, because the two speeds are likely to have the same angle with the plan of sky. The starting speed of the CME is $\sim$ 300 km s$^{-1}$ and reaches a maximum value of $\sim$ 1000 km s$^{-1}$ at $\sim$13:45:50 UT. The speed of the shock up-stream region is initially of $\sim$ 2000 km s$^{-1}$ (*first segment* as defined in Figure \[Figure6\]); it then progressively decreases to $\sim$ 800 km s$^{-1}$ (*second segment*), before increasing again to $\sim$1800 km s$^{-1}$ (*third segment*) while finally reaching the low value $\sim$ 400 km s$^{-1}$ (*fourth segment*) when the type II emission fades out. To establish a comparison between the speeds of the CME LE and of the type II shock with the ambient Alfvén speed, $V_A$, we applied the procedure developed by @Vrsnak2002 for both the quasi perpendicular and the parallel shocks. In the present case, the direction of propagation of the shock, obtained from the locations of the NRH source, is indicated in Figure \[Figure7\]e by the three purple segments labeled by the numbers S1-S2, and S3-4 (the direction is the same for the segments 3 and 4). During the segment S1, the shock propagates through closed loops (see Figure \[Figure3\]a) and the normal to the shock is quasi perpendicular to the direction of the magnetic field. In the cases of S2 and S3-S4, the coronal extrapolation is too complex to establish a definitive estimate. Thus, we present here the results obtained for both quasi parallel and perpendicular cases. The procedure developed by @Vrsnak2002 shows that one can obtain the Mach number $M_A$ then the Alfvén speed, if the compression ratio of the shock is determined. This ratio, *X*, is given by $X=(f_u/f_l)^2$, where $f_u$ and $f_l$ are the frequencies of the upper and lower bands respectively. For a low plasma parameter $\beta\rightarrow 0$, the Mach number can be written as $M_A\approx(X(X+5)/2(4-X))^{0.5}$ for a quasi perpendicular shock. As the speed of the shock, $V_{shock}$ is known from the NRH source, the ambient Alfvén speed can then be calculated from $V_A=V_{shock}/M_A$. The same procedure can be used for the case of a quasi-parallel shock, in this case the Mach number is given by $M_A=(X)^{0.5}$. The calculated Alfvén speeds for both perpendicular and parallel cases are plotted on Figure \[FigureVA\_MA\_B\]b, green squares for the perpendicular case and violet for the parallel case, together with the CME LE speed. The CME LE reaches a super-Alfvénic speed, for the perpendicular case, at a projected distance of $\sim$1.2 R$_{\odot}$ corresponding to the onset time of the type II shock at $\sim$13:45:59 UT. This agrees with the hypothesis that the shock wave is generated by the CME LE in a piston-driven scenario [@Zimovets2012]. Others ambient medium characteristics {#Sect:Others} ------------------------------------- The Mach number obtained from the compression ratio is plotted in Figure \[FigureVA\_MA\_B\]c. The initial value for the perpendicular case of $\approx 2.1$, decreases to the final Mach number of $\approx1.2$; this is indicative of the shock speed approaching the local Alfvén speed, at this time the type II burst fades and disappears from the dynamic spectra at 13:51 UT. Similar but slightly lower values of the Mach number are found for the parallel shock case. In conclusion, this analysis confirms that the approximation of a quasi perpendicular approach for direction 1 looks reasonable while, no firm conclusion on the type of shock can be given for segments S2 and S3-S4. Still the results of Figure \[FigureVA\_MA\_B\] for those segments are comparable between parallel and perpendicular cases. The strength of the local magnetic field B can be also estimated from the Alfvén speed using the equation: $B=V_A(\mu\rho)^{-0.5}$, where the coronal plasma density is approximately given by $\rho\approx m_p n_e$. The magnetic field values are shown Figure \[FigureVA\_MA\_B\]d for the perpendicular and parallel approaches. We found values for the B-field ranging from 10 Gauss at the beginning of the shock at 13:45:59 UT to 5 Gauss at 1.5 R$_{\odot}$ when the type II burst fades at 13:51 UT. Electron density at the shock location -------------------------------------- The NRH observations at different frequencies allow us to estimate the electron density at the shock location. The only assumption at this stage is that the type II emission is due to harmonic plasma emission. The electron density can be then estimated directly from the emitting frequency $f_p\approx18000 \sqrt{n_e}$. The electron density obtained from the NRH frequency bands at 408, 327, 270, 228, 173 and 150 MHz is plotted against the projected distance of the NRH source in Figure \[Figure\_extra\]. The emission of the upstream component in the low-frequency band (ahead) is plotted with blue diamonds, while the emission of the downstream component (behind) is plotted with red diamonds. This density is compared with the coronal background electron density using a 5xfold (brown dashed lines) and 9xfold Saito (blue dashed lines) density model [@Saito1977]. The position of the radio source for projected distances below $\sim$1.4 R$_{\odot}$ is comparable with a dense 9xfold Saito due to the presence of the active region, while at projected distances over $\sim$1.4 R$_{\odot}$ the density drops and it is comparable with a 5xfold Saito. This is due to the change in direction of the radio burst source as indicated in Figure \[Figure7\]e. The density ‘jump’ from a 9xSaito to a 5xSaito of the radio source is evident from the data (see Figure \[FigureVA\_MA\_B\]). This change in density is due to the transition from the dense plasma in the closed magnetic field topology over the active region to the less dense plasma of the other neighboring closed loops structure. This change in the magnetic field topology and electron density may also indicate the reason of the sudden change of direction of the radio source. An actual estimation of the electron density along the path of the radio source (yellow line on the inset of Figure \[FigureDistance-Time\] can be obtained from the 2-dimensional maps using the method described in @Zucca2014. This method calculates the electron density from emission measure (EM) maps derived using the $SDO$/AIA’s six coronal filters and the method of @Aschwanden2013. The plasma electron density can be calculated from the EM by estimating an effective path length of the emitting plasma along the LOS (see @Zucca2014 for details). The electron density calculated using this method is plotted in Figure \[Figure\_extra\] with a black dashed line. The density profile is compatible with the mean density profile obtained from the NRH and spectral observations. Summary and Discussion {#Sect:Discussion} ====================== Jet and CME: the joint evolution -------------------------------- In the present paper, we have presented an unusual event in which an eruptive jet is involved in the onset of a CME and, then, accompanies its development. First detected in EUV, this event appears as a simple loop system rising in the corona, while it is later identified as a CME when observed, in white light, by SOHO. No EUV plasmoid was detected behind the edge of the CME. The aim of this paper was to understand: i) the role of the eruptive jet and of the ambient medium for setting up this CME; ii) the nature of the two shocks associated with two successive type II bursts, which occur during the CME progression. Our main findings are summarized below. A\) The eruption of the jet marks the beginning of the event; this jet originates from a coronal null point above a positive parasitic polarity embedded inside the trailing negative part of the Active Region (AR). The initiation phase occurs when a sudden brightening appears at its base and extends rapidly toward its western neighboring loops. These large scale loops, which connect the two main polarities of the AR, start to shine along their eastern leg. These observations are indicative of the beginning of a destabilization process of the loops caused by their magnetic interaction with the jet structure. B\) The subsequent eruptive EUV and radio manifestations, preceding or accompanying the onset of the CME, occur in the vicinity of the region where the CME LE is formed. The idea of a magnetic reconnection process between the outer part of the jet and the ambient medium, is confirmed by the presence of radio type III bursts. a\) The first dm type III burst group, which is followed by two other ones, coincides with the appearance of a bright narrow EUV source located above a region of positive polarity. This source persists during several minutes, and subsequent weaker sources are observed in the same region. These observations suggest that the sources, from which the electron beams responsible for the type III radio bursts originate, result from a magnetic reconnection between the western side of the eruptive jet magnetic field and the field lines anchored in this region of positive polarity. This interpretation is consistent with i) the sudden appearance, between these two regions, of a bright bridge, particularly well observed by SDO at high temperature; ii) the positive polarization of the radio emission; iii) the trajectory of the electron beams (Note that the magnetic field lines of positive polarity are not detected by PFFS). The ascending motion of the CME starts soon after the occurrence of the first group of type III bursts. b\) During these dm bursts, two groups of interplanetary type III bursts are also detected. The electron beams producing these bursts result from the interaction between the eastern side of the jet and the open field lines originating from a region of negative polarity. C\) The different following observations seem to stress the role of the erupting jet during the CME progression: a\) Its motion is followed by the onset of the first type II burst. The progressions of the CME and of the burst appear to be closely connected: they follow the same direction and the type-II burst sources are located above the front edge of the CME. Moreover, approximately 1 or 2 minutes later, the source of the type II burst stops its southward motion and becomes westward oriented, while the CME leading edge becomes also slightly westward oriented. We attribute this effect to the encounter of the eastern edge of the CME with the eruptive jet. Let us further remark that: i) its eastern edge becomes no longer discernible from the jet after their encounter; ii) the CME leading edge appears to be split into two parts, its eastern part corresponding in fact to the western branch of the jet, now curved and surrounding the CME. These facts are compatible with the shape of the CME as observed later. b\) While the CME continues its southward progression, its lateral expansion is limited, on one side by the presence of the eruptive jet, and on the other side by the pressure generated by the coronal hole. This interaction is possibly the cause of the second type II burst which, during the same period, appears at decameter wavelengths. We note that, 2 hours later, the expansion of the CME, when observed by LASCO/C2, has remained the same. c\) The last IP bursts, which appear approximately at the time of the approach of the CME with the south pole, probably originate from a magnetic interaction between the CME, or the jet itself, with the open field lines of the polar region. First shock and ambient medium characteristics ---------------------------------------------- In this study, the CME LE and the type II burst kinematics were compared with ambient coronal characteristics such as the Alfvén speed and the B-field, in order to understand the origin of the shock and its progression. These properties were calculated without assuming any model for the coronal density and they were derived from the shock compression ratio; the latter was obtained from the type II split lanes, using a method described in @Vrsnak2002. The CME LE showed a fast initial acceleration, and already reached a super-Alfvénic speed. This was subsequently followed by the production of a type II burst with emission lanes split in two bands. The type II burst also presented a fast initial acceleration leading to a speed faster than the CME LE, so that they progressively separate one from the other. A shock can be a blast wave, in which the energy is supplied by a pressure pulse, or it can be driven by a CME, either in a piston-type or in a bow shock scenario [@Vrsnak2005]. In the case of a piston shock geometry, the shock moves faster than its driving piston and the medium is confined, since it is not able to stream around the CME [@Vrsnak2005; @Warmuth2007]. In our event, as recalled in point C-b above, the lateral expansion of the CME is limited, on one side by the presence of the eruptive jet, and on the other side by the pressure generated by the coronal hole. This confinement, together with the shock propagating faster than the CME LE, strongly suggest that the shock has been driven by the CME in a piston-driven mechanism. Another observation is in line with this interpretation: the type II burst sources are located in front of the CME LE and undergo the same change as the CME in the propagating direction. The radio observational coverage by the NRH allowed us to resolve the location of the split bands of the type II burts. We found that the two components were located ahead of the CME LE and that the higher frequency lane was positioned behind the lower frequency band. This is in agreement with the @Smerd1974 interpretation of the splitting lane emission. In our scenario, the hypothetic shock wave, probably to faint to be detected in EUV, is located between the low and hi band position of the splitted lanes. @Bain2012 and @Zimovets2012 arrived to a similar conclusion, in the study of another dm-metric type II burst which was also imaged by the NRH. Conclusions {#Sect:Conclusions} =========== We have presented the formation and development of an unusual CME as described in Section 4 and summarized below. The CME resulted from the interaction of an eruptive jet with the surrounding medium. The key points are the overall magnetic structure of the ambient medium and the relative position of the jet in this environment. To our knowledge, it is the first time that such an event has been analyzed in some depth. A cluster of eruptive EUV and radio observations, stress the predominant role played by the eruptive jet in the history of this CME: First detected in EUV, this event appears as a simple loop system rising in the corona. These loops start to be destabilized by their magnetic interaction with the jet. This early development of the CME does not show the signatures that could be expected from previous observations (see introduction). Then, a destabilization process of the loops is caused by magnetic reconnection between the outer part of the jet and the ambient magnetic field. This process occurs in the vicinity of the region where the CME LE is formed and when the CME speed is strongly increasing. This is also near this time that the onset of the first type III burst is observed. This is reminiscent of the break-out model [@Antiochos1999] but with reconnection between closed and open magnetic field. The progression of this CME is later observed in white light, up to a distance of 8 solar radii. Two type II bursts were detected. A distinct origin is identified for the two successive shocks, both associated with the CME development. One of the primary finding of this study is related to the first type II burst for which the joint spectral and imaging observations allowed us: - To identify step by step the origin of the spectral fragmentation, in relationship with the CME evolution; - To obtain at each step, without introducing an electronic density model or a MHD simulation, the upstream plasma density, the Alfvénic Mach number for the shock and the magnetic strength. The jet and/or CME are at the origin of interplanetary radio type III bursts; these bursts reveal the injection, in the interplanetary medium, of electron beams along different directions. To conclude, we would like to illustrate, on two specific points, how the data analysis has benefited from particularly favorable conditions: i) Though, the event originated on the solar disk, it was observed by the SOHO/LASCO coronagraph. It allowed us to confirm that this event was a real CME; ii) the polarization measurements of the radio type III bursts was determinant to identify the origin of the dm type three bursts and also showed that electron beams escape along magnetic field lines that were not present in PFSS extrapolation. **Acknowledgements** P. Zucca is supported by a TCD Innovation Bursary and acknowledges the CNES for the financial support he received during his stay in LESIA, Meudon Observatory. We thank E. Pariat for constructive discussions. We would like to thank the referee for the valuable comments and suggestions. We are also grateful to the SDO team for his open data policy. The SOHO LASCO data used here are produced by a consortium of the Naval Research Laboratory (USA), Max-Planck-Institut fur Aeronomie (Germany), Laboratoire d’Astronomie Spatiale (France), and the University of Birmingham (UK). SOHO is a project of international cooperation between ESA and NASA. In France, this work was supported by CNES. 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--- abstract: 'A Kaluza-Klein model, with a matter source associated with Hawking radiation from an evaporating black hole, is used to obtain a simple form for the radion effective potential. The environmental effect generally causes a matter-induced shift of the radion vacuum, resulting in the formation of a radion cloud around the hole. There is an albedo due to the radion cloud, with an energy dependent reflection coefficient that depends upon the size of the extra dimensions and the temperature of the hole.' author: - 'J.R. Morris' title: Radion clouds around evaporating black holes --- Introduction ============ Kaluza-Klein type models involving compactified extra dimensions produce effective 4d theories containing moduli fields (radions) that are associated with scale factors of the compact dimensions. Particle masses typically exhibit a radion dependence, and local matter sources contribute to an effective potential for these scalar fields. If the compactification is *inhomogeneous*, particle masses and charges can have spatial and temporal variations. Soliton-like structures associated with these scalars may result (see, for example, [@DG1; @DG2; @db1; @db2; @db3]), with possibly observable consequences. An extreme condition is considered here, where the matter density from an evaporating microscopic black hole (MBH) may become large enough to give rise to an inhomogeneous compactification, resulting in a radionic cloud around the black hole. The reflective properties of this cloud endow the near horizon region with a radion induced albedo. Scenarios of this type involving a radion coupled to the radiative field around an evaporating black hole have been studied previously in refs. [@GSS] and [@JM]. In [@GSS] rather general conditions were considered, and in [@JM] attention was focused on finite temperature, one-loop quantum corrections due to a partially thermalized medium surrounding an evaporating black hole. The resulting effective potential is quite complicated, and difficult to represent in a simple closed form. Here, however, I take a more classical approach and consider a simple, but explicit, class of potentials $V(\varphi)$ for the radion field $\varphi$. This type of radion potential was studied first by Davidson and Guendelman in [@DG1; @DG2] in the context of a Freund-Rubin compactification, and later by Carroll, Geddes, Hoffman, and Wald (CGHW) in [@CGHW], using an extra dimensional two-form magnetic field. Both treatments result in the same radion potential for the case of two extra dimensions, but the former treatment also holds for an arbitrary number $n$ of extra dimensions. In either case, the resulting potential has a rather simple form for an assumed set of parameter relations. For simplicity and concreteness, I specialize to the case of two extra dimensions, and refer to the radion potential $V(\varphi)$ as the DG-CGHW radion potential, although its generalization for arbitrary $n,$ given by [@DG1; @DG2], is also given below. In addition to functional simplicity, the potential allows the extra dimensions to be stabilized classically, without explicit quantum corrections. Using this simple DG-CGHW model, the full effective radion potential $U(\varphi)$ can be developed which includes a matter-sourced correction due to the Hawking radiation matter field. The matter source contribution to the radion effective potential $U(\varphi)$ depends, in a simple way, upon the local matter density $\rho(\mathbf{r},t)$ and radion mass $m_{\varphi}$. A knowledge of these parameters then allows, in principle, a determination of the spatial and temporal variation of the radion field in the vicinity of the evaporating hole. The matter contribution can induce a shift, or complete destabilization, of the radion vacuum near the horizon, resulting in a radion cloud around the MBH. This cloud has an associated energy-dependent reflectivity, which can result in a distortion of the infrared portion of the Hawking radiation spectrum, as well as a partial reflection of low energy particles incident upon the MBH from the outside. By looking at the forms of the effective potential $U$ near the hole’s horizon and at asymptotic distances, it is suggested that the radion cloud has an evolving size $R$ $\lesssim m_{\varphi}^{-1}$, and a maximal reflection coefficient $\mathcal{R}_{\max}$ that depends upon the matter density $\rho_{hor}$ near the horizon through the ratio $\rho_{hor}/(m_{\varphi}^{2}M_{0}^{2})$, where $M_{0}=1/\sqrt{8\pi G}$ is the reduced Planck mass. Radion effective potential ========================== Effective 4d action ------------------- We start by considering a $D=(4+n)$ dimensional spacetime having $n$ compact extra spatial dimensions endowed with a metric given by $$ds_{D}^{2}=\tilde{g}_{MN}dx^{M}dx^{N}=\tilde{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu }+b^{2}(x^{\mu})\gamma_{mn}(y)dy^{m}dy^{n} \label{e1}$$ where $x^{M}=(x^{\mu},y^{m})$. Here $M,N=0,1,2,3,\cdot\cdot\cdot,D-1$ label all the spacetime coordinates, while $\mu,\nu=0,1,2,3$, label the 4d coordinates, and $m,n$ label those of the compact extra space dimensions. The extra dimensional scale factor $b(x^{\mu})$ is assumed to be independent of the $y$ coordinates and takes the form of a scalar field in the 4d effective theory. The extra dimensional metric $\gamma_{mn}(y)$ depends upon the geometry of the extra dimensional space and is related to $\tilde{g}_{mn}(x,y)$ by $\tilde{g}_{mn}=b^{2}\gamma_{mn}$. As in refs.[@DG1; @DG2; @CGHW], consideration is restricted to extra dimensional compact spaces with constant curvature, with a curvature parameter $k$ defined by$$k=\frac{\tilde{R}[\gamma_{mn}]}{n(n-1)} \label{e1a}$$ The action for the $D$ dimensional theory is$$S_{D}=\int d^{D}x\sqrt{\left\vert \tilde{g}_{D}\right\vert }\left\{ \frac {1}{2\kappa_{D}^{2}}\left[ \tilde{R}_{D}[\tilde{g}_{MN}]-2\Lambda\right] +\mathcal{\tilde{L}}_{D}\right\} \label{e2}$$ where $\tilde{g}_{D}=\det\tilde{g}_{MN}$, $\tilde{R}_{D}$ is the Ricci scalar built from $\tilde{g}_{MN}$, $\Lambda$ is a cosmological constant for the $D$ dimensional spacetime, $\mathcal{\tilde{L}}_{D}$ is a Lagrangian for the fields in the $D$ dimensions, $\kappa_{D}^{2}=8\pi G_{D}=V_{y}\kappa^{2}=V_{y}8\pi G$, where $G(G_{D})$ is the 4d ($D$)-dimensional gravitational constant, and $V_{y}=\int d^{n}y\sqrt{\left\vert \gamma\right\vert }$ is the coordinate volume of the extra dimensional space. A mostly negative metric signature $(+,-,-,\cdot\cdot \cdot,-)$ is used here. The action can be expressed in terms of an effective 4d action (see, for example,[@CGHW; @NDL-JM] for details) which takes the form$$\begin{array} [c]{ll}S & ={\displaystyle\int} d^{4}x\sqrt{-\tilde{g}}\left\{ \dfrac{1}{2\kappa^{2}}[b^{n}\tilde{R}[\tilde{g}_{\mu\nu}]-2nb^{n-1}\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}b-n(n-1)b^{n-2}\tilde{g}^{\mu\nu}(\tilde{\nabla}_{\mu }b)(\tilde{\nabla}_{\nu}b)\right. \\ & \left. +n(n-1)kb^{n-2}]+b^{n}\left[ \mathcal{L}_{D}-\dfrac{\Lambda}{\kappa^{2}}\right] \right\} \end{array} \label{e7}$$ in the 4d Jordan frame (with metric $\tilde{g}_{\mu\nu}$), and I have defined a normalized field Lagrangian, $\mathcal{L}_{D}=V_{y}\mathcal{\tilde{L}}_{D}$. A 4d Einstein frame metric $g_{\mu\nu}$ can be defined:$$\tilde{g}_{\mu\nu}=b^{-n}g_{\mu\nu},\ \ \ \ \ \tilde{g}^{\mu\nu}=b^{n}g^{\mu\nu},\ \ \ \ \ \sqrt{-\tilde{g}}=b^{-2n}\sqrt{-g} \label{e8}$$ The action $S$ in (\[e7\]), in terms of the 4d Einstein metric, takes the form$$\begin{array} [c]{ll}S & ={\displaystyle\int} d^{4}x\sqrt{-g}\left\{ \dfrac{1}{2\kappa^{2}}\left[ R[g_{\mu\nu}]+\dfrac{n(n+2)}{2}b^{-2}g^{\mu\nu}(\nabla_{\mu}b)(\nabla_{\nu}b)+n(n-1)kb^{-(n+2)}\right] \right. \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. +b^{-n}\left[ \mathcal{L}_{D}-\dfrac{\Lambda}{\kappa^{2}}\right] \right\} \end{array} \label{e9}$$ where total derivative terms have been dropped. Furthermore, an effective 4d source, or matter, Lagrangian $\mathcal{L}_{m}$ can be defined in terms of the $D$ dimensional source Lagrangian $\mathcal{L}_{D}=V_{y}\mathcal{\tilde{L}}_{D}$ and the scale factor $b$:$$\mathcal{L}_{m}=b^{-n}\mathcal{L}_{D} \label{e10}$$ A scalar radion field $\varphi$ with a canonical kinetic term is now defined by$$\sqrt{\frac{n(n+2)}{2\kappa^{2}}}\ln\frac{b}{b_{0}}=\varphi,\ \ \ \ b=b_{0}\exp\left( \sqrt{\frac{2}{n(n+2)}}\ \kappa\varphi\right) \label{e11}$$ where $b_{0}$ is some constant, which will be set equal to unity, and $\kappa=\sqrt{8\pi G}=\sqrt{8\pi}/M_{P}=M_{0}^{-1}$ is the inverse of the reduced Planck mass. Radion potential $V$ -------------------- For a simple and concrete example, I use the radion potential studied in refs.[@DG1; @DG2; @CGHW] for the case of $n=2$ extra dimensions having a constant positive curvature parameter $k$. I also adopt the same choices of parameter relations to obtain a simple functional form. The potential $V$ has contributions from the curvature term in (\[e9\]), the cosmological constant term $\Lambda$, and either (1) an extra dimensional magnetic field due to $F_{45}=\sqrt{|\gamma|}F_{0}$, where $F_{0}$ is a constant, in $\mathcal{L}_{D}$ (see[@CGHW]), or (2) a Freund-Rubin term (see [@DG1; @DG2]) of the form$$\mathcal{\tilde{L}}_{D}=-\frac{1}{48}F^{2},\ \ \ F_{MNPQ}=\partial_{\lbrack M}A_{NPQ]},\ \ \ F_{\mu\nu\lambda\sigma}=\sqrt{\lambda}\frac{\sqrt{|\tilde {g}|}}{3b^{n}}\varepsilon_{\mu\nu\lambda\sigma} \label{e11a}$$ The potential $V$ obtained by CGHW for the case of $n=2$ extra dimensions and an extra dimensional magnetic field $F_{45}$ leads to a simple potential (written here in terms of the scale factor $b$, rather than in terms of the radion field $\varphi$) given by$$V(b)=\lambda\left( b^{-6}-2b^{-4}+b^{-2}\right) \ \ \ \ \ \text{(CGHW potential)} \label{e12}$$ with the following relations and parameter choices: (see eqs. (36) and (37) in ref.[@CGHW])$$\varphi=\frac{2}{\kappa}\ln b,\ \ \ b=e^{\frac{1}{2}\kappa\varphi },\ \ \ \ \lambda=\frac{k}{2\kappa^{2}}=\frac{m_{\varphi}^{2}M_{P}^{2}}{16\pi }=\frac{1}{2}m_{\varphi}^{2}M_{0}^{2} \label{e13}$$ where $k=\frac{\partial^{2}V(\varphi)}{\partial\varphi^{2}}\Big|_{\varphi =0}=m_{\varphi}^{2}$ is the mass$^{2}$ of the radion field[@CGHW], $M_{P}$ is the Planck mass, and $M_{0}=1/\kappa=M_{P}/\sqrt{8\pi}$ is the reduced Planck mass. The form of $V$ (see Figure 1; $V$ is also sketched in refs.[@DG1] and [@CGHW]) has one local minimum, which for the case $n=2$ is located at $b=1$ (or $\varphi=0$), with $V(b=1)=0$, followed by a barrier for $b>1$, then an asymptotic decrease with $V\rightarrow0$ as $b\rightarrow\infty$. [f1.eps]{}\ [Figure 1: A plot is shown of ]{}$V(b)/\lambda$ [vs ]{}$b$[. The local minimum occurs at ]{}$b=1$ [(]{}$\varphi=0$[) where ]{}$V=0$[. The local maximum occurs at ]{}$b\approx1.75$ [where ]{}$V/\lambda\approx.15$[. As ]{}$b\rightarrow\infty$[, ]{}$V\rightarrow0$[.]{} For the more general case of $n$ extra dimensions given by [@DG1; @DG2], the Davidson-Guendelman (DG) potential arising from the Freund-Rubin term above is given by (with factors of $\kappa$ reinstated here)$$V(b)=\dfrac{\lambda}{b^{3n}}-\frac{1}{2}\frac{k}{\kappa^{2}}n(n-1)\dfrac {1}{b^{n+2}}+\dfrac{\Lambda/\kappa^{2}}{b^{n}}\text{\ \ \ \ \ (DG potential)} \label{e13a}$$ For the case of $n=2$, requiring the potential to vanish at its minimum leads to the condition$$\Lambda=\frac{k^{2}/\kappa^{2}}{4\lambda} \label{e13b}$$ Imposing the parameter choice $\lambda=k/2\kappa^{2}$ as in (\[e13\]) then gives the same potential as in (\[e12\]) with a local minimum at $b=1$ where $V=0$. For simplicity and concreteness, attention is restricted here to this simple form of the potential $V$  for the particular case of $n=2$ extra dimensions. For our case of $n=2$, the potential $V$, given by (\[e12\]) and (\[e13\]), has a local minimum at $b=1$ ($\varphi=0$) where $V=0$, a local maximum located at $b>1$, and the potential falls off exponentially, $V\rightarrow0$ as $b\rightarrow\infty$. An equation of motion (EoM) is obtained from (\[e9\]) for the radion field $\varphi$,$$\square\varphi+\frac{\partial V}{\partial\varphi}-\left\langle \frac {\partial\mathcal{L}_{m}}{\partial\varphi}\right\rangle =0 \label{e14}$$ where $\mathcal{L}_{m}$ is the matter Lagrangian which depends upon scalar, spinor, and vector matter fields, as well as the radion field. Using field redefinitions, the $\varphi$ dependence of $\mathcal{L}_{m}$ appears in particle masses $m_{A}(\varphi)$ and in gauge coupling constants (see, e.g., [@DP]). The matter Lagrangian will therefore contribute to an effective potential $U(\varphi)$ for the radion. Matter contribution–Hawking radiation ------------------------------------- The matter contribution to the EoM for $\varphi$ (or $b$) comes from the Hawking radiation[@Hawking] from an evaporating black hole with surface temperature $T$ (as seen asymptotically). Denote this matter contribution to (\[e14\]) by $\sigma=\left\langle \frac{\partial\mathcal{L}_{m}}{\partial\varphi}\right\rangle $. The Lagrangian $\mathcal{L}_{m}$ contains the matter and gauge fields, such as fermionic terms like[@NDL-JM; @DP] $\mathcal{L}_{\psi}=\bar{\psi}(i\gamma\cdot\partial-m(\varphi))\psi$ with $m(\varphi)=b^{-\frac{n}{2}}(\varphi)m_{0}$, ($m_{0}$ = const) and gauge field terms, such as that for the photon, $\mathcal{L}_{F}=-\frac{1}{4}b^{n}(\varphi)F_{\mu\nu}F^{\mu\nu}$. For a freely propagating electromagnetic field with $F_{\mu\nu}F^{\mu\nu}=0$, we have no contribution to $\sigma$ from $\mathcal{L}_{F}$. However, particle modes with nonzero rest mass do contribute to $\sigma$ through terms like $-\alpha\left\langle m\bar{\psi}\psi\right\rangle $ where$$\alpha_{A}=\frac{\partial\ln m_{A}(\varphi)}{\partial\varphi} \label{e15}$$ with $A$ labeling the particle species. (In the nonrelativistic flat space limit this $\sigma$ term is proportional to the fermionic energy density, $-\alpha\left\langle m\bar{\psi}\psi\right\rangle \sim-\alpha\rho$. However, we want to consider $\sigma$ terms beyond the nonrelativistic flat space limit. We will conclude that $\sigma\sim-\alpha g_{\mu\nu}\mathcal{T}_{cl}^{\mu\nu}=-\alpha\mathcal{T}_{cl}$ for the more general case where $\mathcal{T}_{cl}^{\mu\nu}$ is a classical stress-energy tensor, and that, classically, $\alpha_{A}=\alpha=$ const., independent of particle species.) Now, for simplicity, rather than using the field theoretic version of the matter Lagrangian $\mathcal{L}_{m}$, let us follow the approach used by Damour and Polyakov[@DP] and treat the matter with a classical description, replacing the field theoretic action with a classical particle action $S_{cl}$. We consider particle modes having a nonzero rest mass $m_{A}(\varphi)$ with $\partial_{\varphi}\mathcal{L}_{A}\neq0$ and write a classical action$$\begin{aligned} S_{cl} & =-\sum_{A}\int ds_{A}m_{A}=-\sum_{A}\int m_{A}\left[ g_{\mu\nu }(x_{A})dx_{A}^{\mu}dx_{A}^{\nu}\right] ^{1/2}\nonumber\\ & =-\sum_{A}\int d^{4}x\int m_{A}\left[ g_{\mu\nu}(x_{A})dx_{A}^{\mu}dx_{A}^{\nu}\right] ^{1/2}\delta^{(4)}(x-x_{A})=\int d^{4}x\sqrt {-g}\mathcal{L}_{cl} \label{e16}$$ and identify $\sqrt{-g}\mathcal{L}_{cl}=-\sum_{A}\int m_{A}\left[ g_{\mu\nu }(x_{A})dx_{A}^{\mu}dx_{A}^{\nu}\right] ^{1/2}\delta^{(4)}(x-x_{A})$.  A field theoretic energy-momentum tensor for the matter fields, defined by $\mathcal{T}^{\mu\nu}=\dfrac{2}{\sqrt{-g}}\dfrac{\partial\left( \sqrt {-g}\mathcal{L}_{m}\right) }{\partial g_{\mu\nu}}$, is replaced by an energy-momentum tensor $\mathcal{T}_{cl}^{\mu\nu}$ for the classical particles, with[^1] $$\mathcal{T}_{cl}^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\partial\left( \sqrt {-g}\mathcal{L}_{cl}\right) }{\partial g_{\mu\nu}}=\frac{1}{\sqrt{-g}}\sum_{A}\int m_{A}u_{A}^{\mu}u_{A}^{\nu}\delta^{(4)}(x-x_{A})d\tau_{A} \label{e17}$$ where $u^{\mu}=dx^{\mu}/d\tau$ satisfies an on-shell constraint $g_{\mu\nu}u^{\mu}u^{\nu}=1$. Taking the trace gives$$\mathcal{T}_{cl}=g_{\mu\nu}\mathcal{T}_{cl}^{\mu\nu}=\frac{1}{\sqrt{-g}}\sum_{A}\int m_{A}\delta^{(4)}(x-x_{A})d\tau_{A}=-\mathcal{L}_{cl} \label{e18}$$ We therefore find$$\sigma=\frac{\partial\mathcal{L}_{cl}}{\partial\varphi}=\sum_{A}\alpha _{A}\mathcal{L}_{cl,A}=-\sum_{A}\alpha_{A}\mathcal{\mathcal{T}}_{cl,A} \label{e19}$$ for $\alpha_{A}=$ const. The constant $\alpha_{A}$ takes a value $\alpha _{A}=\alpha=-\sqrt{\frac{n}{2(n+2)}}\ \kappa$, which for our $n=2$ model becomes $\alpha=-\kappa/2$. This can be seen[@Dicke] by considering the matter action,$$S=-\sum_{A}\int m_{0},_{A}\ d\tilde{s}_{A} \label{new1}$$ where $m_{0},_{A}$ is the constant Jordan frame particle mass for species $A$, and $d\tilde{s}=\sqrt{\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}}$ is the Jordan frame line element, which by (\[e8\]), is related to the Einstein frame line element $ds=\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}$ by $d\tilde{s}=b^{-\frac{n}{2}}ds$. The matter action rewritten in the Einstein frame is$$S=-\sum_{A}\int m_{0},_{A}\ (b^{-\frac{n}{2}}ds_{A})=-\sum_{A}\int m_{A}\ ds_{A} \label{new2}$$ where the Einstein frame mass is$$m_{A}=b^{-\frac{n}{2}}m_{0},_{A}=\exp\left( -\sqrt{\frac{n}{2(n+2)}}\ \kappa\varphi\right) m_{0},_{A} \label{new3}$$ Eqs. (\[e15\]) and (\[new3\]) then give$$\alpha_{A}=\frac{\partial\ln m_{A}(\varphi)}{\partial\varphi}=-\sqrt{\frac {n}{2(n+2)}}\ \kappa\rightarrow-\frac{\kappa}{2}\text{\ \ for \ \ }n=2 \label{new4}$$ From (\[e19\]) we therefore have $\sigma=-\alpha\mathcal{T}_{cl}=\frac{\kappa}{2}\mathcal{T}_{cl}$ for our $n=2$ model. The Hawking radiation (for a neutral nonrotating black hole) is assumed to be a fluid with energy density $\rho=\mathcal{T}_{0}^{0}$ and a normal radial pressure component $p_{r}=-\mathcal{T}_{r}^{r}$, and tangential pressure[^2] $p_{T}=-\mathcal{T}_{\theta}^{\theta}=-\mathcal{T}_{\phi}^{\phi}$ in the center of momentum frame, i.e., the rest frame of the black hole. We also assume the energy density and pressures to be related by the equations of state$$p_{r}=w_{r}\rho,\ \ \ p_{T}=w_{T}\rho,\ \ \ 0\leq w_{r}\leq1,\ \ \ 0\leq w_{T}\leq1 \label{new4a}$$ where $w_{r,T}$ are constants. For an isotropic perfect fluid $p_{T}=p_{r}=p$ and $w_{r}=w_{T}=w$ with $0\leq w\leq1$. Let us define an effective pressure $p$ and a parameter $w$ by $$\ p=\frac{1}{3}(p_{r}+2p_{T}),\ \ \ \ w=\frac{1}{3}\left( w_{r}+2w_{T}\right) ,\ \ \ \ 0\leq w\leq1 \label{new4b}$$ so that an effective equation of state can be written in the form $p=w\rho$, as assumed by Zurek and Page[@Zurek-Page] and by ’t Hooft[@'t; @Hooft], where the Hawking radiation is regarded as a perfect fluid with the constant $w=p/\rho\in\lbrack0,1]$. The trace of the stress-energy tensor becomes$$\mathcal{T}_{cl}=\rho-(p_{r}+2p_{T})=[1-(w_{r}+2w_{T})]\rho=(1-3w)\rho \label{new4c}$$ For an ideal gas of noninteracting massless particles in thermal equilibrium, $w=1/3$. However, we proceed by leaving $w$ as a free parameter, subject to $0\leq w\leq1$ as assumed in [@Zurek-Page] and [@'t; @Hooft]. This allows for a value of $\mathcal{T}_{cl}$ that can be positive, negative, or zero. With (\[e19\]) and (\[new4c\]) we obtain our approximate result $$\sigma=\frac{\partial\mathcal{L}_{cl}}{\partial\varphi}=-\alpha\mathcal{T}_{cl}=-\alpha(1-3w)\rho\label{e21}$$ with the energy density $\rho$ being dominated by relativistic particle modes. A couple of remarks are in order here. First, we note that $\mathcal{T}_{cl}$ and therefore $\rho$ in (\[e21\]) are generated by the particle modes with nonzero rest mass and do not include the energy density $\rho_{0}$ and pressure $p_{0}$ that is due to the massless (e.g. photon) components of the radiation. The total energy density and pressure of the entire fluid would be $\rho_{\text{tot}}=\rho+\rho_{0}$ and $p_{\text{tot}}=p+p_{0}$, respectively. Secondly, it may be that there are multiple components of the fluid corresponding to various particle species, with $\rho=\sum_{A}\rho_{A}$, and each species may have an equation of state $p_{A}=p_{A}(\rho)$ which, in principle, could be complicated. However, in order to study the effects of the radiation, these difficulties are avoided here by our simple assumption that $p/\rho=w=$ const., the same assumption made in the Hawking radiation fluid models of refs.[@Zurek-Page] and[@'t; @Hooft]. This seems palatable for a case where there are very few relativistic massive modes, at least on time scales sufficiently small compared to the black hole evaporation time scale $M/\dot{M}$. The EoM  $\square\varphi+\partial_{\varphi}V-\sigma=0$  for the radion field $\varphi$ becomes$$\square\varphi+\frac{\partial V}{\partial\varphi}+\alpha(1-3w)\rho=0 \label{e22}$$ An effective potential $U(\varphi)$ is now defined by $U=V-\sigma \varphi=V+U_{matter}$, or$$U(\varphi)=V(\varphi)+\alpha(1-3w)\rho\varphi=V(\varphi)+\frac{2}{\kappa }\alpha(1-3w)\rho\ln b\text{\ \ \ \ (effective potential)} \label{e23}$$ where $\ln b=\frac{1}{2}\kappa\varphi$ for our model with two extra dimensions and we set $U(\varphi=0)=0$. The matter contribution to the radion effective potential is$$U_{matter}=-\sigma\varphi=-\frac{2}{\kappa}\sigma\ln b=\frac{2\alpha}{\kappa }\mathcal{T}_{cl}\ln b=-\mathcal{T}_{cl}\ln b=-(1-3w)\rho\ln b \label{new6}$$ where the result $2\alpha/\kappa=-1$ from (\[new4\]) has been used. The sign of this matter term is controlled by the parameter $w$, and in the special case $w=1/3$ then $\sigma\rightarrow0$ and matter term does not contribute to the radion effective potential. We note that for $w<1/3$ then $U_{matter}$ is a decreasing function for $b>1$, while for $w>1/3$ we have that $U_{matter}$ is an increasing function for $b>1$. Radion effective potential, $U$ ------------------------------- With (\[e12\]), (\[e13\]), (\[e23\]), and (\[new6\]) we can now write an explicit, but simple, effective potential in the form $U(b)=V(b)-\mathcal{T}_{cl}\ln b$, or \[e29\]$$\begin{aligned} \frac{1}{\lambda}U(b) & =\left( b^{-6}-2b^{-4}+b^{-2}\right) -\frac{\mathcal{T}_{cl}}{\lambda}\ln b\label{e29a}\\ & =\left( b^{-6}-2b^{-4}+b^{-2}\right) -\zeta\ln b \label{e29c}$$ where $$\zeta\equiv\frac{\mathcal{T}_{cl}}{\lambda}=\frac{2\mathcal{T}_{cl}}{m_{\varphi}^{2}M_{0}^{2}}=\frac{(1-3w)\rho}{\lambda} \label{e30}$$ Here, the parameter $\zeta$ is dimensionless and $\zeta=\zeta(r,t)$ is a function of radial distance $r$ from the black hole since $\rho=\rho(r,t)$. The assumed range of $w$ allows a value of $\zeta$ in the range $-2\rho /\lambda\leq\zeta\leq\rho/\lambda$. Figure 2 gives a representation of $U(b)/\lambda$ for various *positive* values of $\zeta$ ($p\leq\rho/3$). The asymptotic vacuum value of $b$ occurs at $b_{0}=1$ for $\zeta=0$, but for values of $0<\zeta\lesssim.5$ the vacuum value of $b$ is shifted to larger values, $b>1$, and the minimum of $U$  becomes more negative. For values $\zeta\gtrsim.5$ the local minimum disappears, the vacuum state is completely destabilized, and $U(\varphi)$ is a monotonically decreasing function whose slope depends on $\zeta$. \[h\] [f2.eps]{} [Figure 2: Plots of ]{}$U(b)/\lambda$ [vs. ]{}$b$ [are shown for positive values of ]{}$\zeta$[. The dotted curve, with a minimum at ]{}$U=0$ [and ]{}$b=1$[, has ]{}$\zeta=0$ [and corresponds to the radion potential ]{}$V(b)/\lambda$[. The solid curves have ]{}$\zeta=.3,.5,1,2$[, with the more negatively sloped curves corresponding to bigger ]{}$\zeta$[. The vacuum value of ]{}$b$ [occurs at ]{}$b_{0}=1$ [for ]{}$\zeta=0$[, but for ]{}$0<\zeta \lesssim.5$ [the vacuum value of ]{}$b$ [is shifted to larger values, ]{}$b_{vac}>1$[, and the minimum of ]{}$U$ [completely disappears for ]{}$\zeta\gtrsim.5$[.]{} Figure 3 gives a representation of $U(b)/\lambda$ for various *negative* values of $\zeta$  ($p\geq\rho/3$). The asymptotic vacuum value of $b$ occurs at $b_{0}=1$ for $\zeta=0$, but for larger values of $|\zeta|$ the vacuum value of $b$ is shifted to smaller values, $b<1$, and the minimum of $U$ becomes more negative. \[h\] [f3.eps]{} [Figure 3: Plots of ]{}$U(b)/\lambda$ [vs. ]{}$b$ [are shown for negative values of ]{}$\zeta$[. The dotted curve, with a minimum at ]{}$U=0$ [and ]{}$b=1$[, has ]{}$\zeta=0$ [and corresponds to the radion potential ]{}$V(b)/\lambda$[. The solid curves have ]{}$\zeta=-.5,-1,-2,-3$[, with the lower minimum curves corresponding to bigger ]{}$|\zeta|$[. The vacuum value of ]{}$b$ [occurs at ]{}$b_{0}=1$ [for ]{}$\zeta=0$[, but for larger values of ]{}$|\zeta|$[the vacuum value of ]{}$b$ [is shifted to smaller values, ]{}$b_{vac}<1$[, and the minimum of ]{}$U$ [becomes more negative.]{} Thus the vacuum values $b_{vac}$ and $\varphi_{vac}$ become $r$ dependent in general. Far from the hole, $\zeta\rightarrow0$ and $\varphi_{vac}\rightarrow0$, $b_{vac}\rightarrow1$. Near the hole, where $\zeta\neq0$, then $\varphi\neq0$ $\ $and $b\neq1$. Therefore $\varphi$ interpolates between a positive or negative value $\varphi\neq0$ near the horizon to $\varphi=0$ at asymptotic distances. As the hole evaporates and $|\zeta|$ increases, any vacuum state near the horizon gets further shifted to smaller or larger values, depending on the sign of $\zeta$. For $\zeta>0$ a stable vacuum eventually disappears and the radion rolls to larger values until the hole’s explosive end. Radion cloud and black hole albedo ================================== Radion cloud ------------ An energy momentum tensor $\mathcal{T}_{\mu\nu}$ has been defined for the matter portion $\mathcal{L}_{m}$ of the effective 4d Lagrangian $$\mathcal{L}=\mathcal{L}_{\varphi}(\varphi)+\mathcal{L}_{m}(\varphi,\sigma ,\psi,\cdot\cdot\cdot)=\frac{1}{2}(\partial\varphi)^{2}-V(\varphi )+\mathcal{L}_{m}(\varphi,\sigma,\psi,\cdot\cdot\cdot) \label{e31}$$ and an energy momentum tensor $S_{\mu\nu}$ can be written for the pure radion part $\mathcal{L}_{\varphi}$:$$S_{\mu\nu}=\partial_{\mu}\varphi\partial_{\nu}\varphi-g_{\mu\nu}\mathcal{L}_{\varphi}=\partial_{\mu}\varphi\partial_{\nu}\varphi-g_{\mu\nu }\left[ \tfrac{1}{2}\partial^{\alpha}\varphi\partial_{\alpha}\varphi -V(\varphi)\right] \label{e33}$$ The energy density part for the radion field,$$S_{00}=\frac{1}{2}(\partial_{0}\varphi)^{2}-\frac{1}{2}g_{00}g^{rr}(\partial_{r}\varphi)^{2}+g_{00}V(\varphi) \label{e34}$$ vanishes asymptotically, but becomes nonzero near the evaporating black hole where $\varphi$ develops a nonzero vacuum value $\varphi_{vac}$ due to the environmental effects of a nonzero $\zeta$ in the effective potential $U$. For a nonradiating black hole (i.e., a matter-vacuum solution with $\mathcal{L}_{m}=0$, $T_{\mu\nu}=0$, $\sigma=0$) we have the Einstein equation $R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\kappa^{2}S_{\mu\nu}$ along with the radion EoM $\square\varphi+V^{\prime}(\varphi)-\sigma=0$. The minimal energy radion solution ($S_{\mu\nu}=0$) is given by the trivial solution $\varphi=0$ ($b=1$). Therefore the gravitational field alone of the black hole has no effect on the radion and there is no radion cloud in this case. However, for a radiating black hole with nonzero values of $\sigma$ and $\zeta$ outside the horizon, $\varphi=0$ is not a solution of the radion EoM and the radion field $\varphi(r,t)$ must interpolate between a value of $\varphi_{hor}\neq0$ near the horizon and $\varphi=0$ asymptotically. Since $\varphi\neq0$ is not at the minimum of $V(\varphi)$, then $V(\varphi)>0$, contributing a positive contribution to $S_{00}$. There are nonnegative gradient terms contributing to $S_{00}$ as well. So near the horizon, $S_{00}>0$, and asymptotically $S_{00}\rightarrow0$. The energy density of the scalar field $\varphi$ is concentrated near the MBH, where the gradient terms are large, and the radion field forms a cloud around it. The exact structure of this cloud requires a knowledge of the solution $\varphi(r,t)$ to the EoM   $\square\varphi+U^{\prime}(\varphi)=0$. A crude estimate of the extent of this cloud of energy can be obtained by considering a thin shell of Hawking radiation at a radius $r\gg r_{S}$ where $r_{S}=2GM$ is the radius of a Schwarzschild black hole. (Relativistic radiation with speed $v\sim1$ is assumed, as a higher energy density is carried by relativistic modes.) At this asymptotic distance the mass $\delta M$ in the spherical shell is approximately constant as it propagates outward, so that $\delta M\approx4\pi r^{2}\rho(r,t)\delta r\sim P(t)\delta t$ where $P(t)=-\dot{M}(t)>0$ is the power output of the radiation from the evaporating MBH and for the sake of simplicity I have neglected any time retardation effects in $P(t)$. This gives a crude estimate for the matter density$$\rho(r,t)\sim\frac{P(t)}{4\pi r^{2}} \label{b1}$$ At a fixed instant $t$ the matter density drops off as $r^{-2}$, while at a fixed distance $r$ the density increases with time as described by the power output $P(t)$. An outer edge of the evolving radion cloud, i.e. the cloud radius $R(t)$, can be defined as a radial distance where the density $\rho$ assumes a sufficiently small constant value, i.e., $\eta\equiv\rho/\lambda\ll1$ is a small constant, so that $\rho\approx0$ outside this radius. From (\[b1\]) this cloud radius, where $\eta$ is a constant, is given by$$R^{2}(t)\sim\frac{P(t)}{4\pi\rho}=\frac{P(t)}{4\pi\eta\lambda}=\frac {P(t)}{2\pi\eta m_{\varphi}^{2}M_{0}^{2}} \label{b2}$$ The ordinary Steffan-Boltzmann law for a perfect blackbody (no gravitational greybody effects assumed, for simplicity) with emitting surface area $4\pi r_{S}^{2}$ gives $P(t)=4\pi r_{S}^{2}\sigma_{SB}T^{4}(g/2)$ where $g=(N_{B}+\frac{7}{8}N_{F})$ is the effective number of degrees of freedom of relativistic particles and $\sigma_{SB}=\pi^{2}/60$ is the Steffan-Boltzmann constant. Using $r_{S}=2GM=\frac{M}{4\pi M_{0}^{2}}$ and $T=\frac{1}{8\pi GM}=\frac{M_{0}^{2}}{M}$ (where $M_{0}=1/\kappa=1/\sqrt{8\pi G}$ is the reduced Planck mass) leads to$$P(t)=\frac{(g/2)\pi}{240}\frac{M_{0}^{4}}{M^{2}(t)} \label{b3}$$ From this, the cloud radius is given by$$R(t)\sim\sqrt{\frac{g/2}{480\eta}}\left( \frac{M_{0}}{M(t)}\right) m_{\varphi}^{-1}\sim\left( \frac{M_{0}}{M(t)}\right) m_{\varphi}^{-1} \label{b4}$$ where, for simplicity, I have set $480\eta/(g/2)\sim1$ and $M(t)$ is the black hole mass. The radion cloud grows in size as the hole shrinks, with $\dot {R}/R\sim-\dot{M}/M$. From (\[b4\]) $R$ approaches an upper limit $R_{\max }\sim m_{\varphi}^{-1}$ near the explosive end of the MBH as $M\rightarrow M_{0}$. The requirement that $r_{S}/R\ll1$ implies that $m_{\varphi}\ll \frac{4\pi M_{0}^{3}}{M^{2}}\lesssim M_{0}$. Provided that the rather natural condition $m_{\varphi}\ll M_{0}$ is satisfied, then so is the requirement that $R\gg r_{S}$. The energy dependent reflection coefficient (see below) $\mathcal{R}(\omega)$ and transmission coefficient $\mathcal{T}(\omega)=1-\mathcal{R}(\omega)$ for a particle of energy $\omega$ will depend upon the variation in $\varphi$ (or $b$) from the near-horizon region to the asymptotic region, along with the radius $R$ of the cloud. The reflection coefficient approaches a maximal value $\mathcal{R}_{\max}$ as particle energy $\omega$ approaches a minimal value[@NDL-JM]. Radion reflectivity and black hole albedo ----------------------------------------- Basic features expected of particle reflectivity by the radion field – a radion induced black hole albedo – can be obtained from the (flat space) results of [@NDL-JM] and other studies of particle reflection from ordinary (nonradionic) scalar field domain walls (see, for example,[@VSbook; @Everett; @EDW1; @EDW2] ). In the case of photons, the radion cloud, treated as a scalar modulus domain wall of thickness $\sim R$, has a maximal reflection coefficient given by[@NDL-JM]$$\mathcal{R}_{\max}=\dfrac{\left( b_{1}^{2}-b_{2}^{2}\right) ^{2}}{\left( b_{1}^{2}+b_{2}^{2}\right) ^{2}}=\dfrac{\left( b_{hor}^{2}-1\right) ^{2}}{\left( b_{hor}^{2}+1\right) ^{2}} \label{a1}$$ where $b_{1}$, $b_{2}$ are the values of the scale factor $b$ on the two different sides of a modulus wall. The two sides of our domain wall are the black hole horizon where $b=b_{hor}$ and the asymptotic region approximately a distance $R$ away where $b\rightarrow1$. The transmission coefficient is $\mathcal{T}(\omega )=1-\mathcal{R}(\omega)$ and for photons of energy $\omega$ we have the thin wall limit with $\mathcal{R}(\omega)\rightarrow\mathcal{R}(0)=\mathcal{R}_{\max}$ in the infrared limit $\omega\rightarrow0$, where the photon wavelength $\lambda_{\gamma}\gg R$. From[@NDL-JM] it was found from numerical calculations that $\mathcal{R}(\omega)/\mathcal{R}_{\max}$ typically begins to become significant for energies $\omega\lesssim1/R$, i.e., the thin wall limit. Note that $\mathcal{R}_{\max}\sim1$ when $b_{hor}\gg1$, or $b_{hor}\ll1$. We may have $b_{hor}\gg1$ for $\zeta_{hor}\gtrsim.5$, at which point the near-horizon vacuum of $U(b)$ completely destabilizes. For a near-horizon value of $0<\zeta_{hor}<.5$, one expects $\mathcal{R}_{\max}\ll 1$, as there is a minimum of $U$ that is not far removed from $b=1$. On the other hand, for negative $\zeta$ with $|\zeta|\gg1$ we have $b_{hor}\rightarrow0$ as $\zeta\rightarrow-\infty$, in which case $\mathcal{R}_{\max }\rightarrow1$ again. Since $|\zeta|$ increases with black hole temperature $T$, one expects $\mathcal{R}_{\max}$ to increase with increasing $T$. These considerations lead us to expect a possible alteration of the infrared portion of the transmitted Hawking radiation, as well as a partial reflection from near-horizon regions of low energy particles incident upon the black hole from outside. (The details of the spectral distortion, however, will depend upon the structure of the radion cloud.) On the other hand, for high energy photons with $\omega\gg1/R$, the cloud becomes transparent (thick-wall limit)[@NDL-JM] with $\mathcal{R}\rightarrow0.$ Similar qualitative statements are expected for massive particle modes. The black hole therefore has an energy-dependent albedo associated with the radion cloud, which in turn, is due to an inhomogeneous compactification of the extra dimensions near the horizon. The above deductions are based upon reflection and transmission characteristics in flat space. The effects of curved space would alter the gradient terms appearing in the $\square\varphi$ portion of the radion EoM, and therefore the gradient nature of the solution $\varphi(r,t)$. The exact expressions for $\mathcal{R}(\omega)$ and $\mathcal{T}(\omega)$ would depend on the exact solution $\varphi$, but the basic qualitative features mentioned above for $\mathcal{R}(\omega)$ are not expected to be significantly affected. $\mathbf{\zeta}$ **near the horizon** -------------------------------------  In the limit of a static, ideal fluid in thermodynamic equilibrium, the local total energy density[^3] is[@Zurek-Page] $\rho_{\text{tot}}\sim T^{\ast(w+1)/w}$, where $T^{\ast}(r)=T/\sqrt{g_{00}(r)}=(\sqrt{g_{00}}8\pi GM)^{-1}$ is the blue-shifted Hawking temperature. In this limit $\zeta=(1-3w)\rho/\lambda\sim(1-3w)T^{\ast(w+1)/w}/\lambda$ can become quite large or divergent near the horizon (or would-be horizon). (There can be significant backreactions on the metric, and the studies[@Zurek-Page],[@'t; @Hooft] suggest that the horizon could be removed by a (static) Hawking atmosphere, with $\rho$ remaining finite.) If the horizon is not removed by backreactions, the local energy density can diverge on the horizon, due to the diverging blue-shifted local temperature[@Barbon]. Furthermore, quantum field effects such as vacuum polarization[@DFU],[@CF],[@FZ] are expected to play important roles and may contribute to the $\sigma=\left\langle \partial\mathcal{L}_{m}/\partial\varphi\right\rangle $ term in the effective potential. In any case, whether $\rho$ diverges or remains finite near the black hole, the local value of $|\zeta|$ and $\sigma$ may become extremely large in the near horizon region, possibly leading to either $b_{hor}\gg1$ or $b_{hor}\ll1$. In either of these cases $\mathcal{R}_{\max}\rightarrow1$, indicating an infrared radionic reflectivity. Summary ======= A Kaluza-Klein model with two spherically compactified extra dimensions, studied previously by Davidson and Guendelmann[@DG1; @DG2] and by Carroll, Geddes, Hoffman, and Wald[@CGHW], is examined here with attention focusing on the development of a radion cloud around an evaporating neutral, nonrotating MBH. The cloud owes its existence not to the gravitational field alone (a Schwarzschild solution is accompanied by a trivial radion solution $\varphi=0$), but arises in response to the environmental effect of the Hawking radiation. The radiation is modeled as a fluid with an effective energy density $\rho(r,t)$ contributing to the radion equation of motion. An effective pressure $p$ is assumed to be related to $\rho$ through an equation of state $p/\rho=w=$ const. with $0\leq w\leq1$, resembling the perfect fluid Hawking atmosphere models of refs. [@Zurek-Page] and [@'t; @Hooft]. For the particular case $w=1/3$, as is expected for a fluid of noninteracting masseless particles in thermal equilibrium, there is no environmental effect on $\varphi$. However, it is not assumed here that the fluid is in equilibrium, and the particle modes contributing to the energy density $\rho=\rho_{\text{tot}}-\rho_{0}$ (where $\rho_{0}$ is due to massless particle modes) are those associated with particles of nonzero rest mass. The matter contribution to the radion effective potential is $U_{matter}=-\mathcal{T}_{cl}\ln b=$ $-\lambda\zeta\ln b$, which can be positive, negative, or zero, depending on the sign of $\zeta=(1-3w)\rho/\lambda$. A classical description has been used to estimate the $\sigma=\left\langle \partial\mathcal{L}_{m}/\partial\varphi\right\rangle $ term in the radion equation of motion, but near the horizon quantum field effects such as vacuum polarization[@DFU],[@CF],[@FZ] are expected to be important and may contribute to a shift in the radion vacuum. The radion $\varphi$ approaches a normal vacuum value $\varphi\rightarrow0$ ($b\rightarrow1$) asymptotically, where the Hawking radiation energy density vanishes, but near the MBH the radion is shifted to a value $\varphi\neq0$ ($b\neq1$) for any $\zeta\neq0$. Gradients of $\varphi$ and a nonzero radion potential $V(\varphi)$ then give rise to a radion cloud with nonvanishing energy density around the MBH. This radion cloud has an estimated size $R(t)\sim\left( \frac{M_{0}}{M(t)}\right) m_{\varphi}^{-1}$ and an energy dependent reflection coefficient $\mathcal{R}(\omega)$ as studied in[@NDL-JM]. This reflection coefficient has a maximum value $\mathcal{R}_{\max}$ (given by (\[a1\]) for the case of electromagnetic radiation in the flat space limit), which depends upon the parameter $\zeta$ near the horizon. $\mathcal{R}(\omega)/\mathcal{R}_{\max}$ begins to become significant for particle energies $\omega\lesssim R^{-1}(t)$. Since the asymptotic compactification radius for the extra dimensions is $\sim m_{\varphi}^{-1}$, the size of the cloud compared to that of the extra dimensions in asymptotic space is $R(t)/m_{\varphi}^{-1}\sim M_{0}/M(t)$ which is initially small, but becomes of order unity at the end stages of the evaporation. An infrared portion of the Hawking spectrum detected by an external observer will be suppressed if $\mathcal{R}_{\max}\rightarrow1$, and some low energy particles incident upon the MBH from the outside will be reflected back. The amount of reflectivity depends upon the temperature $T$ of the MBH (and therefore the parameter $\zeta$ near the horizon) and particle energy $\omega$. For high energy particles ($\omega\gg R^{-1}$) the radion cloud is transparent. The Hawking radiation contributes heavily to the effective potential $U(\varphi)$ for large $|\zeta|$, in which case $\mathcal{R}_{\max}$ may approach unity. For $\zeta<0$ one may have a vacuum with $b_{hor}\ll1$, while for positive values $\zeta\gtrsim.5$, the effective potential $U(\varphi)$ is completely destabilized, i.e., a local minimum disappears. In this case the slope of $U$ is negative, and the radion rolls outward in time with $b(t)$ increasing. In either case, when $\zeta\neq0$, a radion cloud must develop, since $\varphi=0$ and $b=1$ is not a solution of the radion EoM. For $\zeta\neq0$ the radion infrared albedo effect increases as the MBH evaporates. Since transparency begins to set in at particle energies $\omega\gtrsim R^{-1}\sim\left( \frac{M}{M_{0}}\right) m_{\varphi }\gtrsim m_{\varphi}$, the energy range of observable albedo effects ($\omega\lesssim m_{\varphi}$) will be very sensitive to the radion mass $m_{\varphi}$ and therefore the size of the extra dimensions. **Acknowledgement:**  I thank Eduardo Guendelman and an anonymous referee for useful comments. [99]{} A. Davidson and E.I. Guendelman, Phys. Rev. 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--- abstract: 'CUORE, an array of 988 TeO$_2$ bolometers, is about to be one of the most sensitive experiments searching for neutrinoless double-beta decay. Its sensitivity could be further improved by removing the background from $\alpha$ radioactivity. A few years ago it was pointed out that the signal from $\beta$s can be tagged by detecting the emitted [Cherenkov]{}light, which is not produced by $\alpha$s. In this paper we confirm this possibility. For the first time we measured the [Cherenkov]{} light emitted by a CUORE crystal, and found it to be 100 eV at the $Q$-value of the decay. To completely reject the $\alpha$ background, we compute that one needs light detectors with baseline noise below 20 eV RMS, a value which is 3-4 times smaller than the average noise of the bolometric light detectors we are using. We point out that an improved light detector technology must be developed to obtain TeO$_2$ bolometric experiments able to probe the inverted hierarchy of neutrino masses.' author: - '[N. Casali]{}' - '[M. Vignati]{}' - '[J.W. Beeman]{}' - '[F.Bellini]{}' - '[L. Cardani]{}' - '[I. Dafinei]{}' - '[S. Di Domizio]{}' - '[F. Ferroni]{}' - '[L. Gironi]{}' - '[S. Nagorny]{}' - '[F. Orio]{}' - '[L. Pattavina]{}' - '[G. Pessina]{}' - '[G. Piperno]{}' - '[S. Pirro]{}' - '[C. Rusconi]{}' - '[K. Schäffner]{}' - '[C. Tomei]{}' bibliography: - 'cherenkov\_bolo\_ref.bib' title: 'TeO$_2$ bolometers with [Cherenkov]{} signal tagging: towards next-generation neutrinoless double beta decay experiments' --- Introduction ============ Neutrinoless double beta decay ([0$\nu\beta\beta$]{}) is a process that violates the lepton number conservation law by two units, in which a parent nucleus decays into a daughter nucleus and emits two $\beta$ particles. Unlike the process accompanied by the emission of two neutrinos, allowed by the Standard Model and observed in several nuclei, [0$\nu\beta\beta$]{} has not yet been observed. Its discovery would reveal physics beyond the Standard Model: it would tell us that neutrinos, unlike all other elementary fermions, are Majorana particles, and would point to Leptogenesis as the origin of the matter-antimatter asymmetry after the Big Bang (for a recent review see for example [@Bilenky:2012qi] and references therein). The experimental signature is very clear, a peak in the sum energy spectrum of the $\beta$s at the $Q$-value of the decay. Bolometers proved to be good detectors to search for [0$\nu\beta\beta$]{}, thanks to the high variety of isotopes that can be studied, the excellent energy resolution, and the low background they can achieve. The [CUORE]{} experiment [@ACryo; @Artusa:2014lgv] will search for the [0$\nu\beta\beta$]{} of [$^{130}\mathrm{Te}$]{} with an array of 988 [$\mathrm{TeO}_2$]{} bolometers, cryogenic calorimeters working at a temperature around 10[$\mathrm{~mK}$]{}. Each bolometer weighs 750[$\mathrm{~g}$]{}, for a total active mass of 741[$\mathrm{~kg}$]{}, 206[$\mathrm{~kg}$]{} of which are [$^{130}\mathrm{Te}$]{} (34.2% natural abundance [@Fehr200483] in tellurium). The energy resolution at the $Q$-value of the decay, 2528[$\mathrm{~keV}$]{} [@Redshaw:2009zz], is expected to be 5[$\mathrm{~keV~FWHM}$]{}. [CUORE]{} is under construction at Laboratory Nazionali del Gran Sasso (LNGS) in Italy, and will start to take data in 2015. The technology of [$\mathrm{TeO}_2$]{} bolometers has been demonstrated by [Cuoricino]{}, a 40[$\mathrm{~kg}$]{} tower of 62 bolometers that, with $19.75{\ensuremath{\mathrm{~kg\,y}}}$ of [$^{130}\mathrm{Te}$]{} data, set a lower limit to the decay half-life of $2.4\cdot 10^{24}{\ensuremath{\mathrm{~y}}}$ at 90% C.L. [@Andreotti:2010vj]. The analysis of the data pointed out that the main source of background in the energy region of interest (ROI) for the [0$\nu\beta\beta$]{} consisted in $\alpha$ particles generated by natural radioactivity of the copper structure holding the crystals. To reduce it, the [CUORE]{} collaboration developed techniques to clean the copper and to assemble the detector in ultra radiopure environments. The success of this effort has been recently demonstrated by the CUORE-0 experiment, an array of 52 bolometers that reached an $\alpha$ background index of $0.019\pm 0.002{{\ensuremath{\mathrm{~counts/(keV\,kg\,y)}}}}$, a factor 6 less than [Cuoricino]{} [@Aguirre:2014lua]. The background in [CUORE]{}, however, is still foreseen to be dominated by $\alpha$ particles, limiting the sensitivity to the [0$\nu\beta\beta$]{} half-life to around $10^{26}$ years in 5 years of data taking. This corresponds to an effective neutrino Majorana mass that ranges, depending on the choice of the nuclear matrix element, from $40$ to $100{\ensuremath{\mathrm{~meV}}}$, values that are quite far from covering the entire interval of masses corresponding to the inverted hierarchy scenario, that ranges from $10$ to $50{\ensuremath{\mathrm{~meV}}}$ [@Bilenky:2012qi]. The background can be reduced by detecting the small amount of [Cherenkov]{} light that is emitted by interacting particles in [$\mathrm{TeO}_2$]{} crystals. In fact, at the energy scale of interest for [0$\nu\beta\beta$]{}, the $\beta$s (signal) are above threshold for [Cherenkov]{} emission, while $\alpha$ particles (background) are not [@TabarellideFatis:2009zz]. In a previous paper [@Beeman:2011yc] we operated a 117[$\mathrm{~g}$]{} [$\mathrm{TeO}_2$]{} bolometer surrounded by a 3M VM2002 reflecting foil, monitoring a crystal face with a germanium bolometer acting as light detector. In coincidence with the heat released in the [$\mathrm{TeO}_2$]{} we were able to detect the light emitted by $\beta/\gamma$ particles, which amounted to 173[$\mathrm{~eV}$]{} at 2528[$\mathrm{~keV}$]{}. The crystal was doped with natural samarium, which contains $^{147}$Sm, an $\alpha$-unstable isotope with $Q=2310{\ensuremath{\mathrm{~keV}}}$. The light detected from these decays was compatible with zero, confirming that at the [0$\nu\beta\beta$]{} energy scale no light is emitted by $\alpha$s. Finally, room temperature tests confirmed that the light emitted by particles interacting in [$\mathrm{TeO}_2$]{} can be ascribed to the sole [Cherenkov]{} emission, excluding a contribution from the scintillation [@Casali:2013bva]. In this paper we present the results of a test conducted on a CUORE bolometer, i.e. a 750[$\mathrm{~g}$]{} crystal, 6 times larger than that used in our previous work and without samarium doping. The results confirm that the $\alpha$ discrimination in [CUORE]{} is possible, but the light signal is small and requires light detectors with higher sensitivity than that provided by bolometers. Experimental setup ================== The [$\mathrm{TeO}_2$]{} crystal comes from samples of the [CUORE]{}  batches used to check the radiopurity and the bolometric performances during the production [@Alessandria:2011vj], and therefore is identical to the crystals that are currently being mounted in [CUORE]{}. The crystal is a $5\times5\times5{\ensuremath{\mathrm{~cm^3}}}$ cube with translucent faces, two opposite of which have a better polishing quality, close to optical polishing grade. All faces are surrounded by the VM2002 light reflector except for an optical one that is monitored by a 5[$\mathrm{~cm}$]{} in diameter, $300{\ensuremath{\mathrm{~\mu m}}}$ thick germanium light detector (LD) [@Beeman:2013zva] (Fig. \[fig:setup\]). Both the [$\mathrm{TeO}_2$]{} crystal and the germanium are operated as bolometers, using a neutron transmutation doped germanium (NTD-Ge) thermistor as temperature sensor [@Itoh]. The detectors are held in a copper structure by means of teflon (PTFE) supports, anchored to the mixing chamber of a dilution refrigerator. The setup is operated in the CUORE/LUCIFER R&D cryostat, in the Hall C of LNGS [@Pirro:2006mu]. ![The [$\mathrm{TeO}_2$]{} crystal in the copper holder, surrounded by a 3M VM2002 light reflector and monitored by the germanium bolometric light detector.[]{data-label="fig:setup"}](fig1.pdf){width="48.00000%"} As in Ref. [@Beeman:2011yc], the read-out of the thermistor is performed using the [Cuoricino]{} electronics [@Arnaboldi:2004jj]. The analog signals are filtered by 6-pole active Bessel filters [@Arnaboldi:2010zz] and then fed into an 18-bit National Instrument PXI analog-to-digital converter (ADC), the same system being used in CUORE-0. The filter cutoff and the ADC sampling frequency are set to 12[$\mathrm{~Hz}$]{} and 125[$\mathrm{~Hz}$]{} for the [$\mathrm{TeO}_2$]{}, respectively, and to 120[$\mathrm{~Hz}$]{} and 2000[$\mathrm{~Hz}$]{} for the LD, respectively. The trigger is software generated on each bolometer. When it fires, waveforms 5[$\mathrm{~s}$]{} long on the [$\mathrm{TeO}_2$]{} and 250[$\mathrm{~ms}$]{} long on the LD are saved on disk. Additionally, when the trigger fires on the [$\mathrm{TeO}_2$]{}, the waveform on the LD is acquired irrespective of its own trigger. To maximize the signal to noise ratio, the waveforms are processed offline with the optimum filter algorithm [@Radeka:1966; @Gatti:1986cw]. On the [$\mathrm{TeO}_2$]{} the pulse is identified with a peak finder algorithm, and the amplitude is evaluated as the maximum of the peak. On the LD, to eliminate noise artifacts at the threshold, the pulse amplitude is evaluated at the characteristic time delay of the LD response with respect to the pulse on the [$\mathrm{TeO}_2$]{}, which is estimated in calibration runs using events generated by particles interacting in both detectors (for more details see Ref. [@Piperno:2011fp]). The light detector is exposed to a permanent $^{55}$Fe source, providing 5.9 and 6.5[$\mathrm{~keV}$]{} calibration X-rays. The typical rise and decay times of the pulses are 2.6 and 6[$\mathrm{~ms}$]{}, respectively, while the energy resolution at the iron peaks and at the baseline is 135 and 72[$\mathrm{~eV~RMS}$]{}, respectively. To calibrate the [$\mathrm{TeO}_2$]{} and to generate events in the [0$\nu\beta\beta$]{} region, the setup is illuminated by a [$^{232}\mathrm{Th}$]{} source placed outside the cryostat. The rise and decay times of the [$\mathrm{TeO}_2$]{} pulses are 40 and 532[$\mathrm{~ms}$]{}, respectively, values that are similar to the CUORE-0 ones [@Aguirre:2014lua]. The energy resolution at the 2615 keV $^{208}$Tl peak from the thorium source is 11.5[$\mathrm{~keV~FWHM}$]{}, worse than the 5.7[$\mathrm{~keV}$]{} FWHM obtained averaging all the CUORE-0 bolometers. This might be due to the different working temperature, which was chosen higher than in CUORE-0 (20 mK instead of 10 mK) in order to improve the energy resolution of the light detector (see Ref. [@Beeman:2013zva] for details). The worse energy resolution of the [$\mathrm{TeO}_2$]{}  bolometer does not affect our results, since the attention is focused on the light signal. Results {#sec:results} ======= The energy spectrum acquired from the [$\mathrm{TeO}_2$]{} bolometer in 6.86 days of data taking is shown in Fig. \[fig:energyspectrum\]. The peak around 5400[$\mathrm{~keV}$]{} is due to the $\alpha$-decay of [$^{210}$Po]{}, a natural contamination of the [$\mathrm{TeO}_2$]{} crystal observed also in the 117[$\mathrm{~g}$]{} detector and in CUORE-0. The remaining peaks are $\gamma$s from the [$^{232}\mathrm{Th}$]{} source, except for the peak at 1461[$\mathrm{~keV}$]{}, which is a $\gamma$ from [$^{40}$K]{} contamination of the cryostat. Both the single (SE) and double escape (DE) peaks of the 2615 keV $\gamma$ from [$^{208}\mathrm{Tl}$]{} are visible. The presence of the DE peak is of particular interest because it is a single site production of a $e^-$ and of a $e^+$, a process similar to the [0$\nu\beta\beta$]{}. ![Energy spectrum acquired by the [$\mathrm{TeO}_2$]{} crystal. All the labeled peaks are $\gamma$s, except for the single and double escape peaks of the 2615[$\mathrm{~keV}$]{} $\gamma$ from [$^{208}\mathrm{Tl}$]{}, which are $e^- + e^+ +\gamma$ and $e^- + e^+$ events, respectively, and for the events around 5.4[$\mathrm{~MeV}$]{}, which are generated by the $\alpha$-decay of [$^{210}$Po]{} in the crystal.[]{data-label="fig:energyspectrum"}](fig2.pdf){width="48.00000%"} The light detected versus calibrated heat in the [$\mathrm{TeO}_2$]{} crystal is shown in Fig. \[fig:lightvsheat\]. The distribution of the light corresponding to each peak in Fig. \[fig:energyspectrum\] (blue dots in the figure) is fitted with a Gaussian, the mean of which is overlaid onto the figure. The mean light from the $\alpha$-decay of [$^{210}$Po]{}  is found to be $<L_\alpha> = -3.9\pm 14.5{\ensuremath{\mathrm{~eV}}}$, i.e. compatible with zero. The mean light from the $\gamma$ peaks is fitted with a line $<L_{\beta/\gamma}> = {\rm LY}\cdot ({\rm Energy-E_{th}})$, with ${\rm E_{th}} = 280\pm 60{\ensuremath{\mathrm{~keV}}}$ and ${\rm LY} = 45\pm 2{\ensuremath{\mathrm{~eV/MeV}}}$. The standard deviations of the light distributions are found compatible with the baseline noise of the LD, which therefore appears as the dominant source of fluctuation, hiding any possible dependence on the position of the interaction in the [$\mathrm{TeO}_2$]{} crystal or statistical fluctuations of the number of photons. As in our previous work, the light from the DE peak is compatible with the light from $\gamma$s, indicating that the fitted line can be used to predict the amount of light detectable from [0$\nu\beta\beta$]{} events. We compute $101.4\pm 3.4$[$\mathrm{~eV}$]{} of light for a $\beta/\gamma$ event with [0$\nu\beta\beta$]{} energy, 72[$\mathrm{~eV}$]{} less than the light detected at the same energy in the 117[$\mathrm{~g}$]{} detector. ![(Color online) Detected light versus calibrated heat in the [$\mathrm{TeO}_2$]{} bolometer for all the acquired events (gray) and for the events belonging to the peaks labeled in Fig \[fig:energyspectrum\] (blue). The mean light is clearly energy dependent for the $\gamma$ peaks (red circles below 3 MeV) and compatible with zero for the $\alpha$-decay of the [$^{210}$Po]{} (pink circle at 5.4 MeV). []{data-label="fig:lightvsheat"}](fig3.pdf){width="48.00000%"} The detected light at the [0$\nu\beta\beta$]{} is small, at the same level of the LD noise, and does not allow one to perform an event by event rejection of the $\alpha$ background. As indicated in Ref. [@TabarellideFatis:2009zz], the emitted [Cherenkov]{} light amounts to several hundreds of eV, a much higher value than what we detect. To increase the light collection efficiency, we applied different modifications to the setup: 1) we changed the VM 2002 light reflector to aluminum foils. Aluminum is expected to have higher reflectivity in the UV band, the region where the [Cherenkov]{} emission is more intense. Nevertheless, the amount of light detected is 25% less than in the case of VM 2002; 2) we removed the VM 2002, which is a specular light reflector, and wrapped the crystal with teflon tape, which is a light diffusor. The amount of light detected is compatible with the VM 2002 measurement; 3) we changed the LD to an identical one, but we coated the side faced to the [$\mathrm{TeO}_2$]{} with 60[$\mathrm{~nm}$]{} of SiO$_2$. It has been demonstrated, in fact, that in the red/infrared band this layer enhances the light absorption by up to 20% [@Beeman:2012cu; @ZnSe2013]. In our application, however, the amount of light detected does not change significantly; 4) we added a second LD, monitoring opposite faces with two different light detectors. The amount of light detected from each LD is found to be the 50% of the amount detected with a single LD. This causes an overall decrease of the signal to noise ratio, because each LD adds its own noise; 5) we replaced the crystal with a cylindrical one, 4[$\mathrm{~cm}$]{} in diameter and in height. Again the amount of light detected does not change. Summarizing, none of the above trials succeeded in providing a significant increase of the light collection efficiency, indicating that most of the light is absorbed by the [$\mathrm{TeO}_2$]{} crystal. This is due to the high refractive index of the [$\mathrm{TeO}_2$]{} crystal ($n\sim 2.4$ [@PhysRevB.4.3736]): many photons are reflected internally several times up to absorption. This effect is confirmed by the higher light yield obtained with the small (117 g) crystal and by preliminary results from simulations of the light collection. The setup providing the highest light signal, around 100[$\mathrm{~eV}$]{} at the [0$\nu\beta\beta$]{}, consists in a single LD with the crystal surrounded by the VM 2002 reflector or wrapped with teflon tape. Perspectives ============ The recent CUORE-0 result restricted the prediction of the amount of $\alpha$ background in [CUORE]{} from $B_\alpha = 0.01-0.04$ to $B_\alpha=0.01{{\ensuremath{\mathrm{~counts/(keV\,kg\,y)}}}}$ [@Aguirre:2014lua], while the ultimate source of background, due to $\beta/\gamma$ radioactivity from the setup, still amounts to $B_{\beta/\gamma} = 0.001{{\ensuremath{\mathrm{~counts/(keV\,kg\,y)}}}}$. From these numbers and from the specs of [CUORE]{} we perform toy [Monte Carlo]{} simulations to estimate the 90%[$\mathrm{~C.L.}$]{} sensitivity of [CUORE]{} equipped with light detectors. The outcome of a toy experiment is fitted in energy with a flat probability density function (pdf) for the background and a Gaussian pdf for the signal, and is simultaneously fitted in light with Gaussians pdfs for the $\alpha$ and $\beta/\gamma$ light distributions. The posterior pdf of the signal events is obtained integrating over the nuisance parameters and assuming a flat prior. The sensitivity of a single experiment ($N_{90}$) is computed as the number of signal events corresponding to the 90% of the posterior cumulative distribution. Several ($\sim 3000$) experiments are generated, and the median of $N_{90}$ is used as estimator of the sensitivity. The entire procedure is repeated while varying the signal to noise ratio in the light detector (Fig. \[fig:sensitivity\]). From the figure one sees that the application of light detectors to [CUORE]{} would increase its 90% C.L. sensitivity to the half-life of [$^{130}\mathrm{Te}$]{} to $2.7\cdot 10^{26}{\ensuremath{\mathrm{~y}}}$, a factor 3 higher than [CUORE]{} without light detectors. To achieve this goal one needs a signal to noise ratio in the light detector greater than 5, a value that is far from that featured by the setup in this work, equal to $101/72=1.4{\ensuremath{\mathrm{~eV/eV}}}$. From the results presented the increase of the light signal is difficult, and therefore to upgrade [CUORE]{} light detectors able to provide a noise level below $20{\ensuremath{\mathrm{~eV~RMS}}}$ are needed. Other than trying to improve the NTD technology, there are at least two possible alternatives. The use of phonon-mediated transition edge sensors (TES), as in the CRESST dark matter experiment [@Angloher:2011uu], or the use of phonon-mediated kinetic inductance detectors (KID), as recently proposed in Ref. [@CalderLTD13]. The TES technology has already proved to reach very good noise levels, but the implementation of 988 light detectors implies a complicated readout, mainly because of the cryogenic SQUID amplifiers that are employed. KIDs already proved to be a highly multiplexable technology in astrophysical applications (up to $400$ channels on the same readout line [@Bourrion:2012td]) but the required energy resolution in our application still needs to be demonstrated. ![90% C.L. sensitivity to the half-life of [$^{130}\mathrm{Te}$]{} as a function of the signal to noise ratio of the light detectors, under the reasonable hypothesis of an $\alpha$ background index in [CUORE]{} of $0.01{{\ensuremath{\mathrm{~counts/(keV\,kg\,y)}}}}$. The sensitivity of the experiment without light detectors corresponds to ${\rm S/N = 0}$. When ${\rm S/N > 5}$ the $\alpha$ background is hidden by the unreducible background predicted from $\gamma$ interactions, amounting to $0.001{{\ensuremath{\mathrm{~counts/(keV\,kg\,y)}}}}$, and the sensitivity is maximal. The performance of the light detectors used in this work, ${\rm S/N} = 1.4$, is clearly too low. []{data-label="fig:sensitivity"}](fig4.pdf){width="48.00000%"} Conclusions =========== We tested the possibility to discriminate the $\alpha$ background in [CUORE]{} by tagging the signal from $\beta$ particles through the detection of [Cherenkov]{} light. The detected light at the [$^{130}\mathrm{Te}$]{} $Q$-value is around 100[$\mathrm{~eV}$]{} for $\beta/\gamma$ particles and no light is detected from $\alpha$ interactions, confirming the validity of this technology. However, the signal is small at the same level of the noise of the bolometric light detectors we are using, and does not allow us to perform an event by event discrimination of the background. We tested modifications of the setup, by using different light reflectors or multiple light detectors, but the light yield did not increase. We are working on simulations to estimate the fraction of emitted light that escapes the crystal and is eventually absorbed by the light detector. Critical parameters are the index of refraction and the absorbance of [$\mathrm{TeO}_2$]{}, which unfortunately are not available in the literature for low temperatures. To this end we are setting up a dedicated measurement. Given the results obtained so far, we conclude that, to remove completely the $\alpha$ background in [CUORE]{}, light detectors with a noise of $20{\ensuremath{\mathrm{~eV~RMS}}}$ are needed, a factor 3-4 times better than the bolometric light detectors we used in this work. Changing the technology to TES or KID devices could be an alternative, provided that the present readout and sensitivity limits are overcome. [CUORE]{} without $\alpha$ background would reach a 90% C.L. sensitivity to the [0$\nu\beta\beta$]{} half-life of more than $3\cdot 10^{26}{\ensuremath{\mathrm{~y}}}$, a factor 3 better than the upcoming experiment. Combining the light readout with an enrichment in [$^{130}\mathrm{Te}$]{} from the natural 34% to $\sim90\%$ would push the half-life sensitivity by another factor $\sim 3$. Depending on the choice of the nuclear matrix elements, this corresponds to an effective neutrino mass sensitivity in the range $14 -35 {\ensuremath{\mathrm{~meV}}}$, down into the inverted hierarchy of neutrino masses. Acknowledgements {#acknowledgements .unnumbered} ================ The authors thank the CUORE collaboration for providing the [$\mathrm{TeO}_2$]{} crystal. This work was supported by the European Research Council (FP7/2007-2013) under contract LUCIFER no. 247115 and by the Italian Ministry of Research under the PRIN 2010- 2011 contract no. 2010ZXAZK9.

--- abstract: 'Photon coincidence spectroscopy relies on detecting multiphoton emissions from the combined atom-cavity system in atomic beam cavity quantum electrodynamics experiments. These multiphoton emissions from the cavity are nearly simultaneous approximately on the cavity lifetime scale. We determine the optimal time for the detection window of photon pairs in two-photon coincidence spectroscopy. If the window time is too short, some photon pairs will not be detected; if the window time is too long, too many nearly coincident independent single photons will be falsely interpreted as being a photon pair.' address: ' Department of Physics, Macquarie University, Sydney, New South Wales 2109, Australia ' author: - 'L. Horvath, B. C. Sanders and B. F. Wielinga' title: 'Non-simultaneity in two-photon coincidence spectroscopy' --- Introduction {#intro} ============ Cavity quantum electrodynamics (CQED) has continued to develop rapidly, driven both by recent experimental successes and by the promise of exciting new applications. Advances in atom cooling techniques, as well as development of high-Q optical cavities with large dipole coupling, have enabled testing of the strong-coupling regime of CQED [@Thompson98]. Single-atom experiments are now feasible [@Hood98; @Mabuchi98], and the possibility of trapping atoms in optical cavities is tantalisingly close [@Parkins]. Potential applications include quantum logic gates [@Turchette95]. Applications of CQED rely critically on the quantum effects, namely the entanglement between the field degree of freedom and the internal electronic state of the atom [@Carmichael96; @Brune96]. This entanglement is not only challenging to achieve, it is also difficult to probe. In the optical regime of CQED, photon coincidence spectroscopy (PCS) has been proposed as a feasible and unambiguous method for detecting genuine quantum effects in CQED. This technique employs a multichromatic driving field acting on the combined atom-cavity system and detects multiphoton decays by detecting photon coincidences in the cavity output field [@Carmichael96; @Sanders97; @Horvath99]. A difficulty arises in determining whether emitted photons are coincident or not. Let us consider a single two-level atom (2LA) coupled to a single mode of an optical cavity, and $\omega$ is the angular frequency of both the cavity mode and the 2LA. Multi-atom effects can be ignored provided that the atomic density is sufficiently small [@Carmichael99]. In the electric dipole and rotating-wave approximations, the Hamiltonian is $$\label{JCH} H(g) = \omega (\sigma_z + a^{\dagger} a) + i g ( {\bf r} ) (a^{\dagger} \sigma_- - a \sigma_+ )$$ with $\bf r$ the position of the atom, $g({\bf r})$ the position-dependent dipole coupling strength, $a$ and $a^{\dagger}$ the annihilation and creation operators for photons in the cavity field, $\sigma_+$, $\sigma_-$, and $\sigma_z$ the 2LA raising, lowering and inversion operators, respectively, and $\hbar = 1$. The spectrum for this Hamiltonian is depicted in Fig. \[fig:ladder\] and is the well-known Jaynes-Cummings spectrum, or ‘ladder’ [@Jaynes63]. The ‘dressed states’ of the combined atom-cavity system are designated by the lowest-energy state $$\label{dressed:ground} |0\rangle \equiv |0 \rangle_{\rm cav} \otimes |{\tt g} \rangle_{\rm atom} ,$$ and, for $n$ a positive integer, $$\label{dressed:excited} |n \rangle_{\pm} \equiv \frac{\imath}{\sqrt{2}} \left( |n-1\rangle_{\rm cav} \otimes |{\tt e}_{\rm atom} \rangle \pm \imath |n\rangle_{\rm cav}\otimes |{\tt g}\rangle_{\rm atom} \right),$$ where $ |n\rangle $ is the Fock state of the cavity mode and $ |{\tt g}\rangle \, ( |{\tt e}\rangle ) $ is the ground (excited) state of the 2LA. Here we are concerned with two-photon coincidence spectroscopy (2PCS) which proceeds, first by driving the atomic beam with a bichromatic field which causes two-photon excitation to the second couplet of the JC ladder, followed by two-photon decay from the atom-cavity system. The objective is to count photon pairs emitted from the cavity as the frequencies of the driving field are varied. When the sum frequency is $2\omega\pm \sqrt2 g$, we expect to see a resonantly enhanced two-photon count rate (2PCR). Of course, $g$ is a random variable due to beam fluctuations, and this leads to inhomogeneous broadening. Despite these difficulties, 2PCS appears to be a feasible method for detecting the characteristic splitting in the JC model [@Carmichael96; @Sanders97]. However, improvements in the procedure are important to ensure that the detection process is optimised. In the following analysis we investigate the appropriate window time for counting photon pairs. Photon pairs are not emitted from the cavity simultaneously due to the randomness of photoemission from an optical cavity. The detection of a photon pair thus depends on identifying a window time $\tau_{\rm w}$ such that, for two photons detected with temporal separation $t$ such that $t < \tau_{\rm w}$, the two photons are deemed to be members of a pair, and, if $t > \tau_{\rm w}$, are deemed to be independent single photons (not members of a pair). Here we determine the optimal window time $\tau_{\rm opt}$ which maximises the counting rate of genuine pairs relative to the rate of false pair counts. The Master Equation {#sec:master} =================== The Hamiltonian (\[JCH\]) for the combined atom-cavity system ignores the driving field emission of photons from the system. The Hamiltonian is useful in so far as it yields the spectrum for the combined atom-cavity system, but the full quantum master equation is necessary to calculate the quantities relevant to experiments, namely the two-photon count rate (2PCR). The experiment proceeds by measuring the 2PCR as a function of the bichromatic driving field’s scanning field frequency. Two-photon excitation is provided by driving the atom directly with a bichromatic field, characterised by the time-dependent variable $${\cal E}(t) = {\cal E}_1 e^{-i\omega_1 t} + {\cal E}_2 e^{-i\omega_2 t} .$$ The angular frequency $\omega_1$ is fixed and resonantly excites the atom-cavity system from the ground state $|0\rangle$ to the excited state $|1\rangle_-$ for the subensemble $g=g_f$. That is, provided that $$\label{gf} g_f = \omega - \omega_1 ,$$ the bichromatic driving field will resonantly excite the subensemble of atom-cavity systems for which $g=g_f$. Of course subensembles for which $g \neq g_f$ can also be excited, but these excitations are non-resonant and hence less significant. The second frequency, $\omega_2$, is scanned over a particular frequency range. The purpose of the second component of the bichromatic field is to excite to one of the two states in the second couplet of the Jaynes-Cummings ladder, namely $|2\rangle_{\pm}$. Thus, the range of scanning frequencies for $\omega_2$ must include the $|1\rangle_- \longleftrightarrow |2\rangle_{\pm}$ transition frequencies, $$\omega \pm ( \sqrt{2} \mp 1 ) g ,$$ respectively. The amplitudes of the two chromatic components must be large enough to ensure sufficient occupation of the excited state but not large enough that significant Stark shifting or nonnegligible occupation of the higher-order states occurs. Enhanced rates of photon pair detection are then sought as the scanning frequency $\omega_2$ is varied. The enhanced 2PCR occurs at the resonances shown in Fig. \[fig:ladder\]. In addition to a driving term, loss terms must also be included. The Born-Markov approximation is applied to both radiation reservoirs: the reservoir for the field leaving the cavity and the reservoir for direct fluorescence of the 2LA from the sides of the cavity. The cavity damping rate is $\kappa$, and the emission rate into free space is $\gamma$, where $\gamma$ is the inhibited spontaneous emission rate due to the restriction of modes by the cavity. The master equation [@Sanders97] can be expressed as $\dot{\rho} = {\cal L}\rho$ for $\cal L$ the Liouvillean superoperator. More specifically the Liouvillean superoperator can be expressed as the sum of a time-independent Liouvillean operator, a time-dependent Liouvillean operator and a ‘jump’ term. By defining $\delta=\omega_2-\omega_1$, in the rotating picture of ${\cal E}_1$ the master equation is $$\label{master} \dot{\rho}(t;\delta,g) = \left[ {\cal L}(g) + {\cal L}(t;\delta) + {\cal J} \right] \rho(t;\delta,g)$$ for $$\begin{aligned} \label{Heff} H_{\rm eff}(g) &=& \left( \omega - \omega_1 \right) (\sigma_z + a^{\dagger} a) + i g (a^{\dagger} \sigma_- - a \sigma_+ ) \nonumber \\ &&+ i {\cal E}_1 ( \sigma_+ - \sigma_- ) - i\kappa a^{\dagger} a - i(\gamma/2) \sigma_+ \sigma_-\end{aligned}$$ a non-Hermitian Hamiltonian, the first term is $${\cal L}(g) \rho = -i \left[ H_{\rm eff}(g) \rho - \rho H_{\rm eff}^{\dagger} (g) \right],$$ the time-dependent term is $$\label{L(t)} {\cal L}(t;\delta) \rho = {\cal E}_2 \left[ e^{-i\delta t} \sigma_+ - e^{i\delta t} \sigma_- , \rho \right],$$ and the jump term is $$\label{jump} {\cal J} \rho = \gamma \sigma_- \rho \sigma_+ + 2 \kappa a \rho a^{\dagger} .$$ The atom-field coupling strength $g$ depends on the atomic position ${\bf r}$. Provided that the atoms move sufficiently slowly through the cavity [@Carmichael96; @Sanders97], the atom can be treated as if it were at rest at some randomly located position ${\bf r}$. The interaction of the atom with the cavity is then described by the master equation in the asymptotic large time ($t \longrightarrow \infty$). As the position ${\bf r}$ is a randomly varying quantity, the value of the coupling strength $g$ itself is also random. Hence, a coupling strength distribution $P(g)$ can be constructed [@Sanders97]. The resultant density matrix is given by $$\label{density:matrix} \rho(t;\delta) = \int_{Fg_{\rm max}}^{g_{\rm max}} P(g) \rho(t;\delta,g) dg$$ where $g_{\rm max}$ is the coupling strength at a cavity node and $Fg_{\rm max}$ is the effective lower bound cut-off for the coupling $(0<F<1)$. The effect of averaging over $P(g)$ is an inhomogeneous spectral broadening. This broadening is due to atomic position variability. In Fig. \[fig:P(g)\] two typical distributions $P(g)$ are depicted, one for the case of a uniformly distributed atomic beam entering the cavity and the second for an atomic beam initially passing through a rectangular mask [@Sanders97]. In both cases we assumed a single-mode cavity supporting a TEM$_{00}$ mode. For a bichromatic driving field, the density matrix (\[density:matrix\]) does not settle to a steady state value. The time-dependence of the density matrix in the long-time limit can be treated by making a Bloch function expansion of the density matrix [@Sanders97]. In the Bloch function expansion, the density matrix is written as the sum $$\label{Bloch} \rho(t) = \sum_{N=0}^{\infty} \rho_N(t) e^{\imath N \delta t}$$ with $\rho_N(t)$ time-dependent matrices. In the long-time limit, $\dot{\rho}_N \approx 0$ and $\rho_N$ can thus be regarded as time-independent. As the photocount integration time is expected to be long compared to the frequency $\delta$, it is reasonable to approximate $\rho(t)$ by truncating the expansion (\[Bloch\]). The Two-Photon Count Rate {#sec:2PCR} ========================= The two-photon count rate (2PCR) can be obtained in more than one way. Ideally one would have a perfectly efficient photodetector which detects all photons leaving one side of the cavity. The photodetector would then provide a complete record of photon emissions from the cavity as a function of $t$. A perfect coincidence would then arise as two simultaneously detected photons at some time $t$. However, there are two problems. One problem is that there does not exist a perfectly efficient photodetector. Therefore, some pairs of photons are observed as single-photon emissions because one member of the pair escapes observation. In fact some pairs are missed altogether because both photons escape detection. The other problem concerns the detection of two simultaneously created photons. Although created simultaneously, the emission from the cavity is not simultaneous due to the randomness of the emission time which is characterised by the cavity lifetime $1/\kappa$. A better and more accurate way to describe two-photon detections is to employ the 2PCR. To begin with, we consider two photons to be coincident provided that they arrive within a time interval $\tau_{\rm w}$, the ‘window time’. The choice of window time is not obvious, and it is our aim here to determine what the window time should be. As the two simultaneous photons can be separated by a time of order $\kappa^{-1}$, as discussed above, the window time $\tau_{\rm w}$ might be expected to be on the order of $\kappa^{-1}$. However, our purpose here is to consider the choice of $\tau_{\rm w}$ in detail and to identify the optimal choice of window time $\tau_{\rm w}$ which will produce the best measure of the 2PCR. The choice of optimal window time is further complicated by the method of detecting nearly simultaneous photons. In the ideal case discussed above of a perfect photodetector yielding a record of all photon emissions from the cavity, one can then define a two-photon event as taking place if a second photon arrives between times $t_0$ and $t_0 + \tau_{\rm w}$, [*conditioned*]{} on a photodetection at time $t_0$. We refer to this rate as the [*conditional*]{} 2PCR and define this rate to be $$\begin{aligned} \label{2PCR:con} \Delta^{(2)}_{\rm con}(\delta,g,\tau_{\rm w}) &\equiv& \lim_{t_0 \rightarrow \infty}\frac{1}{\tau_{\rm w}} \int_{t_0}^{t_0+\tau_{\rm w}} dt \nonumber \\ && \times \left\langle : \hat{n}(t_0) \hat{n}(t_0+t) : \right\rangle (\delta,g).\end{aligned}$$ The number operator in eq (\[2PCR:con\]) is defined as $\hat{n}(t) \equiv \hat{a}^{\dagger}(t) \hat{a}(t)$, and ‘$: \, :$’ refers to normal ordering. The averaging is performed for the density matrix of eq (\[master\]). The conditional 2PCR for a window time $\tau_{\rm w} = \kappa^{-1}$ was used in the quantum trajectory analysis of PCS in Ref. [@Carmichael96]. Another natural way to measure the 2PCR is by counting all photon pairs defined as being separated by an interval less than $\tau_{\rm w}$. This 2PCR is referred to as the [*unconditional*]{} 2PCR and does not rely on starting the count for the second photon conditioned on detecting the first photon. The definition of the unconditional 2PCR is $$\begin{aligned} \label{2PCR:unc} \Delta^{(2)}_{\rm unc}(\delta,g,\tau_{\rm w}) &=& \lim_{t_0 \rightarrow \infty} \frac{2}{\tau_{\rm w}^{2}} \int_{t_0}^{t_0+\tau_{\rm w}} dt^\prime \int_{t_0}^{t^\prime}dt \nonumber \\ && \times \left\langle : \hat{n}(t)\ \hat{n}(t^\prime) : \right\rangle (\delta,g). \end{aligned}$$ As shown in Appendix A, this expression can be simplified to read $$\begin{aligned} \label{two:result} \Delta^{(2)}_{\rm unc}(\delta,g,\tau_{\rm w}) = \frac{2}{\tau_{\rm w}^{2}} \int_0^{\tau_{\rm w}} & du & \int_0^{u} dw \nonumber \\ & \times & \left \langle : \hat{n}(0)\hat{n}(w) : \right \rangle (\delta,g). \end{aligned}$$ We solve analytically for two extreme cases in Appendix A. The window time can be extremely long $(\kappa \tau_{\rm w} \gg 1)$, yielding expression (\[long:time\]), or extremely short $(\kappa \tau_{\rm w} \ll 1)$, yielding expression (\[short:time\]) for both conditional and unconditional 2PCR. The short window time ($\tau_{\rm w} \longrightarrow$ 0) was the basis of the analysis of 2PCS in Ref. [@Sanders97]. In this treatment both the conditional and unconditional 2PCR at time $t$ is approximated by $\langle:\hat{n}^2(t):\rangle$. In the long-time limit the 2PCR is dominated by Poissonian statistics. The Optimal Window Time {#sec:opt} ======================= The choice of optimal window time $\tau_{\rm opt}$ depends on the technique for observing two-photon coincidences, but another factor must also be considered. The purpose of 2PCS is to observe two-photon decay resonances from the combined atom-cavity system. As explained in Refs [@Carmichael96; @Sanders97], there are three peaks in the 2PCR as a function of the scanning frequency $\delta$. These peaks are shown in Fig. \[adadaa\] as a function of the normalised scanning field frequency $$\label{norm:delta} \tilde\delta=\frac{\omega_2-\omega}{\omega-\omega_1}.$$ The choice of $\tau_{\rm w}$ will depend on which peak is being observed. However, the best peak for observing a two-photon decay resonance occurs for $$\label{delta:peak} \tilde\delta = 1+\sqrt2.$$ This resonance corresponds to the transition $|1)_- \! \longleftrightarrow~\!|2)_+$. This peak does not occur in a semiclassical description of intensity correlations in the cavity output field and therefore serves as a signature of a genuine quantum field effect in CQED. Moreover, this peak can be observed without the added complication of having to perform the experiment twice, once with a bichromatic field and once again with a monochromatic field, in order to perform the signal enhancement technique of background subtraction [@Carmichael96; @Sanders97]. Finally detection of this peak is the most feasible of the three dominant peaks in the two-photon spectrum. Hence, we consider specifically $\tau_{\rm w}$ for this peak at $\tilde\delta$ given by eq (\[delta:peak\]). In Fig. \[adadaa\] we observe that the 2PCR peak sits on a background 2PCR which is largely independent of $\tilde\delta$ and occurs at $\overline{\langle : n^2 : \rangle} \approx 2.1\times 10^{-5}$. Let us characterise the quality of this 2PCR peak by the ratio of the peak height to the height of the background 2PCR. We can understand this ratio in terms of signal to noise, where signal is the 2PCR from the sought-for two-photon decay events, and the background noise corresponds to two-photon decays arising from unwanted off-resonance two-quantum excitations and decay events. The peak-to-valley ratio (PVR) is determined by the height of the peak to the height of the background (or valley) 2PCR. The optimal window time $\tau_{\rm w} = \tau_{\rm opt}$ is defined such that the PVR for this 2PCR is maximal. That is, either a larger or a smaller choice of the window time would reduce the value of the PVR making the peak more difficult to detect. There are other concerns besides the PVR in choosing the window time. For example, choosing a much shorter window time could improve the PVR but also lengthen the run time of the experiment in order to accumulate enough signal. That is, the absolute height of the peak is also a matter of concern in determining the feasibility of the experiment and is determined by the allowable timescale of the experiment. The minimum height would need to be on the order of $T^{-1}$ for $T$ the timescale of the data collection. The PVR is obtained numerically. The matrix continued fraction method is used to solve the master equation to determine the peak height. The background, or valley, can be solved analytically though by treating the detuning of the scanning field as large. The details are provided in Appendix B. The 2PCR for large $\tilde\delta$ is given by expressions (\[offres:c2PCS\]) and (\[offres:2PCS\]). The peak-to-valley ratio 2PCR is thus $$\label{PVR} {\rm PVR_{\xi}} = \frac{ \Delta^{(2)}_{\xi} (\tilde\delta, g, \tau_{\rm w}) } { \left (\Delta^{(2)}_{\rm o} \right )_{\xi} (g, \tau_{\rm w} ) }$$ where $\xi\in\{\rm con, \rm unc\}$. In Fig. \[fig:surfPVR\] surface plots of the PVR [*vs*]{} $g$ and $\tau_{\rm w}$ reveals that the PVR increases as $g$ decreases. This is due to the background signal of two-photon coincidences for $\tilde\delta$ large becoming negligible as shown in Fig. \[adadaa\]. Although the PVR improves as $g$ decreases, the signal of two photon coincidences within the window time $\tau_{\rm w}$ decreases. This decrease is due to the fact that the resonant frequency for the transition $|0)\longleftrightarrow |1)_-$ is $\omega-g$, but we have constrained the pump field frequency by (\[gf\]). Hence, as $g$ decreases, the pump field drives the system more and more off resonance. The window time $\tau_{\rm w}$ for achieving the optimal PVR is an order of magnitude smaller than $\kappa^{-1}$. The optimal time $\tau_{\rm opt}$ exhibited in Fig. \[fig:optvsg\] is a function of coupling strength and in Fig. \[fig:resopt\] is a function of $\gamma/\kappa$ (averaged over $P(g)$). We are concerned specifically with the 2PCR and the PVR for the system with a coupling constant distribution based on the TEM$_{00}$ mode [@Carmichael96; @Sanders97]. The coupling strength distribution is depicted in Fig. \[fig:P(g)\], and we treat the masked beam case which enhances the large-coupling effect. In Fig. \[fig:avgPVR\] we present the PVR for the density matrix $\rho$ of expression (\[density:matrix\]), averaged over $P(g)$. There is a peak of the PVR for each $g/\kappa$ given by $\tau_{\rm w}=\tau_{\rm opt}$ (the optimal time window for observing the peak (\[delta:peak\])). These values of $\tau_{\rm opt}$ are plotted in Fig. \[fig:optvsg\] for $0\leq g/\kappa\leq 10$ for a range of values of $\gamma$. Of particular interest here is the very weak dependence of $\tau_{\rm opt}$ on $\gamma$ where $\gamma$ is varied by a factor of $20$. Moreover, $\tau_{\rm opt}$ is generally decreasing as $g/\kappa$ increases. Discussion ========== In Fig. \[fig:avgPVR\] we observe a maximum of the PVR for each of the assumed $P(g)$ in Fig. \[fig:P(g)\]. This peak occurs at $\kappa\tau_{\rm opt}\approx 0.111$ for the conditional 2PCR and at $\kappa\tau_{\rm opt}\approx 0.135$ for the unconditional 2PCR. We can understand the location of these peaks by referring to Fig. \[fig:optvsg\]. In Fig. \[fig:optvsg\](a) the values of $\kappa\tau_{\rm opt}$ for the conditional 2PCR are predominantly between $0.11$ and $0.16$, but the value of $\kappa\tau_{\rm opt}$ in the vicinity of $g/\kappa=9$ is between $0.05$ and $0.11$. Due to the resonance condition (\[gf\]) this region of the $\kappa\tau_{\rm opt}$ [*vs*]{} $g/\kappa$ curve is more significant. Hence, the dependence of $\tau_{\rm opt}$ on $\gamma$ as depicted in Fig. \[fig:resopt\] is dominated by the $g/\kappa=9$ region of Figs. \[fig:optvsg\]. Similarly, we can estimate the precise value of  $\tau_{\rm opt}$ for the unconditional 2PCR from the dashed line of Fig. \[fig:resopt\] in the context of the values of $\tau_{\rm opt}$ in Fig. \[fig:optvsg\](b). The linear dependence of $\tau_{\rm opt}$ on $\gamma$, as seen in Fig. \[fig:resopt\] yields a correlation of 0.9983 for the conditional 2PCR, and a correlation of 0.9995 for the unconditional 2PCR. In the conditional case $$\label{av:cTopt} \kappa \tau_{\rm opt}\approx-(1.4\times 10^{-3})\gamma/\kappa+ 0.11 \, (\rm con),$$ and in the unconditional case $$\label{av:Topt} \kappa \tau_{\rm opt}\approx-(2.1\times 10^{-3})\gamma/\kappa+ 0.14 \,(\rm unc).$$ The low values of the slopes are indicative of the weak dependence on $\gamma$. Formulae (\[av:cTopt\]) and (\[av:Topt\]) can be used to fine-tune the choice of optimal window time, and the linear relationship simplifies the task of interpolating to obtain $\tau_{\rm opt}$. Conclusions {#sec:conclusions} =========== We have determined expressions for the optimal window time $\tau_{\rm opt}$ for both conditional and unconditional 2PCR. These expression provide an optimal PVR for the 2PCR peak at $\tilde\delta=1+\sqrt{2}$ corresponding to the $|1)_-\longrightarrow |2)_+$ transition. Although we have determined $\tau_{\rm opt}$ for certain parameters and for the coupling-strength distribution $P(g)$ (solid line of Fig. \[fig:P(g)\]), the algorithm presented here is sufficiently general to allow calculation of $\tau_{\rm opt}$ for other parameters and other coupling-strength distributions. In general $\tau_{\rm opt}$ is smaller than $\kappa^{-1}$ by an order of magnitude for both the conditional and unconditional 2PCR. There is some dependence on $\gamma$, but this dependence is weak and is close to linear in the cases studied here. Analyses of optimal window times are aided by studies of $\tau_{\rm opt}$ for particular values of $g$, that is, for the coupling-strength distribution $P(g)$ corresponding to $\delta(g-g_0)$ for some $g_0$. These calculations provide good estimates of the optimal window time for general $P(g)$. A longer window time may be desirable, however, if the timescale for collecting enough data is not experimentally feasible. A compromise between the two objectives of optimising the PVR and of accumulating sufficient data to produce a large peak height may be necessary. An important technique discussed in Refs. [@Carmichael96; @Sanders97] was background subtraction. The principle behind this method is to remove the unwanted two-photon off-resonance excitation to the second couplet. Background subtraction is particularly important to improve the PVR for 2PCR peaks. However, we choose to study detection of the most promising peak experimentally (\[delta:peak\]). In our simulations we determine the PVR for this peak both by performing background subtraction and without background subtraction, and we obtain plots in agreement with Figs. \[fig:surfPVR\] and \[fig:avgPVR\]. These results confirm the assertion in Ref. [@Sanders97] that “the resonance frequency lies outside the inhomogeneous line and the resonance should be resolved even in the presence of the two-photon background”. Hence, background subtraction is not necessary to obtain optimal window times for resolving the peak (\[delta:peak\]). Acknowledgements {#acknowledgements .unnumbered} ================ Dr S. M. Tan has provided Matlab$^{\rm TM}$ programs to us which was used initially to double check our simulations of the monochromatically driven Jaynes-Cummings system. We have benefited from valuable discussions with H. J. Carmichael, J. D. Cresser, Z. Ficek, K.-P. Marzlin, and S. M. Tan. This research has been supported by Australian Research Council Large and Small Grants and a Macquarie University Research Grant. A: The conditional and unconditional two-photon count rate (2PCR) {#app:rate .unnumbered} ================================================================= In the long-time limit, the conditional two-photon count rate (2PCR) is given by $$\begin{aligned} \label{two:photo:b} \Delta^{(2)}_{\rm con} (\delta, g, \tau_{\rm w}) = \frac{1}{\tau_{\rm w}}\int_{0}^{\tau_{\rm w}} dt \left \langle : \hat{n}(0) \hat{n}(t): \right \rangle(\delta, g).\end{aligned}$$ If the time window $\tau_{\rm w}$ is large, compared to $\kappa^{-1}$ (the cavity lifetime), the two photons are highly decorrelated, and we can approximate $$\label{long:time:exp} \left\langle : \hat{n}(0) \hat{n}(t) : \right\rangle(\delta,g) \longrightarrow \left\langle \hat{n}(0) \right\rangle^{2}(\delta,g).$$ Thus, $$\label{long:time} \Delta^{(2)}_{\rm con}(\delta,g, \tau_{\rm w}) \longrightarrow \left\langle \hat{n}(0) \right\rangle^{2}(\delta,g).$$ This count rate reflects the Poissonian nature of the count statistics for long window times. On the other hand, for $ \kappa \tau_{\rm w} \ll 1$, the correlation between photon pairs cannot be neglected. Hence, the count rate reduces to $$\begin{aligned} \label{short:time} \Delta^{(2)}_{\rm con}(\delta,g, \tau_{\rm w})\longrightarrow \left \langle : \hat{n}^2(0) : \right\rangle (\delta,g)\end{aligned}$$ which is the approximation employed in Ref [@Sanders97]. Similarly, in the long-time limit, the unconditional 2PCR is $$\label{two:photo:a} \Delta^{(2)}_{\rm unc}\left( \delta,g,\tau_{\rm w} \right) = \frac{2}{\tau_{\rm w}^{2}} \int_0^{\tau_{\rm w}} dt^\prime \int_0^{t^\prime}dt \left\langle : \hat{n}(t)\hat{n}(t^\prime) : \right \rangle \left( \delta, g \right).$$ This expression can be simplified as we show below. First we make the substitution $u_{\pm} = (t^\prime \pm t)/\sqrt{2}$. We also introduce the notation $d^2 u=du_- du_+$ and let $\nu$ be the union of the two regions $\{0 < u_- < \tau_{\rm w}/\sqrt2, 0 < u_+ < u_-\}$ and $\{\tau_{\rm w}/\sqrt2 < u_- < \sqrt2 \tau_{\rm w}, 0 < u_+ < \sqrt2\tau_w-u_-\}$ This substitution transforms the above double integral into the sum of two double integrals: $$\begin{aligned} \label{two:arrange} \Delta^{(2)}_{\rm unc}(\delta,g,\tau_{\rm w}) &=& \frac{2}{\tau_{\rm w}^{2}} \int\int_{\nu} \left\langle : \hat{n} \left(\frac{u_{+}-u_{-}}{\sqrt{2}}\right) \hat{n} \left(\frac{u_{+}+u_{-}}{\sqrt{2}}\right) : \right\rangle (\delta,g) \nonumber \\ & = & \frac{2}{\tau_{\rm w}^{2}} \int\int_{\nu} \left \langle : \hat{n}(0) \hat{n}(\sqrt{2}u_{+}) : \right\rangle (\delta,g). \end{aligned}$$ The advantage of this expression is that the two-time photon number correlation depends on only one term in the double integral instead of both terms in the double integral. Greater simplification is possible and desirable for studying the short and long window time $\tau_{\rm w}$. Substituting $u_{\pm}= w_{\pm}/\sqrt{2}$ transforms eq (\[two:arrange\]) to $$\begin{aligned} \label{two:arrange:one} \Delta^{(2)}_{\rm unc}(\delta,g,\tau_{\rm w}) & = & \frac{1}{\tau_{\rm w}^{2}} \Bigg[ \int_0^{\tau_{\rm w}} dw_- \int_0^{w_-} dw_+ \nonumber \\ &&+ \int_{\tau_{\rm w}}^{2\tau_{\rm w}} dw_- \int_0^{2\tau_{\rm w}-w_-} dw_+ \Bigg] \nonumber \\ && \left\langle : \hat{n}(0) \hat{n} (w_+) : \right\rangle (\delta,g)\end{aligned}$$ which reduces to $$\begin{aligned} \label{two:res} \Delta^{(2)}_{\rm unc}(\delta,g,\tau_{\rm w}) & = & \frac{2}{\tau_{\rm w}^{2}} \int_0^{\tau_{\rm w}} du \int_0^{u} dw \nonumber \\ & & \times \left\langle : \hat{n}(0) \hat{n} (w) : \right\rangle (\delta,g) .\end{aligned}$$ For large ($\tau_{\rm w} \gg \kappa^{-1}$) and small ($\tau_{\rm w} \ll \kappa^{-1}$) window times $\Delta^{(2)}_{\rm unc}$ reduces identically to $\Delta^{(2)}_{\rm con}$ as shown in equations (\[long:time\]) and (\[short:time\]). B: Background of conditional and unconditional 2PCR {#b-background-of-conditional-and-unconditional-2pcr .unnumbered} =================================================== For the scanning field far off resonance ($\delta$ large), the time-dependent component of the Liouvillean (\[L(t)\]) can be ignored. In the interaction picture, the master equation can be written as $ \dot{\rho} = {\cal L} \rho $ with $\cal L$ time-independent. If $\rho$ is expressed as a vector, then $\cal L$ can be expressed as a complex matrix with $ \{ - \lambda_n | n \in {\cal Z}_{N^2} \} $ the set of eigenvalues for $N$ the number of levels in the Jaynes-Cummings ladder retained after truncation. The density matrix can be approximated by the sum $$\label{rhosum} \rho(t) = \sum_{n=1}^{N^2} \rho_n e^{- \lambda_n (t-t_0)}$$ for $ \{ \rho_n \} $ a set of time-independent $N \times N$ matrices. Thus, the conditional 2PCR (\[2PCR:con\]) can be written as $$\begin{aligned} \label{offres:c2PCS} \left (\Delta^{(2)}_{\rm o}\right )_{\rm con}(g, \tau_{\rm w}) & = & c_0 (g)+ \frac{1}{\tau_{\rm w}} \int_0^{\tau_{\rm w}} dt \nonumber \\ & \times & \sum_{n=1}^{N^2} c_n(g) \exp{\left[ -\lambda_n (g) t \right]} \nonumber \\ & = & \! c_0(g) \! \! + \! \sum_{n=1}^{N^2} \! \frac{ c_n (g) } { \mu_n(g) }\! \left \{ \! 1- e^{ -\mu_n (g) \!} \right \} \! \! .\end{aligned}$$ with $N^2$ scalar constants $\left \{ c_n(g) \right \}$ and $N^2$ constants $\left \{ \mu_n(g) \right \}$ where $$\mu_n (g) = \lambda_n (g) \tau_{\rm w} \neq 0, \hspace{0.2cm} {\rm Re}\left \{\lambda_n(g)\right \} \geq 0$$ as well. In the long window time limit, we equate $$\label{limit:Delta} c_0 (g) = \lim_{\tau_{\rm w} \longrightarrow \infty} \left(\Delta^{(2)}_{\rm o} \right )_{\rm con} (g,\tau_{\rm w}).$$ Expansion (\[offres:c2PCS\]) provides a useful method for calculating $\left (\Delta^{(2)}_{\rm o} \right)_{\rm con}(g,\tau_{\rm w})$. The function $\left ( \Delta^{(2)}_{\rm o} \right)_{\rm con}(g,\tau_{\rm w})$ is monotonically increasing because $\partial \left (\Delta^{(2)}_{\rm o} \right )_{\rm con} (g,\tau_{\rm w})/\partial \tau_{w} > 0$ if $\tau_{\rm w} \longrightarrow \infty$. Thus, $\partial \left(\Delta^{(2)}_{\rm o}\right)_{\rm con} (g,\tau_{\rm w})/\partial \tau_{w} \longrightarrow 0$ as the function approaches the limit given by (\[limit:Delta\]). In the same way, the unconditional 2PCR can be obtained: $$\begin{aligned} \label{offres:2PCS} \left(\Delta^{(2)}_{\rm o} \right)_{\rm unc}(g, \tau_{\rm w}) & = & c_0 (g)+ \frac{2}{\tau_{\rm w}^{2}} \int_0^{\tau_{\rm w}} du \int_0^{u} dw \nonumber \\ & &\times \sum_{n=1}^{N^2} c_n(g) \exp{\left[ -\lambda_n (g) w \right]} =c_0(g) \nonumber \\ && + \! 2 \! \sum_{n=1}^{N^2} \frac{ c_n (g) } { \mu_n (g) }\! \left\{ \! \frac{ e^{ -\mu_n (g) } - 1}{ \mu_n (g) }+ 1 \! \right\}\! . \end{aligned}$$ Thus, as in (\[limit:Delta\]), the long time limit reduces (\[offres:2PCS\]) to $c_0(g)$. R. J. Thompson, Q. A. Turchette, O. Carnal and H. J. Kimble, Phys. Rev. A, [**57**]{}, 3084 (1998). C. J. Hood, M. S. Chapman, T. W. Lynn and H. J. Kimble, Phys. Rev. Lett. [**80**]{}, 4157 (1998). H. Mabuchi, J. Ye and H. J. Kimble, J. Appl. Phys. B unpublished (1998). A. S. Parkins and H. J. Kimble unpublished. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi and H. J. Kimble, Phys. Rev. Lett. [**75**]{}, 4710 (1995). H. J. Carmichael, P. Kochan, and B. C. Sanders, Phys. Rev. Lett. [**77**]{}, 631 (1996). M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. [**76**]{}, 1800 (1996). B. C. Sanders, H. J. Carmichael, B. F.  Wielinga, Phys. Rev. A [**55**]{}, 1358 (1997). L. Horvath, B. C. Sanders and B. F. Wielinga, J. Opt. B: Quant. and Semiclassical Opt. (accepted). H. J. Carmichael and B. C. Sanders, Phys.  Rev. A, (accepted). E. T. Jaynes and F. W. Cummings, Proc. IEEE [**51**]{}, 89 (1963).

--- abstract: 'We present the results from the timing and spectral study of Mrk421 based mainly on the *Swift* data in the X-ray energy range obtained during the time interval 2015December–2018April. The most extreme X-ray flaring activity on the long-term, daily and intraday timescales was observed during the 2-month period which started in 2017December when the 0.3–10keV flux exceeded a level of 5$\times$10$^{-9}$ergcm$^{-2}$s$^{-1}$, recorded only twice previously. While the TeV-band and X-ray variabilities mostly were correlated, the source often varied in a complex manner in the MeV–GeV and radio–UV energy ranges, indicating that the multifrequency emission of Mrk421 could not be always generated in a single zone. The longer-term flares at X-rays and $\gamma$-rays showed a lognormal character, possibly indicating a variability imprint of the accretion disk onto the jet. A vast majority of the 0.3–10keV spectra were consistent with the log-parabolic model, showing relatively low spectral curvature and correlations between the different spectral parameters, predicted in the case of the first and second-order Fermi processes. The position of the synchrotron spectral energy distribution (SED) peak showed an extreme variability on diverse timescales between the energies $E_{\rm p}$$<$0.1keV and $E_{\rm p}$$>$15keV, with 15% of the spectra peaking at hard X-rays and was related to the peak height as $S_{\rm p}$$\varpropto$$E^{\alpha}_{\rm p}$ with $\alpha$$\sim$0.6, which is expected for the transition from Kraichnan-type turbulence into the hard-sphere  one. The 0.3–300GeV spectra showed the features of the hadronic contribution, jet-star interaction and upscatter in the Klein-Nishina regime in different time intervals.' author: - 'B. Kapanadze, A. Gurchumelia, D. Dorner, S. Vercellone, P. Romano, P. Hughes$^{5}$, M. Aller$^{5}$, H. Aller$^{5}$, O. Kharshiladze$^{6}$' title: 'SWIFT OBSERVATIONS OF MRK421 IN SELECTED EPOCHS. III. EXTREME X-RAY TIMING/SPECTRAL PROPERTIES AND MULTIWAVELENGTH LOGNORMALITY IN 2015DECEMBER–2018APRIL' --- INTRODUCTION ============ Blazars (BL Lacertae objects and flat-spectrum radio quasars) form the most violently variable class of active galactic nuclei (AGNs), with timescales ranging from a few minutes (in the keV–TeV energy range) to several years (radio to optical frequencies). Moreover, BL Lacertae sources (BLLs) are characterized by featureless spectra, variable radio–optical polarization, compact radio-structure and superluminal motion of some components and very broad continuum extending over the radio to the very high-energy (VHE, $E$$>$100GeV) $\gamma$-ray energy ranges. The bolometric luminosity occasionally can reach a level of 10$^{48}$ergs$^{-1}$, particularly, during the strong outbursts, by the $\gamma$-ray emission (see @f14). Consequently, BLLs are the most frequently detected class of the extragalactic TeV sources (65 out of 82, with redshifts $z$=0.03–0.61[^1]) and form one of the most important constituents of LAT 4-year Point Source Catalog (3FGL; @ace15). It is widely agreed that the extreme physical properties of BLLs are due to the beamed, non-thermal emission from a relativistic jet which is closely aligned with the observer$\textquotesingle$s direction (estimated viewing angles $\theta$$<$10) and characterized by the bulk Lorentz factor $\Gamma $$\sim$10, which occasionally attains values as high as $\Gamma$$ \sim$50 [@b08]. In the $\log\nu$–$\log\nu F_{\rm \nu}$ plane, BLLs generally demonstrate a double-humped, broadband SED. There is a consensus that the lower-energy component (extended over radio to UV–X-Ray frequencies) is produced by synchrotron emission of ultra-relativistic electrons [@c08]. A sub-class of the high-energy-peaked BLLs (HBLs, peaking at UV–X-ray frequencies; @p95 and references therein), are particularly important due to the disputed particle acceleration and cooling processes: their X-ray budget  should be filled by synchrotron photons from the highest-energy leptons (electrons and, possibly, positrons), while the radiative lifetime at these energies are very short [@m04]. Consequently, a detailed study of the timing and spectral behaviour of these sources on diverse timescales reveals the most plausible acceleration mechanisms and allows us to draw conclusions about the physical properties of the jet emission region. Moreover, since the synchrotron and inverse Compton (IC) cooling are expected to be extremely important at these energies, the intense X-ray timing/spectral study of the nearby, bright HBLs may provide us with very important clues about the injection and radiative evolution of the freshly-accelerated particles. In this regard, the X-ray Telescope onboard the satellite *Swift* (*Swift*-XRT; @b05) makes an outstanding contribution by performing a regular monitoring of selected BLLs in their visibility  periods, particularly, during the densely-sampled Target of Opportunity (TOO) observations[^2]. Owing to the excellent instrumental characteristics, good photon statistics and low background counts of *Swift*-XRT, we are able to search for flux and spectral variability on diverse time-scales (minutes to years), obtain high-quality spectra and derive different spectral parameters for bright HBLs even for exposures lasting a few hundred seconds. Period Dates MJD -------- ---------------------------------- ------------- -- 1 2015December8 to 2016June16 57364–57555 1a 2015December8 to 2016February4 57364–57422 1b 2016February6 to 2016June16 57424–57555 2 2016November25 to 2017June27 57717–57931 2a 2016November25 to 2017January29 57717–57782 2b 2017January31 to 2017June27 57784–57931 3 2017December3 to 2018April8 58090–58216 3a 2017December16 to 2018February19 58103–58168 3b 2018February21 to 2018April8 58170–58216 : \[perdiv\] The intervals and sub-intervals referred throughout the paper. 1.5pt ---------- ------------------------------ ----------- ------ --------------- ------------- ------------- ------------- ------------- ------------- ------------- ObsID Obs. Start – End MJD Exp. CR UVW1 UVW1 UVM2 UVM2 UVW2 UVW2 (UTC) (s) (ctss$^{-1}$) (mag) (mJy) (mag) (mJy) (mag) (mJy) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 35014240 2015-12-08 10:15:58–11:22:39 57364.431 1015 7.11(0.09) 11.53(0.11) 21.79(0.79) 11.72(0.04) 15.65(0.28) 11.51(0.10) 18.31(0.46) 35014241 2015-12-11 11:38:58–12:43:43 57367.487 959 21.14(0.15) 11.78(0.10) 17.29(0.63) 11.90(0.04) 13.25(0.28) 11.80(0.03) 14.00(0.28) 35014242 2015-12-14 09:56:58–11:02:04 57370.416 1069 49.04(0.22) 11.47(0.11) 23.03(0.79) 11.67(0.04) 16.39(0.28) 11.47(0.10) 18.99(0.46) 35014243 2015-12-17 11:20:58–12:26:26 57373.475 1015 43.58(0.21) 11.54(0.11) 21.59(0.79) 11.76(0.04) 15.09(0.28) 11.49(0.10) 18.65(0.46) 35014245 2015-12-18 04:50:58–23:29:49 57374.204 5892 29.34(0.14) 11.49(0.10) 22.61(0.74) 11.66(0.10) 16.55(0.38) 11.50(0.10) 18.48(0.41) ---------- ------------------------------ ----------- ------ --------------- ------------- ------------- ------------- ------------- ------------- ------------- The nearby (*z*=0.031), TeV-detected HBL source Mrk421 provides an unique X-ray space laboratory due to the features as follows [@b16; @k16; @k18a; @k18b]): (i) high brightness (with the *Swift*-XRT 0.3–10keV count rates CR$>$100ctss$^{-1}$ during strong flares, corresponding to de-absorbed fluxes $F_{\rm 0.3-10 keV}\gtrsim$2.5$\times$10$^{-9}$ergcm$^{-2}$s$^{-1}$; ; (ii) exceptionally strong outbursts (e.g., in 2013April; @p14 [@k16]); (iii) very large and fast timing/spectral variability on timescales down to a few hundred seconds; (iv) extremely hard spectra during strong flares with the photon index smaller than 1.6 and synchrotron SED peak shifting beyond 10keV etc. Moreover, the source is also bright in other spectral ranges, making it a frequent target of densely-sampled MWL campaigns (@m95, @a12 [@a15a; @a15b], @b16 etc.), which are crucial for checking the viability of models explaining the origin of the higher-energy SED component via the inter-band correlation study: (1) synchrotron self-Compton (SSC, scattering of synchrotron photons by their parent  lepton population; @m85); (2) external Compton (EC), with the low-energy photons from the accretion disc (AD), dust torus, narrow/broad line clouds upscattered by the jet ultra-relativistic particles [@d92]; (3) hadronic models, which embody a generation of the keV–TeV emission by relativistic protons, either directly (synchrotron-proton scenario; @ab11) or indirectly (e.g., synchrotron radiation by the electron population, produced by a cascade induced by the interaction of high-energy protons with the ambient photons; @m93). Despite the large number of publications related to the aforementioned observations, the details of the physical processes underlying the higher-energy SED component remain mainly unknown owing to (i) sparse multiwavelength (MWL) data during long periods; (ii) moderate or low sensitivity in the hard X-ray and $\gamma$-ray energy ranges in the past; (iii) many previous MWL campaigns were triggered in the epochs of enhanced X-ray and $\gamma$-ray activity, and, consequently, these studies are biased towards the high states of the source, while the distinct physical processes may play a dominant role during the moderate and** lower brightness states.  For the aforementioned reasons, we performed a detailed study of X-ray spectral and flux variability in the MWL context, focussed on the rich archival data obtained with XRT during 2005March–2015June [@k16; @k17a; @k18a; @k18b]. This campaign revealed extreme X-ray flares by a factor of 3–20 on the timescales of a few days–weeks between the lowest historical state and that corresponding to the *Swift*-XRT rate higher than 200ctss$^{-1}$. The source was characterized by extreme spectral and intraday flux variability, particularly during the strong flares. In the latter case, we were able to extract the 0.3–10keV spectra for the time intervals of 50–100 seconds and explore the observational features predicted in the framework of the various particle acceleration and emission scenarios. The distribution of different spectral parameters and their cross-correlations hinted at the importance of the first and second-order Fermi accelerations, changes in the turbulence type, possible hadronic contribution to the MeV–GeV emission etc. In this paper, we present the results of our detailed study of the timing/spectral behaviour of Mrk421 during the period 2015December–2018April, which was characterized by a very strong X-ray flare in the time interval 2017December–2018February, and showed an intensive flaring activity also in other parts of the here-presented period, revealed by the densely-sampled *Swift*-XRT observations. Using XRT observations, we checked correlations between the 0.3–10keV flux variability and those observed with different instruments: the Ultraviolet-Optical Telescope (UVOT; @r05) and the Burst Alert Telescope (BAT; @ba05) onboard *Swift*, the Large Are Telescope (LAT) onboard *Fermi* [@at09], MAXI [@m09], the First G-APD Cherenkov Telescope (FACT; @a13), the 40-m telescope of Owens Valley Radio Observatory (OVRO; @r11), the optical telescopes of Steward Observatory [@s09]. The paper is organized as follows: Section2 encompasses the description of the data processing and analyzing procedures. The results of the X-ray and MWL timing study, as well as those from the X-ray spectral analysis are presented in Section3. We provide a discussion, based on our results, and the corresponding conclusions in Section4. Finally, the summary of our study is given in Section5. -------- ---------------- ------- ------------- ------------------------- --------------- ------- ------------- ------------------------- --------------- ------- ------------- ------------------------- -- -- -- -- -- Per. $CR_{\rm max}$ $\Re$ Mean $100\times F_{\rm var}$ $F^{\rm max}$ $\Re$ Mean $100\times F_{\rm var}$ $F_{\rm max}$ $\Re$ Mean $100\times F_{\rm var}$ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) 1 95.72(0.32) 13.5 31.53(0.02) 50.5(0.1) 15.85(0.18) 12.3 7.19(0.01) 41.6(0.1) 13.93(0.38) 59.5 3.55(0.01) 75.0(0.2) 1a 67.83(0.28) 9.5 37.98(0.03) 35.4(0.1) 12.76(0.26) 9.9 7.60(0.01) 30.3(0.1) 7.96(0.24) 34.0 3.31(0.01) 44.5(0.3) 1b 95.72(0.32) 11.4 26.56(0.03) 60.3(0.1) 15.85(0.18) 8.3 6.62(0.01) 55.9(0.1) 13.93(0.38) 46.3 38.90(0.01) 95.9(0.4) 2 63.75(0.30) 9.0 25.81(0.02) 44.5(0.1) 12.02(0.14) 7.9 5.54(0.01) 34.2(0.1) 12.25(0.33) 47.5 38.24(0.01) 58.8(0.2) 2a 55.70(0.18) 7.9 21.40(0.03) 62.4(0.1) 9.66(0.15) 6.4 5.45(0.01) 39.7(0.1) 9.33(0.25) 36.2 37.70(0.01) 65.8(0.4) 2b 63.75(0.30) 4.2 28.43(0.03) 33.5(0.1) 12.02(0.14) 4.2 5.61(0.01) 30.1(0.1) 12.25(0.33)) 12.7 38.62(0.01) 53.9(0.3)) 3 162.88(0.70) 8.5 60.89(0.03) 51.0(0.1) 26.85(0.30) 13.2 14.29(0.01) 38.8(0.01) 29.04(0.79) 31.2 11.80(0.02) 63.4(0.2) **3a** 162.88(0.70) 7.1 71.26(0.04) 45.9(0.1) 26.85(0.30) 6.1 15.60(0.01) 33.5(0.01) 29.04(0.79) 22.1 13.30(0.02) 55.6(0.2) 3b 61.10(0.26) 3.2 43.33(0.05) 27.2(0.1) 19.45(0.27) 9.6 8.52(0.01) 38.5(0.02) 10.38(0.32) 11.1 5.15(0.02) 40.4(0.4) Per. $F_{\rm max}$ $\Re$ Mean $100\times F_{\rm var}$ $F_{\rm max}$ $\Re$ Mean $100\times F_{\rm var}$ $F_{\rm max}$ $\Re$ Mean $100\times F_{\rm var}$ (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) 1 29.83(1.05) 2.4 22.8(0.4) 20.36(0.07) 21.22(0.41) 2.1 15.19(0.03) 21.3(0.2) 26.24(0.74) 2.4 18.10(0.05) 23.0(0.3) 1a 28.75(0.97) 1.7 22.86(0.13) 12.5(0.6) 20.64(0.28) 1.7 15.88(0.04) 16.5(0.3) 26.24(0.74) 1.9 20.09(0.07) 15.1(0.4) 1b 29.83(1.05) 2.4 18.15(0.09) 26.5(0.5) 21.22(0.41) 2.1 14.55(0.03) 25.0(0.3) 25.06(0.64) 2.3 16.16(0.06) 26.1(0.4) 2 18.11(0.63) 2.1 11.33(0.04) 18.4(0.3) 11.11(0.19) 2.7 7.73(0.02) 20.0(0.2) 14.13(0.28) 1.9 9.90(0.02) 14.9(0.2) 2a 18.11(0.63) 2.1 12.8(0.07) 19.9(0.6) 11.11(0.19) 2.7 7.56(0.03) 28.2(0.4) 14.13(0.28) 1.9 10.65(0.04) 17.1(0.4) 2b 13.71(0.50) 1.5 10.38(0.04) 7.9(0.04) 10.51(0.19) 1.9 7.84(0.02) 13.1(0.3) 13.00(0.28) 1.7 9.42(0.03) 10.1(0.3) 3 25.50(0.64) 2.1 19.40(0.07) 20.1(0.4) 18.82(0.28) 2.0 13.6(0.03) 17.2(0.2) 22.43(0.56) 2.0 17.16(0.04) 19.0(0.3) 3a 25.50(0.64) 1.4 21.82(0.10) 7.2(0.5) 18.82(0.28) 1.6 14.92(0.04) 10.8(0.3) 22.43(0.56) 1.4 19.29(0.06) 7.1(0.3) 3b 19.14(0.63) 1.6 14.10(0.10) 14.8(0.7) 15.08(0.28) 1.6 11.17(0.04) 12.1(0.4) 17.64(0.41) 1.6 12.99(0.06) 13.1(0.5) -------- ---------------- ------- ------------- ------------------------- --------------- ------- ------------- ------------------------- --------------- ------- ------------- ------------------------- -- -- -- -- -- DATA SETS, REDUCTION and ANALYSIS ================================== X-Ray Data ---------- We retrieved the raw *Swift*-XRT data from *NASA’s Archive of Data on Energetic Phenomena*[^3] (HEASARC). The Level1 event files were reduced, calibrated and cleaned via the `XRTPIPELINE` script (included in the package `HEASOFT` v.6.26) by applying the standard filtering criteria and the latest calibration files of XRT CALDB v.20190412. The events with 0–-2 grades are selected for those observations performed in the Windowed Timing (WT) mode. The selection of the source and background extraction regions was performed with `XSELECT`, using the circular area with radii of 25–50 pixels depending on the source brightness and position in the XRT field-of view (FOV), as well as on the exposure length. We produced a pile-up correction for the count rate CR$\gtrsim$100ctss$^{-1}$ by excluding the central area with radii of 1–3 pixels from the source extraction region, following the recipe provided by [@ro06]. Afterwards, the light curves were corrected using the task `XRTLCCORR` for the resultant loss of the effective area, bad/hot pixels, pile-up, and vignetting. Moreover, the corrections on the point-spread function losses, different extraction regions, vignetting and CCD defects were done by generating the ancillary response files (ARFs) using the `XRTMKARF` task. Due to the high X-ray brightness of the source, it generally was not observed in the Photon Counting (PC) regime. However, Mrk421 was accidentally targeted in this regime three times in the here-presented period (ObsIDs35014255[^4], 34228023 and 34228026). In those cases, we used the events with 0-–12 grades for our analysis. The pile-up correction was done according to the prescription of @m05. The radius, below which the model overproduced the data, was accepted as a region affected by pile-up (11–15 pixels for the particular observation). The source events were extracted from an annular region with the inner radius encircling the pileup area and the outer radius of 50–60 pixels. The loss of counts caused by the inner hole in the source region, vignetting and bad pixels were corrected by generating the corresponding ARF-file. The background counts were extracted from a surrounding annulus with radii of 80 and 120 pixels. From the publicly available, daily-binned BAT[^5] and MAXI[^6] data, we used only those corresponding to the target’s detection with a minimum significance of 5$\sigma$ to study a variability of the 15–150keV and 2–20keV fluxes, respectively. $\gamma$-ray Observations ------------------------- The reduction of the *Fermi*–LAT data was performed with `ScienceTools` (version v11r5p3), adopting the instrument response function `P8R3_SOURCE_V2` and the unbinned maximum likelihood method `GTLIKE`. We selected the 0.3–300GeV energy range for extraction of the photon flux and spectral information, since the effective area of the instrument is larger ($>$0.5m$^2$) and the angular resolution relatively good (the 68% containment angle smaller than 2deg) in that case [@at09]. Consequently, we obtain smaller systematic errors and the spectral fit is less sensitive to possible contamination from unaccounted, transient neighbouring sources [@ab11]. The events of the diffuse class (`evclass`=128, `evtype`=3), i.e, those with the highest probability of being photons, from a region of interest (ROI) with the 10-deg radius centered at the location of Mrk421 were included in our analysis. Moreover, we discarded the events at zenith angles $>$100deg (to avoid a contamination from the Earth-albedo photons, generated by cosmic rays interacting with the upper atmosphere) and those recorded when the spacecraft rocking angle was larger than 52deg (greatly reducing the contamination from Earth-limb photons). The background model `gll_iem_v07.fits` was created to extract the $\gamma$-ray signal from (i) the Galactic diffuse-emission component; (ii) an isotropic component, which is the sum of the extragalactic diffuse emission and the residual charged particle background (parameterized by the file `iso_P8R3_SOURCE_V2_v1.txt`); (iii) all $\gamma$-ray sources from the *Fermi*-LAT 4-yr Point Source Catalog (3FGL, @ace15) within 20deg of Mrk421. For the spectral modelling of our target, we adopted a simple power law, similar to the 3FGL catalogue. The spectral parameters of the sources within the ROI were left free during the minimization process while those outside of this range were held fixed to the 3FGL catalog values. The normalizations of both components (i)–(ii) in the background model were allowed to vary freely during the spectral fit. The photon flux and spectral parameters were estimated using the unbinned maximum-likelihood technique [@m96]. When the target’s detection significance is less than 5$\sigma$ (i.e., the corresponding test-statistics TS$<$25) and/or the number of the model-predicted counts $N_{\rm pred}$$\lesssim$10, such detections are not robust. For example, even a small change in the time-bin width can result in significantly different values of the photon flux and spectral parameters. In such cases, we calculated the upper limit to the photon flux[^7]. We used the user-contributed tool `likeSED`[^8] to construct the 300MeV–300GeV SED of Mrk421. In that case, the photon indices of the sources were frozen to the best-fit values obtained from the full spectral analysis when performing unbinned likelihood fits in differential energy bins (following the recipe provided in @a15a). The source was observed with FACT at VHE energies during 363 nights for a total of 1408hr in the period 2015December8–2018April8. For the timing study, we have used only the nightly-binned TeV excess rates corresponding to detection significances higher than 3$\sigma$[^9], since more than 98% of these data are taken with a zenith distance small enough to not influence significantly the energy threshold of the analysis (see @d15 for the data reduction and analysis details). More than 84% of the same data are taken under light conditions not increasing the analysis threshold. This results in 190 nights for which the nightly observation time ranges of 0.66–7.32hr. In the case of the 20-min binned data, the source was detected 456-times in the here-presented period, and the corresponding rates were used in searching for the intraday brightness variability. UV, Optical and radio data -------------------------- The source was targeted with *Swift*-UVOT in the ultraviolet bands *UVW1*, *UVM2*, and *UVW2* simultaneously with XRT. Generally, Mrk421 was not observed with UVOT in the optical *V–U* bands due to the presence of very bright stars in the telescope’s FOV. The absolute photometry for the sky-corrected images was performed by means of the `UVOTSOURCE` tool (distributed within `HEASOFT`) and the calibration files included in the CALDB v.20170922. The measurements were done using a 20arcsec radius due to the target’s high UV-brightness. When the source was brighter than 12mag, a pile-up was estimated and the corresponding correction was performed using the recipe provided in [@pa13]. According to the latter, a systematic uncertainty of $\pm$0.1mag should be added to the measurements. The magnitudes were then corrected for the Galactic absorption adopting $E(B-V)$=0.028mag (see @k18a), and the $A_\lambda/E(B-V)$ values derived from the interstellar extinction curves [@fi07]. For this purpose, we used the effective wavelength of each filter adopted from [@p08]. Finally, the magnitudes were converted into milli-Janskys by adopting the latest photometric zero-points for each band provided in [@b11], and the host contribution was removed by subtracting the values of 0.09mJy, 0.05mJy and 0.06mJy for the *UVW1, UVM2, UVW2* bands, respectively [@c12]. The publicly available *V* and *R*-band magnitudes, obtained with the 2.3m Bock and 1.54m Kuiper telescopes of Steward observatory[^10] (see @s09 for details), were de-reddened and converted into milli-Janskys according to [@b79]. In both bands, the host contribution was subtracted following [@n07] and [@f95]. The 15GHz radio fluxes, obtained with the OVRO 40-m telescope, were retrieved from the corresponding website[^11] (see @r11 for the data reduction and calibration steps). The sharp spikes or drops in the light curve, associated with less-favourable observing conditions, were not included in our analysis. 1.3pt ------------- ------------------- -------------------------- ------------------------- --------------------------- ------------------------- ------------------------- ------------------------- --------- -- ObsID(s) MJD/$\Delta$T(hr) $\chi^2_{\rm r}$/dof/bin $100\times F_{\rm var}$ $ a$ or $\Gamma$ $ b $ $ E_{\rm p}$ (keV) *HR* Remark (1) (2) (3) (4) (5) (6) (7) (8) (9) 35014242 57370.416/0.30 3.41/8/120s 2.1(0.5) 2.20(0.02)$-$2.24(0.02) 0.38(0.04)$-$0.44(0.04) 0.50(0.06)$-$0.58(0.07) 0.41(0.01)$-$0.44(0.02) 1 34014243$-$ 57373.475/17.98 2925/1/Or 26.3(0.3) 2.17(0.02)$-$2.31(0.02) 0.20(0.05)$-$0.27(0.05) 0.21(0.04)$-$0.48(0.06) 0.38(0.01)$-$0.51(0.02) 1,3 31202245Or1 35014245 57374.204/18.66 28.63/5/Or 3.4(0.3) 2.28(0.02)$-$2.36(0.02)LP 0.13(0.04)$-$0.23(0.04) 0.05(0.02)$-$0.24(0.05) 0.38(0.01)$-$0.45(0.02) 1,2,4,5 2.31(0.02)PL 35014245Or1 57374.204/0.48 2.82/18/90s 2.7(0.5) 2.31(0.02)$-$2.36(0.02) 0.14(0.04)$-$0.23(0.04) 0.05(0.02)$-$0.21(0.04) 0.38(0.01)$-$0.42(0.02) 1,2,4 ------------- ------------------- -------------------------- ------------------------- --------------------------- ------------------------- ------------------------- ------------------------- --------- -- Analysis Methods ---------------- The 0.3–10keV spectra, extracted from the XRT observations and corrected for the different effects (see Section2.1), were further reduced as follows: using the `GRPPHA` task, we combined the instrumental channels to include at least 20photons per bin, making a spectrum valid for the $\chi^2$-statistics. The reduced spectra were fitted with three different models, generally adopted for the blazar X-ray spectra (by fixing the Hydrogen column density to the Galactic value $N_{\rm H}=1.90\times10^{20}$ cm$^{-2}$, obtained within the Leiden/Argentine/Bonn (LAB) survey; @k05): (1) log-parabolic model [@m04] $$F(E)=K(E/E_{\rm 1})^{-(a+b log(E/E_{\rm 1}))}, \vspace{-0.1cm}$$ with $E_{\rm 1}$ fixed to 1keV; $a$, the photon index at the energy $E_{\rm 1}$; $b$, the curvature parameter; $K$, the normalization factor. The position of the synchrotron SED peak was calculated as $E_{\rm p}$=10$^{(2-a)/2b}$keV; (2) simple power-law $ F(E)=KE^{-\Gamma}$, with $\Gamma$, the photon index throughout the entire 0.3–10keV energy range; (3) broken power-law $$\begin{array}{c l} F(E)=KE^{-\Gamma_1}, ~ E\leq E_{\rm br} \\ F(E)=KE^{\Gamma_2-\Gamma_1}_{\rm br} (E/1keV)^{-\Gamma_2}, ~ E>E_{\rm br} , \end{array} \vspace{-0.1cm}$$ with $E_{\rm br}$: break point for the energy in keV, $\Gamma_1$: photon index for $E\leq E_{\rm br}$, $\Gamma_2$: photon index for $E\leq E_{\rm br}$. The model validity was determined using the reduced chi-squared ($\chi^2_{\rm r}$), distribution of the residuals, and F-test. The high X-ray brightness of Mrk421 allowed us to extract the spectra from separate orbits of the particular ObsID (especially important when it is impossible to use the same source and/or background extraction regions for all orbits, or the source is variable), or even from the separated segments of a single orbit in the medium and higher brightness states. The unabsorbed 0.3–2keV, 2–10keV and 0.3–10keV fluxes and their errors (in logarithmic units) were derived using the task `EDITMOD`. The hardness ratio (HR) was determined as HR=$F_{\rm 2-10\,keV}/F_{\rm 0.3-2\,keV}$ where the symbols $F_{\rm 2-10\,keV}$ and $F_{\rm 0.3-2\,keV}$ stand for the de-absorbed 2–10keV to 0.3–2keV fluxes, respectively. In order to study the statistical properties of different spectral parameters, we constructed a histogram and normalized cumulative distribution for the values of each parameter. A Kolmogorov-Smirnov (K-S) test was adopted to compare the distributions of the particular parameter in different time intervals, defined in Table1, and to measure the distance $D_{\rm K-S}$ between the normalized cumulative distributions of parameters corresponding to the two different periods, following the recipe provided by [@m11a]. Since our samples are not statistically complete, a comparison of the corresponding distributions can be affected by biases and there is a risk of obtaining a large $D_{\rm K-S}$ value between the selected cumulative distributions indicating that they are different, simply due to the lack/absense of the data in some particular bins of each histogram. To check the significance of the results provided by the K-S test, we adopted a method based on the Monte Carlo simulations (developed by @m11a) to account for this effect and estimate its relevance: firstly, we performed the K–S test and derived the $D_{\rm K-S}$ quantity for two normalized cumulative distributions. Afterwards, we randomly simulated two distributions for both data sets with the same number of components, adopting two different shapes for the simulated distributions: log-uniform and lognormal - the former having simply the same maximum and minimum values of the observed distribution, while the latter with the same variance, the same median of the observed distribution and spanning the same range of values. We measured the $D_{\rm K-S,simul}$ distance between the simulated distributions, repeated simulations at least 30000 times and built a distribution of the obtained $D_{\rm K-S,simul}$. Finally, we estimated the probability of obtaining the observed $D_{\rm K-S}$ randomly, that provides the confidence level of our K-S test. For each variability instance, we calculated the fractional variability amplitude and its error according to [@v03] $$\begin{array}{c l} F_{\rm var}=(S^2-\overline{\sigma^2_{\rm err}})^{1/2} /{\overline{F}} \\ err(F_{\rm var})=\Biggl\lbrace \left ( \sqrt{1\over 2N} {\overline{\sigma^2_{\rm err}} \over {\overline{x}}^2 F_{\rm var}} \right )^2+\left ({\sqrt{\overline{\sigma^2_{\rm err}} \over N}} {1 \over \overline {x}} \right )^2 \Biggr\rbrace ^{1/2} , \end{array}$$ with $S^2$, the sample variance; $\overline{\sigma^2_{\rm err}}$, the mean square error; $\overline{F}$, the mean flux. In order to investigate the possible quasi-periodical behavior of the flux variations, we first constructed the Lomb-Scragle (LSP) periodogram [@l76; @s82], which is an improved Fourier-based technique suitable for unevenly-sampled time series $g_n$ without interpolation for the data gaps [@v18]: $$\begin{array}{c l} P(f)={A^2\over 2}\left(\sum\limits_{n}{g_n cos(2\pi f[t_n-\tau]} \right)^2+\\ {A^2\over 2}\left(\sum\limits_{n}{g_n sin(2\pi f[t_n-\tau]} \right)^2, \end{array} \vspace{-0.2cm}$$ where *A*, *B*, and $\tau$ are arbitrary functions of the frequency *f* and observing times $\{t_i\}$. The LSP yields the most significant spectral power peak, and estimates its significance level by testing the false alarm probability of the null hypothesis. ----------------------- --------------- --------------- -------------- ------------------------- MJD Band $\Delta$T(hr) $\chi^2$/dof $100\times F_{\rm var}$ (1) (2) (3) (4) (5) 57380.79$-$57381.56 *UVW2* 17.47 12.46/1 8.5(1.8) 57391.56$-$57392.21 *UVW1* 12.26 7.77/2 10.3(2.3) 57425.25$-$57426.23 *UVM2* 22.37 23.24/1 7.2(1.1) 57426.18$-$57426.49 *UVW2* 6.38 12.38/1 8.0(1.7) 57430.17$-$57430.35 *UVM2* 3.29 14.88/1 4.6(0.9) 57477.97$-$57478.21 *UVM2* 4.61 14.67/1 7.0(1.4) 57477.97$-$57478.21 *UVW2* 4.58 49.42/1 18.5(1.9) 57724.44$-$57725.29 *UVM2* 19.39 12.31/1 5.4(1.2) 57726.44$-$57727.29 *UVW1* 19.44 29.94/1 19.1(2.5) 57726.44$-$57727.29 *UVW2* 19.44 27.79/1 11.0(1.5) 57728.49$-$57729.28 *UVW1* 17.88 19.70/1 14.4(2.4) 57728.49$-$57729.28 *UVM2* 17.98 78.32/1 14.4(1.2) 57728.49$-$57729.28 *UVW2* 17.81 24.44/1 10.6(1.6) 57755.22$-$57756.20 *UVM2* 22.32 22.63/1 9.0(1.4) 57784.17$-$57785.08 *UVM2* 20.66 53.39/1 13.6(1.3) 57784.17$-$57785.08 *UVW2* 20.69 16.47/1 9.5(1.7) 57785.03$-$57785.54 *UVW1* 11.14 4.17/7 4.1(1.2) 57785.03$-$57785.54 *UVM2* 11.12 13.94/7 6.9(0.7) 57785.03$-$57785.54 *UVW2* 11.11 15.95/7 7.7(0.8) 57785.37$-$57786.27 *UVW1* 20.76 6.04/3 7.1(1.7) 57785.37$-$57786.27 *UVM2* 20.74 5.56/1 3.5(0.9) 57785.37$-$57786.27 *UVW2* 20.72 5.67/1 4.4(1.1) 57839.93$-$57840.37 *UVM2* 9.50 10.74/5 5.3(0.8) 57839.93$-$57840.37 *UVW2* 9.53 6.48/5 5.3(0.9) 57843.91$-$57844.41 *UVM2* 9.51 21.80/1 8.3(1.3) 57873.87$-$57874.12 *UVM2* 4.85 12.68/1 6.1(1.3) 57873.87$-$57874.12 *UVW2* 4.86 40.44/1 14.3(1.6) 57926.42$-$57927.40 *UVM2* 22.27 14.22/1 6.191.2) 58103.44$-$58104.34 *UVM2* 20.66 28.82/1 8.9(1.2) 58103.44$-$58104.34 *UVW2* 20.64 12.50/1 7.8(1.6) 58112.36$-$58113.35 *UVM2* 8.28 38.20/1 13.3(1.5) 58123.91$-$58124.48 *UVM2* 12.65 26.45/1 9.3(1.3) 58123.91$-$58124.48 *UVW2* 12.64 12.46/1 7.7(1.6) 58137.05$-$58137.55 *UVW1* 11.05 13.41/1 11.7(2.4) 58146.68$-$58147.32 *UVW1* 14.50 29.12/1 18.2(2.4) 58146.68$-$58147.32 *UVW2* 14.49 46.25/1 16.2(1.7) 58147.217$-$58148.12 *UVM2* 19.03 14.88/1 6.2(1.2) 58199.81$-$58200.06 *UVW1* 4.73 19.56/1 13.5(2.2) 58139.60$-$58140.40 *V* 19.20 14.19/2 7.2(1.7) 58139.60$-$58140.40 *R* 19.21 7.59/2 7.2(2.2) 58158.00$-$58158.98 0.3$-$300 GeV 23.52 11.62/1 57.6(12.8) 58135.059$-$58136.058 VHE 23.58 2.82/16 23.4(5.4) 58135.236$-$56136.073 VHE 20.09 4.44/5 41.1(10.1) 58136.250$-$58137.141 VHE 21.38 4.22/9 32.3(7.8) ----------------------- --------------- --------------- -------------- ------------------------- : \[idvuvottable\] The optical–UV and $\gamma$-ray IDVs during 2015December–2018April. [@f96] introduced the weighted wavelet Z-transform (WWZ) method, which is a periodicity analysis technique in both the time and frequency domains. Note that WWZ is suited for discovering variability timescales and is robust against missing data. It is defined as follows $$WWZ={(N_{\rm eff}-3)V_y\over{2(V_{\rm x}-V_{\rm y})}}, \vspace{-0.1cm}$$ with $N_{\rm eff}$, the so-called effective number of data points; $V_{\rm x}$ and $V_{\rm y}$, the weighted variation of the data $x(t)$ and model function $y(t)$, respectively. WWZ is based on the Morlet wavelet [@g84] $ f(z)=e^{-cz^2}(e^{iz}-e^{-1/4c})$ where the constant $e^{-1/4c}$ is inserted so that the wavelet’s mean value is zero. Throughout the paper, the errors are quoted at the 90% confidence level for the one parameter of interest, unless otherwise stated.  RESULTS ======= X-Ray Variability ----------------- Table\[xrt\] provides a summary of the *Swift*-XRT and UVOT observations in 2015December8–2018April8. The source was targeted 299-times, with the net exposure time (sum of the Good Time Intervals, GTIs) of 314.5ks. Based on the target’s visibility for *Swift*, the observations were performed during 2015December–2016June, 2016December–2017June and 2017December–2018April, denoted as Interval1, Interval2 and Interval3, respectively. Each interval was split into two sub-periods (e.g., intervals 1a and 1b), according to the flaring activity of Mrk421 in the *Swift*-XRT band. The source was highly variable during all Intervals1–3 (see Table\[persum\] for the corresponding $F_{\rm var}$ and maximum-to-minimum flux ratios in different bands), although it showed significantly stronger X-ray flaring activity in Interval3 compared to the previous ones (Figure\[flares\]a): the mean 0.3–10keV count rate was a factor $\sim$2-2.5 higher; during the eleven XRT pointings, the brightness exceeded a level of 100ctss$^{-1}$ (see Table2), which never was recorded in Intervals1–2; a generally higher state was superimposed by fast, strong flares by a factor of 2.2–3.8 (lasting 4–12d) which were considerably fewer in the previous periods. In Interval3, the most extreme flare occurred during 2018January14–30 with a 2-min binned count rate of 180.9$\pm$2.8ctss$^{-1}$ (to the level observed on 2008June12; see Figure\[flares\]b) and the highest levels were recorded only during the giant X-ray outburst in 2013 April 10–17 [@k16; @k18a]. The source underwent fast brightness variations by 30%–45% within 13–35ks during the declining phase of the flare (Figure\[idvext1\], top panel). Note that the flux doubling time ($\tau_{\rm d}$$\approx$92ks, defined as $\tau_{\rm d}$=$\Delta t \times ln(2)/ln(F_2/F_1)$; @sa13) was faster than the halving one ($\tau_{\rm d}$$\approx$113ks), and a similar situation was seen during the next flare when an intraday flux doubling with $\tau_{\rm d}\approx$64ks was recorded (Figure\[flares\]b). These events were preceded and followed by strong 0.3–10keV flares by a factor of $\sim$2.5 and peak fluxes exceeding a level of 100ctss$^{-1}$ (Figures\[flares\]c–\[flares\]c). They were parts of a well-defined long-term flare lasting more than 2months (MJD58103–58170). Other strong XRT-band flares are presented in Figures\[flares\]f–\[flares\]i, with the intraday flux-doubling/halving events within 4.8–18.9hr. Among these flares, the 2015December29–2016January8 event was characterized by a two-peak maximum, possibly related to the propagation of forward and reverse shocks after the collision of two blobs  in the blazar jet (see @b10). 3.5pt ------------------ ------------ ------------ ------------- -------------- ----------------- ---------------------- ---------------------- ------------------------ ------------ ObsId $a$ $b$ $E_{\rm p}$ 10$\times K$ $\chi^2/d.o.f.$ $F_{\rm 0.3-2\,keV}$ $ F_{\rm 2-10\,keV}$ $ F_{\rm 0.3-10\,keV}$ HR (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 35014240 2.92(0.02) 0.27(0.08) 0.02(0.01) 0.41(0.01) 1.04/145 16.71(0.27) 2.34(0.14) 19.05(0.35) 0.14(0.01) 35014241 2.49(0.01) 0.48(0.04) 0.31(0.05) 1.31(0.01) 1.10/244 43.35(0.40) 11.59(0.32) 54.95(0.50) 0.27(0.01) 35014242S1(267s) 2.22(0.02) 0.38(0.04) 0.51(0.06) 3.12(0.04) 1.07/230 94.62(1.08) 40.46(1.19) 135.21(1.55) 0.43(0.01) 35014242S2(267s) 2.24(0.02) 0.40(0.04) 0.50(0.06) 3.19(0.04) 1.00/236 97.05(1.11) 39.90(1.18) 136.77(1.57) 0.41(0.01) ------------------ ------------ ------------ ------------- -------------- ----------------- ---------------------- ---------------------- ------------------------ ------------ The source also underwent extremely fast instances of the 0.3–10keV intraday variability (IDV: brightness change within a day, detected by means of the chi-squared statistics; see Table\[idvtable\] for details) at the 99.9% confidence level. Namely, the brightness showed a rise by 5%–18% (taking into account the associated measurement errors) in 180–600 seconds in Figures\[idv1ks\]A1 (panels 1–2), \[idv1ks\]C2–\[idv1ks\]C4, \[idv1ks\]D3, \[idv1ks\]E2 and \[idv1ks\]E3–\[idv1ks\]E4 . Figs \[idv1ks\]A–\[idv1ks\]C belong to the epoch of the strongest X-ray flaring activity in 2018January14–30. Note that the states with CR$\gtrsim$100ctss$^{-1}$ sometimes were associated with very fast and large drops just at the start of the XRT orbit, which were related to instrumental effects. The source also showed very fast drops by 6–16 per cent within 180–840 seconds (Figures\[idv1ks\]B1, \[idv1ks\]C2, \[idv1ks\]D1–\[idv1ks\]D3, \[idv1ks\]F1–\[idv1ks\]F3). Finally, entire brightness rising and dropping cycles with 6%–13% within 420-960 seconds were also observed (Figures\[idv1ks\]A2, \[idv1ks\]B2, \[idv1ks\]C1, \[idv1ks\]E1). On the contrary, Mrk421 sometimes showed a slow, low-amplitude variability during some densely-sampled XRT observations in lower X-ray states (see Figure\[idvext2\]). Moreover, tens of 0.3–10keV IDVs were detected by us, whose details are reported in Table\[idvtable\]. MAXI detected the source 78 and 61 times with 5$\sigma$ significance in the 2–20keV and 2–6keV energy ranges, respectively (Figure\[maxi\]a–b). Owing to lower instrumental capabilities, a strong, long-term flare is evident only in Interval3. Significantly fewer detections with 5$\sigma$ significance are found from the daily-binned BAT data, which show a strong flare during the highest XRT and MAXI-band states (Figure\[maxi\]c). ------------- ------------ ------------ ------------ ---------- -- Par. Min. Max. Mean Skewness (1) (2) (3) (4) (5) 2015–2018 $b$ 0.07(0.04) 0.48(0.04) 0.20(0.01) 0.66 $a$ 1.63(0.02) 2.92(0.02) 2.12(0.01) 0.58 $\Gamma$ 1.79(0.02) 2.91(0.02) 2.12(0.02) 0.98 $HR$ 0.14(0.01) 1.23(0.03) 0.64(0.01) 0.28 $E_{\rm p}$ 0.50(0.06) 7.50(0.76) 1.85(0.09) 1.65 Int1 $b$ 0.07(0.04) 0.48(0.04) 0.25(0.01) 0.34 $a$ 1.77(0.02) 2.92(0.02) 2.29(0.01) 0.31 $\Gamma$ 1.99(0.02) 2.91(0.02) 2.12(0.04) 1.37 $HR$ 0.14(0.01) 0.89(0.03) 0.45(0.01) 0.91 $E_{\rm p}$ 0.50(0.06) 2.03(0.14) 1.02(0.18) 0.94 Int1a $b$ 0.09(0.04) 0.48(0.04) 0.25(0.01) 0.61 $a$ 1.98(0.02) 2.92(0.02) 2.30(0.01) -0.37 $HR$ 0.14(0.01) 0.68(0.02) 0.42(0.01) 0.98 Int1b $b$ 0.08(0.04) 0.42(0.05) 0.25(0.01) 0.25 $a$ 1.77(0.02) 2.91(0.02) 2.26(0.01) 0.41 $HR$ 0.14(0.01) 0.89(0.03) 0.49(0.01) 0.33 Int2 $b$ 0.07(0.04) 0.42(0.05) 0.20(0.01) 0.58 $a$ 1.74(0.02) 2.82(0.02) 2.12(0.01) 1.02 $\Gamma$ 1.92(0.02) 2.54(0.02) 2.13(0.03) 1.22 $HR$ 0.17(0.01) 1.09(0.03) 0.63(0.01) -0.20 $E_{\rm p}$ 0.51(0.06) 6.31(0.65) 1.46(0.22) 2.17 Int2a $b$ 0.10(0.04) 0.40(0.04) 0.22(0.01) 0.48 $a$ 1.74(0.02) 2.82(0.02) 2.14(0.01) 0.84 $HR$ 0.17(0.01) 1.09(0.03) 0.61(0.01) -0.23 Int2b $b$ 0.07(0.04) 0.40(0.04) 0.18(0.01) 0.12 $a$ 1.80(0.02) 2.50(0.02) 2.10(0.01) 0.47 $HR$ 0.29(0.01) 1.06(0.04) 0.65(0.01) 0.14 Int3 $b$ 0.07(0.04) 0.36(0.04) 0.18(0.01) 0.53 $a$ 1.63(0.02) 2.64(0.01) 2.01(0.01) 0.43 $\Gamma$ 1.79(0.02) 2.60(0.02) 2.08(0.02) 0.05 $HR$ 0.28(0.01) 1.23(0.03) 0.75(0.01) 0.19 $E_{\rm p}$ 0.50 7.50(0.76) 2.16(0.17) 1.30 Int3a $b$ 0.07(0.04) 0.30(0.04) 0.17(0.01) 0.34 $a$ 1.63(0.02) 2.52(0.02) 1.98(0.01) 0.54 $HR$ 0.28(0.01) 1.23(0.03) 0.79(0.01) -0.04 Int3b $b$ 0.08(0.04) 0.36(0.04) 0.21(0.01) 0.16 $a$ 1.95(0.02) 2.39(0.02) 2.11(0.01) 0.78 $HR$ 0.36(0.02) 0.80(0.03) 0.60(0.01) -0.19 ------------- ------------ ------------ ------------ ---------- -- : \[distrtable\] Distribution of spectral parameters in different periods: minimum and maximum values (Cols (2) and (3), respectively), mean value (Col.(4)) and skewness (last column). Multiwavelength Variability on Various Timescales ------------------------------------------------- Similar to the XRT-band, the strongest FACT VHE $\gamma$-ray flaring activity of Mrk421 occurred in Period3a (Figure\[subper\]a). Namely, two strong flares peaking at MJD58111 and MJD58135, as well as lower-amplitude ones with the peaks at MJD58141, 58145 and 58164 accompanied the X-ray counterparts. The UVOT-band behaviour was also predominantly correlated with the 0.3–10keV flaring activity, while the source demonstrated different timing properties in the LAT-band, showing slower variability and only one peak at the end of this sub-interval, coinciding with the X-ray and VHE peaks. The three subsequent peaks in the FACT-band light curve were evident during the strongest 0.3–10keV activity of the source in Interval3b (Figure\[subper\]b). No LAT-band GTIs were obtained during the second half of this period and a correlated behaviour with X-ray variability was not observed in the first half. The optical–UV light curves showed a decline at the beginning of the sub-interval, followed by the weak variability which was not strongly correlated with the VHE–X-ray activity. However, a stronger correlation was observed during Period1a, in the three consecutive peaks having X-ray and VHE counterparts (Figure\[subper\]c). Similarly, the LAT-band behaviour was more correlated with those in the XRT and FACT bands, compared to that shown in Interval3. The majority of Interval1b also was characterized by a correlated X-ray and VHE flaring activity (Figure\[subper\]d). The peak days of the X-ray–VHE fluxes also coincided with enhanced activity of Mrk421 in the LAT-band. However, the source was not targeted with *Swift* and FACT during the strongest GeV-band flare peaking on MJD57441. Initially, the source was flaring at UV-optical frequencies (correlated with the higher-energy activity), while it showed a declining trend and low states afterwards, during the four consecutive keV–TeV flares. A similar situation was observed in Interval2a when the source exhibited enhanced VHE and LAT-band activities during the long-term X-ray flare (see Figure\[subper\]e). Finally, Mrk421 underwent strong VHE flares along with the X-ray ones in Interval2b, while the latter showed fewer correlation with the LAT and UVOT-band fluctuations (Figure\[subper\]f). The source did not show flares at the radio frequencies during any sub-interval(see the bottom panels of Figure\[subper\]). Although Mrk421 frequently showed 8–16 detections a night with 3$\sigma$ significance **in** the 20-min binned FACT data (particularly, in the time interval 2018January–February), only 3 instances of a VHE IDV at the 99.9% confidence level were detected[^12], belonging to the epoch of the strongest X-ray flaring activity in the period presented here (see Figure\[idvvhe\]a and Table\[idvuvottable\]). While no correlated X-ray–VHE variability was evident during MJD58136.1–58136.3, the subsequent VHE data exhibit a brightness decline similar to the XRT ones. In the same period, the source showed one instance of a LAT-band IDV with a brightness decline by 42% (taking into account the associated errors) and accompanied by similar behaviour in X-rays (Figure\[idvvhe\]b). Finally, three ultraviolet IDVs, showing a brightness increase nearly simultaneously with that in the 0.3–10keV energy range, are presented in Figures\[idvvhe\]c–\[idvvhe\]d. The details of another 35 optical–UV IDVs are provided in Table\[idvuvottable\]. Spectral Variability -------------------- ### Curvature Parameter Similar to the period 2005March–2015June [@k16; @k17a; @k18a; @k18b], a vast majority of the 0.3–10keV spectra of Mrk421 (886 out of 980) show a significant curvature and are well fitted with the log-parabolic model. The corresponding results are presented in Table\[lp\]. The distribution of values of the curvature parameter $b$ (corresponding to the curvature detection significance of 3$\sigma$ and higher) for different periods are provided in Figures\[figdistr\]A1–\[figdistr\]A4 and the corresponding properties (minimum, maximum and mean values, distribution skewness) are listed in Table\[distrtable\]. Figure\[figdistr\]A1 demonstrates that the source was characterized by a relatively low curvature in the period 2015December–2018April: 98.1% of the values of the parameter $b$ were smaller than $b$=0.4 (conventional threshold between the lower and higher curvatures). Moreover, 46.2% of the spectra shows $b$$<$0.2 (see Section4.3 for the corresponding physical implication). The lowest curvatures were observed in Interval3: there were no spectra with $b$$>$0.4; 62.3% of the values were lower than 0.2; the mean value $\overline{b}$=0.18$\pm$0.01 is significantly smaller than that recorded in Interval1 (see Table\[distrtable\]). The latter was characterized by the majority of the spectra showing curvatures with $b$$>$0.4 (5 out of 9, including the highest value for the entire 2.3-yr period presented here). Interval2 was different from both cases and characterized by intermediate  properties of the parameter $b$ between Intervals1 and 3. This situation is clearly evident from the cumulative distributions of the curvature parameter presented in Figure\[figdistr\]A3 and Table\[ks\], providing the results of the K-S test and distances between the corresponding distributions. According to Figure\[figdistr\]A3 and Table\[ks\], the distribution of the parameter $b$ shows differences between the different parts of Intervals2–3. The lowest curvatures are found for Interval3a ($\overline{b}$=0.17$\pm$0.01, 66% of the spectra with $b$$<$0.2, and $b_{\rm max}$0.30$\pm$0.04), while Intervals 1a and 1b are not significantly different from each other. The curvature parameter showed a weak positive correlation with the photon index at 1keV and an anti-correlation with the position of the synchrotron SED peak, as observed in different sub-intervals (see Figures\[figcor\]a–\[figcor\]b, as well as Table\[cortable\] for the corresponding values of the Spearman correlation coefficient $\rho$). Moreover, this parameter showed an anti-correlation with the de-absorbed 0.3–10keV flux in Intervals1–2(Figure\[figcor\]c). The sixth column of Table\[idvtable\] demonstrates that the parameter $b$ was variable 36-times at the 3$\sigma$ confidence level during 0.3–10keV IDVs, showing $\Delta b$=0.14(0.05)–0.31(0.06) within 0.38–23.92hr (Figures \[idvext1\]–\[idvext2\]). The fastest instances incorporated a curvature increase by 0.19(0.06)–0.22(0.06 in $\sim$1ks (Figure\[idvext1\], at MJD58135.93; Figure\[idvext2\], at MJD57375.4). Moreover, the curvature parameter varied with 2$\sigma$ and 1$\sigma$ significances (capable of causing a significant flux change) 43 and 22 times, respectively. ### Photon Index and Hardness Ratio Although the photon index at 1keV showed a very wide range of values ($\Delta a$=1.29) with the hardest spectrum yielding $a$=1.63$\pm$0.02 in the period presented here, this range is narrower compared to that recorded in the time interval 2009–2012 ($\Delta a$=1.51). In the latter, the source showed the hardest ($a$=1.51–1.61) and the softest spectra ($a$=2.93–3.02) since the start of the *Swift* operations. On the other hand, narrower ranges were found in the time intervals 2005–2008, 2013 April and 2013November–2015June (1.01, 0.87 and 0.94, respectively). Similar to the previous periods, the source clearly demonstrated a harder-when-brighter spectral behaviour during the 0.3–10keV flares. Figure\[figcor\]d exhibits a strong anti-correlation between the parameter $a$ and the de-absorbed 0.3–10keV flux (see also Table\[cortable\]). This trend was evident in Intervals1–3 and sub-intervals separately, although with different strengths and slopes of the corresponding scatter plot. Moreover, Figure\[partiming\]b shows some short time intervals when the opposite spectral trend was observed (during MJD57370.42–57374.47, 57426.45–57428.31 etc.). ------------- ----------- ----------- ----------- ------------- ------------- ------------- ----------- ----------- ----------- ------------- ------------- ------------- -- -- -- -- Quant. Int1–Int2 Int1–Int3 Int2–Int3 Int1a–Int1b Int2a–Int2b Int3a–Int3b Int1–Int2 Int1–Int3 Int2–Int3 Int1a–Int1b Int2a–Int2b Int3a–Int3b (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) *b* 1 1 1 0 1 1 0.25 0.38 0.15 - 0.34 0.27 *a* 1 1 1 1 1 1 0.49 0.61 0.26 0.32 0.19 0.47 $\Gamma$ 0 0 0 - - - - - - - - - HR 1 1 1 1 0 1 0.50 0.62 0.25 0.31 - 0.55 $E_{\rm p}$ 0 1 1 - - - - 0.37 0.28 - - - ------------- ----------- ----------- ----------- ------------- ------------- ------------- ----------- ----------- ----------- ------------- ------------- ------------- -- -- -- -- 294 spectra were harder than $a$=2 (conventional threshold between the hard and soft X-ray spectra), amounting to 33.2% of all log-parabolic spectra (see Figure\[figdistr\]B1). Note that this percentage is smaller than in the periods 2005–2008, 2009–2012, 2013January–May (39%–46%) and higher than those shown in 2013November – 2015June (20%; see @k16 [@k17a; @k18a; @k18b]). On average, the hardest spectra with the mean value $\overline{a}$=1.98$\pm$0.01 were observed in Interval3a versus the softest spectra belonging to Interval1 ($\overline{a}$=2.29$\pm$0.01). Figures\[figdistr\]B2–\[figdistr\]B5 and Table\[ks\] clearly show that the distribution properties of the parameter $a$ varied not only from interval to interval, but also among the sub-intervals. Similar to the curvature parameter $b$, the photon index *a* showed an extreme variability on diverse timescales. It varied at the 3$\sigma$ confidence level 69 times with $\Delta a$=0.08(0.03)–0.31(0.02) within 0.27–23.97hr along with the X-ray IDVs (see Col.(5) of Table\[idvtable\] and Figures \[idvext1\]–\[idvext2\]). Among them, 5 subhour instances were recorded: hardenings by $\Delta a$=0.08(0.03)–0.23(0.03) within 0.13–0.28hr. On longer timescales, the largest variabilities with $\Delta a$=0.66–1.07 in 3.1–27.6d were observed along with the strong 0.3–10keV flares (Figure\[partiming\]b). Quantities $\rho$ $p$ --------------------------------------------------------- ----------------- ---------------------- 2015Dec–2018Apr $a$ and $b$ 0.33(0.08) $6.23\times10^{-6}$ $b$ and $E_{\rm p}$ $-$0.34(0.09) $1.04\times10^{-6}$ $b$ and $F_{\rm 0.3-10\,keV}$ $-$0.33(0.10) $3.42\times10^{-6}$ $a$ and $F_{\rm 0.3-10\,keV}$ $-$0.80(0.03) $3.08\times10^{-14}$ $\Gamma$ and $F_{\rm 0.3-10\,keV}$ $-$0.75(0.07) $7.42\times10^{-13}$ $HR$ and $F_{\rm 0.3-10\,keV}$ 0.79(0.04) $4.19\times10^{-14}$ $E_{\rm p}$ and $F_{\rm 0.3-10\,keV}$ 0.69(0.08) $4.34\times10^{-12}$ $\log E_{\rm p}$ and $\log S_{\rm p}$ 0.65(0.08) $3.39\times10^{-11}$ $\Gamma_{\rm 0.3-2\,GeV}$ and $\Gamma_{\rm 2-300\,GeV}$ 0.39(0.09) $1.99\times10^{-8}$ Int1 $a$ and $b$ 0.29(0.09) $1.00\times10^{-5}$ $a$ and $F_{\rm 0.3-10\,keV}$ $-$0.70(0.04) $6.56\times10^{-12}$ $\Gamma$ and $F_{\rm 0.3-10\,keV}$ $-$0.73(0.907) $3.04\times10^{-6}$ $HR$ and $F_{\rm 0.3-10\,keV}$ 0.64(0.06) $1.24\times10^{-10}$ $E_{\rm p}$ and $F_{\rm 0.3-10\,keV}$ 0.73(0.08) $2.51\times10^{-10}$ $\log E_{\rm p}$ and $\log S_{\rm p}$ 0.64(0.07) $8.99\times10^{-10}$ Int2 $a$ and $b$ 0.23(0.08) $3.34\times10^{-4}$ $b$ and $E_{\rm p}$ $-$0.30(0.10) $8.60\times10^{-5}$ $b$ and $F_{\rm 0.3-10\,keV}$ $-$0.31(0.10) $9.11\times10^{-6}$ $a$ and $F_{\rm 0.3-10\,keV}$ $-$0.88(0.03) $<10^{-15}$ $\Gamma$ and $F_{\rm 0.3-10\,keV}$ $-$0.70(0.10) $7.66\times10^{-5}$ $HR$ and $F_{\rm 0.3-10\,keV}$ 0.88(0.03) $<10^{-15}$ $E_{\rm p}$ and $F_{\rm 0.3-10\,keV}$ 0.69(0.09) $7.15\times10^{-9}$ $\log E_{\rm p}$ and $\log S_{\rm p}$ 0.62(0.07) $6.02\times10^{-10}$ Int3 $a$ and $b$ 0.32(0.08) $6.23\times10^{-12}$ $b$ and $E_{\rm p}$ $-$0.32(0.09) $4.48\times10^{-5}$ $b$ and $F_{\rm 0.3-10\,keV}$ $-$0.30(0.11) $8.19\times10^{-5}$ $a$ and $F_{\rm 0.3-10\,keV}$ $-$0.86(0.03) $1.03\times10^{-15}$ $\Gamma$ and $F_{\rm 0.3-10\,keV}$ $-$0.85(0.05) $2.47\times10^{-14}$ $HR$ and $F_{\rm 0.3-10\,keV}$ 0.89(0.03) $<10^{-15}$ $E_{\rm p}$ and $F_{\rm 0.3-10\,keV}$ 0.85(0.06) $1.69\times10^{-14}$ $\log E_{\rm p}$ and $\log S_{\rm p}$ 0.83(0.05) $5.54\times10^{-14}$ : \[cortable\] Correlations between spectral parameters and 0.3–10keV flux in different periods. In Cols(2)–(3), $\rho$ and $p$ stand for the Spearman coefficient and the corresponding p-chance, respectively. The hardness ratios, derived from the log-parabolic and power-law spectra, showed a wide range of values ($\Delta HR$=1.09) with 68.7% of the spectra with $HR>$0.5, and 90 spectra (9.2%) showing $HR>$1 (see Figure\[figdistr\]D1 and Table\[distrtable\]) when the de-absorbed 2–10keV flux is higher than the 0.3–2keV one. A vast majority of the latter (90%) belong to Interval3a, characterized by the highest mean value $\overline{HR}$=0.79$\pm$0.01 (versus $\overline{HR}$=0.42–0.65 in other sub-intervals; Figures \[figdistr\]D2–\[figdistr\]D3). A positive $F_{\rm 0.3-10 keV}$–HR correlation was observed in all intervals, demonstrating a dominance of the harder-when-brighter spectral evolution during X-ray flares, although this trend was significantly weaker in Interval1 (see Figure\[figcor\]f) and Table\[cortable\]). The long-term behaviour of the hardness ratio followed that of the parameter $a$: during the largest variability of the photon index, HR increased by a factor of 3.5–6.4 and showed 75 IDVs by 11%–88% per cent (see Col.(8) of Table\[idvtable\]). . --------------------- ------------ -------------- ----------------- ----------------------- ---------------------- ------------------------ ------------ ObsId $\Gamma$ 10$\times K$ $\chi^2/d.o.f.$ $ F_{\rm 0.3-2\,keV}$ $ F_{\rm 2-10\,keV}$ $ F_{\rm 0.3-10\,keV}$ HR (1) (2) (3) (4) (5) (6) (7) (8) 35014245Or5 2.31(0.02) 1.73(0.02) 0.95/186 57.68(0.66) 28.12(0.83) 85.90(1.18) 0.49(0.02) 35014246Or4 2.43(0.02) 1.51(0.02) 0.88/182 52.60(0.60) 20.70(0.52) 73.28(0.67) 0.39(0.01) 34228001Or1S2(410s) 2.15(0.02) 1.64(0.02) 1.09/229 51.88(0.71) 33.81(0.77) 85.70(0.98) 0.65(0.02) 34228001Or2S2(285s) 2.14(0.02) 2.32(0.02) 1.08/233 73.45(0.84) 48.42(1.10) 121.90(1.40) 0.66(0.02) 34228023 2.26(0.03) 1.77(0.03) 0.91/139 58.34(1.06) 31.05(1.26) 89.33(1.43) 0.53(0.02) --------------------- ------------ -------------- ----------------- ----------------------- ---------------------- ------------------------ ------------ ### The Position of the Synchrotron SED Peak During the period 2015December–2018April, 463 spectra (52.3% of those showing a curvature) are characterized by 0.5$\leqslant$$ E_{\rm p}$$\leqslant$8keV when the position of the synchrotron SED peak is well-constrained by the XRT data (see @k18b). In the case of 407 spectra (46%), $E_{\rm p}$$<$0.5keV when the synchrotron SED peak position, derived via the X-ray spectral analysis, should be assumed as an upper limit to the intrinsic peak position (not used by us for the construction of the scatter plots and distributions). Note that such instances amounted to 86.6% and 68.9% of all curved spectra from Intervals 1a and 1b, respectively (versus 25.4% in Interval3a). Moreover, the majority of the spectra with $E_{\rm p}$$<$0.1keV (when the synchrotron SED peak is situated in the UV energy range) belonged to these sub-intervals. On the other hand, for the spectra with $E_{\rm p}$$>$8keV, the synchrotron SED peak is poorly constrained by the observational data and such $E_{\rm p}$ values should be considered as lower limits to the intrinsic position (see @k18b). During the period 2018December–2018April, the source showed 14 spectra with 8.25$\leqslant$$E_{\rm p}$$\leqslant$15.85keV, mostly from the observations corresponding to the highest X-ray states in Interval3a. The $E_{\rm p}$ values from the range 0.5–8keV mainly belong to Interval3 (62.9%), and their mean is significantly higher than those from Intervals1–2 (2.16keV versus 1.02–1.46keV in Intervals1–2; see Figures\[figdistr\]D2 and Table\[distrtable\]). Note that the K-S test and related Monte Carlo simulations did not show a significant difference between the distributions corresponding to Intervals 1 and 2 (see Table\[ks\]). For the entire 2015December–2018April period, the parameter $E_{\rm p}$ showed a positive correlation with $F_{\rm 0.3-10 keV}$, which was the strongest in Interval3 (Figure\[figcor\]g and Table\[cortable\]. Moreover, a positive correlation between $E_{\rm p}$ and $S_{\rm p}$ (the height of the synchrotron SED peak) was detected in all three time intervals (see Figure\[figcor\]h, Table\[cortable\] and Section4.3 for the corresponding physical implication). Note that the latter quantity was calculated for each spectrum as [@m04] $$S_{\rm p}=1.6\times 10^{-9}K10^{(2-a)^2/4b} ~~~ \mbox{ erg cm$^{-2}$ s$^{-1}$}. \vspace{-0.1cm}$$ On intraday timescales, the parameter $E_{\rm p}$ varied 56 times at the 3$\sigma$ confidence level, observed during the 0.3–10keV IDVs (see Col.(7) of Table\[idvtable\]). The most dramatic changes were observed during the extreme flare in 2018January14–30 (see the bottom panels of Figures \[idvext1\] and \[idv1ks\]A1–\[idv1ks\]A2): $E_{\rm p}$ sometimes showed shifts by several keV within 0.1–9.5hr to higher energies and moved back in comparable timescales. . CONCLUSIONS AND DISCUSSION ========================== X-Ray and MWL Flux Variability ------------------------------ ### Variability Character The 0.3–10keV brightness of the source reached the highest level in the time interval 2018January–February and Mrk 421 was brighter only during the giant outburst in 2013April. A similar situation was observed in the VHE energy range: the highest VHE states were recorded in 2018January (coinciding those in the XRT band), while the strongest VHE flare was observed in 2013April. The TeV-band variability mostly showed a good correlation with the X-ray one, although there were some exceptions (see Figure\[subper\] and Section4.2). Conversely, there were significantly fewer detections with 5$\sigma$ significance and/or lower fluxes in the BAT-band, compared to the periods 2005December–July, 2008March–July, 2009October–November, 2010January–May, 2011September (see a more detailed discussion in Section4.3.5). In other spectral ranges, Mrk421 exhibited a relatively different behaviour. Namely, the highest 0.3–**300**GeV flux from the weekly-binned LAT data was (2.2$\pm$1.8)$\times$10$^{-7}$phcm$^{-2}$cm$^{-1}$ (in 2016February, not coinciding with the highest X-ray states). Contrary to the X-ray and VHE observations, significantly higher levels were recorded in 2012July–August (with the highest historical MeV–GeV level) and the comparable states - in 2013 March and 2014April. There was a frequent absence or weakness of the correlation between the LAT-band and X-ay variability [@k16; @k17a]. In the UVW1–UVW2 bands, the highest states, corresponding to the de-reddened and host-subtracted fluxes of 23.3–29.8mJy, were observed during 2016January–February, and they were significantly lower 2yr later when the source showed its highest X–ray activity. The higher UV states were observed in 2010June–2011April, 2012April–May, 2012December–2013April (the highest historical UV brightness, preceding the giant X-ray outburst), 2013November–2014April, 2015February–June. A nearly-similar situation was found in the optical *V–R* and OVRO bands. We checked the MWL data of Mrk421 for periodicity during 2015December–2018 April. As an example, Figure\[lsp\] presents the LSP and WWZ plots from the XRT observations performed in Intervals1–3. No clearly expressed quasi-periodic variations are found in this energy range, similar to the radio–UV and GeV–TeV ranges. Periodic brightness variations have not been found also by different authors (see, e.g., @c17 [@s17]). As in past years, the source showed a double-humped behaviour in the plane $\log\nu$–$F_{\rm var}$ during the entire 2015December–2018April period and its separate intervals (with $F_{\rm var}$, calculated using the entire data set obtained in the given spectral band during the particular period; see Figures\[idvs\]A1–\[idvs\]A4 and @k16 [@k17a; @k18a; @k18b]). We used the 1-d binned XRT, BAT, FACT, OVRO and optical-UV data in our study, while the 3-d binned LAT was used in the 0.3–300GeV energy range (similar to the light curves provided in Figure\[subper\]). Although the latter binning and the cuts at 3$\sigma$–5$\sigma$ detection significances can lead to some undersampling in the corresponding $F_{\rm var}$ values, a two-humped shape with the synchrotron and higher-energy peaks situated at the X-ray and VHE frequencies, respectively, seems to be inherent for HBLs and are frequently reported for these sources. For our target, a similar result was reported by various authors from various MWL campaigns [@a15a; @a15b; @a16; @abey17], favouring the one-zone SSC model (predicting the correlated X-ray–VHE variability) for Mk421 in different shorter-term periods and indicating that the electron energy distribution is most variable at the highest energies [@a15b]. Although the optical data points in Figures\[idvs\]A1–\[idvs\]A2 violate the general trend of increasing variability power from radio to hard X-ray frequencies, this result can be related to the data sampling: some *V* and *R*-band observations were carried out at Arizona Observatory in those time intervals when the source was not targeted by UVOT, although it was showing a strong variability. The presence of lower VHE peaks in Figures\[idvs\]A1–\[idvs\]A3, compared to the peak in the synchrotron frequency, is difficult to explain via the upscattering of synchrotron photons in the Thomson regime (when a squared relation is expected). However, this result can be related to the use of the FACT excess rates instead the linear fluxes in our study. ### Flux Lognormality We also checked whether the X-ray and MWL fluxes of Mrk421, observed in the 2015 December-2018 April time interval, showed lognormal distributions. According to [@mc08], a lognormal flux behavior in blazars can be indicative of the variability imprint of the accretion disk onto the jet. Moreover, the lognormal fluxes have fluctuations, that are, on average, proportional to the flux itself, and are indicative of an underlying multiplicative, rather than additive physical process. Consequently, the excess variance $\sigma^2_{\rm excess}$=$\sqrt{S^2-\overline{\sigma_{\rm err}}^2}$ (with the quantities $S$ and $\overline{\sigma_{\rm err}^2}$ defined in Equation(3)) plotted versus the mean flux for all flares, should show an increasing linear trend [@ch19]. For BLLs, the lognormality in different spectral ranges and time intervals were reported for PKS2155-304, BL Lac and 1ES1011+496 [@ch19; @g09; @si17]. In the case of Mrk421, the lognormality was studied by [@si17] using the radio, optical–UV and LAT-band data obtained during 2009–2015. The lognormal fit to the histograms were clearly preferred for most of the bands, leading to the suggestion that the flux variability in the source can be mainly attributed to changes in the particle spectrum rather than to the variability of the jet physical parameters such as the magnetic field or Doppler factor (see @si16). In order to investigate lognormality, we fit the histograms of the MWL fluxes with the Gaussian and lognormal functions (similar to the aforementioned studies). Figure\[logn\]A1 demonstrates, that the distribution of the de-absorbed 0.3–10keV flux from the entire 2015December–2018April period is closer to the lognormal shape than the Gaussian one (similar to the MAXI observations; Figure\[logn\]B). A lognormal behaviour is confirmed by the corresponding scatter plot in the $\sigma^2_{\rm excess}$–$\overline{F_{\rm 0.3-10 keV}}$ plane where the data points, corresponding to the XRT-band flares during 2015December–2018April, show an increasing linear trend with higher mean flux (Figure\[logn\]H1). Note that this result is mainly due to the observations performed in Intervals 1 and 3, while the data from Interval2 are closer to the Gaussian function (Figures\[logn\]A2–\[logn\]A6). The longer-term flares may result from the propagation and evolution of relativistic shocks through the jet (see @s04 and references therein). The shock appearance can be related to the instabilities occurring in the accretion disk, which may momentarily saturate the jet with extremely energetic plasma with much higher pressure than the steady jet plasma downstream [@s04]. Consequently, a lognormal flaring activity of the source on longer timescales may indicate a variability imprint of the accretion disk onto the jet. However, the fluxes, corresponding to **the** highest X-ray states in Intervals 1 and 3, produce outliers from the lognormal distributions. These states generally were recorded during fast flares superimposed on the long-term one. These flares could be triggered **by** the shock interaction with the jet inhomogeneities whose origin was related to the jet instabilities (e.g., strong turbulent structures; @m14). Therefore, no lognormal distribution of those fluxes is expected in that case, owing to the absence of the AD variability imprint. Moreover, Interval2 clearly shows a better fit to the Gaussian function and since this period was characterized, on average, by lower X-ray states (see Section3.1 and Table\[persum\]), this result could be related to the propagation of weaker shocks through the jet. On the other hand, a shock weakness possibly was due to the lower AD variability in that period, which resulted in a fewer imprint onto the jet of Mrk421. In each interval, the FACT and LAT-band fluxes clearly showed a better fit of the lognormal function with the corresponding distributions (Figures\[logn\]C1–\[logn\]D4). Although the same is shown by the *R*-band histogram constructed for the entire 2015December–2018April period, it is impossible to draw a firm conclusion related to the lognormal behaviour of the source in this band, since we could not construct the corresponding $\overline{F_{\rm R}}$–$\sigma^2_{\rm excess}$ scatter plot due to the sparse sampling of the optical flares (see Figure\[subper\]). In contrast, the OVRO data do not show a good fit between the corresponding histogram and lognormal function. This result implies the radio-contributions from various emission regions with different physical conditions. Finally, the UVOT-band histograms show a better (but not good) fit with the lognormal function compared to the Gaussian one only in Interval2 (see the bottom row of Figure\[logn\]). ### Intraday Variability In the 2015 December-2018 April period, the source showed three intraday flux-doubling and three flux-halving events with $\tau_{\rm d}$=14.2–17.8hr and $\tau_{\rm h}$=4.8–18.9hr, respectively. Note that the flux doubling time can be used for constraining the upper limit to the emission zone as follows (@sa13 and references therein) $$R_{\rm em}\leqslant {{c\tau_{\rm d} \Gamma_{\rm em}}\over{1+z}}, \vspace{-0.2cm}$$ with $R_{\rm em}$ and $\Gamma_{\rm em}$, the size and Lorentz-factor of the emission zone, respectively. Adopting the typical value of the bulk Lorentz-factor for the emission zone $\Gamma_{\rm em}$=10 [@f14], we obtain upper limits of 5.0$\times$10$^{15}$–1.9$\times$10$^{16}$cm for the emission zones with intraday flux-doubling instances. Mrk421 exhibited more extreme behaviour during the densely-sampled *Swift*-XRT campaign in 2009May22–27 (net exposure time of 23.5ks, 59 orbits) $\tau_{\rm h}$=1.4–1.5hr and $\tau_{\rm d}$=6.5–12.4hr [@k18b]. A series of the brightness halving and doubling events with $\tau_{\rm d,h}\sim$1.1hr were recorded on 2009February17, and a similar variability occurred also during 2017February2–3 (see Section3.1 and Figure\[flares\]i). Note that the successive large brightness drop and rise events can be explained as a consequence of a shock passage through two inhomogenous areas with strong magnetic fields, which are separated by a region with significantly weaker field and lower particle density (yielding the generation of fewer X-ray photons). The most extreme behaviour was observed during the giant X-ray outburst in 2013April, with several events showing $\tau_{\rm d}$=1.2–7.2hr and $\tau_{\rm h}$=1.0–3.5hr [@k16]. The distribution of the 0.3–10keV fluxes extracted from those *Swift*-XRT pointings which showed IDVs in the period 2015December–2018April is not well fitted with the lognormal function, in contrast to the longer-term flares. This result hint at the absence of an AD variability imprint onto these events. Moreover, Figures\[idvs\]B1–\[idvs\]B3 show a rare occurrence of these events in low X-ray states[^13] when the IDVs caused by the instability in AD or in the inner jet regions should be more easily detectable: the variable emission from these regions will not be overwhelmed  by the huge flux produced near the shock front, propagating through the jet and causing longer-term flares (see @mang93 and references therein). On the other hand, most of these events are detected from short XRT exposures and the entire cycle (brightness increase and drop) is generally not recorded, in contrast to the longer-term flares. Therefore, it is not possible to draw a firm conclusion about the absence of lognormality. However, the 0.3–10keV flux values from those IDVs whose complete variability cycles were observed do not exhibit a lognormal flux distribution. Therefore, these results favour the shock-in-jet scenario where IDVs are triggered by the interaction of a shock front with small-size jet inhomogeneities [@s04; @m14; @miz14]. The duty cycle of the 0.3–10keV IDVs (i.e., the fraction of total observation time during which the object displayed IDVs; see @r99 and references therein) amounted to 58.4%, which is higher than that shown by the source in 2009–2012 and 2013November–2015June (43–46 per cent; @k17a [@k18b]), but significantly lower than compared to the periods 2005–2008 and 2013January–May (about 84%; @k16 [@k18a]). Note that this result can be partially related to significantly more densely sampled campaigns in some parts of the latter periods. For example, Mrk421 was observed very densely during 2006June15–25 with a total exposure of 119ks distributed over 124 *Swift*-XRT orbits. On the other hand, the source exhibited an unequal 0.3–10keV activity on intraday timescales in different epochs. Note that the XRT pointings without IDVs were mainly performed during the lower X-ray states. However, we also discerned some flares with a clear lack of intraday activity. For example, such flaring activity was observed during MJD57521–548 with $CR_{\rm max}$=95.73$\pm$0.32ctss$^{-1}$, which is a factor of 2.5 higher than the mean rate during the whole period 2015December–2018April. This event was preceded by the long time interval (MJD57400-57520) when the source had showed the lowest mean rate (23.24$\pm$0.01ctss$^{-1}$) and duty cycle of IDVs (36%) for the entire 2015–2018 period. A similar situation was observed in 2010June–July when the source demonstrated low 0.3–10keV states with CR$\sim$10ctss$^{-1}$ and insignificant variability over 1.5months, possibly related to the absence of strong shock waves, which could cause a long-term flare in different spectral bands [@k18b]. Moreover, the source also showed significantly slower and weaker IDVs during those densely-sampled XRT observations presented in Figure\[idvext2\]. These results lead to the conclusion that the source underwent weaker variability on intraday timescales during the low X-ray states and some longer-term flares. The last column of Table\[idvtable\] provides a list of the variable spectral factor presumably making the dominant contribution to the particular 0.3–10keV IDV. The most common factor in the observed fast variability was a change of the photon index (i.e., in the slope of the Particle Energy Distribution, PED). Nevertheless, some events were related to the transition from a log-parabolic PED into a power-law one and/or vice versa. The most extreme log-parabolic/power-law/log-parabolic transitions occurred within the 1-ks exposures during ObsID31202174 (MJD58135.39), 4-th orbit of ObsID31202175 (MJD58136.33) and first orbit of ObsID31202178 (MJD58137.11), in the epoch of the strongest X-ay flaring activity of the source during 2015–2018. Such extreme events can be related to very fast variability of the magnetic field properties within the jet regions which are smaller than $R$=10$^{14}$cm (for $\delta$=10): a transition from a magnetic field characterized by decreasing confinement efficiency with rising electron’s gyroradius or strong turbulence (both yielding a log-parabolic PED) into a volume without these properties and vice versa (see below). A number of the IDVs were related also to the variability of the curvature parameter, the position and height of the synchrotron SED peak. The changes of the latter two quantities imply an intraday and even a subhour variability of the physical parameters (magnetic field, Doppler factor, $\gamma^2_{\rm 3p}$: the peak of the function $n(\gamma)\gamma^3$) included in the relations as follows (@r79 and references therein) $$E_{\rm p} \varpropto \gamma^2_{\rm 3p}B\delta,~~ S_{\rm p} \varpropto N\gamma^2_{\rm p}B^2\delta^4.$$ Multiwavelength Correlations ----------------------------- During the time interval presented in this work, the TeV-band variability showed the strongest correlation with the X-ray one: strong VHE flares or enhanced activity mostly coincided with strong X-ray flares (see Section3.2). Consequently, Figure\[cor1\]a shows a positive $F_{\rm 0.3-10 keV}$–$F_{\rm VHE}$ correlation in each interval, although it was relatively weak in Interval3 compared to that in Interval1 (the difference between the values of the coefficient $\rho$ was larger than the corresponding error ranges; see Table\[cortable1\]). In fact, some data points from Interval3 make the largest exceptions from the general trend. Such situation was particularly evident during MJD58145–153 (Interval3a): the VHE flux showed a decline by a factor of $\sim$4 and subsequent low states during the fast X-ray flare with one of the highest states during the 2015–2018 period (see Figure\[subper\]a). Two data points in the right lower corner of the $F_{\rm 0.3-10 keV}$–$F_{\rm VHE}$ plane belong to the X-ray flare around MJD58136 when the 0.3–10keV flux reached its highest values, while the TeV-band state was the highest on the previous day. This situation is hard to reconcile with one-zone SSC scenarios [@bl05]. However, Fig\[idvvhe\]a demonstrates that no strictly simultaneous XRT observation was performed during those 20-min FACT pointings detecting the source in the highest VHE states. Therefore, we cannot draw any firm conclusion for this case. The source was not detectable with 3$\sigma$ or showed low VHE states when it had undergone short-term X-ray flares during MJD(57)413–422 (Figure\[subper\]c), MJD(57)425–438 and MJD(57)473–491 (Figure\[subper\]d), MJD(57)723–730 and MJD(57)801–838(Figure\[subper\]e). Similar instances were reported from the FACT observations of Mrk421 performed in previous years [@k16; @k17a]. Moreover, the MWL campaign in 2002December–2003January revealed a strong X-ray flare by a factor of 7 within 3d, not accompanied by a comparable TeV-counterpart [@r06]. The TeV flux reached its peak days before the X-ray flux during the giant flare in 2004 that was impossible to explain via the standard one-zone SSC model, and [@bl05] suggested this as an instance of the orphan TeV flare. [@ac11] also found high X-ray states, not accompanied by TeV flaring and vice versa in 2006–2008. We conclude that the broadband SED can not always be modelled using one-zone SSC scenarios, although they were acceptable for Mrk421 during the majority of the X-ray flares (corroborated by the appearance of a two-hump structure in the $F_{\rm var}$–log$\nu$ plane, with the peaks at X-ray and TeV frequencies, respectively). Quantities $\rho$ $p$ ------------------------------------------------ ----------------- ---------------------- 2015Dec–2018Apr $F_{\rm 0.3-2\,keV}$ and $F_{\rm 2-10\,keV}$ 0.93(0.02) $<10^{-15}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm FACT}$ 0.61(0.08) $5.44\times10^{-10}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW2}$ 0.30(0.11) $4.07\times10^{-5}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW1}$ 0.25(0.11) $8.34\times10^{-4}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm LAT}$ 0.49(0.10) $1.03\times10^{-8}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm 15 GHz}$ -0.53(0.09) $8.89\times10^{-9}$ $F_{\rm UVM2}$ and $F_{\rm UVW2}$ 0.94(0.02) $<10^{-15}$ $F_{\rm UVW1}$ and $F_{\rm UVW2}$ 0.96(0.01) $<10^{-15}$ $F_{\rm UVW2}$ and $F_{\rm LAT}$ 0.32(0.11) $7.58\times10^{-5}$ $F_{\rm FACT}$ and $F_{\rm LAT}$ 0.55(0.09) $1.24\times10^{-9}$ $F_{\rm LAT} $ and $ F_{\rm 15 GHz}$ -0.49(0.14) $3.55\times10^{-4}$ $F_{\rm 0.3-2\,GeV}$ and $ F_{\rm 2-300\,GeV}$ 0.75(0.06) $5.02\times10^{-13}$ Int1 $F_{\rm 0.3-2\,keV}$ and $F_{\rm 2-10\,keV}$ 0.88(0.03) $<10^{-15}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm FACT}$ 0.62(0.07) $1.28\times10^{-11}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm LAT}$ 0.44(0.10) $6.10\times10^{-8}$ $F_{\rm FACT}$ and $F_{\rm LAT}$ 0.45(0.10) $3.71\times10^{-8}$ Int1a $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW2}$ 0.56(0.11) $5.01\times10^{-5}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW1}$ 0.48(0.12) $7.79\times10^{-5}$ Int1b $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW2}$ -0.46(0.11) $6.77\times10^{-5}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW1}$ -0.43(0.11) $3.40\times10^{-5}$ Int2 $F_{\rm 0.3-2\,keV}$ and $F_{\rm 2-10\,keV}$ 0.93(0.02) $<10^{-15}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm FACT}$ 0.51(0.09) $7.76\times10^{-9}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW2}$ -0.45(0.09) $1.08\times10^{-6}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW1}$ -0.47(0.09) $2.03\times10^{-7}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm LAT}$ 0.34(0.12) $3.17\times10^{-4}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm 15 GHz}$ -0.71(0.09) $3.33\times10^{-9}$ $F_{\rm FACT}$ and $F_{\rm LAT}$ 0.64(0.08) $6.10\times10^{-11}$ Int2a $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW2}$ -0.75(0.09) $1.92\times10^{-8}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW1}$ -0.74(0.09) $4.56\times10^{-8}$ Int3 $F_{\rm 0.3-2\,keV}$ and $F_{\rm 2-10\,keV}$ 0.93(0.03) $<10^{-15}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm FACT}$ 0.45(0.11) $9.01\times10^{-8}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW2}$ 0.39(0.10) $4.07\times10^{-5}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm UVW1}$ 0.38(0.10) $9.95\times10^{-5}$ $F_{\rm 0.3-10\,keV}$ and $ F_{\rm LAT}$ 0.39(0.11) $5.26\times10^{-4}$ : \[cortable1\] Correlations between the MWL fluxes (denoted by $F_{i}$ for the particular $i$-band in Column1) during different periods. In Cols(2)–(3), $\rho$ and $p$ stand for the Spearman coefficient and the corresponding p-chance, respectively. The 0.3–10keV variability showed a weak positive correlation with that observed in the MeV–GeV energy range (see Figure\[cor1\]b and Table\[cortable1\]). The source did not undergo comparable LAT-band activity or exhibited lower states during the most extreme X-ray behaviour (MJD58115–58160; Figure\[subper\]a). Moreover, no credible detections or low fluxes were recorded in the time intervals MJD(58)190–215 and (57)364–381 (Figures\[subper\]b–\[subper\]c), MJD(57)731–746 (Figures\[subper\]e). An even weaker $F_{\rm 0.3-10 keV}$–$F_{\rm 0.3-300 GeV}$ correlation was observed during 2009–2012 and 2013January–June, while no significant correlation was found for the period 2013November–2015June [@k16; @k17a; @k18b]. We extracted the 0.3–2GeV and 2–300GeV photon fluxes from the LAT observations to check their cross-correlation and search for possible contributions from different electron populations to the LAT-band emission. We used 2-weekly binned observations to ensure $N_{\rm pred}$$\geqslant$10 (Table\[lat\]). Figure\[latgamma\]a exhibits a strong $F_{\rm 0.3-2 GeV}$–$F_{\rm 2-300 GeV}$ correlation, demonstrating a predominantly common origin for the soft and hard LAT-band photons. This result is in contrast to that obtained for 1ES1959$+$650 during the period 2016August–2017November, with a weak correlation between the softer and harder LAT-band fluxes that hints at uncorrelated behaviour and possible contribution from different particle populations [@k18e]. Note that the LAT-band flux was also correlated weakly with the UVOT- and FACT-band fluxes, respectively (see Figures\[cor1\]d–\[cor1\]e). This result can be explained as an IC uppscatter of the UV photons to the MeV–GeV energies in the Thomson regime (a similar relation between the X-ray and VHE photons). Another source of the LAT-band emission can be an upscatter of X-ray photons in the Klein-Nishina (K-N) regime. An upscatter of the BAT-band photons in the same regime could be the case during those time intervals when the source showed low VHE states with the absent $F_{\rm 0.3-10 keV}$–$F_{\rm VHE}$ correlation or was not detectable in the TeV energy range (owing to strong suppression of the $\gamma$-ray emission in this regime; see @t09). Both LAT and XRT-band fluxes showed an anti-correlation with the 15GHz emission which was particularly strong in Interval2 in the case of the X-ray emission (see Figures\[cor1\]f–\[cor1\]g and Table\[cortable1\]). A similar situation is evident from the scatter plot $F_{\rm 0.3-10 keV}$–$F_{\rm UVW2}$ constructed for Interval2 and Interval1b (Figure\[cor1\]c and the aforementioned table). Such a feature was reported also by [@a15b] and explained through a hardening of the electron energy distribution that can shift the entire synchrotron bump to higher energies. Consequently, the emission of the synchrotron SED in the radio–UV energy range is expected to decrease with rising X-ray brightness. Note that such MWL behaviour is expected for the stochastic acceleration of electrons within the specific initial conditions (see Section4.3). No $F_{\rm 0.3-10 keV}$–$F_{\rm UVW2}$ correlation at the 99% confidence level was detected in Interval1, and a very weak positive correlation occurred in Interval3. An anti-correlation was observed also during 2009–2012, and uncorrelated variabilities occurred in 2005–2008 and 2014February–2015June [@k17a; @k18b]. A stronger positive correlation was observed during 2013January–June and 2013November–2014January [@k16; @k17a]. We have not found a correlation between the UV and VHE fluxes, hinting at an insignificant role for the upscatter of the UV photons to VHE frequencies in the Thomson regime (Figure\[cor1\]h). Finally, the UVW1–UVW2 fluxes showed very strong cross-correlations, as in previous periods, reflected in the absence of the UV colour variability and demonstrating the generation of these photons by the same electron population via the synchrotron mechanism in each period (Figures\[cor1\]i–\[cor1\]j). Spectral Properties and Particle Acceleration Processes ------------------------------------------------------- ### First-Order Fermi Acceleration A positive $a$–$b$ correlation (see Figure 9a), detected in Intervals1–3, shows an importance of the first-order Fermi acceleration since this correlation was predicted for those jet regions where particles are confined by a magnetic field at the shock front, whose confinement efficiency is declining with increasing gyration radius (i.e., particle’s energy; so-called EDAP: energy-dependent acceleration probability process; @m04). Consequently, the probability $p_i$ that a particle undergoes an acceleration step $i$, with the corresponding energy $\gamma^{q}_i$ and energy gain $\varepsilon$, is given by $p_i$=$g/\gamma^{q}_i$, where $g$ and $q$ are positive constants. Consequently, the probability of the particle’s acceleration is lower when its energy increases, and the differential energy spectrum is given by $N(\gamma)\sim{\gamma/\gamma_0}^{-s-r\log{\gamma/\gamma_0}}$, with a linear relationship between the spectral index and curvature terms (*s* and *r*, respectively) $s=-r(2/q)\log{g/\gamma_0}-(q-2)/2$. The synchrotron emission produced by this distribution is given by $$P_S (\nu)\varpropto (\nu/\nu_0)^{-(a+b\log(\nu/\nu_0))}, \vspace{-0.1cm}$$ with $a=(s-1)/2$ and $b=r/4$ [@m04]. However, the detected $a$–$b$ correlation was weak in each interval. This result can be explained due to the sub-samples having different slopes in the *a*–*b* plane, leading to a large scatter of the data points during the entire period 2015December–2018April. Note that some sub-samples showed even a negative *a*–*b* trend (e.g., those corresponding to the short-term flares recorded during MJD57364–375, 57395–57408, 57527-57544 in Interval1), which is expected when $g>\gamma_0$, i.e., there were electron populations with a very low initial energy $\gamma_0$ in the emission zone. Moreover, the co-existence of stochastic (second-order Fermi) acceleration also could weaken the aforementioned correlation, since the latter is not expected within the stochastic mechanism. The Monte Carlo simulations of [@k06] revealed that electrons can be accelerated at the shock front via EDAP and continue gaining energy via the stochastic mechanism into the shock downstream region (after escaping the shock front). After some time, a particle will be able to re-enter the shock acceleration region and repeat the acceleration cycle. Consequently, such combined acceleration will not result in the strong $a$–$b$ correlation. A stronger positive a–b correlation was found for our target during the XRT observations in 2005–2008 and 2015February ($\rho$=0.41–0.57; @k17a [@k18a]). An even stronger correlation was reported by [@m04] from the*BeppoSAX* observations in 1997–2000. On the other hand, this correlation was weaker during 2009–2012 ($\rho$=0.21$\pm$0.07) and absent in 2013January–2014June [@k16; @k17a]. \[ht!\] ------------------------- ------- ------------ ------------- ------------ ------- ---------------- --------------- ------------ -------- Dates TS $N_{pred}$ Flux $\Gamma$ TS $N_{\rm pred}$ Flux $\Gamma$ Remark (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 2015-12-08$-$08-22 55.0 143.5 6.05(0.85) 1.63(0.14) 175.8 24.6 2.14(0.29) 1.69(0.09) 3 2015-12-22$-$2016-01-05 113.1 380.1 7.67(0.88) 1.37(0.10) 639.9 55.9 3.07(0.30) 1.56(0.05) 2 2016-01-05$-$01-19 80.5 274.8 9.39(1.15) 1.65(0.12) 481.4 40.4 3.21(0.33) 1.63(0.06) 3 2016-01-19$-$02-02 80.5 256.1 7.86(0.96) 1.62(0.12) 400.8 37.6 2.95(0.31) 1.60(0.06) 3 2016-02-02$-$02-16 62.7 183.7 4.56(0.62) 1.22(0.16) 482.6 47.0 3.05(0.31) 1.69(0.06) 2 2016-02-16$-$03-01 103.0 280.5 8.28(1.02) 2.00(0.15) 257.0 28.6 1.79(0.22) 1.77(0.09) 3 2016-03-01$-$03-15 51.4 143.9 4.64(0.60) 1.72(0.14) 247.3 22.4 1.62(0.21) 1.66(0.08) 3 2016-03-15$-$03-29 43.2 108.5 3.65(0.69) 1.49(0.15) 178.0 20.1 1.39(0.18) 1.98(0.11) 2 2016-03-29$-$04-12 57.7 136.5 4.41(0.71) 1.39(0.14) 153.8 18.0 1.13(0.16) 1.80(0.10) 2 2016-04-12$-$04-26 61.9 144.0 5.40(0.76) 2.26(0.21) 95.4 16.2 1.04(0.20) 2.10(0.13) 2 2016-04-26$-$05-10 58.6 166.9 6.02(0.72) 1.96(0.12) 126.2 13.4 1.08(0.18) 1.59(0.09) 1 2016-05-10$-$05-24 60.5 179.6 7.39(0.91) 1.81(0.15) 274.3 23.3 2.27(0.27) 1.58(0.08) 3 2016-05-24$-$06-07 80.4 208 6.41(0.82) 2.03(0.18) 216.3 28.1 1.73(0.23) 2.22(0.12) 3 2016-06-07$-$06-21 110.6 353.7 9.64(1.04) 1.94(0.14) 334.2 33.4 2.30(0.26) 1.70(0.07) 1 2016-06-21$-$07-05 44.3 101.1 6.14(1.18) 2.18(0.20) 111.0 14.3 1.54(0.25) 1.91(0.12) 1 2016-07-05$-$07-19 46.7 99.8 5.97(1.20) 2.56(0.24) 144.9 13.5 1.30(0.20) 1.85(0.11) 1 2016-07-19$-$08-02 64.8 156.6 5.57(0.80) 1.77(0.15) 177.9 17.2 1.17(0.18) 1.47(0.08) 1 2016-08-02$-$08-16 37.7 106.2 6.54(1.08) 2.30(0.22) 66.4 7.7 1.54$^{\ast}$ - 2016-08-16$-$08-30 39.7 109.5 3.84(0.69) 1.64(0.15) 86.7 11.2 0.87(0.19) 1.68(0.10) 3 2016-08-30$-$09-13 36.7 70.2 2.92(0.68) 1.77(0.17) 118.2 13.9 0.88(0.15) 2.21(0.12) 2 2016-09-13$-$09-27 25.9 31.5 2.12(0.50) 2.37(0.26) 38.4 6.7 1.32$^{\ast}$ - 2016-09-27$-$10-11 28.0 56.7 4.15(1.00) 2.32(0.24) 6.1 3.2 1.56$^{\ast}$ - 2016-10-11$-$10-25 22.9 75.7 2.98(0.63) 1.26(0.20) 65.5 8.3 1.61$^{\ast}$ - 2016-10-25$-$11-08 32.1 55.3 3.45(0.77) 2.35(0.24) 81.6 9.9 0.81(0.17) 1.97(0.13) 1 2016-11-08$-$11-22 67.0 168.4 5.52(0.78) 1.83(0.15) 223.4 23.2 1.51(0.20) 1.71(0.09) 3 2016-11-22$-$12-06 55.0 201.8 5.12(0.67) 1.54(0.12) 103.4 13.4 1.00(0.19) 2.15(0.12) 2 2016-12-06$-$12-20 35.1 69.5 6.25(1.37) 2.72(0.28) 87.6 10.5 1.41(0.28) 2.79(0.24) 3 2016-12-20$-$2017-01-03 75.1 187.2 6.57(0.87) 1.89(0.15) 175.1 21.6 1.48(0.20) 1.81(0.10) 3 2017-01-03$-$01-17 94.8 246.7 7.90(1.01) 2.02(0.17) 317.0 33.0 2.15(0.24) 1.87(0.08) 3 2017-01-17$-$01-31 61.7 186.3 6.14(0.86) 1.75(0.15) 155.7 15.4 1.22(0.16) 1.54(0.08) 3 2017-01-31$-$02-14 89.7 205.9 7.19(0.99) 1.90(0.15) 262.1 29.9 1.95(0.23) 2.23(0.12) 2 2017-02-14$-$02-28 69.3 217.0 5.72(0.74) 1.91(0.15) 393.7 43.7 2.74(0.28) 1.72(0.07) 3 2017-02-28$-$03–14 46.2 111.8 4.02(0.78) 1.86(0.17) 72.3 10.7 0.73(0.15) 1.97(0.13) 3 2017-03-14$-$03-28 63.6 195.6 6.31(0.80) 1.73(0.13) 90.2 10.9 0.85(0.15) 2.24(0.14) 2017-03-28–04-11 73.2 210.9 4.45(0.58) 1.72(0.13) 212.2 24.4 1.19(0.16) 1.82(0.09) 2 2017-04-11$-$04-25 100.5 301.6 8.03(0.92) 1.69(0.12) 331.6 34.1 2.17(0.25) 1.80(0.08) 3 2017-04-25$-$05-09 58.4 164.5 4.98(0.69) 1.72(0.14) 176.1 17.6 1.20(0.16) 1.64(0.09) 2 2017-05-09$-$09-23 46.1 112.5 4.25(0.72) 1.63(0.15) 123.7 15.5 1.15(0.18) 1.74(0.10) 3 2017-05-23$-$06-06 48.5 111.2 5.94(0.98) 2.05(0.19) 70.0 9.8 0.93(0.20) 2.15(0.14) 3 2017-06-06$-$06-20 50.5 90.2 3.98(0.78) 2.19(0.22) 216.3 22.3 1.35(0.18) 1.83(0.09) 2 2017-06-20$-$07-04 26.7 39.5 2.80(0.66) 2.26(0.27) 35.9 5.2 1.42$^{\ast}$ - 2017-07-04$-$07-18 31.1 98.6 4.87(0.97) 1.37(0.12) 71.3 6.4 1.82$^{\ast}$ - 2017-07-18$-$08-01 75.8 171.7 6.24(0.83) 1.96(0.15) 194.0 23.2 1.46(0.19) 2.06(0.12) 3 2017-08-01$-$08-15 49.6 107.7 4.06(0.78) 2.02(0.18) 76.6 11.4 0.73(0.15) 2.45(0.20) 2 2017-08-15$-$08-29 49.6 121.7 4.84(0.80) 1.89(0.17) 112 12.7 0.98(0.18) 1.71(0.10) 3 2017-08-29$-$09-12 52.1 141.5 4.22(0.62) 1.29(0.11) 149.4 16.6 1.11(0.16) 1.53(0.08) 2 2017-09-12$-$09-26 83.8 195.2 7.14(0.90) 1.85(0.14) 126.4 17.0 1.14(0.17) 1.69(0.10) 3 2017-09-26$-$10-10 49.0 114.7 5.35(0.88) 2.34(0.18) 223.0 23.1 1.42(0.18) 1.73(0.08) 1 2017-10-10$-$10-24 58.0 134.7 4.87(0.80) 1.94(0.17) 98.5 13.8 1.15(0.22) 2.38(0.19) 2 2017-10-24$-$11-07 67.4 173.9 5.43(1.18) 1.81(0.15) 280.7 29.9 1.91(0.24) 1.87(0.09) 3 2017-11-07$-$11-21 84.1 147.2 6.02(0.84) 2.41(0.17) 216.2 24.2 1.31(0.17) 2.01(0.11) 1 2017-11-21$-$12-05 96.1 284.1 7.36(0.84) 1.54(0.12) 353.6 38.2 2.38(0.26) 1.87(0.07) 2 2017-12-05$-$12-19 63.3 248.0 7.22(0.98) 1.38(0.11) 152.5 15.8 1.50(0.21) 2.15(0.14) 2 2017-12-19$-$2018-01-02 73.8 206.6 6.74(0.87) 1.56(0.12) 387.7 36.2 2.69(0.28) 1.83(0.06) 2 2018-01-02$-$01-16 98.8 264.9 7.18(0.88) 1.66(0.13) 360.5 41.6 2.42(0.26) 1.62(0.06) 3 2018-01-16$-$01-30 86.6 279.7 10.71(1.24) 1.99(0.15) 321.5 29.0 2.82(0.33) 1.59(0.06) 1 2018-01-30$-$01-13 73.8 255.8 12.46(1.52) 1.97(0.15) 386.0 37.3 4.95(0.53) 1.87(0.06) 3 2018-02-13$-$02-27 161.5 587.7 12.77(1.21) 1.60(0.10) 748.9 65.5 4.17(0.40) 1.61(0.04) 3 2018-02-27$-$03-13 86.4 208.6 7.02(1.01) 1.95(0.15) 434.1 42.5 2.71(0.28) 1.73(0.06) 1 ------------------------- ------- ------------ ------------- ------------ ------- ---------------- --------------- ------------ -------- ### Stochastic Acceleration and Turbulence As in previous years, the source mostly showed low curvatures ($b\sim$0.3 or smaller for a vast majority of the log-parabolic spectra; see Section3.3.1), i.e., wider synchrotron SED, expected in the case of efficient stochastic acceleration [@m11b]. This result is related to the inverse proportionality of the PED curvature $r$ to the diffusion coefficient *D* in the Fokker-Plank kinetic equation: $r\propto D^{-1}$. On the other hand, $r\propto \varepsilon/(n_{\rm s}\sigma^2_{\varepsilon})$ [@m04], with $n_{\rm s}$, the number of the acceleration steps; $\sigma^2_{\varepsilon}$, the variance of the energy gain $\varepsilon$. Consequently, the detection of low spectral curvatures implies higher values of $n_{\rm s}$ and diffusion coefficient, which needs strongly developed turbulence in smaller acceleration region. In fact, the relativistic magnetohydrodynamic simulations of [@miz14] showed that shock propagation can strongly amplify the turbulence in the shocked jet material due to its interaction with higher-density inhomogeneities existing in the pre-shock medium; the more frequent detection of the 0.3–10keV IDVs in flaring X-ray states (compared to quiescence periods; see Section4.1.3) demonstrates the viability of this scenario for Mrk421 during the time interval presented in this work. Moreover, our detection of the anti-correlation $F_{\rm 0.3-10 keV}$–$b$ (see Figure 9c), i.e., a dominance of lower curvatures in higher X-ray states, favours the shock-in-jet scenario and strongly-developed turbulence during those states. The simulations of @miz14 also demonstrated that the higher-energy photons (including those having 0.3–10keV energies) are expected to originate in the smallest jet regions, which contain the strongest magnetic field and yield the most rapid time variability. In fact, the fastest IDVs, occurring within a few hundred seconds, were observed mostly in the highest X-ray states in Interval3a, and such instances can be related to the interaction of the relativistic shock front with the smallest-scale turbulent regions, embodying stronger magnetic fields (according to light-travel argument). Note also that this sub-interval was remarkable for the lowest mean curvature observed in the whole 2015-2018 time interval, implying the existence of the most efficient stochastic acceleration in that sub-interval. Along with the flux, the 0.3–10keV spectral parameters varied on intraday timescales. Sometimes, these changes were extremely fast, within 1-ks observational runs: curvature risings by 0.19–0.22; hardenings by $\Delta a$=0.08–0.23; shifts of the synchrotron SED peak by several keV to higher or lower energies; transitions from the log-parabolic PED into the power-law one and/or vice versa. Such behaviour could be related to the passage of a shock front through the regions with spatial scales $l$$\lesssim$10$^{14}$cm and significantly stronger turbulence, separated by the region with less extreme physical properties (magnetic field strength, particle number density, bulk Lorentz factor etc.). Our detections show the viability of the simulations of [@miz14], yielding a strong turbulence amplification on extremely small spatial scales in the shocked jet medium. Stochastic acceleration scenarios predict the presence of the $E_{\rm p}$–$b$ anti-correlation [@t09]. However, the latter was not detected at the 99% confidence level in Interval1 and was weak during Intervals1–3 (see Table\[cortable\] and Figure\[figcor\]b). This correlation is deduced from the relation $ \ln E\propto 2\ln \gamma_{\rm p}+3/(5b)$ [@t09], with $\gamma_{\rm p}$, the PED peak energy. Note that the significant difference in the $\gamma_{\rm p}$ values, corresponding to the different X-ray flares, may result in a large scatter of the data points in the $E_{\rm p}$–$b$ plane or even a destruction of the anti-correlation. Furthermore, this correlation is also predicted for EDAP through $ \log E\sim 3/(10b)$ [@c14], and since this relation shows a different slope compared to the stochastic case, the joint operation of these mechanisms can yield a large scatter in the $E_{\rm p}$–$b$ plane and weaken the anti-correlation. We found a positive $F_{\rm 0.3-10 keV}$–$E_{\rm p}$ correlation during Intervals1–3, i.e., a trend of shifting the synchrotron SED peak to higher energies with rising X-ray flux (see Figure\[figcor\]g). [@t09] demonstrated that as the peak energy of the emission increases, the cooling timescale shortens and can compete with the acceleration timescales. It is then possible to observe a bias in the $E_{\rm p}$–$b$ relation (weakening the anticorrelation), since the cooling timescale is shorter than that of EDAP or stochastic acceleration. Note that the anti-correlation between the 0.3–10keV and radio variabilities, discussed above and explained as resulting from the shifting of the PED peak with a rising X-ray flux (corroborated by our finding of the positive $F_{\rm 0.3-10 keV}$–$E_{\rm p}$ correlation), is expected in the framework of stochastic acceleration with a narrow initial energy distribution, having the mean energy significantly higher than the equilibrium energy [@k06]. Presumably, such a physical condition was not the case for some X-ray flares when no declining radio brightness was observed. Moreover, since there were some flares with a negative $a$–$b$ trend, this result hints at the low initial energies of the accelerating particles during those events. In the case of the low spectral curvature, the electron volume density $n_{\rm e}$ is expected to be higher, yielding a brighter IC peak within the SSC scenario [@m11b]. Since the source generally shows its IC peak at the TeV frequencies (see, e.g., @ac11), lower VHE states are expected along with the high spectral curvatures. In fact, during the majority of the XRT observations with $b$$>$0.4, Mrk421 was not detectable with 3$\sigma$ significance with FACT or showed excess rates lower than the mean value during 2015December–2018April. However, there were two exceptions showing higher VHE states along with $b$$>$0.4 (MJD57388 and 57548). On the other hand, the source was not detectable or showed low VHE states during some time intervals with predominantly low curvatures (e.g., MJD57408-57486, 58147–58162). These instances demonstrate that the one-zone SSC scenario was not always acceptable for our target in the here-presented period. ### Spectral Loops In the Bohm limit, the first-order Fermi mechanism yields an electron acceleration timescale $\tau_{\rm FI}$$\approx 10200(c/v^2_{\rm sh})\sqrt{\gamma^2-1}(B/1G)^{-1}$ , with $v_{\rm sh}$ – the shock speed [@tam09]. For a 1-G field, relativistic shock ($v_{\rm sh}\rightarrow c$) and $\gamma$$\lesssim$10$^4$, this timescale will be a few milli-seconds or shorter. In that case, the acceleration and injection of electrons into the emission zone will be instantaneous and a clockwise (CW) evolution of the X-ray flare in the plane $F_{\rm 0.3-10\,keV}$–HR is expected, making the spectrum progressively harder in the brightening phase of the source, due to the emergence of a flaring component starting at hard X-rays [@t09]. Although the de-absorbed soft 0.3–2keV and hard 2–10keV fluxes showed strong or very strong cross-correlations during Intervals1–3 (Figure\[hyster\]a and Table\[cortable1\]), the latter underwent a higher variability in each interval (see Table\[persum\] for the corresponding $F_{\rm var}$ and $\Re$ values) and the hysteresis patterns were clearly evident in the $F_{\rm 0.3-10\,keV}$–HR plane. The CW loops, expected in the case of EDAP, are evident in Figures \[hyster\]c–\[hyster\]n, \[hyster\]p, \[hyster\]r–\[hyster\]u during the various short and longer-term flares discussed in Section3.1. Two CW-loops were detected also on intraday timescales (Figures \[hyster\]w and\[hyster\]y). However, EDAP can not be considered as instantaneous in the case of significantly weaker magnetic fields, frequently inferred from the one or multi-zone SSC modelling of Mrk421 ($B\lesssim$0.05G; see, e.g., @abey17 [@a12]). Furthermore, no instantaneous injection is expected for the hadronic content in the emission zone whose acceleration time-scales are $\sim$1000-times longer than for electrons [@tam09]. Note that the latter is more naturally compatible with the hard $\gamma$-ray spectra characterized by the photon index $\Gamma\lesssim1.8$ [@sh16; @m93], and such spectra frequently were recorded during the LAT observations in the here-presented period (as hard as $\Gamma_{\rm min}$=1.26$\pm$0.13). Consequently, EDAP will be more gradual than instantaneous, X-ray flares propagate from low energies to high energies and counter-clockwise (CCW) spectral evolution should be observed [@tam09]. Such behaviour was also frequently observed in Intervals1–3 (see Figures\[hyster\]b–\[hyster\]e, \[hyster\]g–\[hyster\]j, \[hyster\]o–\[hyster\]t, \[hyster\]v). Note that the slow, gradual acceleration and CCW-loops are also expected during the stochastic acceleration in the jet region with low magnetic field and high matter density [@vv05]. On the contrary, this time-scale can be much shorter and even instantaneous in the purely or mainly lepton plasma if the matter density is low and high magnetic fields are presented. In such situation, CW-type loops can develop, although this requires quite ideal turbulence conditions with particle-scattering waves moving in opposite directions over a sufficiently long length-scale [@tam09]. ### Power-law Spectra Along with EDAP and stochastic mechanisms, there could be other competing  processes acting in the emission zone and weakening the observed $E_{\rm p}$–$b$ correlation. Namely, the first-order Fermi process can yield a power-law PED when the magnetic field properties are variable and its confinement efficiency becomes independent of the particle’s energy for some time intervals. Note that 9.6% of the 0.3–10keV spectra showed a simple power-law distribution of the photons with frequency, and they were observed mostly in higher X-ray states. This percentage was unprecedented high during 2005–2008 (27.5%) and the power-law spectra were observed most frequently during the densest XRT campaign in 2006June15–25, although they were recorded in any brightness states shown by the source in that period [@k18a]. A higher percentage (13%) and no clear trend with brightness was observed also during 2013January–May [@k16]. On the contrary, the periods 2009–2012 and 2013November–2015June were characterized by significantly fewer occurrence of power-law spectra (4.9%–6.7%; @k17a [@k18b]). Hard power-law PEDs with slopes $p$$<$2 can be established by relativistic magnetic reconnection, expected to operate efficiently in highly-magnetized plasma with the magnetization parameter $\sigma$$\gtrsim$10 [@sir14]. However, the simulated broadband SEDs, obtained by [@p16] for the cases $\sigma$=10–50, differ significantly from those of Mrk421 constructed using the data obtained during the different MWL campaigns (@abey17 [@b16; @a12] etc.), particularly, in the MeV–TeV energy range. ### The Position of the Synchrotron SED Peak and Variable Turbulence Spectrum During 2015December–2018April, only 15% of the spectra showed $E_{\rm p}$$>$2keV, i.e., peaking at the hard X-ray frequencies (taking into account the error ranges). This percentage is significantly lower than that shown by Mrk421 during 2005–2008 (24%; @k18a) and a higher occurrence of such spectra were recorded also during 2009–2012 (17%; @k18b). Note that the spectra with $E_{\rm p}>$2keV were mostly concentrated in the sub-periods 2006April–July, 2008March–June, 2009November, 2010January–May. Due to the position of the synchrotron SED peak at higher frequencies, BAT detected the source with 5$\sigma$ confidence frequently in the aforementioned time intervals: the BAT-band photons are generally of synchrotron origin in the HBL sources and no significant contribution from the IC photons are found, in contrast to the low-energy-peaking BLLs (LBLs; e.g., OJ287; see @k18c). On the other hand, the periods 2013January–May and 2013November–2015June were characterized by a significantly lower occurrence of hard X-ray peaking spectra (2% and 5%, respectively; see @k16 [@k17a]). Note that the percentage of the spectra with $E_{\rm p}$$>$2keV was significantly higher for 1ES1959+650 in 2016January–August (48%) and 2016August–2017November (28%) [@k18d; @k18e]. The highest value of this parameter was 12.80$\pm$0.86keV. However, more extreme case with 94% spectra with the synchrotron peaks in X-rays was recorded for Mrk501 during the extended X-ray flaring activity in 2014March–October [@k17b]. In that period, the maximum value $E^{max}_{\rm p}$=20.96$\pm$2.81keV and unprecedented spectral behaviour when the synchrotron SED peak position underwent a shift by at least two orders of frequency and moved beyond 100keV [@tav01]. For our target, the most extreme SED position was observed on 2006April22 with $E^{max}_{\rm p}$=26$^{+19}_{-8}$keV, obtained by [@t09] from the joint fit of the log-parabolic model with the XRT and BAT spectra. Although the same authors reported more extreme cases $E_{\rm p}$$>$100keV from the 2006April–June observations, our thorough analysis of these spectra showed insignificant spectral curvature and a good fit with a simple power-law [@k18a]. Fig.\[figcor\]h demonstrates a positive $\log E_{\rm p}$–$\log S_{\rm p}$ correlation with a slope of 0.63$\pm$0.08 – the value of the exponent $\alpha$ in the relation $S_{\rm p} \varpropto E^{\alpha}_{\rm p}$. This relation was predicted by the simulations of [@t11] corresponding to the case when the momentum-diffusion coefficient $D$ is variable during stochastic acceleration of the X-ray emitting electrons. Consequently, there should be a transition from the Kraichnan spectrum of the turbulence with the exponent $Q$=3/2 into the hard spherespectrum ($Q$=2). In the latter regime, the scattering and acceleration timescales are independent of the particle energy. During the transition, the synchrotron SED follows the expectation of a lower curvature for the harder turbulence spectra [@t11]. On average, the lowest curvatures were observe in Interval3 which shows the strongest $\log E_{\rm p}$–$\log S_{\rm p}$ correlation during 2015December–2018April (see Table\[cortable\]). This results can serve as another confirmation of the stochastic acceleration of particles in that period. ### LAT-Band Spectral Properties and Possible Jet-Star Interaction The photon indices corresponding to the softer $\Gamma_{\rm 0.3-2 GeV}$ and harder $\Gamma_{\rm 2-300 GeV}$ LAT-bands showed a weak cross-correlation during 2015December–2018April (see Figure\[latgamma\]b and Table\[cortable\]). On some occasions, the index $\Gamma_{\rm 2-300 GeV}$ was lower than the 0.3–2GeV one, which can be related to the soft gamma ray excess at energies of several hundred MeV. One of the possible explanations consists in a star-jet interaction, expected in the blazars hosted by elliptical galaxies (including Mrk421; see @s00). These galaxies may have a population of red giants surrounding the blazar jet and can carry large wind-blown bubbles into the jet, leading to gamma-ray emission through bubble-jet interactions [@t19]. Note that those instances, characterized by a spectral hardening with energy, are mostly observed in the period 2016April–August, while no opposite spectral trend was observed during that time (see Figure\[latgamma\]e and Table\[lat\]). The simulations of [@t19] have shown that the IC emission, resulting from the jet-bubble interaction, is negligible ($L_{\rm IC}\sim$10$^{40}$ergs$^{-1}$), while that generated by the synchrotron mechanism can make a significant contribution to the MeV energy budget ($L_{\rm IC}\sim$10$^{44}$ergs$^{-1}$; see @a12 for comparison). In the latter case, the equipartition value of the magnetic field and acceleration efficiency $\xi\gtrsim0.1$ are required. In 17 cases, the index $\Gamma_{\rm 2-300 GeV}$ was higher than its lower-energy counterpart (the data points situated below the red dashed line in Figure\[latgamma\]b; see Figure\[latgamma\]f for the corresponding SEDs). These cases could be related to the up-scatter of X-ray photons to the 2–300GeV energy range in the K–N regime, yielding a steepening of the corresponding photon spectrum with respect to that established in the 0.3–2GeV range by means of the Thompson up-scattering of the optical–UV photons [@k18e]. Finally, 8 out 23 LAT-band SEDs, where the difference between the $\Gamma_{\rm 0.3-2 GeV}$ and $\Gamma_{\rm 2-300 GeV}$ photon indices did not exceed the error ranges, were very hard and their origin could be related to the hadronic contribution to the 0.3–300GeV energy range (as suggested by @sh16). Summary ======= In this paper, we have presented the spectral and timing results obtained during the intensive *Swift*-XRT and MWL observations of Mrk421 in 2015December–2018April. The main results of our study are as follows: - Similar to the previous years, the source exhibited strong and erratic X-ray variability (without any quasi-periodicity). The most extreme behaviour was recorded during 2017December–2018February when a long-term flare, lasting more than 2months, was superimposed by short-term ones during which the 0.3–10keV flux exceeded a level of 5$\times$10$^{-9}$ergcm$^{-2}$s$^{-1}$, similar to that recorded on 2008June12, and even higher states were observed during the giant outburst in 2013April. This period was also characterized by several intraday flux doubling and halving events with $\tau_{\rm d}$=4.8–18.9hr, as well as by numerous lower-amplitude 0.3–10keV IDVs with $F_{\rm var}$=0.20–0.42, including extremely fast brightness fluctuations by 5%–18% within 180–600 seconds. Six another long-term flares of comparable durations but with lower amplitudes were evident during other parts of the period presented in this work. - The highest VHE states were recorded in 2018January (coinciding with those in the XRT band) and the TeV-band variability mostly showed a good correlation with the X-ray one, although there were several exceptions when the VHE flux showed a decline or low states during the fast X-ray flare, or the X-ray and VHE peaks were separated by the time interval of $\sim$1d or longer, posing problems for one-zone SSC scenarios. In other spectral ranges, Mrk421 exhibited a relatively different behaviour: there was only a weak positive $F_{\rm XRT}$–$F_{\rm LAT}$ correlation and the highest 0.3–300GeV states were recorded about 2-yr earlier than the X-ray ones, while the source exhibited only a moderate LAT-band flaring activity along with the strongest X-ray flares recorded in Interval3a. Similarly, the highest optical–UV states were observed during 2016January–February, and they were significantly lower 2 yr later when the source showed its highest X–ray activity. Consequently, the latter was anti-correlated with the UVOT-band variability, and a similar $F_{\rm 0.3-10 keV}$–$F_{\rm 15 GHz}$ relation was observed during the entire 2015December–2018April period. Such MWL variability favours some earlier simulations of the second-order Fermi process, when a population of the accelerating electrons are characterized by a narrow initial distribution of energy, having a mean value significantly higher than the equilibrium energy. - During Intervals 1 and 3, the distributions of the de-absorbed 0.3–10keV flux showed lognormality features, which could be indicative of the variability imprint of AD on the jet. However, the data from Interval2 and the highest X-ray states did not show the same property. Since Interval2 clearly shows a better fit with the Gaussian function and since this period was characterized, on average, by lower X-ray states (see Section3.1 and Table\[persum\]), a lack of the lognormality could be related to weaker shocks through the jet compared to other periods (possibly, due to weaker AD instabilities). The FACT and LAT-band fluxes showed lognormality features in all intervals, in contrast to the radio-UV observations. The 0.3–10keV IDVs were observed significantly more frequently during higher X-ray states and did not exhibit a lognormality. This result favours the shock-in-jet scenario. The longer-term flares may result from the propagation and evolution of relativistic shocks through the jet. The shock appearance could be related to an abrupt increase of the collimation rate at the jet base owing to some processes in the accretion disc, yielding a lognormal flaring behaviour on longer timescales. - Along with the strong flux variability, the source also exhibited an extreme spectral behaviour. The 0.3–10keV spectra generally showed their best fits with the log-parabolic model, yielding wide ranges of the curvature parameter $b$=0.07(0.05)–0.48(0.04) and photon index at 1keV $a$=1.63(0.03)–2.92(0.02). The position of the synchrotron SED peak underwent extreme variability on various timescales between the energies $E_{\rm p}$$<$0.1keV (the UV frequencies) and $E_{\rm p}$$>$15keV, with 15% of the spectra peaking at hard X-rays. The synchrotron SED showed a positive correlation with the 0.3–10keV flux: it shifted by several keV to higher energies during the flaring phases and moved back along with brightness drops, exhibiting the most violent intraday variability by several keV during the strongest X-ray flares. 33% of the spectra were harder than $a$=2, and the energy spectral shape generally followed a harder-when-brighter trend (except for some short time intervals with the opposite trend, explained by the emergence of a new soft X-ray component in the emission zone). The photon index varied on diverse timescales with variations from $\Delta a$=0.08–0.23 within 0.13–0.28hr to $\Delta a$=0.66–1.07 in 3–27d. 9.6% of the spectra were fitted well with a simple power-law, with photon indices $\Gamma$=1.79-2.91 and strongly following the harder-when-brighter trend. The source mostly showed a low spectral curvature ($b\sim$0.1–0.3) and an anti-correlation $E_{\rm p}$–*b*, as predicted for the efficient stochastic acceleration of X-ray emitting electrons by the magnetic turbulence. Moreover, the source showed a positive $a$–$b$ correlation, expected within the EDAP scenario, although it was weak, possibly due to the competition with other types of the acceleration mechanisms and cooling processes, not displaying the same correlation. - The 0.3–10keV spectra showed a relation $S_{\rm p}$$\varpropto E^{\alpha}_{\rm p}$, with $\alpha$$\sim$0.6 which demonstrates a transition from the Kraichnan-type turbulence spectrum into the hard sphere  one, due the variability of the momentum-diffusion coefficient. This result corroborates the importance of stochastic acceleration in the here-presented period. Our study of the spectral hysteresis patterns in the flux–HR plane shows the patterns of both the instantaneous injection and the gradual acceleration of X-ray emitting electrons, owing to first and second-order Fermi processes. - The source frequently showed very hard 0.3–300GeV spectra, predicted for a hadronic contribution to the HE emission. On some occasions, the corresponding SED showed a soft $\gamma$-ray excess, possibly owing to the jet interaction with a wind-blown bubble from a nearby red giant. This suggestion is corroborated by the fact that the MeV-excess SEDs mostly belong to the period 2016April–August. On the contrary, there was a softening in the 2–300GeV energy range compared to the 0.3–2GeV spectrum, possibly due to the upscatter of X-ray photons in the 2–300GeV energy range in the K-N regime. This may yield a steepening of the corresponding photon spectrum with respect to the 0.3–2GeV range, corresponding to the Thompson up-scatter of the optical–UV photons. PR acknowledges the contract ASI-INAF I/004/11/0. We acknowledge the use of public data from the *Swift* data archive. This research has made use of the `XRTDAS` software, developed under the responsibility of the ASDC, Italy, and the data from the OVRO 40-m monitoring program which is supported in part by NASA grants NNX08AW31G and NNX11A043G, and NSF grants AST-0808050 and AST-1109911. We thank the FACT collaboration for making their analysis results publicly available. Finally, we thank the anonymous referee for his/her useful comments and suggestions that helped to improve the quality of the paper. Abdo, A. A., et al. 2011, , 736, 131 Abeysekara, A. U., et al. 2017, , 834, 2 Acciari, V. A., et al. 2011, , 738, 25 Acero, F., et al. 2015, , 218, 23 Ahnen, M. L., et al. 2016, , 593, 91 Alecsic, J., et al. 2012, , 542,100 Alecsic, J., et al. 2015a, , 578, 22 Alecsic, J., et al. 2015b, , 576, 176 Anderhub, H. et al. 2013, Journ. of Instr., 8, article id. P06008 Atwood, W. 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[^5]: http://swift.gsfc.nasa.gov/results/transients/weak/Mrk421/ [^6]: http://maxi.riken.jp/ [^7]: See fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/upper\_limits.html [^8]: https://fermi.gsfc.nasa.gov/ssc/data/analysis/user/ [^9]: See http://www.fact-project.org/monitoring/ [^10]: See http://james.as.arizona.edu/ psmith/Fermi/ [^11]: see http://www.astro.caltech.edu/ovroblazars/ [^12]: Note that the FACT results are still provided in the form of the excess rates, and this result should be considered with caution. [^13]: The de-absorbed 0.3–10keV flux values, corresponding to the low X-ray states of the source, were below the thresholds of about 4.0$\times$10$^{-10}$cgs, 4.5$\times$10$^{-10}$cgs and 9.0$\times$10$^{-10}$cgs in Intervals 1,2 and 3, respectively.

--- abstract: 'We present the results of a *Herschel* survey of 21 late-type stars that host planets discovered by the radial velocity technique. The aims were to discover new disks in these systems and to search for any correlation between planet presence and disk properties. In addition to the known disk around GJ 581, we report the discovery of two new disks, in the GJ 433 and GJ 649 systems. Our sample therefore yields a disk detection rate of 14%, higher than the detection rate of 1.2% among our control sample of DEBRIS M-type stars with 98% confidence. Further analysis however shows that the disk sensitivity in the control sample is about a factor of two lower in fractional luminosity than for our survey, lowering the significance of any correlation between planet presence and disk brightness below 98%. In terms of their specific architectures, the disk around GJ 433 lies at a radius somewhere between 1 and 30au. The disk around GJ 649 lies somewhere between 6 and 30au, but is marginally resolved and appears more consistent with an edge-on inclination. In both cases the disks probably lie well beyond where the known planets reside (0.06-1.1au), but the lack of radial velocity sensitivity at larger separations allows for unseen Saturn-mass planets to orbit out to $\sim$5au, and more massive planets beyond 5au. The layout of these M-type systems appears similar to Sun-like star + disk systems with low-mass planets.' author: - | G. M. Kennedy[^1]$^1$, G. Bryden$^2$, D. Ardila$^{3,4}$, C. Eiroa$^5$, J.-F. Lestrade$^6$, J. P. Marshall$^7$, B. C. Matthews$^{8,9}$, A. Moro-Martin$^{10}$, M. C. Wyatt$^{11}$\ $^1$ Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK\ $^2$ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA\ $^3$ NASA Herschel Science Center, California Institute of Technology, MC 100-22, Pasadena, CA 91125, USA\ $^4$ The Aerospace Corporation, M2-266, El Segundo, CA 90245, USA\ $^5$ Dpto. Fisica Teorica, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain\ $^6$ Observatoire de Paris - LERMA, CNRS, 61 Av. de l’Observatoire, 75014, Paris, France\ $^7$ Academia Sinica, Institute of Astronomy and Astrophysics, Taipei 10617, Taiwan\ $^8$ National Research Council of Canada Herzberg Astronomy & Astrophysics Programs, 5071 West Saanich Road, Victoria, BC, V9E 2E7, Canada\ $^9$ Department of Physics & Astronomy, University of Victoria, 3800 Finnerty Road, Victoria, BC, V8P 5C2, Canada\ $^{10}$ Space Telescope Science Institute, 3700 San Martin Dr, Baltimore, MD 21218, USA\ $^{11}$ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK title: 'Kuiper Belt Analogues in Nearby M-type Planet-host Systems' --- planetary systems: formation — circumstellar matter — stars: individual: GJ 433 — stars: individual: GJ 649 Introduction {#s:intro} ============ It is now well established that planet formation processes are robust, and proceed around stars of a wide range of masses. At the higher mass end, planets have been discovered around evolved stars with masses up to three times the Sun’s . At the lower mass end the results have been equally impressive, with planets discovered around objects ten times less massive than the Sun, and whose luminosity is a thousand times weaker [e.g. @2016Natur.533..221G; @2016Natur.536..437A]. This wide mass range provides a unique way to study planet formation processes, and has shown that while the occurence rate of giant planets increases towards higher mass stars , the converse is true for the frequency of Earth to Neptune-mass planets [@2015ApJ...798..112M]. In tandem with these searches, observations that seek to detect the building blocks of these planets have also been conducted. These mid and far-infrared (IR) surveys detect ‘debris disks’, the collections of small dust particles that are seen to orbit other stars (the ‘dust’ comprises various constituents, such as silicates, ice, and organic compounds). Since their discovery in the 1980’s, a growing body of evidence has shown that they can be interpreted as circumstellar disks made up of bodies ranging from $\sim$$\mu$m to many km in size; while the observations only detect $\mu$m to mm-size particles, the lifetime of these particles is commonly shorter than the age of the host star, leading to the conclusion that they must be replenished through the collisional destruction of a mass reservoir of larger planetesimals [e.g. @1993prpl.conf.1253B]. For main-sequence stars this paradigm is generally accepted, so in terms of the dust having an origin in collisions between larger bodies, debris disks can be genuinely thought of as analogues of the Solar System’s Asteroid and Kuiper belts. A key unknown is how the planetesimals acquire high enough relative velocities for their collisions to be destructive; while it is possible that planets excite these velocities [@2009MNRAS.399.1403M], it may be a natural outcome upon emergence from the gas rich phase of evolution, or the planetesimals may ‘stir’ themselves [e.g. @2004AJ....127..513K], in which case planets are not necessarily needed in order for debris disks to exist. However, it is well known that the Solar System planets play an important role in sculpting the Asteroid and Kuiper belts, two examples being the presence of the Kirkwood gaps and the capture of Pluto into 2:3 mean motion resonance by Neptune [@1993Natur.365..819M]. In attempts to make analogous link in other planetary systems, hypotheses that connect the properties of the disks and planets have been developed, and vary in complexity. The most basic is that some systems are simply ‘better’ at forming large bodies (whether those bodies be planetesimals or planets), and more detailed models suggest that the outcomes depend on whether planetary instabilities occurred . As with planets, merely detecting these belts is challenging, so quantifying the connection between the planets and disks in these systems is typically limited to searching for correlations between their basic properties . Ultimately, these searches yielded a significant correlation between the presence of radial velocity planets and the brightness of debris disks around Sun-like stars [@2014prpl.conf..521M]. This trend is unfortunately not strong, so while splitting the sample to look for trends among sub-samples (e.g. as a function of planet mass) yields tentative trends [e.g. @2012MNRAS.424.1206W] it also lowers the significance. Thus, while there is evidence that some Sun-like stars are indeed better at forming disks and planets than other, the origin of this correlation remains unclear. In the case of low-mass stars the challenge of finding connections between the planet and disk populations is even greater; for disks at the typical radial distances of a few tens of astronomical units, the low stellar luminosities do not heat the dust to temperatures greater than about 50K. While the Stefan-Boltzmann law therefore limits the luminosity of these disks, the low temperatures further hinder detection because discoveries must be made at far infrared and millimeter wavelengths . Thus, it is not particularly surprising that efforts to discover debris disks around late-type stars at mid-infrared wavelengths have often been unsuccessful . Further, the sensitivity of surveys is normally such that the non-detections are not sufficiently constraining to rule out disks that have similar properties to those that are known to orbit Sun-like stars . In this paper, we present far infrared *Herschel*[^2] observations that aim to detect Kuiper belt analogues around a sample of 21 nearby late K and M-type stars that host planets discovered by the radial velocity technique. The primary aim is to search for a correlation between the presence of planets and the brightness of disks, and secondary aims are to detect new disks that may be amenable to further detailed investigation, and to obtain more sensitive observations than were possible with larger surveys. We present the sample and observations in section \[s:obs\], discuss the results in section \[s:disc\], and summarise and conclude in section \[s:conc\]. Sample and Observations {#s:obs} ======================= Name ObsIDs OD Reps ----------- ---------------- ------ ------ GJ 176 1342250278/279 1202 6 GJ 179 1342250276/277 1202 6 GJ 317 1342253029/030 1245 6 GJ 3634 1342257175/176 1310 6 GJ 370 1342256997/998 1308 6 GJ 433 1342257567/568 1316 6 GJ 1148 1342247393/394 1138 6 GJ 436 1342247389/390 1138 6 GJ 9425 1342249877/878 1194 6 GJ 9482 1342248728/729 1170 6 HIP 79431 1342262219/220 1355 6 GJ 649 1342252819/820 1244 6 GJ 1214 1342252011/012 1237 6 GJ 674 1342252841/842 1244 6 GJ 676 A 1342243794/795 1058 6 GJ 849 1342246764/765 1121 6 : PACS observations of 16 targets taken as part of our programme (OT2\_gbryden\_2). OD is the Herschel Observing Day, and Reps is the number of repeats of a standard PACS mini scan-map used to reach the desired sensitivity.[]{data-label="tab:obs"} Our sample comprises nearly all low-mass planet-host stars within 20pc. Most stars are M spectral type, but we include three that are late K types (GJ 370, GJ 9425 and GJ 9482). Not all systems in the final sample were known to host planets at the time the observations were proposed (2011 September), but some in which planets were subsequently discovered were observed by the volume-limited DEBRIS Key Programme . The final sample has 21 stars, 16 of which were observed by *Herschel* in this programme, and which are listed in Table \[tab:obs\]. Five more targets, GJ 15 A, GJ 581, GJ 687, GJ 842, and GJ 876, were observed by the DEBRIS survey so are also included in our sample . The sample does not include the planet host Proxima Centauri [@2016Natur.536..437A], as it was not observed by *Herschel*. While it has been suggested to host excess emission arising from a debris disk [@2017arXiv171100578A], these observations use the Atacama Large Millimeter Array (ALMA) and this system is therefore not easily integrated into our sample. Two of our targets are possible wide binaries; GJ 15 A (NLTT 919) is a common proper motion pair at a projected separation of 35 with NLTT 923 [@2004ApJS..150..455G], and GJ 676 A has a wide common proper motion companion (GJ 676 B) at a projected separation of 50[@1994RMxAA..28...43P]. We do not expect the planetary systems to be affected seriously by these companions, so retain them in our sample. {width="27.50000%"} {width="27.50000%"} {width="27.50000%"} {width="27.50000%"} {width="27.50000%"}\ {width="27.50000%"} {width="27.50000%"} {width="27.50000%"} {width="27.50000%"} {width="27.50000%"} The targets were observed using the Photodetector Array Camera and Spectrometer , using the so-called ‘mini-scan map’ mode. A series of ten parallel scans with a separation of 4 are taken to make a single map, which is repeated six times to build up the signal. One such sequence coresponds to a single observation ID number, or ObsID. The observatory is then rotated by 40$^\circ$, and the sequence repeated, to provide some robustness to striping artefacts and low-frequency noise. The total integration time for each source is 56 minutes. For our observations the noise level at 100$\mu$m was typically 1mJy, while observations carried out by DEBRIS (integration time of 15 minutes) had fewer repeats and a noise level nearer 2mJy. The images used in the analysis are the standard ‘level 2.5’ observatory products obtained from the Herschel Science Archive,[^3] which combine the two observing sequences (ObsIDs) into a single image. Photometry $F_{\rm obs}$ for each source was extracted using point spread function (PSF) fitting. Observations of the calibration star $\gamma$ Dra, again level 2.5 observatory products, were used as PSFs, which were rotated to a position angle appropriate for each observation. The fitting was done at 100 and 160$\mu$m simultaneously, so the four free parameters in each fit were a position common to both wavelengths, and two fluxes (i.e. $F_{100}$ and $F_{160}$). Uncertainties $\sigma_{100}$ and $\sigma_{160}$ were estimated by measuring the flux in apertures at hundreds of random locations near the center of the images; this method was found to be more reliable and provide more realistic flux distributions than attempting to fit PSFs at random locations. The apertures were chosen to be those optimal for source extraction (5 and 8 for 100 and 160$\mu$m respectively, derived using calibration observations). In the case of GJ 649 there is evidence that the source (i.e. disk) is marginally resolved (see Figure \[fig:im\]), so the flux for this source at 100$\mu$m is measured using an aperture radius of 10, and the uncertainty estimated as above but with 10apertures. The results of the source extraction are summarised in Table \[tab:fluxes\], and the results for a few problematic sources are described in more detail below. To assess whether each star shows the infrared excess that is indicative of a debris disk requires an estimate of the flux density expected from the stellar photosphere $F_\star$ at the PACS wavelengths. These estimates are made by fitting stellar photosphere models to optical and near-IR photometry. The method has been described elsewhere, and for example has been used for the DEBRIS survey and shown to provide photospheric fluxes that are sufficiently precise that the detection of excesses is limited by the *Herschel* photometry, not the photosphere models [i.e. $\sigma_{\rm obs} > \sigma_\star$, @2012MNRAS.426.2115K; @2012MNRAS.421.2264K]. While photospheric models for late-type stars are less precise than for earlier types (e.g. because of uncertain molecular opacity), the flux of many of our target stars is predicted to be near our noise level and the models are not a limiting factor. The photospheric predictions at 100 and 160$\mu$m are given in Table \[tab:fluxes\]. The significance of any excess is then given in each PACS bandpass by $\chi = (F_{\rm obs} - F_\star) / \sqrt{\sigma_{\rm obs}^2 + \sigma_\star^2}$, where $\chi>3$ is taken to be a significant excess. To summarise the observational results; in addition to the disk known to orbit GJ 581, we find two new systems that show strong evidence for infrared excesses: GJ 433 and GJ 649, whose images are shown in Figure \[fig:im\]. ------------ --------- -------- ------ ----------------- --------------- -------------------- ------------------ ----------------- --------------- -------------------- ------------------ ----------------------------- GJ HIP no. SpTy Dist $F_{\star,100}$ $F_{\rm 100}$ $\sigma_{\rm 100}$ $\chi_{\rm 100}$ $F_{\star,160}$ $F_{\rm 160}$ $\sigma_{\rm 160}$ $\chi_{\rm 160}$ Notes (pc) (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) GJ 15 A 1475 M2V 3.6 15.3 14.9 2.2 -0.2 5.9 13.2 3.0 2.4 Photosphere at 100$\mu$m GJ 176 21932 M2.5V 9.4 4.1 3.6 1.6 -0.3 1.6 -3.6 6.4 -0.8 No detection GJ 179 22627 M2V 12.4 1.3 -1.6 1.2 -2.5 0.5 -4.7 2.8 -1.9 No detection GJ 317 - M3.5V 15.3 1.1 3.2 1.1 1.9 0.4 4.9 2.2 2.0 No detection GJ 370 48331 K6Vk: 11.3 5.7 6.9 0.9 1.3 2.2 -5.2 3.2 -2.3 Photosphere at 100$\mu$m GJ 3634 - M2.5 19.8 0.7 -0.9 1.2 -1.3 0.3 3.0 2.8 1.0 Detection at 6 SW **GJ 433** 56528 M2V 9.1 3.9 11.9 1.3 6.2 1.5 13.9 4.3 2.9 Excess detection GJ 1148 57050 M4.0Ve 11.1 1.5 1.4 1.0 -0.1 0.6 -2.1 3.3 -0.8 No detection GJ 436 57087 M3V 9.7 2.4 3.4 1.1 0.9 0.9 4.3 2.3 1.5 No detection GJ 9425 63833 K9Vk: 15.9 3.1 -0.7 2.1 -1.8 1.2 -15.1 7.6 -2.1 No detection GJ 9482 70849 K7Vk 23.6 1.0 1.6 1.4 0.5 0.4 -4.1 3.4 -1.3 No detection GJ 581 74995 M3V 6.3 3.8 21.8 1.5 11.8 1.5 22.4 5.0 4.2 Excess - 79431 M3V 14.4 1.7 - - - 0.6 - - - Extended detection at 5 N **GJ 649** 83043 M2V 10.4 3.6 22.6 2.4 7.9 1.4 16.3 5.2 2.9 Excess detection, extended? GJ 1214 - M4.5V 14.6 0.3 0.9 1.1 0.6 0.1 -0.1 2.3 -0.1 No detection, source 10 W GJ 674 85523 M3V 4.5 8.1 - - - 3.1 - - - Extended, high background GJ 676 A 85647 M0V 15.9 2.6 0.9 1.1 -1.6 1.0 0.5 2.2 -0.2 No detection, source 10 SW GJ 687 86162 M3.0V 4.5 10.1 6.1 1.6 -2.5 3.9 0.2 3.4 -1.1 No detection GJ 832 106440 M2/3V 5.0 10.4 12.5 1.6 1.3 4.0 1.2 3.5 -0.8 Photosphere at 100$\mu$m GJ 849 109388 M3.5V 8.8 4.3 4.2 1.2 -0.1 1.7 3.6 1.8 1.1 No detection GJ 876 113020 M3.5V 4.7 8.1 6.5 1.6 -1.0 3.2 6.5 3.5 0.9 Photosphere at 100$\mu$m ------------ --------- -------- ------ ----------------- --------------- -------------------- ------------------ ----------------- --------------- -------------------- ------------------ ----------------------------- Several other targets were also found to have emission at or near the source position, but in these cases we do not believe the emission to be associated with the star in question. These are shown in Figure \[fig:im\]. - GJ 3634: A bright ($\sim$14mJy) source is seen 6 SW of the expected position of GJ 3634. This offset is larger than expected given the $\sim$2 1$\sigma$ pointing accuracy of *Herschel*[^4] and our small sample size. By comparing the positions of several other sources detected in the 100 $\mu$m PACS image with the (optical) DSS2 plates[^5] we found that three were almost perfectly coincident. Thus, we conclude that the 6 offset seen is real, and that the PACS detection near GJ 3634 is not associated with this star. - HIP 79431: Extended structure is seen to the North of the stellar position, but the peak is 5 away. Only one low S/N source was seen to be common between the PACS and DSS2 images, with perfect coincidence. The background as seen in IRAS and WISE images is complex and variable. We conclude that the large offset and high background mean that the detected source is unlikely to be associated with HIP 79431. - GJ 674: The background level around GJ 674 is significantly above zero. At 100 $\mu$m the flux in the image peaks at the position of GJ 674, but if a point source with the photospheric flux of GJ 674 is subtracted the background becomes uniform. Thus, we conclude that the image shows emission from the star GJ 674 superimposed on a non-negligible background, and that there is no evidence for excess emission from the star itself. Discussion {#s:disc} ========== Our survey finds two new excess detections, around the stars GJ 433 and GJ 649. We first consider these detections as part of our sample, and then take a closer look at the architecture of these two systems in more detail. Planet - disk correlation {#ss:corr} ------------------------- One of our goals was to test for a correlation between the brightness of debris disks around low-mass stars and the presence of planets. That is, all stars may host debris disks, but we can only detect those above a given dust level, so we cannot test for a correlation between the ‘existence’ of planets and disks. The same is true for planet detection of course, so we are in fact testing for a correlation between disks above a given brightness threshold and planets above a given semi-major axis vs. mass threshold (acknowledging that the star-to-star sensitivity also varies). These thresholds are discussed below. A significant correlation has been seen among Sun-like stars that host radial velocity planets [@2014prpl.conf..521M], and tentative evidence that this trend is stronger for stars that host low-mass planets was found among a small sample of nearby stars . No clear trends were seen in the volume-limited DEBRIS FGK-type sample considered by @2015ApJ...801..143M, illustrating the tentative nature of the latter trend, and that samples that do not specifically target planet-host stars suffer from small numbers of planet hosts that limit the power to discover trends. Here, our sample comprises 21 planet-hosting low-mass stars that were observed in search of IR excesses by *Herschel*, for which three were found to host disks. Thus, our detection rate is 14%, but clearly suffers from a small number of detections. As a control sample, we consider the volume-limited DEBRIS M-type sample, which comprises 89 nearby stars [@2010MNRAS.403.1089P]. Of these, two were discovered to host debris disks; the planet host GJ 581 and the third star in the very wide Fomalhaut triple system, Fomalhaut C [@2014MNRAS.438L..96K]. We remove GJ 581 and the four other planet-host stars from this sample, leaving 84 stars with one disk detection, a rate of 1.2%. A Fisher’s exact test to determine whether these two populations could arise from the same underlying distribution yields a $p$-value of 0.025, thus showing reasonable evidence that the planet-host stars have a tendency to have more detectable (i.e. brighter) debris disks. The Fomalhaut system is known to be relatively young, at 440Myr [@2012ApJ...754L..20M]; if we were to assume that all of the planet host systems are older than this and exclude Fomalhaut C from the control sample the $p$-value decreases to 0.01. However, we cannot be sure that the planet-host stars are all older than the Fomalhaut system, since for example GJ 674 may also be a relatively young system . {width="48.00000%"} {width="48.00000%"} Thus, we find suggestive evidence that debris disks are more easily detected around M-type stars that also host planets. A further consideration however is whether the observations are biased towards detections for the planet-host sample. This might be expected given that our noise level is about half that of the DEBRIS observations of the control sample, but might also be balanced by the fact that all DEBRIS M-type stars are within 10pc, and thus on average closer than our planet-host stars. The relative sensitivities for the two samples is shown in Figure \[fig:detlims\], where the grey scale shows the number of systems for which disks at a given temperature and above a certain fractional luminosity ($f = L_{\rm disk}/L_\star$) could have been detected. The lowest red contour shows the maximum sensitivity (disks that could have been detected around only one star), the highest shows the level above which disks could have been detected around all stars, and the intermediate contours show where disks could have been detected around 25, 50, and 75% of systems. By comparing the red contours it can be seen that our observations could typically detect disks that are a factor of two to three lower in fractional luminosity than those observed by DEBRIS (as expected from observations that are 2-3 times deeper). While the three disks around planet-host stars could have been detected around 75% of our sample, they could only have been detected around about 30% of the DEBRIS sample. Thus, the evidence for any correlation between planets and debris disk brightness is weaker than suggested by the $p$-value above. The significance of the $p$-value may be further reduced by future radial velocity observations, because an implicit assumption is that the stars in the control sample do not host planets in a similar parameter space range as those around our planet-host sample. This is unlikely to be true because not all systems in our control sample will have been observed in search of planets, and our control sample is best termed ‘stars with no known planets’. If any of the systems in the control sample that do not host disks were in fact found to host planets, the significance of our result would decrease further. If however Fomalhaut C were found to host a planet (and a search may be well motivated by our results), the significance would increase. As noted earlier, it is not yet known whether M-type stars host a disk population that is the same or different to those that orbit Sun-like stars, and a major problem is that obtaining comparably sensitive observations is challenging. This sensitivity difference can be seen by comparing the contours in the right panel of Figure \[fig:detlims\] with those in Figure 4 of Sibthorpe et al. (2017, MNRAS in press), which shows the sensitivity for FGK-type stars observed as part of the DEBRIS survey (and for which an FGK-type disk detection rate of 17% was obtained). The 50% contour for our survey is at best about $f = 5 \times 10^{-6}$, an order of magnitude better than achieved by DEBRIS for M-type stars. In comparison, our survey is about midway between the two in terms of sensitivity. Therefore, with the caveats that the number of detections is small, and that the results could be biased by a planet-disk correlation, the fact that we have here obtained a disk detection rate similar to that seen for Sun-like stars suggests that in surveys of equal sensitivity in fractional luminosity the disk detection rate among Sun-like and M-type stars should be approximately the same. A marginally resolved disk around GJ 649 {#ss:649} ---------------------------------------- {width="48.00000%"} {width="48.00000%"} GJ 649 (HIP 83043, BD+25 3173, LHS 3257) was reported to host a planet with a minimum mass similar to Saturn’s, in an eccentric 598 day (1.1au) orbit [@2010PASP..122..149J]. The age of the star is uncertain, though it was classed as a member of the ‘old disk’ (as opposed to the young disk or halo) based on kinematics [@1992ApJS...82..351L], and noted to be among the 20% most chromospherically active early M-type stars [@2010PASP..122..149J]. Using constraints from the disk temperature and *Herschel* images we can therefore build a picture of the system’s architecture. The flux density distribution for GJ 649 is shown in Figure \[fig:seds\]. The excess flux above the photosphere is modelled using a modified blackbody function, where the disk spectrum is divided by $\lambda/210\mu{\rm m}$ beyond 210$\mu$m. This steeper long-wavelength spectral slope approximates the poor efficiency of dust emission at wavelengths longer than the grain size, though in this case is not constrained and included simply in order to make the extrapolations to millimeter wavelengths more realistic. The main point to take away from this figure is that the dust thermal emission is very cold, so could not have been detected in the WISE observations at 22$\mu$m. The best-fit disk temperature is $50$K with $f = 7 \times 10^{-5}$, but is uncertain because the 160$\mu$m observation is not formally a 3$\sigma$ detection of the disk (i.e. Table \[tab:fluxes\] shows that $\chi_{160}$ for GJ 649 is 2.9). The non-detection of an excess at 22$\mu$m means that the temperature cannot be significantly more than 100K. ![*Herschel* image of GJ 649 after subtracting point sources near the location of GJ 649 (at the white +) and at the bright peak to the SE (at the black +, see Figure \[fig:im\]). The low level residual structure around GJ 649 provides circumstantial, though not conclusive, evidence, that the disk is resolved. The asymmetry in the residuals suggests that the disk position angle is near to North, and that the disk is closer to edge-on than face-on. White contours are at 1, 2, and 3 times the 1$\sigma$ noise level. The center of the image is approximately midway between the plus symbols.[]{data-label="fig:im2"}](figs/im100-1-starsub.eps){width="45.00000%"} Given a stellar luminosity of 0.044$L_\odot$ the best fit temperature of 50K corresponds to a radial distance of 6au if the disk material behaves as a blackbody, while a temperature of 100K yields a distance of about 2au. Given that most debris disks are comprised of dust small enough to have super-blackbody temperatures, the disk around GJ 649 would be expected to be larger than blackbody estimates, by a factor of several at least [e.g. @2012ApJ...745..147R; @2013MNRAS.428.1263B; @2014ApJ...792...65P; @2016ApJ...831...97M]. This factor was found to be 6-20 for GJ 581 , with the large uncertainty arising because the disk radius depends on the square of the temperature. At a distance of 10.4pc the GJ 649 disk may therefore have an angular diameter large enough to be resolved. This extent may be confirmed by the *Herschel* images, which at 100$\mu$m show some extended residual emission after PSF subtraction (see Figure \[fig:im2\]). The fact that these residuals are extended in a non-axisymmetric pattern suggests that the disk may be nearer to edge-on than face-on, as might be expected given in the case of a planet detection with the radial velocity technique. Given that most of the residual contours are only 1$\sigma$ however, we consider that these residuals provide circumstantial evidence that the disk is resolved, in which case the disk diameter would be similar to the PACS beam size of 6. We therefore conclude that the disk radius could lie in the range 2-50au, but is more likely to be a few tens of au. The system layout is shown in Figure \[fig:sys\], where the planet GJ 649 b is indicated by the dot, and the error bar indicates the range of radii covered by the eccentric orbit. The solid line shows limits estimated based on the radial velocity residuals once the best-fit planet orbit is subtracted,[^6] indicating that planets more massive than Saturn that orbit beyond about 5au would not have been detected. The range of estimated disk locations is shown by the hatched region, where we have taken the marginally resolved image to indicate that the disk has a radius between 10-30au. The basic conclusion is that while the separation between the planet and disk is probably large, it is possible that this gap is occupied by one or more undetected planets. A further conclusion is that lower mass planets at smaller radii could have been detected, though the sensitivity is a factor of two poorer than for the other systems discussed below. An unresolved disk around GJ 433 {#ss:433} -------------------------------- GJ 433 (HIP 56528, LHS 2429) was reported to host a low-mass planet GJ 433 b ($M \sin i = 5.8 M_\oplus$) on a 7.4 day period at 0.058 au . They detected an additional significant signal with a much longer period of 10 years (3.6au), but based on the variation of activity indices on a similar timescale , concluded that a magnetic cycle of the star was a more likely origin. The same signals were recovered by @2014MNRAS.441.1545T, who considered the second signal to be a candidate planet. Given the uncertain nature of the outer planet we do not include it here. The age of GJ 433 is uncertain, but the dynamical, x-ray, and Ca II emission properties show that the star is not young . As above we can constrain the disk location relative to the planet’s, but in the case of GJ 433 there is no clear evidence that the disk is resolved with *Herschel*. The best fit disk temperature is $30$K (see Figure \[fig:seds\], but again the temperature is poorly constrained by a weak detection at 160$\mu$m, and could be as warm as 100K. The fractional luminosity is also poorly constrained, but is approximately $2.5 \times 10^{-5}$. For the stellar luminosity of 0.033$L_\odot$ a disk temperature range from 100 to 30K yields a blackbody radius range of about 1 to 16au, or about 0.2 to 3.5diameter at the 9.1pc distance of the system. As for GJ 649, the disk structure as seen at 100$\mu$m can constrain the disk extent to less than the PACS beam size, but as with GJ 649 only limits the disk radius to less than about 30au, and does not constrain the inclination or position angle. The system layout is shown in Figure \[fig:sys\]. While the observational limits on the disk radius are poor, a radius of 1au would make GJ 433 host to an unusually small disk [@2007ApJ...658..569W], so it seems most likely that the disk extent is similar to that expected for GJ 649. If this is indeed the case, there is again space for undetected planets in the region between the known planet and the disk. Summary of system architectures {#ss:arch} ------------------------------- ![Mass semi-major axis diagrams showing the GJ 433, GJ 581, and GJ 649 planets (dots), the approximate RV sensitivity (lines), and the possible range of disk locations (hatched regions, showing the disk extent in the case of GJ 581). GJ 581 e lies below the sensitivity curve because the RV amplitude (1.7 m s$^{-1}$) is smaller than the RMS (2.12 m s$^{-1}$) reported by @2014Sci...345..440R. In each case, with the possible exception of GJ 433, there remains room in the detection space for sizeable planets that reside between the known planets and the disk, but that could not have been detected with the current RV observations.[]{data-label="fig:sys"}](figs/sys.eps){width="48.00000%"} Figure \[fig:sys\] summarises the architecture of the planet-host systems in our sample, and includes the multi-planet system GJ 581. The number of planets residing in this system is contentious, and stellar activity has been proposed as the cause of some of the periodic signals seen; here we show the three planets proposed by @2014Sci...345..440R, and the hatched disk region shows the extent of the disk derived by . As with GJ 433 and GJ 649, there is space for undetected planets in the intervening region. Given the lack of strong evidence for any correlation between the presence of planets and debris disk brightness, we should not necessarily expect clear trends when looking at plots such as Figure \[fig:sys\]. We might however note trends that are glossed over by a simple disk brightness metric, such as tendencies for systems to show particular architectures or scales. Again noting that a disk as small as 1au around GJ 433 would be very unusual, the radii of the disks is consistent with being a few tens of au. However, this size is also inferred for the disk that orbits Fomalhaut C [@2014MNRAS.438L..96K], so there is no evidence that this preference is related to the presence of planets. Indeed, this radius range is also preferred for disks around FGK-stars, independent of whether planets are known (Sibthorpe et al. 2017). There is no obvious link between the disks and the layout of the planets that orbit closer in, but in each case there remains room in the detection space for sizeable planets that reside between the known planets and the disk, but that could not have been detected with the current RV observations. In this regard the M-type planet + disk systems appear to be analogues of Sun-like planet + disk systems such as HD 20794, HD 38858, and 61 Vir [@2012MNRAS.424.1206W; @2015MNRAS.449.3121K]. This similarity may however simply reflect that detecting long period planets takes time, and that small debris disks grind down to undetectable levels more rapidly than large ones, and that these biases are present regardless of the mass of the host star. That is, there may be differences in the architectures of planetary systems across different spectral types, but that this difference is in the type or existence of planets that reside near 10au. For further discussion of planet formation scenarios, we refer the reader to @2012MNRAS.424.1206W, @2015MNRAS.449.3121K, and @2017MNRAS.469.3518M. The very cool disk temperatures shown in Figure \[fig:seds\] make it clear that progress in our understanding of these disks, and the links with the planets, can only be made by far infrared and millimeter-wave observations. The present observations are hindered by the low spatial resolution of *Herschel*, which means that we are constrained to estimating disk locations. With no far infrared missions on the near horizon, and an expectation of sub-mJy disk flux densities, observations with the Atacama Large Millimeter Array (ALMA) are the main avenue for progress. These will be challenging, but necessary to obtain further discoveries, and in cases such as GJ 433, GJ 581, and GJ 649 could provide higher resolution images that instead of yielding disk location estimates, will allow the discussion of disk structure. Conclusions {#s:conc} =========== This paper presents the results of a *Herschel* survey of 21 nearby late-type stars that host planets discovered by the radial velocity technique. These observations were obtained with the aim of discovering new debris disks in these systems, and in search of any correlation between planet presence and disk brightness. We report the discovery of two previously undetected disks, residing at a few tens of au around the stars GJ 433 and GJ 649. The disk around GJ 649 appears marginally resolved and more consistent with being viewed edge-on. Despite uncertainty in their radii these disks orbit well beyond the known planets, and it is possible that other as-yet undetected planets reside in the intervening regions. The layout of these systems therefore appears similar to star + disk systems around Sun-like stars such as HD 20794, HD 38858, and 61 Vir. Estimating the ages of M-type stars is challenging, but neither star shows evidence of youth, so there is no evidence that the ages of these stars are special compared to the rest of the sample. Including the previously known disk around GJ 581, our sample comprises three planet hosts with disks, a detection rate of 14%. While this rate is higher than for a control sample of M-type stars without reported planets observed by the DEBRIS survey (1 out of 84 stars), the difference is only significant at 98% confidence. This evidence is further shown to be optimistic, because the observations of the planet-host sample were somewhat more sensitive to debris disks than those in the control sample, and because not all systems in the control sample have been searched for planets (or reported not to have planets above some detection threshold). Though this survey represents an improvement over previous surveys of M-tye stars, the fractional luminosity sensitivity achieved remains about a factor of three poorer than similar surveys of Sun-like stars. Nevertheless, the fact that we find disks around 14% of M-type stars, in comparison to 17% of Sun-like stars, provides circumstantial evidence that there is no difference in their disk populations. Acknowledgements ================ We thank the referee for a useful report. GMK is supported by the Royal Society as a Royal Society University Research Fellow. This work was supported by the European Union through ERC grant number 279973 (GMK & MCW). The Digitized Sky Survey was produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. 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--- abstract: 'Since its initial discovery, the Fast radio burst (FRB) FRB 121102 has been found to be repeating with millisecond-duration pulses. Very recently, 14 new bursts were detected by the Green Bank Telescope (GBT) during its continuous monitoring observations. In this paper, we show that the burst energy distribution has a power law form which is very similar to the Gutenberg-Richter law of earthquakes. In addition, the distribution of burst waiting time can be described as a Poissonian or Gaussian distribution, which is consistent with earthquakes, while the aftershock sequence that exhibits some local correlations. These findings suggest that the repeating FRB pulses may originate from the starquakes of a pulsar. Noting that the soft gamma-ray repeaters (SGRs) also exhibit such distributions, the FRB could be powered by some starquake mechanisms associated with the SGRs, including crustal activity of a magnetar and solidification-induced stress of a new-born strangeon star. These conjectures could be tested with more repeating samples.' author: - 'Weiyang Wang, Rui Luo, Han Yue, Xuelei Chen, Kejia Lee, Renxin Xu' title: 'FRB 121102: a star quake-induced repeater?' --- Introduction ============ Fast Radio Bursts (FRBs) are mysterious millisecond-duration radio flashes with high flux densities and prominent dispersive features [@lorimer07; @Keane12; @thornton13; @mas15; @ravi16; @Bannister17; @Bhandari17; @Caleb17]. The observed large values of dispersion measure (DM) are in the range of $\sim100-2600\rm{\,pc\,cm^{-3}}$, which indicate that FRBs are probably of extragalactic or even cosmological origins (e.g., @Katz16a [@sch16]). These transient phenomena stimulate interests of astrophysicist significantly, especially FRB 121102 which is the only repeater that has been detected so far, with an estimated burst energy $\sim10^{37-38}$erg [@Spitler14; @Spitler16]. The optical counterpart of the repeater, has been identified as a host faint star-forming dwarf galaxy which is at a redshift of $z = 0.193$ [@chatterjee17; @kok17; @tendulkar17]. A persistent radio source which is thought to be associated with the repeater, was identified at a distance of $\lesssim40\,$pc from the FRB location [@chatterjee17; @marcote17]. [@ofek17] also found 11 source candidates with luminosities of $\nu L_{\nu}> 3\times10^{37}\,\rm{erg\,s^{-1}}$ which are spatially associated with disks or star-forming regions of galaxies rather than be in galactic center. The persistent radio source is likely to be a pulsar wind nebula [@2017ApJ...843L..26B; @dai17; @kas17]. With an active pulsar producing bursts repeatedly, ejecta or ultra-relativistic electron/positron pair winds may sweep up and heat the nebula that emits synchrotron radio emissions. Additionally, [@wax17] calculated some stringent constraints on the persistent source’s age. The local environment of FRB source would be tested by the variation of DM which has not shown significant evolution [@yang17]. It is proposed that FRBs are highly likely to be associated with pulsars and more than a few efforts have been made to understand their origins. For instance, FRB is supposed to result from a pulsar’s magnetosphere suddenly combed by a nearby cosmic plasma stream [@zhang17]. Also, [@dai16] suggested that the repeater is originated from a highly magnetized pulsar traveling through asteroid belts. Alternatively, in a neutron star (NS)-white dwarf binary system, the accreted magnetized materials may trigger magnetic reconnection that is accounting for FRBs [@Gu16]. Other possibility is that the radio emission is produced by the interaction between a highly relativistic flow and nebula which is powered by a new-born millisecond magnetar [@murase16; @2017ApJ...843L..26B; @dai17]. This process might couple with a long gamma-ray burst or an ultraluminous supernova [@Metzger17]. FRB are also interpreted by the model of supergiant pulse or giant flare from young pulsar or magnetar [@pop10; @kul14; @cordes16; @Katz16b]. Very recently, 14 bursts above threshold of 10 sigma in two 30-minute scans were detected by Breakthrough Listen Digital Backend with the C-band receiver at the Green Bank Telescope [@ATel17]. In this paper, we propose that this repeating burst may arise from a pulsar’s starquake. The burst energy and waiting time distributions as well as the time decaying of the seismicity rate, are shown in Section 2. The scenarios of possible origins will be discussed in Section 3. In Section 4, we make our conclusions. The Earthquake-like behaviors of FRB 121102 =========================================== ---------- ----------------- -------------------- No. MHD Energy Density ($\rm{Jy\,\mu s}$) 1 57991.577788085 114.2 2 57991.580915232 24.8 3 57991.581342500 112.5 4 57991.581590370 61.0 5 57991.581720752 54.6 6 57991.584516806 144.5 7 57991.586200359 25.3 8 57991.586510463 27.7 9 57991.589595602 29.3 10 57991.590822338 26.5 11 57991.594435069 49.6 12 57991.599814375 32.4 13 57991.607200359 49.4 14 57991.616266551 25.7 \[tab1\] ---------- ----------------- -------------------- : 14 bursts of FRB 121102 in continuous observations by Green Bank Telescope [**Notes.**]{} Data are quoted from [@ATel17], where the event 11E and 11F are actually a same burst (see @Katz17b, for a review of close burst pairs). The energy of this burst is calculated to the average value of 11E and 11F. ![[]{data-label="fig1"}](number-E.eps){width="48.00000%"} ![[]{data-label="fig2"}](time-N.eps){width="48.00000%"} ![[]{data-label="fig3"}](sequence.eps){width="48.00000%"} Considering the unity of telescope selection and wavebands ($4-8$GHz), and the completeness of the continuous observations (290mins), here we adopt the parameters of repeating bursts from latest continuous monitoring GBT observations [@ATel17] rather than including the Arecibo events [@Spitler16] and previous GBT events [@sch16]. Parameters of these 14 bursts are shown in Table \[tab1\]. We make statistics on the observational parameters and it turns out that the burst rate as function of burst energy is consistent. Considering the binning width can affect the fitting result, here we calculate the cumulative distribution to avoid this problem. The burst energy is proportional to the observed energy density. With a power law distribution for the number distribution of burst energy $N(E)\propto E^{-\alpha}$, the cumulative distribution can be obtained, $$\label{eq3} N(>E)\propto \int^{\infty}_{E}E^{-\alpha}dE\propto E^{-\alpha+1}.$$ The fluctuations of events for the cumulative distribution are assumed to follow a random statistic, $\sigma(E)=\sqrt{N(>E)}$. The cumulative energy distribution of events for each burst energy (energy density) is well fitted by a power law with an index $\alpha_{E}=\alpha-1=1.16\pm0.24$, shown in Figure \[fig1\]. This power law distribution is consistent with the Gutenberg-Richter power law (i.e., $N(E)\propto E^{-2}$, @Gutenberg56) which is a well-known earthquake law. Furthermore, the statistics of waiting times contain much significant informations about occurrences and correlations of events. The waiting time $\Delta t$ is defined as the interval time between the adjacent detected FRB events in the continuous monitoring observation. For a simple Poisson process, the cumulative distribution of waiting time can be described by a simple exponential function, $$\label{eq2} N(>\Delta t)\propto e^{-\lambda \Delta t},$$ where $\lambda$ is the burst rate, which is constant. Also, with an assumed Gaussian distribution of the number distribution $N(\Delta t)$, the cumulative distribution of the waiting time is, $$\label{eq3} N(>\Delta t)\propto \int^{\infty}_{\Delta t}\mathrm{exp}^{-\frac{(\Delta t-\tau)^2}{\sigma^2}}d(\Delta t)\propto 1-\mathrm{erf}(\Delta t).$$ As shown in Figure \[fig2\], the cumulative distribution of the waiting time are plotted, fitted by the exponential function with $\lambda=(3.05\pm0.48)\times10^{-3}\,\rm{s^{-1}}$ and equation (\[eq3\]) with $\tau=(1.13\pm0.20)\times10^3$s and $\sigma=(1.03\pm0.16)\times10^{3}$s. The waiting time distribution can be represented by a simple Poissonian or Gaussian distribution, which are extracted from statistic of earthquakes [@pep94; @Leonard2001]. There may be additional burst events with detection threshold of around 7 over full bandwidth of 4 GHz. However, they are not listed in [@ATel17] because they are relatively weak and need more analysis. These bursts might have narrow frequency spread and thus do not show high signal to noise ratio. This may not affect our model significantly while affect the fitting parameters. Earthquakes from different regions are regarded as random processes of independent and uncorrelated events, while aftershocks which occur in shorter time intervals are correlated. Additionally, combining with magnitudes of burst, the burst rates are plotted with different binning widths of time and fitted by a power law, shown in Figure \[fig3\]. For short-range temporal correlations between earthquakes, the time decaying of the seismicity rates of an aftershock sequence can be interpreted by an empirical relationship, i.e. Omori law [@omori94; @Utsu61; @Utsu95]. By the Omori law, the seismicity rates decay with time, can be expressed by a power law, $$\label{omori} n(t)=\frac{K}{(C+t)^p},$$ where $n(t)$ is the seismicity rate and the time decaying rate of seismicity is controlled by the third constant $p$, which typically falls in the range of 0.9-1.5 [@Utsu95]. For an aftershock sequence, the variation of $p$ may be controlled by the structural heterogeneity, stress and temperature [@Utsu95]. We fixed the parameter $C$ at 1min, and then the number rate at $t=0$ is close to $K$ which is the peak value of the number rate curve. Thus only one parameter $p$ which controls the decaying rate is estimated. With the occurrence time of 14 FRB events, the burst rates can be well fitted by the power law determined a index $p$ value of $1.42\pm0.24$, which is consistent with the index for earthquakes. Therefore, the bursts are followed by an earthquake-like aftershock sequence. Possible origins of FRB 121102 ============================== As we see from Section 2, the FRB repeater exhibits several features commonly found in earthquakes. In fact, nonlinear dissipative systems always show self-organized criticality (SOC) behaviors. With a solid crust or stiff equation of state (EOS) of pulsar, a star can build up stresses that makes the crust cracks and adjust stellar shape to reduce its deformation. The characters for these processes of statistical independence, nonlinear coherent growth and random rise times, are consistent with a SOC system [@SOC]. According to the SOC theory, NS quakes are giant catastrophic events like earthquakes and probably accompanied by global seismic vibrations or oscillations. Hence, starquakes in a pulsar, share similar statistical distributions with earthquakes. Here we present two possible scenarios as following. One scenario is that a normal NS with a solid crust and a superfluid core, and with a strong toroidal magnetic field (i.e., magnetar). NS can form a solid crust quickly after its birth. The stellar shape changes from oblate to spherical configuration, as well as thermal and dynamic responses, will induce stresses in the crusts. When stress buildup in the dense solid crust beyond a yield point, a starquake would happen with sudden energy release that supports FRB. This process also brings a slight change of moment of inertia with an abrupt jump of angular frequency that is the so-called pulsar glitch [@ruderman69; @Link96]. For a highly magnetized NS, the stellar crust is coupling with the magnetosphere. In this scenario, starquakes induce the magnetic curl or twist ejected into the magnetosphere from crust in a few milliseconds [@Thompson02; @Thompson17]. Electrons in the magnetosphere are suddenly accelerated to ultra-relativistic velocity by magnetic reconnection [@zhang11], move along the magnetic field lines, resulting in producing curvature radiation. The characteristic frequency of the curvature radiation is $$\label{eq5} \nu_{\mathrm c}=\frac{3c\gamma^3}{4\pi R_{\rm c}}=7.16\times(\frac{\gamma}{100})^3(\frac{10\,\rm km}{R_{\rm c}})\,\rm GHz,$$ where $R_{\rm c}$ is the curvature radius with a typical value of $\sim10$km and $\gamma$ is the Lorentz factor of electrons. With detected FRB frequency $\nu_{\rm c}=4-8$GHz, a $\gamma\approx50-100$ is required. If generated by such stress-induced reconnections (e.g., @Lyutikov15), the FRB has an estimated duration timescale, $$\label{eq6} t_{\rm {rec}}\sim\frac{L}{v_{\rm A}}\sim1-10\,\rm ms,$$ where the scale of the reconnection-unstable zone $L\sim 1-10$km, and the Alfv$\acute{\rm e}$n velocity is $v_{\rm A}\simeq B/(4\pi \rho_{\rm c})^{1/2}\sim0.01c$, in which $\rho_{\rm c}\simeq10^{14}\,\rm{g\,cm^{-3}}$ is the average mass density of the crust. The sudden elastic and magnetic energy release in the crustal stress is estimated, $$\label{eq11} \delta E_{\rm{cru}}=4\pi R^2h_{\rm c}\sigma\delta \varepsilon,$$ where $h_{\rm c}\simeq R/30\simeq 0.3$km is the crustal thickness [@Thompson17], $\varepsilon$ is the shear strain and the total stress including crustal shear stress and Maxwell stress in magnetosphere, is $$\label{eq12} \sigma=\sqrt{(\mu\varepsilon)^2+(\frac{BB_z}{4\pi})^2},$$ in which $\mu$ is the shear modulus, $B$ is the surface magnetic field and $B_z$ is the component of magnetic field perpendicular to the direction of plastic flow. Within the crust, the force balance $\mu\varepsilon\simeq BB_z/(4\pi)$ implies that $$\label{eq13} \delta E_{\rm{cru}}\simeq4.2\times10^{46}(\frac{BB_z}{10^{30}\,\rm{G^2}})(\frac{R}{10\,\rm{km}})^2\delta\varepsilon\,\rm{erg}.$$ $\delta\varepsilon$ is smaller than $\sim10^{-2}$ (e.g., @hof12). Here, the energy release can meet energy requirements of FRBs while not for soft gamma repeaters (SGRs). However, a plastic flow can be initiated when the elastic crustal deformation exceeds a critical value, launching a thermo-plastic wave that dissipates the magnetic energy inside the crust [@Bel14]. This mechanism might bring much more energy from the inner crust in which it stores a SGR-required magnetic energy of $\gtrsim10^{47}$erg with a interior magnetic field of $\sim10^{16}$G [@Lander16]. The Ohmic dissipation in this process can be neglected because of the long timescale [@2014MNRAS.445.2777F]. The timescale for the local energy release is $t_{\rm tw}\sim4\pi\eta/BB_{z}$, where $\eta$ is the viscosity. If the energy releases quickly enough for FRB, a viscosity of $\sim10^{26}\,\rm{erg\,s\,cm^{-3}}$ is required. In addition, the transition to hydromagnetic instability of the magnetar core may offer larger energy [@Thompson17] supporting SGRs. The other scenario is that a new-born strangeon star (SS), which has a stiff EOS [@lai17a] and could release more elastic energy than that of the solid crust of a normal NS. At early age, SS may shrink its volume abruptly by solidification-induced stress, and a bulk-variable starquake happens. Basically, there are two kinds of quakes in a solid star: bulk-invariable (type I) and bulk-variable (type II) starquake [@zhou04; @zhou14]. Here, type II starquake is more likely to be dominant because the elastic energy accumulation of type I quake is not sufficient to produce such short time interval quakes in this quake sequence (see equation (39) in @zhou14 for a test of $t=100$s). The elastic and gravitational energy release during type II starquake is $$\label{eq4} \begin{split} \delta E_{\rm{g}}=\frac{3GM^2}{5R}\frac{\delta R}{R}=\frac{3GM^2}{10R}\frac{\delta \Omega}{\Omega}\\ \simeq10^{53}(\frac{M}{1.4M_{\odot}})^2(\frac{10\,\mathrm{km}}{R})(\frac{\delta\Omega}{\Omega})\,\mathrm{erg}, \end{split}$$ where $G$ is the gravitational constant, $M$ is the stellar mass, $\delta R/R$ is the strain and $\delta \Omega/\Omega$ is the amplitude of a glitch (e.g., $10^{-9}-10^{-6}$, @Alpar94 [@Alpar96]). This energy is large enough to support a FRB and possibly associating with SGR. Bulk-variable starquakes are accompanied by the change of electrostatic energy [@Katz17a] and some electrodynamic activities in magnetosphere. A giant quake can power energetic relativistic outflow to produce the observed prompt emission of short-hard GRBs, and some aftershocks result in following X-ray flares observed [@xu06]. Starquakes may also lead to the magnetic reconnection that accelerates electrons, and these charges move along the magnetic field lines, emitting curvature radiation. The duration timescale of the magnetic reconnection in this scenario can be obtained from equation (\[eq6\]). Besides, these short time interval quakes might be motivated by an initial shock which is type I quake dominate. The waiting time of the initial shock can be obtained [@zhou04], $$\label{eq7} t_{\rm i}=\frac{\sigma_{\rm c}}{\dot\sigma},$$ where $\sigma_{\rm c}$ is the critical stress and $\dot\sigma$ can be denoted as, $$\label{eq8} \dot\sigma=\frac{3\pi I\dot P}{R^3P^3}\approx 9.42\times10^{27}(\frac{\dot P}{1\,\rm{s\,s^{-1}}})(\frac{1\,\rm{s}}{P})^3,$$ in which $I\approx10^{45}\,\rm{erg\,s^2}$ is the moment of inertia, $P$ is the rotation period and $\dot P$ is period derivative of the star. Then, the waiting time of the initial shock can be written as, $$\label{eq9} \log({\frac{t_{\rm i}}{1\,\rm s}})=\log({\frac{\sigma_{\rm c}}{1\,\rm erg\,cm^{-3}}})+3\log({\frac{P}{1\,\rm s}})-\log({\frac{\dot P}{1\,\rm s\,s^{-1}}})-28.$$ The rotation period $P\sim10$ms for a new-born rapidly rotating SS. From equation (\[eq9\]), the critical stress is estimated to $10^{19-22}\,\rm{erg\,cm^{-3}}$ which is consistent with strangeon stars [@zhou04]. Starquakes are magnetically powered in a NS while elastically and gravitationally powered in a SS. In these scenarios, the toroidal oscillation, which might be derived from starquakes [@Bas07], propagates into the magnetosphere and changes its charge density that brings an induced electric potential [@lin15]. The electric potential [@ruderman75; @chen93] is estimated, $$\label{V} \Delta V\simeq2.1\times10^{12}(\frac{\Omega_{\rm{osc}}}{10\,\rm{kHz}})^{1/7}(\frac{B}{10^{14}\,\rm{G}})^{-1/7}(\frac{R}{10\,\rm{km}})^{4/7}\,\rm V,$$ where stellar oscillation frequency is estimated as $\Omega_{\rm {osc}}\sim c/R\sim30$kHz that enlarges the size of radio beam. Within this picture, a magnetar is most likely to produce electron/positron pair plasma. The electron/positron pair plasma production due to the electric potential is the necessary condition for coherent radio emission. The potential enhances voltage along the gap which accelerates electrons to higher Lorentz factors emitting curvature radiation. Then, the pulsar becomes radio loud (i.e., beyond the pulsar death line) until oscillations damp out in which the magnetosphere becomes inactive and radio emissions evaporate. Therefore, FRB may be the “oscillation” of a dead pulsar at near pulsar death line. Conclusion and dicussion ======================== Basically, we found that the behaviors of the repeating FRB 121102 are earthquake-like. The distribution of burst energy exhibits a Gutenberg-Richter power law form which is a well-known earthquake distribution. And the distribution of waiting time, can be characterized as a Poissonian or Gaussian distribution, which are consistent with earthquakes as well as the local correlated aftershock sequence. The possible origins of the repeater are discussed including crustal activity of a magnetar and solidification-induced stress of a new-born SS. Both possible origins might be associated with SGRs which are difficult to detect at cosmological distance. Statistic distributions of burst energy and duration time show that FRB 121102 is very similar to SGR $1806-20$ [@JCAP17]. Also, SGR $1806-20$ share some distinctive properties with earthquake that indicates SGRs are indeed powered by starquakes [@cheng96], and the giant flares of SGRs are suggested to be quake-induced [@xu06]. In addition, the observed continuous bursts with the modeled occurrence rate $\lambda=11.0\,\rm{hr}^{-1}$ for Poissonian, while $\tau^{-1}=3.2\,\rm{hr}^{-1}$ for Gaussian, may originate from some uncorrelated quakes. In that case, these quakes are suggested foreshocks, storing energy and motivate a main quake. Then an aftershock sequence, which may be caused by some local coherent deformations before a new equilibrium sets up, occurs. The motivated shocks are non-Poissonian and not rotation-powered dominate, while the type I starquake may lead the initial shock begins when the stresses exceed a certain threshold. Hence, the next repeating FRB might be waiting for $\sim10^6$s because a long time to store elastic energy is needed [@zhou14]. And, the latest FRB volumetric rate including all of repeating bursts is calculated as $R_{\rm FRB}\sim10^{-5}\,\rm{Mpc^{-3}\,yr^{-1}}$ out to redshift of 1 [@law17]. A pulsar, which has a solid crust or stiff EOS, would be natural to have glitches as the result of starquakes. From statistics of pulsar glitches, the number of glitches per year can be interpreted as [@esp11], $$\label{glitch} \dot N_{\rm g}\simeq0.003\times(\frac{\dot \nu}{10^{-15}\,\rm{Hz\,s^{-1}}})^{0.47}\,\rm{yr^{-1}},$$ where $\dot \nu$ is rotational frequency derivative. For a typical millisecond pulsar with $P\sim10$ms and $\dot P\sim10^{-21}\,\rm{s\,s^{-1}}$, the number of glitches per year can be evaluate to $\sim3\times10^{-4}\,\rm{yr^{-1}}$. A total number of glitches $N_{\rm g}\sim3\times10^3$ can be estimated during the pulsar life-time $\sim10$Myr. It is hypothesized that the hydrogen-poor superluminous supernovae (SLSNe-I) are powered by millisecond magnetars. The volumetric birth rate of SLSNe-I is $R_{\rm {SLSN}}=10^{-8}\,\rm{Mpc^{-3}\,yr^{-1}}$ [@gal12]. Therefore, we estimate the FRB volumetric rate $R_{\rm {FRB}}\simeq N_{\rm g}R_{\rm {SLSN}}\sim3\times10^{-5}\,\rm{Mpc^{-3}\,yr^{-1}}$. This inferred event rate from FRB/SLSNe-I associated events is consistent with the observational FRB events. Starquakes associating with some X-ray or Gamma-ray bursts in a normal NS share similar behaviors with that in a SS. While the X-ray spectra might be different in these scenarios. In a SS atmosphere, thermal X-rays from the lower layer of a normal NS atmosphere are prohibited, relatively more optical/UV photons and a energy cutoff at X-ray bands are exhibited [@2017arXiv170503763W]. Considering a NS at 1Gpc with $2-8$keV flux of $\sim2\times10^{-16}\,\rm{erg\,cm^{-2}\,s^{-1}}$ which is consistent with the X-ray limit of [*Chandra*]{} and [*XMM-Newton*]{}, the luminosity is calculated to $\sim10^{39}\,\rm{erg\,s^{-1}}$. Such distant source is too faint to be detected by current X-ray telescopes except with a supper-Eddington luminosity. The normal NS and SS have different EOS that are most likely to be tested by gravitational wave and electromagnetic radiation from mergers of compact stars further [@GW; @Lai17b]. We expect to detect more repeating events. More constraints on the mysterious origin of FRB will be given by the statistics growing samples. The earthquake-like behaviors, including distributions of energy and waiting time for the repeater, are expected to be tested by more continuous monitoring observations. We are grateful to Stephen Justham at National Astronomical Observatories, Chinese Academy of Sciences, for discussions. This work is supported by National Key R&D Program of China (No. 2017YFA0402602), the National Natural Science Foundation of China (11373030, 11673002, 11633004 and U1531243), Frontier Science Key Project (QYZDJ-SSW-SLH017) of Chinese Academy of Sciences, and the Ministry of Science and Technology (2016YFE0100300). natexlab\#1[\#1]{} Abbott B. P., et al., 2017, Phys. Rev. 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--- author: - Mirza Galib - 'David T. Limmer' title: 'Elucidating the mechanism of reactive uptake of N$_2$O$_5$ in aqueous aerosol' --- [**Nearly one third of all nitrogen oxides are removed from the atmosphere through the reactive uptake of ${\mathrm{N}_2\mathrm{O}_5}$ into aqueous aerosol. The primary step in reactive uptake is the rapid hydrolysis of ${\mathrm{N}_2\mathrm{O}_5}$, yet despite significant study, the mechanism and rate of this process are unknown. Here we use machine learning-based reactive many body potentials and methods of importance sampling molecular dynamics simulations to study the solvation and subsequent hydrolysis of ${\mathrm{N}_2\mathrm{O}_5}$. We find that hydrolysis to nitric acid proceeds through the coordinated fluctuation of intramolecular charge separation and solvation, and its characteristic rate is 4.1 ns$^{-1}$, orders of magnitude faster than traditionally assumed. This large rate calls into question standard models of reactive uptake that envision local equilibration between the gas and the bulk solution. We propose an alternative model based on interfacial reactivity that can explain existing experimental observations and is corroborated by explicit simulations.** ]{} The heterogeneous hydrolysis of ${\mathrm{N}_2\mathrm{O}_5}$ plays a key role in establishing the oxidative power of the troposphere, and is a major factor in determining air quality and climate.[@sein_atmospheric_2016; @crutzen_areps_1979] In night time air, NO and NO$_2$ are oxidized by O$_3$ to form NO$_3$ and ${\mathrm{N}_2\mathrm{O}_5}$. [@brown_csr_2012] Around 20% of that atmospheric ${\mathrm{N}_2\mathrm{O}_5}$ is thought to be subsequently removed by hydrolysis to ${\mathrm{H N}\mathrm{O}_3}$ in aqueous aerosol.[@holmes2019role] However, a molecular level understanding of the reactive uptake of ${\mathrm{N}_2\mathrm{O}_5}$ is lacking, frustrating attempts to rationalize variations in field measurements. [@davis_acp_2008; @bertram_acpd_2009; @abbatt_csr_2012; @chang_ast_2011; @mcduffie2018heterogeneous] Using state of art computational tools, including machine learning based [@singraber2019library; @wang_cpc_2018] reactive force fields and methods of importance sampling molecular dynamics simulations, we have studied the reactive uptake in pure water. We have determined that the hydrolysis of ${\mathrm{N}_2\mathrm{O}_5}$ in aqueous aerosol is fast, occurring in less than 1 ns on average, and subsequently that interfacial processes dominate its reactive uptake. This finding is inconsistent with traditional models of reactive uptake, which assume reaction-limited bulk hydrolysis and equilibrium solvation.[@bertram_acpd_2009; @chang_ast_2011; @davidovits_cr_2006; @poschl_ar_2011] Rather, we show with explicit simulations that reactive uptake can be understood as a result of competition between interfacial hydrolysis and evaporation. As an important reactive intermediate in the atmospheric chemistry of nitrogen oxides and nitrate aerosol, the heterogeneous chemistry of ${\mathrm{N}_2\mathrm{O}_5}$ has been the subject of intense study.[@mozurkewich_jgra_1988; @bertram_acpd_2009; @abbatt_csr_2012; @chang_ast_2011; @davidovits_cr_2006] Experimentally, only the overall mass transfer of ${\mathrm{N}_2\mathrm{O}_5}$ gas to aqueous aerosol can be easily measured, precluding a detailed understanding of the physical and chemical processes that underpin it.[@bertram_acpd_2009; @davidovits_cr_2006; @poschl_ar_2011] Under standard conditions, mass transfer is determined by the reactive uptake coefficient, $\gamma$, which is the fraction of ${\mathrm{N}_2\mathrm{O}_5}$ molecules that collide with an aerosol surface that are irreversibly lossed through reaction. Measurements of $\gamma$ in pure water vary between $0.01 \lesssim \gamma \lesssim 0.08$.[@bertram_acpd_2009; @chang_ast_2011] Uptake on pure water aerosol represents a speed limit for typical atmospheric aerosol, as contributions from surface active organics and soluble inorganic salts tend to suppress uptake.[@bertram_acpd_2009; @park_jpca_2007; @davis_acp_2008; @ryder2015role] The size of $\gamma$ and its dependence on solution composition and thermodynamic state is currently rationalized with simplified kinetic models.[@chang_ast_2011; @davidovits_cr_2006; @akimoto_arc_2016] Unfortunately, the basic physical and chemical properties of ${\mathrm{N}_2\mathrm{O}_5}$, like its solubility and hydrolysis rate constant, that are needed to validate assumptions made in such models are not available. Therefore a model capable of directly interrogating the molecular dynamics that transfer an initially gaseous ${\mathrm{N}_2\mathrm{O}_5}$ molecule into its solution hydrolysis products is needed. Molecular simulations can in principle be used to gain microscopic insight into the reactive uptake of atmospheric gases into solution, but traditional theoretical methods are insufficient to reach the broad range of length and time scales required.[@davidovits_cr_2006] Classical force fields have been used to study the physical solvation of ${\mathrm{N}_2\mathrm{O}_5}$,[@li_jcp_2018; @hirshberg_pccp_2018] where it is computationally tractable to employ enhanced sampling methods and represent large inhomogeneous systems. However, existing potentials are not suitable to model chemical reactions, precluding a study of the hydrolysis reaction. *Ab initio* molecular dynamics has been used to study hydrolysis and halide substitution reactions of ${\mathrm{N}_2\mathrm{O}_5}$ in water clusters.[@hammerich_pccp_2015; @rossich2019microscopic; @mccaslin_sa_2019; @karimova_jpca_2019; @mcnamara_pccp_2000] However, it is not typically feasible to study systems large enough to represent inhomogeneous systems or to evolve systems long enough to study rare events. To overcome these limitations, we have employed machine learning techniques to fit a high dimensional reactive potential to *ab initio* training data. The combination of novel potential representations and algorithms to fit them has recently enabled the use of machine learning based force fields for a range of complex chemical problems.[@singraber2019library; @wang_cpc_2018] The resultant potential allows us to access larger length and time scales than typical *ab initio* simulations, but with comparable accuracy. In so doing we are able to employ advanced simulation methods to uncover a complete picture of the thermodynamics and reactive dynamics that lead to the uptake of ${\mathrm{N}_2\mathrm{O}_5}$. Results and Discussion {#results-and-discussion .unnumbered} ====================== In order to simulate the hydrolysis of ${\mathrm{N}_2\mathrm{O}_5}$ in liquid water, we have developed a reactive force field capable of describing a broad ensemble of solvation and bonding configurations. Specifically, we constructed a model using *ab initio* reference data fit to a flexible artificial neural network functional.[@wang_cpc_2018] We used supervised and active learning procedures on a range of condensed phase and reactive path structures.[@SI] The artificial neural networks are trained on reference energies and forces computed from density functional theory,[@zhang_prl_1998; @grimme_jcp_2010] which provides an accurate description of aqueous solution structure and thermodynamics,[@bankura_jpcc_2014; @galib_jcp_2017; @morawietz_pnas_2016; @marsalek_jcp_2017] and which we have additionally benchmarked for ${\mathrm{N}_2\mathrm{O}_5}$ gas phase dissociation energies.[@SI] All of our studies are at ambient conditions with temperature $T=$ 298K and pressure $p=1$ atm. The resultant reactive force field accurately represents the *ab initio* potential-energy surface of water and ${\mathrm{N}_2\mathrm{O}_5}$, but at a significantly reduced computational cost, enabling the systematic study of the thermodynamics and kinetics of solvated ${\mathrm{N}_2\mathrm{O}_5}$ and its hydrolysis products. Thermodynamics of solvation and hydrolysis. {#thermodynamics-of-solvation-and-hydrolysis. .unnumbered} ------------------------------------------- Shown in Fig. \[Fi:1\]a) is a characteristic snapshot of ${\mathrm{N}_2\mathrm{O}_5}$ and its surrounding solvation environment generated from our neural network force field. The intramolecular structure of the solvated ${\mathrm{N}_2\mathrm{O}_5}$ is characterized by large fluctuations in the position of the center oxygen.[@SI] These fluctuations manifests the tendency of ${\mathrm{N}_2\mathrm{O}_5}$ to spontaneously undergo intramolecular charge separation, localizing excess positive charge in an emergent NO$_2^{\delta+}$ moiety and excess negative charge in an NO$_3^{\delta-}$ moiety, as an transient precursor to dissociation.[@hirshberg_pccp_2018; @mcnamara_pccp_2000] Despite the transient charge separation, we find that ${\mathrm{N}_2\mathrm{O}_5}$ is relatively weakly solvated on average. Water forms less than one hydrogen bond with the outer oxygens on average, and even fewer with the nitrogens and bridging oxygen, resulting in an unstructured solvation shell. This is because the localization of the charge is primarily on the nitrogens, which are typically sterically inaccessible. ![Solvation and hydrolysis thermodynamics in bulk water. a) Representative snapshots of solvated ${\mathrm{N}_2\mathrm{O}_5}$ and ${\mathrm{H N}\mathrm{O}_3}$ in bulk water from molecular dynamics simulations. The red spheres denote oxygens, the blue nitrogens, and the white hydrogens. b) Free energy profile for ${\mathrm{N}_2\mathrm{O}_5}$ hydrolysis reaction as a function of intramolecular nitrogen-nitrogen distance. c) Free energy profile for the dissociation of ${\mathrm{H N}\mathrm{O}_3}$ as a function of a continuous coordination number between an O in the NO$_3$ moiety and a hydrogen, $n_h$. A characteristic snapshot of dissociated ${\mathrm{H N}\mathrm{O}_3}$ from molecular dynamics trajectory, where the excess proton is highlighted in yellow. []{data-label="Fi:1"}](Fig1.pdf){width="8.5cm"} The observed hydration structure is consistent with ${\mathrm{N}_2\mathrm{O}_5}$ being sparingly soluable in water. To quantify the driving force for dissolving ${\mathrm{N}_2\mathrm{O}_5}$ in water, we have computed the solvation free energy using thermodynamic perturbation theory.[@SI] The resultant solvation free energy, $\Delta F_s$, was determined to be $\Delta F_s = -1.3 \pm 0.5$ kcal/mol implying a Henry’s law constant of $H= 0.4\pm$ 0.1 M/atm. To our knowledge this is the first *ab initio* estimate of the solubility of ${\mathrm{N}_2\mathrm{O}_5}$. It is much smaller than has been inferred from previous mass uptake experiments which range from 1-10 M.[@sander_acp_2015; @mentel_pccp_1999] However, the interpretation of such experiments is difficult due to the inability to separate solvation of ${\mathrm{N}_2\mathrm{O}_5}$ from subsequent hydrolysis. For a molecule with a dipole this solubility is relatively low, though it is similar to other weakly solvated gases like $\mathrm{SO}_3$. The low solubility reflects a subtle interplay between favorable long range electrostatic energetics and a large unfavorable cavity formation entropy. The hydrolysis of ${\mathrm{N}_2\mathrm{O}_5}$ in liquid water is thermodynamically favorable. We have calculated the free energy for dissociating ${\mathrm{N}_2\mathrm{O}_5}$ using umbrella sampling. Specifically, we have computed the free energy as a function of the intramolecular nitrogen-nitrogen distance, $R$, as $ F(R) = -{k_\mathrm{B}}T \ln \langle \delta (R -\hat{R} )\rangle $ where the hat denotes a fluctuating quantity, $\langle .. \rangle$ denotes a canonical ensemble average, ${k_\mathrm{B}}$ is Boltzmann’s constant, and $\delta$ is Dirac’s delta function. The free energy is shown in Fig. \[Fi:1\]b), and exhibits a narrow minimum at $R=2.6 \mathrm{\AA}$ and a broad plateau for $R>4 \mathrm{\AA}$, separated by a barrier at $R=3 \mathrm{\AA}$. The minimum at $R=2.6 \mathrm{\AA}$ reflects the intact ${\mathrm{N}_2\mathrm{O}_5}$ molecule, as shown in Fig. \[Fi:1\]a), while the plateau for $R>4 \mathrm{\AA}$ manifests its dissociation. We find that at large $R$ it is thermodynamically favorable to form two equivalents of ${\mathrm{H N}\mathrm{O}_3}$, also shown in Fig. \[Fi:1\]a). At relatively short separation distances, $4 \mathrm{\AA} \le R \le 6 \mathrm{\AA}$ only one of the two ${\mathrm{H N}\mathrm{O}_3}$ molecules are likely to be dissociated. The barrier region is wide, as large separations are needed to solvate the separated nitrogens. A barrier of nearly 4 kcal/mol implies that hydrolysis is a rare event, and that ${\mathrm{N}_2\mathrm{O}_5}$ can be dynamically distinguished from its eventual hydrolysis products. The free energy difference between the reactant and product basin is -4 kcal/mol. The low solubility of ${\mathrm{N}_2\mathrm{O}_5}$ implies that nearly all solvated ${\mathrm{N}_2\mathrm{O}_5}$ in pure water is transformed to ${\mathrm{H N}\mathrm{O}_3}$. After hydrolysis, it is thermodynamically favorable for the nascent nitric acid to dissociate into an excess proton and $\mathrm{NO}_3^-$. We have computed the free energy to deprotonate ${\mathrm{H N}\mathrm{O}_3}$ by monitoring a continuous coordination number, $n_h$, between the oxygens on the NO$_3$ moiety and a hydrogen.[@SI] The free energy, $F(n_h)$, can be estimated directly from $ F(n_h) = -{k_\mathrm{B}}T \ln \langle \delta (n_h - \hat{n}_h)\rangle $, which is shown in Fig. \[Fi:1\]c). The free energy difference for removing a proton, $F(n_h=0)-F(n_h=1)$ is -1.4 kcal/mol and corresponds to a pKa value of -1.1, which is reasonably close to the experimental value of -1.35.[@mckay_tfs_1956] {width="17cm"} Taken together, the calculated thermodynamics of ${\mathrm{N}_2\mathrm{O}_5}$ solvation and subsequent hydrolysis in water are consistent with experimental observations that its accommodation into aqueous aerosol is largely irreversible.[@davidovits_cr_2006] Though weakly soluble, once in pure water ${\mathrm{N}_2\mathrm{O}_5}$ will undergo hydrolysis to form two ${\mathrm{H N}\mathrm{O}_3}$, which will subsequently deprotonate. Under high nitrate concentrations, or in low humidity droplets, this equilibrium could be shifted back towards an intact ${\mathrm{N}_2\mathrm{O}_5}$ and subsequently its reevaporation. Indeed, low water content droplets are observed to have smaller reactive uptake coefficients, and dissolved nitrate salts can reduce the reactive uptake by over an order of magnitude.[@mentel_pccp_1999; @bertram_acpd_2009] However, due to the effective irreversibility of the reaction, a complete understanding of reactive uptake requires insight into the kinetics of solvation and hydrolysis. Kinetics of ${\mathrm{N}_2\mathrm{O}_5}$ hydrolysis {#kinetics-of-mathrmn_2mathrmo_5-hydrolysis .unnumbered} ---------------------------------------------------- The mechanism of ${\mathrm{N}_2\mathrm{O}_5}$ hydrolysis involves an interplay between intramolecular charge separation and stabilization from the surrounding water. To understand this interplay, we identified a reaction coordinate that encodes the microscopic details relevant to hydrolysis in solution. An appropriate reaction coordinate is one that is capable of both distinguishing the intact ${\mathrm{N}_2\mathrm{O}_5}$ from its dissociation products, as well as characterizing the transition state ensemble of configurations, which are those configurations that have equal probability of committing to either the reactant or product states.[@geissler-jpcb_1999] While the nitrogen-nitrogen distance in Fig. \[Fi:1\]b) is capable of the former, it fails in the latter. Configurations taken at fixed values of $R$ are overwhelmingly committed to either the reactant or product basins of attractions. This is because $R$ lacks direct information about the surrounding water, which is pivotal in describing hydrolysis. We have found that an appropriate reaction coordinate for hydrolysis is a linear combination of the nitrogen-nitrogen distance, $R$, and a continuous coordination number between the nitrogen atoms in ${\mathrm{N}_2\mathrm{O}_5}$ and the surrounding water molecules, denoted $n_w$.[@SI] Figure \[Fi:2\]a) shows the corresponding free energy surface, $F(n_w,R)$, computed from $ F(n_w,R) = -{k_\mathrm{B}}T \ln \langle \delta (n_w - \hat{n}_w)\delta (R - \hat{R})\rangle \, $ using umbrella sampling. The reactant basin with an ${\mathrm{N}_2\mathrm{O}_5}$ solvated in water and product basin are separated by the line $n_w = -3 R + 9.6$, which defines a separatrix distinguishing the two basins of attraction. The direction orthogonal to the separatrix we refer to as the reaction coordinate, $\xi$. For small $R$, the weak hydration structure of ${\mathrm{N}_2\mathrm{O}_5}$ is evident by the low value of $n_w$. The $\mathrm{NO}_3^{\delta -} + \mathrm{NO}_2^{\delta +}$ pair generated at large $R$ but $n_w=0$ are not thermodynamically stable. The hydrolysis products, two equivalents of ${\mathrm{H N}\mathrm{O}_3}$, at large $R$ have an elevated coordination number, $n_w=1$, reflecting the altered bonding arrangement upon abstracting a water molecule. The saddle point of the surface, which we denote $\xi^{*}$, is located at an intermediate coordination number $n_w=0.4$, and intermediate nitrogen-nitrogen distance $R=3.1\, \mathrm{\AA}$, with a free energy barrier $\Delta F(\xi^{*}) =3.8$ kcal/mol. The thermodynamically most likely reactive path follows the simultaneous increase in the nitrogen-nitrogen distance and coordination number. The increasing distance correlates with the lengthening of a N–O bond and accompanying charge reorganization, which is thermodynamically stabilized by a solvent fluctuation that alters the coordination number. The correlated increase in $R$ and $n_w$ is not only thermodynamically favored, but also well characterizes the transition state ensemble for ${\mathrm{N}_2\mathrm{O}_5}$ hydrolysis. We have confirmed the latter by performing a committor analysis,[@pratt_jcp_1986; @geissler-jpcb_1999] whereby the probability of configurations constrained to lie along the separatrix to commit to the product basin, $p_B$, is estimated by integrating an ensemble of trajectories from an initial Maxwell-Boltzmann distribution of velocities. If the dividing surface is a true representation of the transition state ensemble, there should be an equal probability to be committed to either reactant or product basins. Shown in Fig. \[Fi:2\]b) is the distribution of commitment probabilities for configurations taken along the separatrix. The distribution is peaked at $p_B = 0.5$ confirming that the combination of $n_w$ and $R$ is capable of characterizing the dynamics that lead to hydrolysis from ${\mathrm{N}_2\mathrm{O}_5}$. We have employed the Bennett-Chandler method [@chandler_jcp_1978] to quantify the rate constant for hydrolysis. Specifically, we compute the rate, $k_\mathrm{h}(t)$, as a product of the transition state theory estimate, $k^\mathrm{TST}$, and the transmission coefficient, $\kappa(t)$, $k_\mathrm{h}(t)=\kappa(t) k^\mathrm{TST}$. The transition state theory estimate of the rate is computable from $ k^\mathrm{TST}= \nu \exp[-\Delta F(\xi^{*})/{k_\mathrm{B}}T] $ where the prefactor $\nu$ is related to the mean velocity of $\xi$ in the reactant basin. The transmission coefficient corrects transition state theory for dynamical effects at the top of the barrier, and is given by the plateau region of the flux-side correlation function.[@SI] The transmission coefficient is shown in Fig. \[Fi:2\]c) and plateaus to a value of 0.25 within 0.5 ps. Taken together we find the rate of hydrolysis to be $k_\mathrm{h}=4.1 \,\mathrm{ns}^{-1}$, implying an average lifetime of ${\mathrm{N}_2\mathrm{O}_5}$ to be nearly $240$ ps. This time is in excellent agreement with that estimated from 100 individual reactive trajectories propagated with direct dynamics. Figures \[Fi:2\]d and \[Fi:2\]e) show representative snapshots taken along hydrolysis pathways generated from our molecular dynamics trajectories. Subsequent to passing through the transition state, we find that the ensemble of reactive pathways bifurcate resulting in two different product states. In one pathway, Fig \[Fi:2\]d), two nitric acids are formed through the concerted ionization of water and addition of the OH$^-$ to the NO$_2^+$ moiety, followed by proton transfer to NO$_3^-$. In the other pathway, Fig \[Fi:2\]e), one ${\mathrm{H N}\mathrm{O}_3}$ and one NO$_3^{-}$ are formed. As in the first pathway, this process proceeds through the ionization of water and addition of the OH$^-$ to the NO$_2^+$, however, the $\mathrm{H_3O}^+$ generated does not have an existing hydrogen bond wire to enable the subsequent donation of the proton to the NO$_3^{-}$. In our ensemble of 100 trajectories, 20 % of those follow the first pathway and 80 % follow the latter one. These product distributions and the pathways that evolve them are similar to previous calculations in water clusters.[@rossich2019microscopic] During hydrolysis, we find NO$_2^+$ is only formed transiently, with an average lifetime of 4 ps, and is better characterized by a hydrated $\mathrm{H_2 O NO_2}^{+}$ species than a stable intermediate. Once an ${\mathrm{H N}\mathrm{O}_3}$ molecule has its own independent solvation shell, we find that dissociation occurs on average within 60 ps, or that ionization to NO$_3^-$ + H$_3$O$^+$ occurs with a reaction rate of 15.4 ns$^{-1}$. The Grotthuss diffusion of the excess protons are well reproduced with our force field. We find a relative diffusivity of $\mathrm{H_3O}^+$ to $\mathrm{OH}^-$ of 2.2 compared to the 1.9 measured experimentally.[@mills_els_2013] Previous estimates of the hydrolysis rate for ${\mathrm{N}_2\mathrm{O}_5}$ in solution place it on the order of 10$^{-4}\,\mathrm{ns}^{-1}$, or four orders of magnitude slower than our computed rate.[@chang_ast_2011] However, like the solubility of ${\mathrm{N}_2\mathrm{O}_5}$, this rate has been inferred indirectly from mass transfer models and not measured independently. The model most commonly invoked assumes equilibration between the vapor and bulk solution, and is valid when uptake is reaction limited.[@davidovits_cr_2006] Given the short lifetime of ${\mathrm{N}_2\mathrm{O}_5}$ in solution, this equilibrium assumption requires reevaluation. Indeed, the reaction diffusion length, defined as $\ell_\mathrm{D} = \sqrt{D/k_\mathrm{h}}$ where $D$ is the self-diffusion constant of ${\mathrm{N}_2\mathrm{O}_5}$ is small. Within our model $\ell_\mathrm{D}\approx 1 $nm. As a consequence, we expect that ${\mathrm{N}_2\mathrm{O}_5}$ does not diffuse away from the interface before reacting. This suggests that rather than being mediated by bulk solvation and subsequent reaction, reactive uptake of ${\mathrm{N}_2\mathrm{O}_5}$ is determined directly at the air-water interface, through a process of interfacial absorption and reaction. A model for reactive uptake based on such interfacial activity is expanded on below. {width="17cm"} Interfacial model for N$_2$O$_5$ reactive uptake {#interfacial-model-for-n_2o_5-reactive-uptake .unnumbered} ------------------------------------------------ The canonical kinetic model for the reactive uptake of ${\mathrm{N}_2\mathrm{O}_5}$ is the so-called resistor model.[@worsnop1989temperature; @akimoto_arc_2016; @davidovits_cr_2006; @poschl_ar_2011] This model assumes that the gas molecule is first accommodated at the surface, with probability $\alpha$, and then diffuses from the surface to the bulk where the reaction takes place. The bulk reaction with rate $k_\mathrm{h}$, should be slow enough that an equilibrium can be established between the gas and the liquid phase, with concentrations determined by the Henry’s law constant $H$. Under these assumptions for the mass transfer kinetics, the reactive uptake coefficient, $\gamma$, can be estimated from $$\gamma=\left ( \dfrac{1}{\alpha} + \dfrac{v }{4 {k_\mathrm{B}}T H\sqrt{D k_\mathrm{h}}} \right )^{-1}$$ where $v$ is the thermal velocity. Measurements suggest a value of $\alpha \gtrsim 0.4$,[@grvzinic2017efficient] however as discussed above, neither $H$ nor $k_\mathrm{h}$ can be independently measured. Previous work has assumed a value of $H$ to be 5.0 M/atm, taken from extrapolating the known solubilities of a series of other $\mathrm{NO}_x$ compounds.[@mentel_pccp_1999] Inverting the expression for $\gamma$ and setting it equal to the 0.03, which is the middle of the range of experimental estimates, provides an estimate of the reaction rate on the order of 10$^5$ s$^{-1}$ for ${\mathrm{N}_2\mathrm{O}_5}$ hydrolysis. This analysis is internally consistent, as it predicts a reaction-diffusion length much larger than the width of the interface, $\ell_\mathrm{D}\approx 80 $nm, but the solubility and hydrolysis rate are dramatically different from those computed *ab initio*. Using our computed values of $H$ and $k_\mathrm{h} $, we arrive at $\gamma=0.6$, much higher than observed. This inconsistency can be resolved by formulating an alternative to the standard resistor model that envisions the reactive uptake of ${\mathrm{N}_2\mathrm{O}_5}$ as an interfacial process. Specifically, assuming all incoming ${\mathrm{N}_2\mathrm{O}_5}$ stick to the interface and do not diffuse away, the reactive uptake is given by a competition between hydrolysis of ${\mathrm{N}_2\mathrm{O}_5}$ at an interface[@rossich2019microscopic] and its reevaporation back to the gas phase. If $k^\mathrm{s}_\mathrm{h}$ is the reaction rate at the surface and $k_\mathrm{e}$ is the evaporation rate, then the reactive uptake coefficient can be computed from $$\gamma= \frac{k^\mathrm{s}_\mathrm{h}}{k^\mathrm{s}_\mathrm{h}+k_\mathrm{e}}$$ which in the limit that $\gamma$ is small reduces to $\gamma \approx k^\mathrm{s}_\mathrm{h}/ k_\mathrm{e}$.[@hanson1997surface] This competition is illustrated in Fig. \[Fi:3\] with accompanying simulation snapshots, and contrasts it with the processes of solvation and bulk hydrolysis included in the standard resistor model. This interfacial model is analogous to an older perspective on ${\mathrm{N}_2\mathrm{O}_5}$ uptake from Mozurkewich and Calvert[@mozurkewich_jgra_1988]. Using molecular dynamics simulations, we have tested the assumptions of this model and explicitly computed $\gamma$ by estimating $k^\mathrm{s}_\mathrm{h}$ and $k_\mathrm{e}$. We have computed the reaction rate at the air-water interface to be $k^\mathrm{s}_\mathrm{h}=0.95$ ns$^{-1}$ from direct molecular dynamics simulations. The distribution of waiting times for hydrolysis for both the interface and bulk are shown in Fig. \[Fi:3\]. This rate is slower than the corresponding rate in the bulk by a factor of 4, and predominantly follows a pathway that generates two protonated ${\mathrm{H N}\mathrm{O}_3}$ molecules. This is consistent with previous reports of the weaker acidity of ${\mathrm{H N}\mathrm{O}_3}$ at the air-water interface.[@shamay2007water] We have estimated the evaporation rate by first computing the free energy of adsorption to the interface from the vapor using thermodynamic perturbation theory and then assuming that evaporation is barrierless. We obtained a free energy of adsorption of $\Delta F_\mathrm{s} =-3.4$ kcal/mol, which is lower than the corresponding solvation free energy, as shown in Fig. \[Fi:3\]. This indicates that ${\mathrm{N}_2\mathrm{O}_5}$ is preferentially solvated at the interface, which is consistent with previous studies using empirical potentials[@hirshberg_pccp_2018; @li_jcp_2018] and the weak hydration observed in our bulk simulations. From this, we estimate an evaporation rate of $k_\mathrm{e}= 12.5$ ns$^{-1}$. The reactive uptake coefficient computable from these two rate processes yields $\gamma= 0.07$, which is in reasonable agreement with the experimental range.[@bertram_acpd_2009; @chang_ast_2011] An interfacial model of ${\mathrm{N}_2\mathrm{O}_5}$ reactive uptake helps rationalize a number of existing experimental observations, and opens new questions for further examination. For example, it has been noted that the temperature dependence of ${\mathrm{N}_2\mathrm{O}_5}$ uptake is rather weak.[@van_jpc_1990] The similar barrier heights for interfacial hydrolysis and evaporation result in both processes increasing with temperature at about the same rate, leaving $\gamma$ nearly invariant in our model. Further, measurements of the reactive uptake on ice particles are close to those for liquid particles.[@apodaca_acp_2008] The importance of surface processes elucidated in our work clarifies this coincidence, as diffusion into the bulk of the solid is prohibitively slow, and as we have shown hydrolysis can still proceed. Finally, the rapid rate of hydrolysis observed here explains why the uptake coefficient does not strongly depend on reactions with inorganic species in solution[@bertram_acpd_2009] as such reactions cannot kinetically compete for intact ${\mathrm{N}_2\mathrm{O}_5}$. However, it is known that the branching ratio for ${\mathrm{N}_2\mathrm{O}_5}$ decomposition in solutions with halide anions, $\mathrm{X}^{-}$ and excess nitrate strongly favors $\mathrm{XNO}_2$ over ${\mathrm{H N}\mathrm{O}_3}$, beginning at 1 M $\mathrm{X}^{-}$ concentration.[@sobyra_jpca_2019] This seems at odds with the rapid hydrolysis to ${\mathrm{H N}\mathrm{O}_3}$, which reacts with $\mathrm{X}^{-}$ to form $\mathrm{XNO}_2$ only at very low pH. Surface chemistries not viable in the bulk solution such as those catalyzed by enhanced interfacial proton concentrations could be studied to clarify this. Such studies are now possible by employing analogous neural network based simulations as we have developed here. With these tools, many heterogeneous chemistries previously defying explanation can now be systematically studied and understood. Methods {#methods .unnumbered} ======= [**Machine learning *ab initio* potential.**]{} We have used the DeePMD-kit[@wang_cpc_2018] to learn the many body interatomic potential energy and forces generated at the DFT level of theory. The primary data sets for the training were generated from ab initio molecular dynamics simulations using the Gaussian Plane Wave(GPW) implementation in CP2K.[@vandevondele_cpc_2005] All *ab initio* molecular dynamics simulations were carried out in the canonical ensemble at ambient temperature and density using the revised version of PBE functional[@zhang_prl_1998] along with empirical dispersion correction (Grimme D3)[@grimme_jcp_2010]. We used a molopt-DZVP basis set and a plane wave cut-off of 300 Ry. The core electrons were described with GTH pseuodopotential.[@goedecker_prb_1996] We also carried out metadynamics simulations[@laio_rpp_2008] to generate reactive structures along the hydrolysis pathway. With the primary data set generated by molecular dynamics and metadynamics simulations, we first trained two independent machine learned potentials that were then followed by active learning to improve both models. The disagreement in force between the two models was used to select the new data sets for active learning. Final convergences for testing errors in the energy were 0.2 meV/atom. [**Molecular dynamics simulations.**]{} To investigate the hydrolysis reaction of ${\mathrm{N}_2\mathrm{O}_5}$, we carried out molecular dynamics simulations at ambient temperature and pressure with 0.5 fs timestep. The integrator used a Langevin thermostat, with characteristic time constant of 1 ps. The bulk system contained one ${\mathrm{N}_2\mathrm{O}_5}$ molecule solvated by 253 water molecules in a 19.73 x 19.73 x 19.73 Å  box with periodic boundary conditions in all three dimensions. An equilibration molecular dynamics simulations of 5 ns was carried out by classical molecular dynamics that was followed by another equilibration MD simulations for 400 ps with machine learned force field. During the equilibration, the N-N distance was constrained to  2.6 Å  to prevent the hydrolysis reaction taking place. We sampled the initial configurations from a 1 ns constrained molecular dynamics simulation and then carried out unconstrained molecular dynamics simulations for an ensemble of 50 trajectories, each for 1 ns. To investigate the hydrolysis reaction of ${\mathrm{N}_2\mathrm{O}_5}$ at the air-water interface, we prepared a slab model with thickness of 25 x 25 x 25 Å, having free interface and an additional 20 Å  vacuum on each side. We employed periodic boundary conditions in all three dimensions. The slab model included 1 ${\mathrm{N}_2\mathrm{O}_5}$ molecule and 522 water molecules. The $z$-position of the ${\mathrm{N}_2\mathrm{O}_5}$ molecule was constrained at the Gibbs diving surface of the slab. The initial configuration was generated from an equilibrated water box. An equilibration of 10 ns was carried out by classical molecular dynamics simulations with SPC/E water and GAFF force field,[@hirshberg_pccp_2018] which was then followed by another equilibration MD simulations for 500 ps with the machine learned force field. During the equilibration, the N-N distance was constrained to  2.6 Å  to prevent the hydrolysis reaction taking place. We sampled the initial configurations from a 1 ns constrained molecular dynamics simulation and carried out molecular dynamics simulations for an ensemble of 28 trajectories, each for 3 ns. [**Free energy and rate calculations.**]{} We used umbrella sampling [@frenkel2001understanding] to estimate the reaction free energies for the hydrolysis reaction of ${\mathrm{N}_2\mathrm{O}_5}$ in the bulk water. Harmonic biases were employed for both $n_w$ and $R$, and each of 26 windows were simulated for 1 ns. The free energies were then estimated using WHAM.[@kumar1992weighted] In order to calculate the correction to the transition state theory rate we computed the transmission coefficient from an ensemble of 2000 unbiased trajectories. The commitor probability was computed from an ensemble of 1000 unbiased trajectories, each starting from constrained configuration at the dividing surface with a random velocity taken from Maxwell-Boltzmann distribution. We have computed the free energy to dissociated ${\mathrm{H N}\mathrm{O}_3}$ in bulk water from an ensemble of molecular dynamics trajectories having one ${\mathrm{H N}\mathrm{O}_3}$ molecule solvated by 255 water molecule in a 19.73 x 19.73 x 19.73 Å box. Since deprotonation occurs frequently, we have computed the free energy by monitoring $n_h$. We have computed the solvation free energy of ${\mathrm{N}_2\mathrm{O}_5}$ using thermodynamic perturbation theory. For computational efficiency, we first used an empirical nonreactive reference potential and constructed a reversible work path by pulling a molecule of ${\mathrm{N}_2\mathrm{O}_5}$ initially in the vapor through a liquid-vapor interface and into the bulk using a slab geometry. We used the SPC/E water model and a GAFF force field for the ${\mathrm{N}_2\mathrm{O}_5}$ with partial charges parametrized to reproduce the *ab initio* electrostatic potential.[@hirshberg_pccp_2018] We then estimated the free energy difference between the empirical model and our neural network potential model, by linearizing the relative Boltzmann weights collected from 20,000 configurations of the solvated classical model. An analogous calculation was used to compute the absorption free energy at the interface.\ [**Acknowledgments**]{} The authors thank Timothy Bertram, Benny Gerber, Andreas Goetz and Gilbert Nathanson for stimulating discussions and Barak Hirshberg for initial solvated ${\mathrm{N}_2\mathrm{O}_5}$ configurations. This work was funded by the National Science Foundation through the National Science Foundation Center for Aerosol Impacts on Chemistry of the Environment (NSF-CAICE) under Grant No. CHE 1801971. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. DOE Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. 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--- author: - | Daniel Britzger\ Max-Planck-Institut für Physik, Föhringer Ring 6, D-80805 München, Germany\ E-mail: - | \ Institut für Theoretische Physik, Universität Heidelberg,\ Philosophenweg 16, D-69120 Heidelberg, Germany\ and\ ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung,\ Planckstra[ß]{}e 1, D-64291 Darmstadt, Germany\ E-mail: - | Sasha Glazov, Stefan Schmitt\ Deutsches Elektronen-Synchrotron DESY, Notkestra[ß]{}e 85, D-22607 Hamburg, Germany\ E-mail: , - | Otto Nachtmann\ Institut für Theoretische Physik, Universität Heidelberg,\ Philosophenweg 16, D-69120 Heidelberg, Germany\ E-mail: title: 'The Tensor Pomeron and Small-x Deep Inelastic Scattering' --- Tensor pomeron ============== High energy hadronic reactions are dominated by the physics of the pomeron. For example, total cross sections, related to the elastic forward scattering amplitude by the optical theorem, are at high c.m. energy described by the exchange of a pomeron in the $t$-channel. For soft reactions, calculations of the pomeron from first principle are currently not possible, and one has to retreat to Regge models to describe soft high-energy scattering. Here, we want to discuss deep inelastic scattering (DIS) in the context of such a Regge model. This presentation summarizes some of the key results of [@Britzger:2019lvc] to which we refer the reader for further details. Until recently, the spin structure of the pomeron has not received much attention. It is well known that the pomeron carries vacuum quantum numbers with regard to charge, color, isospin and charge conjugation. But what about spin? It has been shown some time ago that the pomeron can be regarded as a coherent superposition of exchanges with spins 2, 4, 6, etc [@Nachtmann:1991ua]. The new aspect that we want to discuss here is the structure of the couplings of the pomeron to external particles. Most models treat these couplings like those of a photon [@Donnachie:1983hf], i.e., as vector couplings. In many fits to DIS data the coupling is not specified at all and only the energy dependence of the pomeron enters the formulae used for fitting the data, see for example [@Donnachie:1998gm]. We argue that the pomeron couplings play an important role, and that they should be treated as tensor couplings. Two of us, in collaboration with M. Maniatis, have constructed an effective theory with such a tensor pomeron and reggeon contributions (and a vector odderon) [@Ewerz:2013kda]. Conceptually, vector-type couplings of the pomeron turn out to be rather questionable. For example, a vector pomeron implies that the total cross sections for $pp$ and $\bar{p}p$ scattering at high energy have opposite sign. But, of course, quantum field theory forbids negative cross sections. A further argument against a vector pomeron is that it does not give any contribution to photoproduction data, as we show in [@Britzger:2019lvc]. One may also ask about the possibility of a scalar coupling of the pomeron to external particles. While possible from the point of view of quantum field theory, such a coupling is experimentally disfavored. In [@Ewerz:2016onn] it was shown that STAR data on polarized elastic $pp$ scattering are compatible with the tensor pomeron but clearly rule out scalar pomeron couplings. DIS in the tensor pomeron model =============================== Good fits of some DIS data in the context of Regge theory have been obtained in the literature, see for example [@Donnachie:1998gm]. However, these do not explicitly treat the couplings of the pomeron (assumed to be a vector) to the photon and proton. In [@Britzger:2019lvc] we have addressed the question whether a Regge model based on the tensor pomeron, treating in full detail its tensor couplings to the photon and proton, can successfully describe DIS and photoproduction data. For making such a comparison to DIS data, we have added a hard pomeron to the model of soft high-energy reactions of [@Ewerz:2013kda]. The reggeon contribution, denoted by $\mathbbm{R}_+$, is dominated by the $f_2$ but can contain a small contribution from the $a_2$. The parameters of the model are summarized in table \[tab1\]. The indices $j = 0,1,2$ always refer to the hard pomeron, the soft pomeron, and the reggeon, respectively. hard pomeron $\mathbbm{P}_0$ soft pomeron $\mathbbm{P}_1$ reggeon $\mathbbm{R}_+$ ---------------------------------------- ------------------------------------ ------------------------------------ ------------------------------------ intercept $\alpha_0(0)=1+\epsilon_0$ $\alpha_1(0)=1+\epsilon_1$ $\alpha_2(0)=1+\epsilon_2$ slope parameter $\alpha'_0$ $\alpha'_1$ $\alpha'_2$ $W^2$ parameter $\tilde{\alpha}'_0$ $\tilde{\alpha}'_1$ $\tilde{\alpha}'_2$ $pp$ coupling parameter $\beta_{0pp}$ $\beta_{1pp}$ $\beta_{2pp}$ $\gamma^* \gamma^*$ coupling functions $\hat{a}_0(Q^2)$, $\hat{b}_0(Q^2)$ $\hat{a}_1(Q^2)$, $\hat{b}_1(Q^2)$ $\hat{a}_2(Q^2)$, $\hat{b}_2(Q^2)$ : Parameters of our two-tensor-pomeron model. \[tab1\] The couplings of the pomerons and the reggeon to the photon contain functions $\hat{a}_j$ and $\hat{b}_j$ which we cannot derive from first principles. In fitting the data we make polynomial ansätze for them. We use the standard variables for DIS, namely the lepton-proton c.m. energy $\sqrt{s}$, the photon virtuality $Q^2$, the c.m. energy $W$ of the photon-proton system, Bjorken $x=Q^2/(W^2+Q^2-m_p^2)$, and $y=(W^2+Q^2-m_p^2)/(s-m_p^2)$. Then, the total cross sections $\sigma_T$ and $\sigma_L$ for transversely and longitudinally polarized photons resulting from our two-tensor-pomeron model are $$\begin{aligned} \label{2.11} &\sigma_T(W^2,Q^2)=4\pi \alpha_{\rm em} \, \frac{W^2-m_p^2}{W^2} \sum_{j=0,1,2} 3 \beta_{jpp} (W^2 \tilde{\alpha}'_j)^{\epsilon_j} \cos\left(\frac{\pi}{2} \, \epsilon_j\right) \\ & \:\: \times \left\{ \hat{b}_j(Q^2) \left[1+ \frac{2Q^2}{W^2-m_p^2} + \frac{Q^2 (Q^2+2m_p^2)}{(W^2 - m_p^2)^2}\right] - 2 Q^2 \hat{a}_j(Q^2) \left[1+ \frac{2Q^2}{W^2-m_p^2} + \frac{Q^2 (Q^2+m_p^2)}{(W^2 - m_p^2)^2} \right]\right\} , \nonumber \\ \label{2.12} &\sigma_L(W^2,Q^2)=\, 4\pi \alpha_{\rm em} \, \frac{W^2-m_p^2}{W^2}\,Q^2 \sum_{j=0,1,2} 3 \beta_{jpp} (W^2 \tilde{\alpha}'_j)^{\epsilon_j} \cos\left(\frac{\pi}{2} \, \epsilon_j\right) \\ & \:\:\times \left\{ 2 \hat{a}_j(Q^2) \left[1+ \frac{2Q^2}{W^2-m_p^2} + \frac{Q^2 (Q^2+m_p^2)}{(W^2 - m_p^2)^2}\right] + \hat{b}_j(Q^2) \frac{2m_p^2}{(W^2 - m_p^2)^2} \right\} . \nonumber\end{aligned}$$ They are related to the structure functions $F_2$ and $F_L$ in the standard way, $$\begin{aligned} \label{2.13} F_2(W^2,Q^2) =& \, \frac{Q^2}{4\pi^2 \alpha_{\rm em}} (1-x) \left[1+2 \delta(W^2,Q^2)\right]^{-1} \left[\sigma_T (W^2,Q^2) + \sigma_L(W^2,Q^2) \right] \\ \label{2.14} F_L(W^2,Q^2) =& \, \frac{Q^2}{4\pi^2 \alpha_{\rm em}} (1-x) \sigma_L(W^2,Q^2) \end{aligned}$$ with $\delta(W^2,Q^2) = 2m_p^2Q^2/(W^2 + Q^2 -m_p^2)^2$. We use standard values for the electromagnetic coupling $\alpha_{\rm em}$ and the proton mass $m_p$. Fit to DIS and photoproduction ============================== Using our model we perform a fit to the available DIS [@Abramowicz:2015mha] and photoproduction data [@Aid:1995bz; @Chekanov:2001gw; @Vereshkov:2003cp; @Caldwell:1978yb] in the kinematic range given by $6 < \sqrt{s} < 318$ GeV, $Q^2 < 50 \,\mbox{GeV}^2$, and $x<0.01$. We fit the reduced cross section $$\label{3.5} \begin{split} \sigma_{\rm red} (W^2,Q^2,y)=& \,\frac{1+(1-y)^2 + y^2 \delta(W^2,Q^2)}{1+(1-y)^2} \left[ F_2(W^2,Q^2) - \frac{\tilde{f}(W^2,Q^2,y) F_L(W^2,Q^2)}{1+2\delta(W^2,Q^2)} \right] \,, \end{split}$$ where $$\label{3.4} \tilde{f}(W^2,Q^2,y) = \frac{y^2[1+2\delta(W^2,Q^2)]}{1+(1-y)^2 + y^2 \delta(W^2,Q^2)} \,.$$ It contains the full experimental information and is the quantity that is actually measured. Some of the couplings and slope parameters in our model are well constrained from other experiments so that we use their known values as default parameters. Formally, our fit has 25 parameters. However, not all of them are equally important. The most important parameters are the intercepts of the pomerons and of the reggeon. Further parameters of significance are the values of the coupling functions $\hat{a}_j(Q^2)$ and $\hat{b}_j(Q^2)$ at $Q^2=0$. The remaining parameters are used to describe the fall-off of the coupling functions $\hat{a}_j$ and $\hat{b}_j$ at large $Q^2$, and these 17 parameters are much less significant for our result. The fit is of very satisfactory quality, for a detailed quantification of the fit quality see [@Britzger:2019lvc]. For the intercepts of the hard pomeron, the soft pomeron, and the reggeon we find, respectively, $$\alpha_0(0) = 1.3008 \,({}^{+73}_{-84})\,, \qquad \alpha_1(0) = 1.0935 \,({}^{+76}_{-64})\,, \qquad \alpha_2 (0) = 0.485 \, ({}^{+88}_{-90})\,.$$ Figure \[fig:comp0\] shows a comparison of our tensor-pomeron fit to the photoproduction cross sections of [@Aid:1995bz; @Chekanov:2001gw; @Vereshkov:2003cp; @Caldwell:1978yb]. ![ Comparison of our fit to the photoproduction cross sections [@Aid:1995bz; @Chekanov:2001gw; @Vereshkov:2003cp; @Caldwell:1978yb], with the reggeon contribution also shown separately. The experimental uncertainties of the fit are indicated as shaded bands. \[fig4\] []{data-label="fig:comp0"}](figure5){width="50.00000%"} We find that photoproduction is dominated by the soft pomeron, while the hard pomeron contribution is compatible with zero here. As an example, we quote the three different contributions for $W=200$ GeV: $$\label{4.12c} \begin{split} 170.4\, {}^{+4.2}_{-4.0} \: \mu{\rm b}& \qquad \mbox{for the soft pomeron } \mathbbm{P}_1\,, \nonumber\\ 0.002\, {}^{+0.086}_{-0.002} \: \mu{\rm b}& \qquad \mbox{for the hard pomeron } \mathbbm{P}_0\,, \nonumber\\ 0.84\, {}^{+0.99}_{-0.58} \: \mu{\rm b}& \qquad \mbox{for the $\mathbbm{R}_+$ reggeon}. \nonumber \end{split}$$ The reggeon contribution, also shown in the figure, is sizable for low $W$. We point out again that a vector pomeron would give zero contribution to the photoproduction cross section. Next, we turn to DIS. Figure \[fig9\] compares of our fit results, shown as a blue band, to the DIS cross sections of [@Abramowicz:2015mha]. ![ Comparison of our fit to DIS cross sections [@Abramowicz:2015mha] at $\sqrt{s}=318\,\text{GeV}$, at low $Q^2<1.5\,\text{GeV}^2$. We also show the soft contribution (soft pomeron plus $\mathbbm{R}_+$ reggeon) and the contribution of the structure function $F_2$ in the reduced cross section. The experimental uncertainties of the fit are indicated as shaded bands. \[fig9\] []{data-label="fig:comp4l"}](figure10){width="90.00000%"} Here we have chosen as an example the data for $\sqrt{s}= 318$ GeV and low $Q^2<1.5\,\text{GeV}^2$. We indicate the soft contribution (soft pomeron plus reggeon) as a purple band in figure \[fig9\]. The thin grey band represents the contribution of the structure function $F_2$ to the reduced cross section, cf. equation . The fit is of similarly good quality as in figure \[fig9\] for the whole kinematic range that we consider. In the full kinematic range we find the following properties. The hard contribution increases with increasing $Q^2$, and the hard and soft contributions are of approximately equal size at $Q^2 = 5\,\mbox{GeV}^2$. But the soft contribution is still visible at $Q^2 = 20\,\mbox{GeV}^2$. The difference between $\sigma_{\rm red}$ and $F_2$, due to the longitudinal cross section $\sigma_L$, is clearly visible at large $W$. As expected, the reggeon contribution becomes very small at large $W$. We have also computed the ratio $R=\sigma_L/\sigma_T$ from our fit results for the longitudinal and transverse cross sections. We compare this to the $R$ data from H1 [@Andreev:2013vha] which are extracted directly from cross section measurements at fixed $Q^2$ but different c.m. energies. We observe that our fit prefers higher values for $R$. The origin of this discrepancy and its significance in view of the sizable error of the $R$ data from H1 are difficult to assess. Measurements of $\sigma_L$ and of $R$ at a future electron-ion collider would be very helpful to improve our understanding of the dynamics of the longitudinal cross section. Summary ======= We have developed a two-tensor-pomeron model and have made a fit to photoproduction data and to small-$x$ DIS data from HERA. We obtain a very satisfactory fit and determine in particular the intercepts of the two pomerons. For DIS, the soft contribution is still clearly visible up to about $Q^2=20 \,\mbox{GeV}^2$. The transition from low to high $Q^2$ is nicely described. We have argued that a vector pomeron, that is a pomeron with vector type couplings, is excluded by general arguments based on quantum field theory. For instance, it would not give any contribution to photoproduction in clear contradiction with the data. A tensor pomeron, on the other hand, is compatible with quantum field theory and permits an excellent description of photoproduction and DIS data, as we have shown. [99]{} D. Britzger, C. Ewerz, S. Glazov, O. Nachtmann and S. Schmitt, [*The Tensor Pomeron and Low-$x$ Deep Inelastic Scattering*]{}, arXiv:1901.08524 \[hep-ph\]. 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--- author: - 'K. Petrovay' - 'M. Talafha' bibliography: - 'SFTparams.bib' date: 'Received ; accepted ' title: Optimization of surface flux transport models for the solar polar magnetic field --- [The choice of free parameters in surface flux transport (SFT) models describing the evolution of the large-scale poloidal magnetic field of the Sun is critical for the correct reproduction of the polar magnetic flux built up during a solar cycle, which in turn is known to be a good predictor of the amplitude of the upcoming cycle. ]{} [ For an informed choice of parameters it is important to understand the effect and interplay of the various parameters and to optimize the models for the polar magnetic field. ]{} [Here we present the results of a large-scale systematic study of the parameter space in an SFT model where the source term representing the net effect of tilted flux emergence was chosen to represent a typical, average solar cycle as described by observations. ]{} [Comparing the results with observational constraints on the spatiotemporal variation of the polar magnetic field, as seen in magnetograms for the last four solar cycles, we mark allowed and excluded regions in the 3D parameter space defined by the flow amplitude $u_0$, the magnetic diffusivity $\eta$ and the decay time scale $\tau$, for three different assumed meridional flow profiles.]{} [Without a significant decay term in the SFT equation (i.e., for $\tau >10$ yr) the global dipole moment reverses too late in the cycle for all flow profiles and parameters, providing independent supporting evidence for the need of a decay term, even in the case of identical cycles. An allowed domain is found to exist for $\tau$ values in the 5–10 yr range for all flow profiles considered. Generally higher values of $\eta$ (500–800 [km$^2/$s]{}) are preferred though some solutions with lower $\eta$ are still allowed.]{} Introduction {#sect:intro} ============ Synoptic maps of the line-of-sight component of the large scale solar magnetic field have been available on a regular basis since the 1970s. [Most of this large-scale flux resides in unipolar areas where one polarity prevails among the magnetic network elements ([@Stix:book]{}).]{} High resolution observations indicate that this field is mostly concentrated in numerous thin, strong flux tubes that are vertically aligned [(as illustrated e.g. in Fig. 2 of [@Orozco2007]{})]{}, so the true photospheric field is assumed to be radial, and its amplitude on the solar surface, $B(\lambda,\phi)$ can be derived by a de-projection of the line-of-sight field as a function of heliographic latitude $\lambda$ and longitude $\phi$. [(A possible smaller contribution to the large-scale field from a slight polarity imbalance in the small-scale mixed polarity internetwork fields would not alter this situation, as the orientation of this field is more isotropic but on average still symmetric around the radial direction.) ]{} In order to interpret the evolution of the photospheric field on these synoptic maps, surface flux transport (SFT) models were developed in the 1980s [([@sheeley1983quantitative]{})]{}. The models described the evolution of $B$ by an advective-diffusive transport equation, advection being ascribed to differential rotation and poleward meridional flow, and diffusion interpreted as due to the mixing action of supergranular flows; a source term was included to represent flux emergence in the form of newly appearing bipolar active regions (ARs). This early age of SFT model development, reviewed by [@Sheeley:LRSP], mostly focused on the overall reproduction of the evolution of the large-scale field during a period of about one solar cycle (Cycle 21, for which data were available at the time). [These models demonstrate that the reversal of the polar field and of the solar dipole moment at the middle of activity cycles and the build-up of a new poloidal field in the late phases of the cycle originates from the systematic latitude-dependent tilt of ARs relative to the azimuthal direction (Joy’s law): as this imparts a meridional component to the AR magnetic field, each AR gives a contribution to the global solar dipole moment and these contributions gradually change the Sun’s polar magnetic field, after the magnetic fields of the trailing parts of decayed active regions are transported to the poles.]{} Interest in SFT models increased again in the 2000s when they were used for the reconstruction of the long-term variation of the total solar magnetic flux and open flux, with a view to reconstructing coronal and interplanetary conditions [([@wang2000long]{}). The long-term solar record clearly indicates that solar cycles vary in amplitude in a manner that appears to be random.]{} As these explorations run SFT models for many solar cycles, it became clear [([@schrijver2002missing]{}) that such variations in cycle amplitude will, for reasons explained in the previous paragraph,]{} result in a random walk of the unsigned amplitude of the solar dipole moment from one cycle to the next, until it reaches values well exceeding the typical total contribution for ARs in a cycle, so polar reversal would cease, quenching the dynamo. As a workaround, [@schrijver2002missing] suggested the introduction of a sink or decay term of the form $-B/\tau$ in the transport equation, [tentatively interpreted as an unobserved process (U-loop emergence).]{} The decay term was [more convincingly]{} interpreted as a simplistic phenomenological representation of a 3D effect by [@Baumann+:decayterm] who argued that, owing to the high aspect ratio of the solar convective zone, the decay time scale of the field due to vertical diffusion should be two orders of magnitude shorter than the decay time scale of horizontal diffusion, implying $\tau=5$–$10$ years. While it has later been suggested that downward directed pumping in the subsurface layers may suppress vertical diffusion, this has not been convincingly demonstrated, nor did it solve the dipole moment drift problem. [Intercycle variations in the meridional flow ([@Wang:flowvar]{}) or in Joy’s law ([@Cameron+:tiltprecursor]{}) have also been suggested to solve the dipole moment drift problem, though the suggested variations are still disputed and the time periods studied were relatively short.]{} A third period of increased activity in SFT modelling started around 2010, with the increased interest in solar cycle prediction following the abrupt change of the long-term level of solar activity with Cycle 24 [([@pesnell2008predictions]{})]{}. As it was realized that the best physical precursor of the amplitude of an upcoming solar cycle was the amplitude of the Sun’s polar magnetic field at the start of the cycle ([@Petrovay:LRSP]{}), the next logical step was the use of SFT models to describe the buildup process of this polar field, in the hope of extending the temporal range of cycle prediction. Any SFT model needs to make assumptions concerning an ill-constrained function, the effective meridional flow profile $u(\theta)$, as well as three free parameters: the flow amplitude $u_0$, the diffusivity $\eta$ and the decay time scale $\tau$. (Other choices, such as differential rotation or the form of the source, are much better constrained by observations.) [For applications where an exact quantitative reproduction of the evolving magnetic field is important the appropriate choice of parameters will be especially crucial.]{} The determination of model parameters is a complex issue, to be discussed in some detail in Section \[sect:motivation\] below. We will argue that a calibration of SFT models to correctly reproduce the characteristics of the evolution of the polar magnetic field and of the solar dipole moment is the method that best fits the objectives of solar cycle prediction. This argument provides the motivation of the research presented in the rest of the paper. In Section \[sect:obsdata\] we determine the criteria that the SFT model of a “typical” or “ideal” solar cycle should satisfy to qualify as a good representation of the actual evolution of the solar polar magnetic fields. Then, in Section \[sect:model\] we perform a systematic study on a large grid of SFT models with different parameter sets and with an idealized but realistic source term representing a “typical” cycle. Comparing the results with the preset criteria we mark out allowed and excluded domains in the parameter space. Results of this analysis are discussed in Section \[sect:discussion\]. Section \[sect:concl\] concludes the paper. Motivation {#sect:motivation} ========== Two general approaches have been taken in the choice of parameter values and flow profiles in SFT models: direct measurements and internal calibrations of the model. Direct measurements of the meridional flow have been made by numerous researchers with three different methods: Doppler shifts of spectral lines, [(e.g., [@hathaway1996doppler]{}]{}), helioseismic inversions, [(e.g., [@schad2011unified]{})]{}, and local correlation tracking of moving magnetic flux concentrations [(e.g., [@iida2016tracking]{})]{}. While a general review of these efforts is far beyond the scope of this paper, we can safely say that, despite some claims to the contrary, the results can hardly be considered concordant. There is not even agreement regarding the latitude range where the overall meridional flow peaks. To cite just a few recent results: in a time–distance helioseismic analysis [@Chen+Zhao:highpeak] find that the polewards surface flow peaks at latitudes of $50$–$60^\circ$, whereas [@Imada+:midpeak] report a peak near $45^\circ$ from magnetic feature tracking, while [@Zhao+:lowpeak] find that the flow peaks at a latitude of $\sim 15^\circ$. Part of the reason for the discrepancy is the very significant cycle dependence of the flow, following a migrating pattern associated with torsional oscillations ([@Lin+Chou:cycledep.flows]{}). [ Determinations of the surface differential rotation based on correlation tracking show that small-scale, short-term correlation tracking leads to results that strongly deviate from the more rigid rotation of large-scale, long-lived magnetic patterns ([@Wilcox+:patternrot]{}, [@Wang+:patternrot]{}). This suggests that small-scale, short-term tracking studies are of limited use for the determination of the transport properties of large-scale fields. A study of the effect of transport processes in the deep convective zone on surface patterns by [@Petrovay+Szakaly:2d.pol] also showed that the evolution of the large-scale surface field is strongly influenced or even dominated by subsurface processes, rather than by the flows localized to the surface. Similar conclusions were reached recently by [@Whitbread+:disconn] who find that the apparent diffusion of surface magnetic fields resulting from AR decay cannot be represented by the nominal surface value of the diffusivity unless these fields are somehow disconnected from the magnetic field in the deep convective zone. All this may well explain why direct determinations of the magnetic diffusivity generally tend to yield lower values than internal calibrations of SFT models (see Table 6.2 in [@Schrijver+Zwaan:book]{}). ]{} In the face of this state of the affairs, internal calibrations of the SFT model seem to be the preferable way to follow in the determination of parameters. Parameter studies and optimization attempts have been made previously by several authors. The first more extensive parameter study of the SFT model was performed by [@baumann2004evolution] who considered the effect of varying each parameter, one at a time, in a model without a decay term, with a fixed form of the meridional flow profile and a simplified source term. More recently, a 3x3 model grid in the $(u_0,\eta)$ plane was presented by [@Virtanen+:SFT] with a source term taken from observations and the resulting butterfly diagram was qualitatively compared to the observed pattern. [@Lemerle1] and [@Whitbread+:SFT] searched for an optimal combination of parameters (including flow profile) using the PIKAIA optimization algorithm. These latter studies were the first to apply a quantitative criterium (the “merit function” of the algorithm) to judge the goodness of the fit between a model result and observations. The merit used in those studies was essentially the $\chi^2$ difference between observed and simulated magnetic butterfly diagrams. These two studies differed in the time period considered and in the way the source term was constructed from observational data, as well as in some further details, but their results were in general comparable. Depending on model details, [@Whitbread+:SFT] find that acceptable values of $u_0$, $\eta$ and $\tau$ can fall in the ranges 7–22 m/s, 220–800 km$^2$/s and 2.5–30 year, respectively. At the same time, the characteristics of polar fields were not particularly faithfully reproduced in the optimized solution. This is due to the fact that the polar area covers a small fraction of the solar surface, having little weight in the determination of the merit applied. However, as discussed above in the Introduction, the ability to correctly reproduce the polar magnetic field is critical if SFT models are to be applied for solar cycle prediction purposes (as opposed to a general reproduction of observed magnetic butterfly diagrams). Therefore, in the present work we perform another SFT model parameter study where the merit is based on a good quantitative reproduction of the spatiotemporal variation of solar polar magnetic fields. Furthermore, while PIKAIA is an extremely powerful tool, it offers little physical insight and yields only limited information on the extent of the allowed domain in parameter space. Therefore, another difference relative to previous work is that instead of using an optimization algorithm or a more limited systematic study, here we perform a detailed mapping of the 3D parameter space defined by $(u_0,\eta,\tau)$ for several alternative meridional flow profiles. This extensive mapping of the parameter space is supported by our choice to employ a source term representing an idealized cycle (as discussed below in more detail) instead of constructing the source based on the observation of a particular solar cycle. ![The hemispherically averaged polar field amplitude from the WSO data set [(black solid)]{} and the global axial dipole moment [(red dot-dashed)]{} as a function of time. The monthly sunspot number series [(blue dashed)]{} is shown for comparison, with an arbitrary rescaling. All curves were smoothed with a 13-month sliding window. Times of sunspot minima are marked by the dashed vertical lines. Global dipole amplitudes were obtained by courtesy of Jie Jiang and represent the average of values computed for all available data sets at the given time ([@jiang2018predictability]{}). []{data-label="fig:polarfieldobs"}](polarfield){width="\columnwidth"} --------------------------------------------------------------------------------------- --------------------------- WSO polar field reversal time counted from sunspot minimum \[years\]: $4.33 \pm 0.36$ WSO polar field strength at cycle minimum / maximum of WSO polar field strength: $0.90 \pm 0.06$ Latitude $\lambda_{1/2}$ of the edge of topknot \[$B(\lambda_{1/2})=B(\theta=0)/2$\]: $70.0^\circ\pm 2.5^\circ$ Global dipole moment reversal time counted from sunspot minimum \[years\]: $3.44 \pm 0.18$ Dipole moment at cycle minimum / maximum of dipole moment: $0.84 \pm 0.12$ --------------------------------------------------------------------------------------- --------------------------- \[table:obsconstraints\] Observational constraints and choice of merit {#sect:obsdata} ============================================= [The amplitude of the Sun’s poloidal magnetic field at sunspot minimum is known to be a good predictor of the amplitude of the subsequent solar cycle ([@Petrovay:LRSP2]{}). SFT models offer a way to to extend the temporal scope of this cycle precursor, provided they are optimized to best reproduced the observed variation of the solar poloidal field. In order to determine what are the salient features of this variation, to be reproduced in the models, in Figure \[fig:polarfieldobs\] we show the temporal variation of two measures of the poloidal field, contrasted with the variation of the sunspot number. In line with standard practice in solar physics (e.g. in determining sunspot cycle maxima and minima), all curves have been smoothed with a 13-month sliding window. The black solid line shows]{} the smoothed amplitude of the polar magnetic field strength measured at Wilcox Solar Observatory (WSO) [averaged over 55–90$^\circ$ latitudes along the central meridian over both poles]{}. The geometry is illustrated in Fig. 1 of [@Svalgaard+:polarfield]. It is thus approximately given by $$\tilde{B}_{l}=\frac{1}{1.8\tilde{N}} \int_{\lambda_0}^{\pi/2} B(\lambda,\phi=\phi_{\rm CM},t) \cos^{2}{\lambda}\, \mathrm{d}\lambda , \label{eq:WSOB}$$ where $\tilde{N}=1-\sin{\lambda_0}$ and $\lambda_0=55^\circ$ is the lowest latitude of the polemost pixel. (The factor 1.8 accounts for magnetograph saturation, cf. [@Svalgaard+:polarfield]{}.) [ The plotted line is the average of the unsigned amplitude at the two poles. ]{} Also shown is an alternative measure of the amplitude of the poloidal field component, the [*axial dipole coefficient*]{}, i.e. the amplitude of the coefficient of the $Y_1^0$ term in a spherical harmonic expansion of the distribution of the radial magnetic field strength over the solar disk: $$D(t) = \frac32 \int_0^{\pi} \overline{B}(\theta,t)\cos\theta\sin\theta\, \mathrm{d}\theta. \label{eq:dipmom}$$ where $\overline B$ denotes the azimuthally averaged radial magnetic field. [In solar physics literature this dipole coefficient is normally referred to as the solar (axial) dipole moment. (Dipole moment, as usually defined in physics, has a different dimension but it is indeed proportional to $D$.) ]{} The behaviour of the curves in Figure \[fig:polarfieldobs\] shows that the times of dipole reversal are usually rather sharply defined, [ (in the sense that the dipole moment values below, say, 10% of the peak are only observed in time intervals that are much shorter than the cycle period). ]{} From the plotted data the overall dipole is found to reverse $3.44 \pm 0.18$ years after the minimum, while the WSO field characterizing the polar contribution to the dipole reverses after $4.33 \pm 0.36$ years. (The formal $1\sigma$ errors here correspond to the [standard deviations]{} calculated for the 4 observed reversals —admittedly a very low number.) The low scatter in these values suggests that the cycle phase of polar dipole reversal may be a sensitive test of SFT and dynamo models. In contrast to reversal times, maxima of the dipole amplitude are much less well defined (occurring $7.27 \pm 1.38$ and $8.33 \pm 1.08$ years after minimum for the two curves). The curves display broad, slightly slanting plateaus covering about 3 to 5 years (@Iijima+:plateau); the dipole amplitude at the time of solar minimum is still typically $84\pm 12\,$% (global dipole) and $90\pm 6\,$% (polar fields) of its maximal value, reached years earlier. A further well-known empirical result is that around solar minimum the poloidal field has a “topknot” structure [ (a field distribution that is much more concentrated toward the poles than a dipole configuration, [@sheeley1989implications]{}). Owing to the limited resolution and projection effects, the latitude dependence of this field can only be determined indirectly, exploiting the annual variation of the tilt of the solar axis relative to the line of sight and comparing the resulting modulation in the longitudinal field with fits of the form $B\sim\sin^n\lambda$. As discussed in detail by [@Petrie:LRSP]{}, various determinations of $n$ have yielded values in the range 8–9. As in our models the field profiles are not necessarily well fitted by $\sin^n\lambda$ we characterize the extent of the topknot with the latitude $\lambda_{1/2}$ where $B$ drops to half its value at the pole. The observational constraints indicating that $n>7$ imply $\lambda_{1/2}>65^\circ$. The upper limit may be less well defined due to the limited resolution; so here we adopt $\lambda_{1/2}<75^\circ$ — a rather generous value compared to the $69^\circ$ that corresponds to $n=10$. ]{} These empirical limits on polar field variation are summarized in Table \[table:obsconstraints\], For use in solar cycle prediction based on the polar precursor method we require that SFT models of a “typical” solar cycle must satisfy the listed 5 constraints. In view of the rather high uncertainty in the value of the mean related to the very low number of observed cycles, in our parameter study we will we extend the allowed domain to $\pm 2\sigma$. (Note that, as $\sigma$ is the [standard deviation of the data]{}, the uncertainty of the mean is only half that -so our study actually allows for a generous $\pm 4\sigma$ uncertainty in the value of the mean.) [ We reiterate that these values were determined for the last four solar cycles (21–24), Figure \[fig:polarfieldobs\], after smoothing the data with a 13-month sliding window, and the errors given in Table \[table:obsconstraints\] are standard deviations calculated for the four cycles 21–24. E.g., the reversal times of the WSO field amplitudes for the individual cycles, counted from the last minimum, are 4.50, 3.92, 4.08, and 4.83 years, while for the dipole moment values they were 3.42, 3.27, 3.33, and 3.75 years, respectively. For $\lambda_{1/2}$ the nominal values shown in the table correspond to a fiducial $\pm 2\sigma$ interval of $65^\circ$–$75^\circ$, adopted as discussed above. ]{} ![Meridional flow profiles used in the paper. []{data-label="fig:circul"}](circul){width="\hsize"} Model {#sect:model} ===== As our intention here is to model “typical” or “average” solar cycles, our source function will not consist of individual ARs but a smooth distribution representing the probability distribution of the emergence of preceding ($p$) and following ($f$) polarities on the solar surface. By its nature this source is then axially symmetric, so our whole SFT model will be reduced to one dimension: $$\begin{aligned} \label{eq33} {\frac{\partial B}{\partial t}} &=& \frac{1}{R\cos{\lambda}}{\frac{\partial }{\partial \lambda}}\left[Bu(\lambda)\cos{\lambda}\right] \nonumber \\ &&+\frac{\eta}{R^2\cos{\lambda}} {\frac{\partial }{\partial \lambda}}\left(\cos{\lambda}{\frac{\partial B}{\partial \lambda}}\right) -\frac{B}{\tau} + S(\lambda,t) \label{eq:transp}\end{aligned}$$ where $R$ is the solar radius and $S$ is the source representing flux emergence. Meridional flow --------------- For the meridional flow we consider three different profiles that have been used in SFT or dynamo models for cycle prediction (Fig. \[fig:circul\]). [*Flow 1:*]{} a simple sinusoidal profile $$\label{eq4} u_{c}=u_{0}\sin\lambda$$ This profile was used e.g. by [@dikpati2006predicting] and [@cameron2007solar], with $u_0=14.5/,$m$/$s, $\eta=300\,$km$^2/$s and $\tau=5.6$ yr. [*Flow 2:*]{} a sinusoidal profile with a dead zone around the poles, $$\label{eq5} u_{c} = \left\{ \begin{array}{ll} u_{0}\sin(\lambda/\lambda_{0}) & \mbox{if } |\lambda| < \lambda_{0} \\ 0 & \mbox{otherwise } \end{array} \right.$$ This profile was used by [@van1998magnetic] and in numerous papers by the MPS/Beijing group ([@Cameron+:tiltprecursor], [@jiang2014magnetic], [@Jiang+Cao] ), with $\lambda_0=75^\circ$, $u_0=11/,$m$/$s, $\eta=250\,$km$^2/$s and $\tau=\infty$. [*Flow 3:*]{} a profile peaking at high latitudes, $$\label{eq6} u_{c}(R,\theta) = \frac{u_{0}}{u_0^\ast} \mbox{erf}\,(V\cos{\lambda})\, \mbox{erf}\,(\sin{\lambda}) \qquad V=7$$ This profile was used by [@Lemerle2]. $u_0^\ast=0.82$ is a normalization factor defined so that $u_{0}$ gives the maximum meridional flow velocity, like in the other profiles. {width="\hsize"} {width="\hsize"} -------------------------------------------- ---------------------------------------------- {width="0.5\hsize"} {width="0.5\hsize"} -------------------------------------------- ---------------------------------------------- Source term {#sect:source} ----------- Our source term is a smooth distribution representing the probability distribution of the emergence of $p$- and $f$- polarities on the solar surface. Similar to the approach of [@dikpati2006predicting] and [@cameron2007solar], we represent the source by a pair of rings of opposite magnetic polarity: $$\begin{aligned} S(\lambda,t)&=& kA_m S_1(t) S_2\left[\lambda;\lambda_0(t)-\Delta\lambda(t),\delta\lambda\right] \nonumber \\ &&-kA_m S_1(t) S_2\left[\lambda;\lambda_0(t)+\Delta\lambda(t),\delta\lambda\right] \nonumber \\ && +kA_m S_1(t) S_2\left[\lambda;-\lambda_0(t)-\Delta\lambda(t),\delta\lambda\right] \nonumber \\ && -kA_m S_1(t) S_2\left[\lambda;-\lambda_0(t)+\Delta\lambda(t),\delta\lambda\right]\end{aligned}$$ where $k=\pm 1$ is a factor alternating between even and odd cycles. $A_m$ is an arbitrary amplitude depending on the flow profile (0.003, 0.015 and 0.0005 for our three profiles, respectively) used to ensure that the resulting polar field amplitude roughly agrees with observations (even though the amplitudes produced in this linear model are not used as a merit in the optimization). The latitudinal profile $S_2\left[\lambda;\lambda_0(t),\delta\lambda\right]$ is a Gaussian with a fixed full width at half-maximum of $2\delta\lambda=6^\circ$, migrating equatorward during the course of a cycle: $$S_2(\lambda;\lambda_0,\delta\lambda) = \exp\left[-(\lambda-\lambda_0)^2/2\delta\lambda^2\right]$$ The latitudinal separation of the rings is a consequence of Joy’s law: $$2\Delta\lambda=0.5\frac{\sin\lambda}{\sin 20^\circ}$$ while their trajectory during the course of a cycle is given by a quadratic fit derived by [@Jiang+:1700a] from many observed solar cycles: $$\lambda_0 [{}^\circ] = 26.4 - 34.2 (t/P) + 16.1(t/P)^2 \label{eq:quadratic}$$ where $P=11\,$year is the cycle period. Finally, the time profile of solar activity in a typical cycle was determined by [@hathaway1994shape] from the average of many cycles as $$S_1(t)= at^{3}_{c}/[\exp(t^{2}_{c}/b^{2})- c]$$ with $a = 0.00185$, $b = 48.7$, $c= 0.71$, where $t_c$ is the time since the last cycle minimum. In addition to the quadratic source term described above, an alternate form of the source term was also considered, to check the sensitivity of the results on this choice. The alternate form had a linear trajectory instead of equation (\[eq:quadratic\]) and a constant latitudinal separation of $1^\circ$ between the flux rings, while in other respects it was identical to the above. Model grid ---------- For each choice of meridional flow profile and source function we computed 1105 models with different parameter combinations. $u_0$ was allowed to vary between 5 and 20 m/s in steps of 2.5; $\eta$ varied from 50 to 750 km$^2/$s in steps of 50, while $\tau$ varied from 2 to 10 years in steps of 1 year, and two additional values ($\tau=20$ and $\tau=100$) were also considered, the latter value being effectively equivalent to a negligible decay term in equation (\[eq:transp\]). The transport equation was solved with a simple explicit code, making sure that the resolution ensures flux conservation to an acceptable degree not to appreciably influence the results. A latitudinal grid spacing of 0.5 degree and a timestep of 6 hours was found to suffice in most cases. Our code is made available on GitHub[^1] to ensure reproducibility of the results. Starting from an arbitrary (dipole) initial state, the calculations were run for 20 solar cycles, until very nearly steady oscillations set in. At that point we evaluated whether a given model complies with the imposed observational constraints listed in Table\[table:obsconstraints\]. Parameter maps {#sect:parammaps} -------------- Figures \[fig:goodplot\], \[fig:badplot\] and \[fig:goodbadbfly\] show examples of a “good” and a “bad” solution. [The left-hand panels of Figures \[fig:goodplot\] and \[fig:badplot\] present the time variation of the actual value of the polar magnetic field strength $B(\theta=0)$, the value measured by WSO \[eq. (\[eq:WSOB\])\] and the global dipole moment \[eq. (\[eq:dipmom\])\]. The right-hand panels in these figures, in turn, display the latitudinal profiles of $B$ at solar minimum and at the instants when the measures of polar field amplitude plotted in the left panels take their maxima. Figure \[fig:goodbadbfly\] displays the spatiotemporal variation of $B$ against time and latitude, i.e. the magnetic butterfly diagram characterizing these simulations. ]{} While at first sight similar, these plots indicate that in the second case the polar topknot is too broad and the polar reversal occurs too early in comparison with the observational constraints listed in section  \[sect:obsdata\], imposed with a generous limit of $2\sigma$ (or $4\sigma$ in terms of the uncertainty of the mean, cf. discussion in Section \[sect:obsdata\]). For the two cases plotted, $\lambda_{1/2}$ takes the value $71^\circ$ and $64^\circ$, respectively, while the WSO field is reversed $4.0$ and $3.4$ years after the minimum. Comparing this with the values in Table \[table:obsconstraints\], both values fall within the allowed $\pm 2\sigma$ interval in the first case, while both fall outside it in the second. The second set of parameter is therefore discarded as incompatible with the observational constraints on the polar field, while the first model is admissible by all five constraints. {width="\hsize"} {width="\hsize"} {width="\hsize"} Such comparisons can be performed for each parameter combination in our model grids, and as a result, admitted and discarded domains in the 3D ($u_0, \eta, \tau$) parameter space can be identified. In order to visualize these domains, we construct 2D maps of slices of the parameter space with fixed values of $\tau$. In these 2D maps the allowed domain is shown unmasked, in more vivid colors, while the excluded domain is shown through a semitransparent grey mask, in pale colors. One set of such maps for flow profile 2 at $\tau=100$ yr is shown in Figure \[fig:maps\_case2\_100\], where the colored panels represent the first three merit parameters listen in Table \[table:obsconstraints\] on the $u_0$–$\eta$ plane. The allowed domains are shown combined in the middle right panel, the darkest gray showing the domain allowed by all three criteria. [ The last two merit criteria related to the global dipole moment are not included in this plot partly for clarity and partly because these criteria were found to be most sensitive to $\tau$, so in slices taken at fixed values of $\tau$ either the allowed or the excluded domain tends to cover the whole plane. This is separately borne out in Figure \[fig:maps\_tauvar\] where slices of the 3D parameter space orthogonal to the slices in Figure \[fig:maps\_case2\_100\] are shown, mapping the distribution of the merit parameters associated with the dipole moment. It is apparent that these parameters are indeed mainly controlled by $\tau$: for shorter values of $\tau$ the global field decays faster, so it reverses earlier and declines more significantly from its maximum to the end of the cycle. ]{} Discussion {#sect:discussion} ========== One immediate conclusion from these studies follows directly from our above mentioned observation that the time variation of the dipole moment primarily depends on $\tau$. In fact we find that for higher values of $\tau$, i.e. [*without a significant decay term in the transport equation, the global dipole moment invariably reverses too late in the cycle.*]{} This is seen in Fig. \[fig:maps\_tauvar\]; the equivalent plots look similar also for other flow types and other values of $u_0$ or $\eta$. In contrast to this, we find that a range of shorter decay times exists where, with our realistic source function, all five merit criteria can be satisfied in some part of the parameter space. This range of $\tau$ lies somewhere between 5 and 10 years for all three flow types considered (though exact limits of the range depend on the flow profile). All this can be taken as evidence for the need of a decay term in the transport equation for the realistic modelling of the time variation of the global dipole moment in an average cycle. As in our models intercycle variations are not present, this line of evidence is independent from the drift in dipole moment that originally motivated the introduction of the decay term [ ([@schrijver2002missing]{} —cf. discussion in Section 1 above). ]{} {width="\hsize"} [For the allowed solution with $\tau$ in the range $5$–$10$ years, higher diffusivity values (500km$^2$/s and above) are generally favoured, i,e. the bulk of the allowed domain lies here. In the case of the flow with a polar dead zone (flow type 2, Fig. \[fig:maps\_case2\_7\]) the value of $u_0$ generally also needs to be high ($\ga 15\,$m/s) or the topknot becomes too broad; however, a narrow band of allowed solutions along a line of constant $\eta/u_0$ ratio also exists. ]{} From this, the parameter combination $u_0=11\,$m/s and $\eta=250\,$km$^2/$s used by [@Cameron+:tiltprecursor] or [@Jiang+Cao] stands out as admissible with a $\tau$ value of about 8 years, while for $\tau\ga 10$ it produces too late dipole reversals in the “average” cycle studied here. {width="\hsize"} {width="\hsize"} Turning our attention now to the pole-reaching flows (type 1 and 3), we find that the allowed domain that exists in the range $\tau\simeq 5$ to $9$ invariably lies in the top left part of our parameters plane, i.e. combinations with higher diffusivity and lower flow speed are preferred. This is mainly due to the topknot width constraint: it was already shown by [@Sheeley+:topknot] that the topknot width for pole-reaching flows scales with $\eta/u_0$, thus becoming too broad for higher values of this parameter. In particular, with the flow profile obtained in the 2$\times$2D dynamo model of [@Lemerle2] polar fields will only show fully solar-like behaviour for $\eta/u_0\ga 150$ (e.g. for $u_0\ga 5\,$m/s $\eta\ga 700\,$km$^2/$s) and $\tau\sim 5$–$9$ yr. In contrast, the values 17, 600 and 10, resulting from their globally optimized solution, yield a too narrow topknot and a delayed dipole reversal (Fig. \[fig:maps\_case3\_10\]). The reason why this parameter set was obtained by [@Lemerle2] as an optimalized solution clearly lies in their different choice of merit and their combined optimization of an SFT model and a coupled dynamo model, instead of using a source term modeled on observations. Overall, the admitted domain is found to be the most extensive for a simple sinusoidal flow profile with $\tau\simeq 7$ (Fig. \[fig:maps\_case1\_7\]): in this case, practically the whole upper left half of the $u_0$–$\eta$ plane is allowed. Taken by themselves, then, the polar field constraints considered in this paper would suggest the use of this simple flow model for cycle prediction purposes with a $\eta/u_0$ ratio exceeding about 50. As mentioned at the end of subsection \[sect:source\], for comparison we also considered a less realistic choice of the source function, with a linear decrease in AR latitudes during the cycle and no accompanying variation in the tilt. Results for this case are available from the GitHub repository. It was found that there exist no parameter combinations where all five merit criteria can be simultaneously satisfied in this case. This shows that satisfying our criteria is non-trivial, and the very existence of admissible solutions indirectly supports the correctness of our choice of the source function. Conclusion {#sect:concl} ========== In this paper we mapped the parameter space of surface flux transport models from the point of view of the spatiotemporal variation of the polar field resulting with a source term representing an average solar cycle, marking the allowed domain compatible with observational constraints. In Section \[sect:motivation\] we argued that other methods of constraining parameter space are less well suited for applications of SFT to solar cycle prediction. A key finding from this work was that without a significant [decay]{} term in the SFT equation (i.e., for $\tau >10$ yr) the global dipole moment reverses too late in the cycle for all flow profiles and parameters. This provides independent supporting evidence for the need of the inclusion of a decay term in SFT models even for identical cycles. The question may arise whether this conclusion is related to our use of a rather restricted range (about $\pm 4$ months) allowed for the dipolar moment reversal time, derived from the average of all available observational datasets. Indeed, considering individual datasets, the dipole reversal often deviates by several months (in rare cases by even a year) from the averaged set. However, our parameter maps for $\tau=100$ (e.g. Fig. \[fig:maps\_case2\_100\]) show that no reasonable extension of the allowed range can account for the delay in the dipole reversal time which in these cases well exceeds two years in all cases. We further note that anecdotal evidence exists in the literature for cases where the dipole reversal time of one individual cycle was apparently well reproduced in SFT models without a decay term [([@devore1985numerical]{}, [@wang1989evolution]{}, [@Jiang+Cao]{})]{}. While none of these studies was comprehensive enough to draw general conclusions beyond the particularities of a given cycle, the roots of this apparent puzzle clearly need to be understood. It is worth noting that recently [@Virtanen+:SFT] also confirmed the need for a decay term in order to attain polar reversal in Cycle 24. Turning now to models with a decay term, an allowed domain was found to exist for $\tau$ values in the 5–10 yr range for all flow profiles considered. Generally higher values of $\eta$ (500–800) are preferred, though some solutions with lower $\eta$ are still allowed. The admitted domain was found to be the most extensive for a simple sinusoidal flow profile with $\tau\simeq 7$ (Fig. \[fig:maps\_case1\_7\]): in this case, practically the whole upper left half of the $u_0$–$\eta$ plane is allowed. Taken by themselves, then, the polar field constraints considered in this paper would suggest the use of this simple flow model with the $\eta/u_0$ ratio exceeding about 50. [ It may be worth comparing these results with those of [@Whitbread+:SFT], who used the PIKAIA algorithm to optimize the SFT models for different regimes. The flow profile used in these models differed from our profiles and was relatively closest to our profile 2. The properties of the source also differed and the merit criterion was to optimize overall agreement with observed magnetograms, rather than with the polar field. Their results, summarized in their Table 1, indicate admissible ranges for $u_0$ and $\eta$ comparable to our results for flow type 2; however, their preferred ranges for $\tau$ are shorter than ours, in the range 1.5–6 years, except for one configuration of the source term where significantly higher values were admitted. It is to be noted that, as we discussed in Section 3 above, the neglect of 3D transport processes renders SFT modelling an essentially phenomenological approach to the description of magnetic flux transport where the effective flows, diffusivities or decay times yielding a good fit to observations do not necessarily correspond to their actual values on the solar surface. In view of this it may come as little surprise that optimization for different merit functions may lead to different results. The choice of parameters will simply need to be adapted to the objectives of the modelling. ]{} A plausible further extension of this work would admit intercycle variations in cycle amplitudes and periods, focusing on how SFT parameter choices influence the interplay of stochastic and nonlinear effects. This research was supported by the Hungarian National Research, Development and Innovation Fund (grant no. NKFI K-128384) and by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 739500. [^1]: <https://github.com/kpetrovay/Polar-SFT>

--- abstract: 'In the current paper we present some new data on the issue of quasi-normal modes (QNMs) of uniform, neutron and quark stars. These questions have already been addressed in the literature before, but we have found some interesting features that have not been discussed so far. We have increased the range of frequency values for the scalar and axial perturbations of such stars and made a comparison between such QNMs and those of the very well-known Schwarzschild black holes. Also addressed in this work was the interesting feature of competing modes, which appear not only for uniform stars, but for quark stars as well.' author: - 'E. Abdalla' - 'D. Giugno' title: 'Shedding Some New Lights upon the Stellar Quasi-Normal Modes' --- Introduction ============ Quasi normal modes (QNM’s) have been studied for quite a long time due to the possibility of gathering information from astrophysical objects in terms of their response to external perturbations, in the same sense as we may study a bell from its sound. For reviews and earlier notes see [@Chand-Ferr], [@Kokkotas], [@Price]. They are particularly useful to grasp also general properties of the metric under consideration - see [@WMA], [@WLA], [@Kar]. Stellar QNM’s have been under considerable scrutiny over the last decades, since they provide not only a test for Einstein’s theory of General Relativity (GR), but also a look into the stellar structure and, indirectly, into the very nature of stellar matter and its properties, such as its equation of state (EOS). Stars have an internal structure which must be accounted for when studying any kind of perturbation, be it a test scalar field or a gravitational perturbation. Among these, the axial perturbations are much easier to compute, since they do not mix with the fluid modes of the star [@Chand-FerrB]. Such axial perturbations (along with the scalar and electromagnetic) can be described by a wave equation of the form $$-\frac{\partial^2W}{\partial t^2}+\frac{\partial^2W}{\partial x^2}=VW,$$ where $W$ stands for the amplitude and $V$, for the perturbative potential. For details, see the Appendix. The polar perturbations mix with the fluid modes and require a completely different approach in their numerical analysis and evolution, and are not dealt with here. Uniform stars, that is, stars possessing a uniform density $\varepsilon=\varepsilon_{0}$ are idealized astrophysical objects, since they cannot exist in nature. Unrealistic as they may sound, though, such stars provide an interesting background in which some insight on the field dynamics in stellar geometries may be gained, since the physical quantities of relevance (mass, pressure and gravitational potentials of the metric) are very straightforward to evaluate. We begin with such stars, but we do not limit ourselves to them. We present some results for the neutron and quark stars and compare these results to those of the well-known Schwarzschild black holes. For the neutron stars, we have dealt with the simplest model available, that of Oppenheimer and Volkov [@OV-39], consisting of a pure Fermi gas of neutrons. For the quark stars, we have also used one of the simplest models available, the MIT Bag Model. In this paper we consider QNMs of uniform, neutron and quark stars and compare with the analogous results for black holes, in an attempt to describe properties inherent of astrophysical objects. In section II we present some results about uniform stars. Within that section, the question of secondary modes in such stars is discussed. In section III, we discuss neutron star QNMs and in section IV we do the same for quark stars. Comparative charts for all QNMs are available in section V and the remarks and conclusions are left for section VI. Concerning units, we have used the geometric system of units, for which $\hbar=c=G=1$. This means that the masses have dimension of length and are measured in metres. The conversion factor from metres to kilograms is $c^2/G$. Before proceeding, we just recall a definition which will be very useful throughout this paper, that of *compactness* of a star. The compactness $c$ for a spherically symmetric star is defined in the literature as $$c=\frac{r_{g}}{R},$$ in which $r_{g}=2M$ is the star’s gravitational radius and $R$ is its actual radius. QNMs of Uniform Stars - Some Results ==================================== We begin with a series of figures - from the data we have tabulated - on the frequencies of the QNMs for uniform stars, with some of the masses we have chosen for neutron and quark stars, namely $M=1048m$, $M=977m$, $M=665m$ and $M=330m$ (the first two were also used for neutron and quark stars), for the sake of comparison. One must bear in mind that such values are not special in any way, being just the results of star integrations for some particular choices of the central density $\varepsilon_{0}$, and even these latter choices are just choices - for more details on the matter, see section III. In what follows, $c$ stands for the compactness of the star, $\ell$ for the multipole index and $\omega_{R}$ and $\omega_{I}$ for the real and the imaginary part of the frequency, respectively. From the data we have compiled, we can make a few initial remarks. First, we have noticed that $\omega\propto\frac{1}{M}$, as in the pure Schwarzschild case. Such a scaling property for $\omega$ in uniform stars was also explored in this work, and more data are left for subsection B. Second, increases of $\omega_{R}$ with $\ell$ (as expected). Moreover, all axial frequencies have smaller real parts than their scalar counterparts, given some $\ell$ (exactly like Schwarzschild). But $-\omega_{I}$ is higher for axial perturbations than for scalar ones (in contrast to the Schwarzschild case). We shall illustrate our data with a set of graphics. We begin with the picture for the scalar $\ell=0$ case, which can be viewed in Fig. (\[L0b\]). In Fig. (\[L0b\]), the tail obeys a power-law, namely $t^{-4}$. A comparison between uniform star and Schwarzschild BH modes for the $\ell=0$ scalar field is provided in Fig. (\[CompBHUNI\]). Notice that the BH-background scalar field oscillates much less than in uniform stars, although the tail decays much in the same way, according to the same power law. QNM Overtones ------------- We have detected, throughout our investigation of compact uniform stars, the presence of secondary modes which decay faster than the dominant ones. Since we have seen a similar behavior in the Schwarzschild BH context (see note at the end of this subsection) and we have checked these Schwarzschild BH data and concluded that they correspond to the first overtones, we can speak of overtones of the fundamental modes in the stellar context, also. These overtones show themselves in the form of wiggles in the envelopes of the Misner curves characterising the QNMs, as if there were modes (actually, the overtones) competing with the dominant ones. This issue of competing modes deserves special attention, especially for - but not limiting to - high compactnesses ($c>0.76$), when these wiggles become clearer. Three examples of competing modes are provided in Figs. (\[competing1\]), (\[competing2\]) and (\[competing3\]). A comment is due on the method of extraction of such secondary modes: we have fitted a damped oscillating function to the original data, so that we could know the frequency $\omega$ of the dominant mode (the one with the weaker damping) - visible at the end of the time evolution in all figures - and subtracted the fitted function from the original data. The remaining mode is the competing mode we have just talked about. For some extra details on the method, see [@dgiugno-06]. Our fittings are not always very precise, so that we give no more than 3 significant figures for them, in the dominant mode. The remaining mode is not always very easy to fit, primarily due to these precision limitations. This remaining mode may look as if it were trembling, but in some cases we can still made a (quite poor) fitting to it. We expect it to have a stronger - usually much stronger - decay, which was indeed the case. In the few cases in which we have seen a somewhat clear secondary mode, we could notice also a slightly faster oscillation. For the modes seen on Fig. (\[competing1\]) we have, for instance, $M\omega=0.351-0.0204i$ for the dominant mode and $M\omega=0.47-0.087i$ for the secondary mode, whereas for those seen on Fig. (\[competing2\]) we have $M\omega=0.401-0.0120i$ for the dominant mode and $M\omega=0.49-0.042i$ for the competing mode, so that the latter has a much stronger decay. The number of significant figures for the latter was also reduced, since it was obtained after a three-figure fitting had been performed. More data are available in Table (\[TableCompeting\]), for $\frac{R}{M}=2.6$. These data simply confirm what we have said about the secondary modes in question: they oscillate somewhat faster (in contrast to their Schwarzschild counterparts) and decay much faster than the dominant ones (similarly to the Schwarzschild BH case). This behaviour is confirmed in Table (\[TableCompeting2\]), where the relation between the real and imaginary parts of $\omega$ for the dominant and the secondary modes is shown (for the scalar $\ell=1$ field and $M=977m$). The oscillations are about 30% faster for the secondary modes and this percentage seems to have a weak - if any - correlation to the compactness $c$, but the damping rates can be at least 4 times higher for the secondary modes, and sometimes up to 9 times, increasing sharply with $c$. For the Schwarzschild BH, at least for the $\ell=2$ and $\ell=3$ axial cases, the decay rate was around 3 times faster for the competing mode (the first overtone). As for the first overtones, we could see that $\omega_{R}\propto\frac{1}{M}$, but for $\omega_{I}$ the data seemed to fluctuate somewhat. This may be due to the fitting precision limitations to which we have referred earlier. See table (\[TableCompeting\]). $Mass(m)$ $field$ $\ell$ $M\omega^{DOM}$ $M\omega^{SEC}_{I}$ ----------- --------- -------- ----------------- --------------------- $1048$ $scal$ $1$ $0.329-0.0102i$ $0.42-0.056i$ $1048$ $axial$ $2$ $0.401-0.0121i$ $0.49-0.046i$ $977$ $scal$ $1$ $0.329-0.0102i$ $0.42-0.053i$ $977$ $axial$ $2$ $0.402-0.0120i$ $0.49-0.050i$ $665$ $scal$ $1$ $0.329-0.0102i$ $0.43-0.055i$ $665$ $axial$ $2$ $0.401-0.0120i$ $0.49-0.042i$ : Comparing dominant ($\omega^{DOM}$) and competing ($\omega^{SEC}$) mode frequencies for several masses of uniform stars with $R/M=2.6$ ($c=0.769$).[]{data-label="TableCompeting"} $c$ $M\omega^{DOM}_{R}$ $-M\omega^{DOM}_{I}$ $M\omega^{SEC}_{R}$ $-M\omega^{DOM}_{I}$ $\frac{\omega^{SEC}_{R}}{\omega^{DOM}_{R}}$ $\frac{\omega^{SEC}_{I}}{\omega^{DOM}_{I}}$ --------- --------------------- ---------------------- --------------------- ---------------------- --------------------------------------------- --------------------------------------------- $0.727$ $0.359$ $0.0262$ $0.48$ $0.11$ $1.3$ $4.2$ $0.741$ $0.351$ $0.0204$ $0.47$ $0.087$ $1.3$ $4.3$ $0.755$ $0.341$ $0.0150$ $0.45$ $0.073$ $1.3$ $4.9$ $0.769$ $0.329$ $0.0102$ $0.43$ $0.055$ $1.3$ $5.4$ $0.784$ $0.316$ $0.00608$ $0.40$ $0.042$ $1.3$ $6.9$ $0.800$ $0.298$ $0.00272$ $0.39$ $0.025$ $1.3$ $9.2$ : Comparing dominant and secondary modes for the $\ell=1$ scalar field, for a uniform star with mass $M=665m$.[]{data-label="TableCompeting2"} A brief remark on the Schwarzschild BH case: it is well-known [@Kokkotas] that in their context, secondary modes (or first overtones) may indeed appear. The same holds for higher overtones. We have searched for them in the same context in order to test the procedure we have adopted to extract secondary modes from uniform compact stars. The data we got for the axial $\ell=2$ axial field, for instance, were compared to those of [@Kokkotas] and we have found a good agreement between our results and theirs, indicating that our method, however simple-minded as it seems, may indeed yield interesting results. To be more precise, for the $n=0$ mode (the fundamental), they got $M\omega=0.37367-0.08896i$, the same as ours. For the first overtone ($n=1$) they had $M\omega=0.34671-0.27391i$ and we had $M\omega=0.352-0.272i$ for $M=1048m$. This agreement is not so good as that for the $n=0$ mode, but seems to be good enough for us, indicating that our numerical procedures are on the right track. No higher overtones ($n>1$) were detected in the present context. Again, see [@dgiugno-06] for closer details. From the last paragraph, we may conclude that the first overtones decay faster in both the Schwarzschild case and the uniform compact star case, though in the latter case the decay rate depends on the compactness, being higher in more compact stars and usually higher than in the Schwarzschild context. One important difference between first overtones in Schwarzschild BHs and uniform stars backgrounds concerns their oscillations, which are slightly slower than in the fundamental mode for the former and somewhat faster (though not much) in the latter. Scaling Properties and Other Comments ------------------------------------- We can also check the dependence of the modes - for a given compactness and perturbation - on the mass, as shown in Fig. (\[ChMass1\]). Besides fig. (\[ChMass1\]), one may find an interesting scaling property for uniform star QNMs. $Mass(m)$ $c$ $field$ $\ell$ $M\omega$ ----------- --------- ---------- -------- ------------------ $1048$ $0.769$ $scal$ $1$ $0.329-0.0102i$ $1048$ $0.769$ $axial$ $2$ $0.401-0.0121i$ $977$ $0.769$ $scal$ $1$ $0.329-0.0102i$ $977$ $0.769$ $axial$ $2$ $0.402-0.0120i$ $665$ $0.769$ $scal$ $1$ $0.329-0.0102i$ $665$ $0.769$ $axial$ $2$ $0.401-0.0120i$ $1048$ $0.740$ $axial$ $2$ $0.422-0.0233i$ $977$ $0.740$ $axial$ $2$ $0.421-0.0234i$ $665$ $0.740$ $axial$ $2$ $0.423-0.0233i$ $977$ $0.769$ $scalar$ $2$ $0.489-0.00440i$ $665$ $0.769$ $scalar$ $2$ $0.489-0.00448i$ : Search for scaling properties of the QNM frequencies - Uniform Stars.[]{data-label="MassScaling1"} The table (\[MassScaling1\]) shows that for a given field, compactness and $\ell$, the quantity $M\omega$ is practically constant. For instance, when $\ell=1$ for the scalar field and $c=0.769$ ($R/M=2.6$), one has $M\omega$ being $0.329-0.0102i$ for both $M=977m$ and $M=665m$, while for $c=0.740$ ($R/M=2.7$) one has $M\omega=0.351-0.0204i$ for $M=665m$, and almost the same ($M\omega=0.349-0.0205i$) for $M=977m$. This can be corroborated for a larger mass spectrum, different compactnesses, fields and $\ell$ values. Neutron Stars ============= We have worked with the simplest model available for neutron stars, that of a noninteracting Fermi gas of neutrons. For those, the maximum mass is $0.72M_{s}$ [@Haensel], and we have selected a few values of neutron star masses to work with. For a brief description of the EOS involved, see [@Weinberg]. Here we shall work on the QNMs and other features of the stellar perturbations. The masses we have selected for comparison with other kinds of stars were $M=1048m$, $M=977m$, $M=665m$ and $M=330m$. Such values are simply a matter of choice, having no special feature. They simply correspond to some particular choices of central density, $\varepsilon_{0}$, namely $4.0\cdot 10^{15}g/cm^3$, $1.5\cdot 10^{15}g/cm^3$, $3.0\cdot 10^{14}g/cm^3$ and $5.0\cdot 10^{13}g/cm^3$, respectively. These values, in turn, are also just a matter of choice. Further values could have been equally taken. Some of our results are available in Figs. (\[MQNnstar1\]), (\[MQNnstar4\]) and (\[MQNnstar5\]). It would be interesting to compare these results to those coming from the Schwarzschild black hole QNM analysis. Some of our data on neutron stars can be seen in tables (\[TabModesNstar2\]) and (\[TabModesNstar3\]). These tables also display $M\omega$, and upon examining them we verify that the simple mass-frequency scaling property seen in the Schwarzschild BH and in the uniform star contexts did not show up here. The product $M\omega$ decreases with decreasing $M$, both for its real part and for the negative of its imaginary part. $M(m)$ $\ell$ $\omega(\times 10^{-4})$ $M\omega$ ----------- -------- -------------------------- ---------------- $1048.25$ $2$ $3.78-2.86i$ $0.396-0.300i$ $1048.25$ $3$ $4.86-3.15i$ $0.509-0.330i$ $1048.25$ $4$ $5.91-3.58i$ $0.619-0.375i$ $1048.25$ $5$ $7.06-3.94i$ $0.740-0.413i$ $977.12$ $2$ $-$ $-$ $977.12$ $3$ $4.04-3.18i$ $0.395-0.311i$ $977.12$ $4$ $4.78-3.41i$ $0.467-0.333i$ $977.12$ $5$ $5.64-3.45i$ $0.551-0.337i$ $846.54$ $2$ $-$ $-$ $846.54$ $3$ $3.55-2.79i$ $0.300-0.236i$ $846.54$ $4$ $4.05-3.28i$ $0.343-0.277i$ $846.54$ $5$ $4.86-3.31i$ $0.411-0.280i$ : Frequency data on Neutron Stars, several masses, scalar field. Dashes indicate lack of reliable data. The $\ell=0,1$ modes showed very few oscillations before decaying to a power-law tail.[]{data-label="TabModesNstar2"} $Mass(m)$ $\ell$ $\omega^{surf}(\times 10^{-4})$ $M\omega$ ----------- -------- --------------------------------- ---------------- $1048.25$ $2$ $2.81-2.49i$ $0.294-0.261i$ $1048.25$ $3$ $4.23-2.76i$ $0.443-0.289i$ $1048.25$ $4$ $5.61-3.45i$ $0.588-0.362i$ $1048.25$ $5$ $6.65-3.44i$ $0.697-0.361i$ $977.12$ $2$ $-$ $-$ $977.12$ $3$ $-$ $-$ $977.12$ $4$ $4.42-3.25i$ $0.432-0.318i$ $977.12$ $5$ $5.70-3.84i$ $0.557-0.372i$ $846.54$ $2$ $-$ $-$ $846.54$ $3$ $-$ $-$ $846.54$ $4$ $3.65-2.96i$ $0.309-0.250i$ $846.54$ $5$ $4.42-3.04i$ $0.374-0.257i$ : Frequency data on neutron stars. Axial field, different masses.[]{data-label="TabModesNstar3"} The corresponding frequencies for the Schwarzschild BH are available in Table (\[kompmqn1\]). We have picked, for these, masses similar to those used for the neutron stars, and the data were taken from [@dgiugno-06]. $field$ $\ell$ $M\omega$ ---------- -------- ----------------------- $scalar$ $2$ $0.483644-0.0967590i$ $axial$ $2$ $0.37367-0.08896i$ $scalar$ $3$ $0.675367-0.0964997i$ $axial$ $3$ $0.599444-0.0927031i$ $scalar$ $4$ $0.867417-0.0963923i$ $axial$ $4$ $0.809180-0.0941643i$ $scalar$ $5$ $1.059614-0.0963337i$ $axial$ $5$ $1.012297-0.0948713i$ : [*QNM frequencies for the Schwarzschild BH, for the sake of comparison. We have chosen to show $M\omega$ due to the mass-frequency scaling property of such Schwarzschild BH modes.*]{}[]{data-label="kompmqn1"} The tables (\[TabModesNstar2\]) and (\[TabModesNstar3\]) show the stellar QNM frequencies obtained at the stellar surface. We must mention that the location of extraction - in the time domain - of the modes mattered in the final result, although it is not visible in tables (\[TabModesNstar2\]) and (\[TabModesNstar3\]), because we have taken average values, that is, the (arithmetic) average of several values measured at different time intervals. For example, for the case of a neutron star with $M=1048m$ and with an $\ell=4$ axial peturbation, we have gotten $\omega_{1}=5.55\cdot 10^{-4}-3.45\cdot 10^{-4}i$, $\omega_{2}=5.47\cdot 10^{-4}-3.32\cdot 10^{-4}i$ and $\omega_{3}=5.53\cdot 10^{-4}-3.39\cdot 10^{-4}i$ for the time intervals $t_{1}=60000-90000m$, $t_{2}=65000-85000m$ and $t_{3}=61000-91000m$, respectively, yielding an average value of $\omega=5.52\cdot 10^{-4}-3.39\cdot 10^{-4}i$. Similar variations were also seen for other $M,\ell$ values and fields. A similar phenomenon also happened for quark stars, although in a less pronounced way. No clear overtones were found in the neutron star context, while - in a few cases only - we could clearly find them in the quark star context (see next section). From tables (\[TabModesNstar2\]) and (\[TabModesNstar3\]), we can draw a few conclusions. First, the axial modes have oscillated slowlier than their scalar counterparts, exactly as in the Schwarzschild BH case. But what has caught our attention, in fact, was the behavior of the oscillation rate ($Re(\omega)$) of the modes as a function of the stellar mass: the smaller the latter, the smaller the former, contrary to what is observed in the Schwarzschild case. More data are needed to clarify the matter. Upon comparing the tables (\[TabModesNstar2\]) and (\[TabModesNstar3\]) to table (\[kompmqn1\]), we may say that neutron star QNMs (at least in this simplified model we have dealt with) oscillate more slowly and decay considerably faster than Schwarzschild’s, at least for the few cases which we were able to study. Quark Stars =========== In what follows, we deal with the issue of searching for and analysing the QNMs of very simple quark stars. The ones we have searched for obeyed a very simple EOS - the so-called MIT Bag Model - in which the pressure $p$ is given by $$p=\frac{1}{3}(\varepsilon-4B),$$ where $B$ stands for the bag constant. For more details on this model see [@Weber04], [@Glendenning] and references therein. For further models of quark stars, see [@Glendenning] and [@Su]. Upon dealing with the present class of stars, first of all, we have searched for values of $\varepsilon_{0}$ and $B$ which yielded masses very near those we had gotten for our neutron stars. For those, our limiting mass was around $0.72M_{s}$ and, according to [@Witten], there is an empirical expression for the maximum mass for a given $B$, namely, $$M_{max}=\frac{1.96M_{s}}{\sqrt{B/B_{c}}},$$ where $B_c = 60 MeV/fm^{3}$. In order to have $M_{max}= 0.72 M_s$, one must choose $B=445MeV/fm^3$. Once we have found that, we just determine - by trial and error - the central densities $\varepsilon_{0}$ which yield the masses we want. Once the stars were integrated, we had all the data on the perturbative potentials, as we did for the uniform and neutron stars, and the QNM-searching routine could be started to find the modes. For these stars, we have some results, which can be seen in Figs. (\[QStarGraph1\]) and (\[QStarGraph2\]). A comparison between quark and neutron star modes is provided in Fig. (\[CompNSQS1\]). The figures above show that quark star modes, for a given mass, field type and $\ell$, oscillate considerably faster than neutron star modes, besides being less damped. Recall that the most massive neutron stars in this very simple model have a maximum mass of $1048m\thickapprox 0.72M_{s}$, and that they experience an increase in their oscillation rates for higher masses - just the opposite of quark stars. Thus, for any mass below this limit, neutron star modes will oscillate less than quark star modes. $M(m)$ $\ell$ $\omega^{surf}(\times 10^{-4})$ $M\omega$ -------- -------- --------------------------------- ---------------- -- $1048$ $1$ $3.47-1.45i$ $0.364-0.152i$ $1048$ $2$ $5.26-1.63i$ $0.551-0.171i$ $1048$ $3$ $7.06-1.78i$ $0.740-0.187i$ $1048$ $4$ $8.85-1.92i$ $0.927-0.201i$ $1048$ $5$ $10.6-2.06i$ $1.11-0.216i$ $977$ $1$ $3.58-1.84i$ $0.350-0.180i$ $977$ $2$ $5.42-2.09i$ $0.530-0.204i$ $977$ $3$ $7.26-2.27i$ $0.709-0.220i$ $977$ $4$ $9.11-2.45i$ $0.890-0.239i$ $977$ $5$ $10.9-2.62i$ $1.06-0.254i$ $846$ $1$ $3.76-2.38i$ $0.318-0.201i$ $846$ $2$ $5.77-2.90i$ $0.488-0.245i$ $846$ $3$ $7.75-3.04i$ $0.656-0.257i$ $846$ $4$ $9.79-3.13i$ $0.828-0.265i$ $846$ $5$ $11.7-3.58i$ $0.990-0.303i$ : Frequency data for $B=445MeV/fm^3$ Quark Stars, several masses, scalar field. Dashes indicate the lack of reliable data.[]{data-label="QS20"} As for our current data on the axial modes, they seemed not to be of good quality and we have yet to find out why. One point should be stressed about the data in Table (\[QS20\]): as we had already commented on the neutron star context, depending on the region of the time domain used to make the fittings to find the QNM frequencies, the results differed a bit (not as much as for the neutron stars). We have decided to rely on the data obtained at the end of the time domain - because any overtone tends to show itself at the beginning of the wave profile - but before the onset of any instability (seen by the ’trembling’ data). After making such a fitting, we decided to subtract it from our original function (as we had done in [@dgiugno-06], in another context). To our surprise, we could find a secondary mode in most cases, though only in a few cases it was possible to make a minimally reliable fitting to it. See Fig. (\[QStarGraph3\]). An extra remark is due on the data in Table (\[QS20\]): we computed also $M\omega$, and we have not seen the same simple scaling property we had seen before for the Schwarzschild BH modes and for the uniform stars. The real part of $M\omega$ decreases with $M$, and the opposite holds for the imaginary part of $-M\omega$. In Fig. (\[QStarGraph3\]), for example, one has a secondary mode with $\omega=0.00105-0.000400i$, compared to $\omega=0.000706-0.000178i$ for the fundamental mode. For the scalar $\ell=5$ field, in Fig. (\[QStarGraph4\]), the figures are $\omega=0.00135-0.000364i$ and $0.00106-0.000206i$, respectively. That is, overtones were a recurrent theme in our quest for stellar modes. The oscillations become slower as the masses increase, and so does the damping rates, much as in the Schwarzschild BH case, and contrary to the neutron star case. Comparative Charts ================== In this section we compare all kinds of stars mentioned in this paper between themselves and to Schwarzschild BHs. $Mass$ $Star/BH$ $\ell$ $\omega_{R}$ $-\omega_{I}$ ----------- ----------- -------- -------------- --------------- $1048.25$ $NS$ $2$ $3.78E-4$ $2.86E-4$ $1048.25$ $NS$ $3$ $4.86E-4$ $3.15E-4$ $1048.25$ $NS$ $4$ $5.91E-4$ $3.58E-4$ $1048.25$ $NS$ $5$ $7.06E-4$ $3.94E-4$ $1048.25$ $QS$ $1$ $3.47E-4$ $1.45E-4$ $1048.25$ $QS$ $2$ $5.26E-4$ $1.63E-4$ $1048.29$ $QS$ $3$ $7.06E-4$ $1.78E-4$ $1048.29$ $QS$ $4$ $8.85E-4$ $1.92E-4$ $1048.25$ $QS$ $5$ $1.06E-3$ $2.06E-4$ $1048.25$ $SCH$ $3$ $6.4428E-4$ $9.206E-5$ $1048.25$ $SCH$ $4$ $8.2749E-4$ $9.196E-5$ $1048.25$ $SCH$ $5$ $1.0108E-3$ $9.190E-5$ $977.12$ $NS$ $1$ $-$ $-$ $977.12$ $NS$ $2$ $-$ $-$ $977.12$ $NS$ $3$ $4.04E-4$ $3.18E-4$ $977.12$ $NS$ $4$ $4.78E-4$ $3.41E-4$ $977.12$ $NS$ $5$ $5.64E-4$ $3.45E-4$ $977.12$ $QS$ $1$ $3.58E-4$ $1.84E-4$ $977.12$ $QS$ $2$ $5.42E-4$ $2.09E-4$ $977.12$ $QS$ $3$ $7.26E-4$ $2.27E-4$ $977.12$ $QS$ $4$ $9.11E-4$ $2.45E-4$ $977.12$ $QS$ $5$ $1.09E-3$ $2.62E-4$ $977.12$ $SCH$ $1$ $2.9980E-4$ $9.9953E-5$ $977.12$ $SCH$ $2$ $4.9497E-4$ $9.9025E-5$ $977.12$ $SCH$ $3$ $6.9118E-4$ $9.8759E-5$ $977.12$ $SCH$ $4$ $8.8773E-4$ $9.8649E-5$ $977.12$ $SCH$ $5$ $1.0844E-3$ $9.8589E-5$ : Comparing neutron stars (NS) to quark stars (QS) to Schwarzschild black holes (SCH), scalar field. Fields measured at the stellar surface, for stars.[]{data-label="CC1"} $846.54$ $NS$ $1$ $-$ $-$ ---------- ------- ----- ------------- ------------- $846.54$ $NS$ $2$ $-$ $-$ $846.54$ $NS$ $3$ $3.55E-4$ $2.79E-4$ $846.54$ $NS$ $4$ $4.05E-4$ $3.28E-4$ $846.54$ $NS$ $5$ $4.86E-4$ $3.31E-4$ $846.54$ $QS$ $1$ $3.76E-4$ $2.38E-4$ $846.54$ $QS$ $2$ $5.77E-4$ $2.90E-4$ $846.54$ $QS$ $3$ $7.75E-4$ $3.04E-4$ $846.54$ $QS$ $4$ $9.79E-4$ $3.13E-4$ $846.54$ $QS$ $5$ $1.17E-3$ $3.58E-4$ $846.54$ $SCH$ $1$ $3.4604E-4$ $1.1537E-4$ $846.54$ $SCH$ $2$ $5.7132E-4$ $1.1430E-4$ $846.54$ $SCH$ $3$ $7.9780E-4$ $1.1399E-4$ $846.54$ $SCH$ $4$ $1.0247E-3$ $1.1387E-4$ $846.54$ $SCH$ $5$ $1.2517E-3$ $1.1380E-4$ : Continuation of the previous table.[]{data-label="CC1b"} $Mass$ $Star/BH$ $\ell$ $\Omega_{R}$ $-\Omega_{I}$ ----------- ----------- -------- -------------- --------------- $1048.25$ $NS$ $2$ $2.81E-4$ $2.49E-4$ $1048.25$ $NS$ $3$ $4.23E-4$ $2.49E-4$ $1048.25$ $NS$ $4$ $5.61E-4$ $3.45E-4$ $1048.25$ $NS$ $5$ $6.65E-4$ $3.44E-4$ $1048.25$ $SCH$ $2$ $3.5647E-4$ $8.4865E-5$ $1048.25$ $SCH$ $3$ $5.7185E-4$ $8.8436E-5$ $1048.25$ $SCH$ $4$ $7.7193E-4$ $8.9829E-5$ $1048.25$ $SCH$ $5$ $9.6570E-4$ $9.0504E-5$ $977.12$ $NS$ $2$ $-$ $-$ $977.12$ $NS$ $3$ $3.25E-4$ $2.46E-4$ $977.12$ $NS$ $4$ $4.42E-4$ $3.25E-4$ $977.12$ $NS$ $5$ $5.70E-4$ $3.84E-4$ $977.12$ $SCH$ $2$ $3.8242E-4$ $9.1043E-5$ $977.12$ $SCH$ $3$ $6.1348E-4$ $9.4874E-5$ $977.12$ $SCH$ $4$ $8.2813E-4$ $9.6368E-5$ $977.12$ $SCH$ $5$ $1.0360E-3$ $9.7093E-4$ $846.54$ $NS$ $2$ $-$ $-$ $846.54$ $NS$ $3$ $-$ $-$ $846.54$ $NS$ $4$ $3.64E-4$ $2.96E-4$ $846.54$ $NS$ $5$ $4.42E-4$ $3.04E-4$ $846.54$ $SCH$ $2$ $4.4141E-4$ $1.0509E-4$ $846.54$ $SCH$ $3$ $7.0811E-4$ $1.0951E-4$ $846.54$ $SCH$ $4$ $9.5587E-4$ $1.1123E-4$ $846.54$ $SCH$ $5$ $1.1958E-3$ $1.1207E-4$ : Comparing neutron stars (NS) to Schwarzschild black holes (SCH), axial field. Fields measured at the stellar surface, for stars.[]{data-label="CC2"} We did not place $M\omega$ in tables (\[CC1\]), (\[CC1b\]) and (\[CC2\]) because the neutron and quark stars do not share the simple $\omega\propto\frac{1}{M}$ property with the Schwarzschild BHs, as seen in their respective sections. From the aforementioned tables we learn that the neutron stars modes are more strongly damped than its quark counterparts and these are, in turn, more damped than Schwarzschild modes. The Schwarzschild modes oscillate faster than the neutron star modes, but slightly less than the quark star modes. Final Remarks and Conclusions ============================= First of all, the oscillating frequencies of the QNMs ($\omega_{R}$) increases with increasing $\ell$, for a given mass and perturbation, as in the black-hole case, irrespective of the star type. The damping $-\omega_{I}$ of the modes, however, decreases with increasing $\ell$, and does so very markedly, for all the masses and fields under study for uniform stars, whereas in the Schwarzschild black-hole (SchBH) case $-\omega_{I}$ has a very slight increase instead for the axial field, while also decreasing (very slightly, though) for the scalar field. This is an interesting contrast. And uniform stars showed stronger dampings for axial fields than for scalar fields, also contrary to SchBH. The neutron stars (NS) and quark stars (QS) modes, in general, showed an increase in $-\omega_{I}$ with increased $\ell$ for scalar perturbations. For the axial modes, a similar trend could be detected for NS, and the only possible anomaly could be the $M=1048m$ case, when the $\ell=4$ mode had practically the same damping ratio as the $\ell=5$ mode, as can be seen at the end of the table (\[CC2\]). There is also a clear dependence of the modes on the uniform star mass mass, for a given compactness and perturbation, as in the Schwarzschild case: the more massive a uniform star is, the slower is the oscillation of the field and the weaker is its damping. The same holds for the overtones we have found for them. As for the compactness $c$ itself, given a fixed mass, an increase in $c$ made both $Re(\omega)$ and $-Im(\omega)$ decrease, that is, more compact stars had slower and (much) less damped oscillations. The case of competing modes (overtones) in stars needed our foremost attention, since it seems not to have been mentioned anywhere so far. We could detect such competing modes in very compact uniform stars, especially when $c\thickapprox 0.77$ onwards. And here comes another contrast: while in the Schwarzschild black hole context the overtones have oscillated more slowly than the fundamental mode (see [@Konoplya03], [@dgiugno-06]), for the uniform stars the opposite was true. A similar feature was detected for some QS modes in which the presence of overtones was detectable, although we could not find clear overtones for NS. In general, leaving aside the uniform stars (UniS) due to their dependency on $c$, given some perturbation type, $M$ and $\ell$, QS modes tend to oscillate a bit faster than SchBH modes for higher masses and slightly slower for smaller masses, and these, in turn, oscillate faster than NS modes. When it comes to damping, though, the Sch BH modes are the least damped of all, followed by the QS modes (at least twice as damped) and by NS modes (the most damped of all). That is, NS modes are the slowest in oscillation rate and the most damped. At least, in the mass range under scrutiny. Nevertheless, if UniS are considered, their modes have slightly higher $\omega_{R}$ than SchBHs, but their $-\omega_{I}$ can be much smaller than their Sch BH counterparts, so that these uniform stellar modes are, in fact, the LEAST damped of all modes, particularly when $c$ increases towards its limiting value of $8/9$. The Sch BH and the uniform stars have a simple scaling property for $\omega$, namely $M\omega=k$, where $k$ is a constant depending on the perturbation, on $\ell$ and, for the uniform stars, $c$. Not so for the NS and QS modes, where we have not found so simple a correlation between $M$ and $\omega$. For QS at least, a decreasing $M$ meant an increasing $\omega$ (especially for $\omega_{R}$). For NS, a curious feature emerged: the larger $M$ is, the higher $\omega$ is, especially $\omega_{R}$, in sharp contrast to what has been seen for SchBH, UniS and QS. At any rate, however, in the mass range we have worked with, even the most massive NS have presented modes which were slower and more damped than the most massive SchBH, UniS and QS modes. We are still in the search for the reason(s) for such a behaviour. Brief Theory of Uniform Stars ============================= Uniform stars are stars possessing a uniform density $\varepsilon_{0}$, and references abound in the literature, as in [@Weinberg]. For these stars, computations are very easy, as we shall see. The star mass function is simply $$m(r)=4\pi\frac{\varepsilon_{0}r^3}{3},$$ if $r\leq R$ and $m(r)=M$ if $r>R$. Hence, in terms of the radius $R$ and the mass $M$ of the star, one has $\varepsilon_{0}=\frac{3M}{4\pi R^3}$. The pressure is determined via the Oppenheimer-Volkov equation, $$\frac{dp}{dr}=-\frac{(p+\varepsilon)(m+4\pi r^3p)}{r^2(1-\frac{2m}{r})},$$ and is given by $$p(r)=\frac{3M}{4\pi R^3}\lbrack\frac{\sqrt{1-\frac{2M}{R}}- \sqrt{1-\frac{2Mr^2}{R^3}}}{\sqrt{1-\frac{2Mr^2}{R^3}}-3\sqrt{1-\frac{2M}{R}}}\rbrack,$$ which approaches zero smoothly as $r\to R$. The $g_{tt}$ term of the metric is given by $$g_{tt}=-\frac{1}{4}(3\sqrt{1-\frac{2M}{R}}-\sqrt{1-\frac{2Mr^2}{R^3}})^2,$$ which reduces to the Schwarzschild $g_{tt}$ for $r>R$. One point is of utmost importance in what follows: the expression for $p$ becomes singular when $M/R>4/9$. Since the star pressure cannot be infinite anywhere, one concludes that there is an upper limit to the degree of compactness ($c=2M/R$) for any star. In fact, since $M/R$ cannot exceed $4/9$, no star can have $c>8/9$, so there’s a gap in $c$ between Schwarzschild black holes and spherically symmetric stars of any kind. This limitation stems from General Relativity itself, not from any particular stellar model. From the existing literature [@Chand-Ferr],[@Chand-FerrB] we have the wave equation satisfied by the scalar and gravitational perturbations, namely $$\frac{\partial^2W}{\partial x^2}-\frac{\partial^2W}{\partial t^2}=VW,$$ where $W$ stands for the perturbation amplitude and $V$, the perturbative potential. The latter is written as $$\begin{aligned} V&=&\frac{1}{4}(3\sqrt{1-\frac{2M}{R}}-\sqrt{1- \frac{2Mr^2}{R^3}})^2[\frac{\ell(\ell+1)}{r^2}+\frac{2\sigma m(r)} {r^3}+4\pi(\varepsilon_{0}-p(r))]\nonumber\\ V&=&(1-\frac{2M}{r})[\frac{\ell(\ell+1)}{r^2}+\frac{2\sigma M}{r^3}],\end{aligned}$$ where $\sigma=1,0,-3$ depending on whether we are dealing with scalar, electromagnetic or axial perturbations. The first of these equations holds inside the star, and the latter holds outside. We can now illustrate some of the potentials, and find out how they change with the compactness. Such an illustration is available in Fig. (\[Vstar\]). The Numerical Method ==================== We have employed a direct numerical method consisting of a grid in the tortise coordinate $x$ and the time coordinate $t$. Since $x$ runs from some finite $x_{0}$ at $r=0$ to $x\to\infty$ when $r\to\infty$, we start by specifying the field $W$ (scalar or axial) and its time derivative at $t=0$ in the region of interest in $x$ (usually a Gaussian wave packet centered around some $x_{1}>x_{0}$). The time evolution of the field is given by $$\begin{aligned} \Psi(t_{0}+\delta t,x_{0})&=&-\Psi(t_{0}-\delta t,x_{0})+(2- \delta^2 xV(x_{0})-\frac{5\delta t^2}{2\delta x^2}) \Psi(t_{0},x_{0})+\nonumber \\ &+&\frac{4\delta t^2\lbrack\Psi(t_{0},x_{0}+\delta x)+ \Psi(t_{0},x_{0}-\delta x)\rbrack}{3\delta x^2}-\nonumber \\ &-&\frac{\delta t^2\lbrack\Psi(t_{0},x_{0}-2\delta x)+\Psi(t_{0},x_{0}+2\delta x)\rbrack}{12\delta x^2}, \label{xtgridf}\end{aligned}$$ in which $\delta x$ is the spacing in $x$ and $\delta t$ is the time step. Given the ratio $m=\frac{\delta t}{\delta x}$ (the so-called mesh ratio), one must have $m<1$ for the sake of convergence. We stop at some $t>t_{0}$ and analyse the data for all $x$ at that time. Reflection may occur at the borders of the grid. [99]{} Chandrasekhar, S. and Ferrari, V., [*Proc. R. Soc. Lond.*]{} **A434** (1991) 449-457. Chandrasekhar, S. and Ferrari, V., [*Proc. R. Soc. Lond.*]{}[**A 434**]{} (1991) 247-279. Kokkotas, K. D. and Schmidt, B. 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--- abstract: | In [@B02] we have showed that the Generalized Grothendieck’s Period Conjecture applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck’s Period Conjecture in the case of 1-motives whose underlying semi-abelian variety is a *non trivial extension* of a product of elliptic curves by a torus. This will imply the introduction of *elliptic integrals of the third kind* for the computation of the periods of $M$ and therefore the Generalized Grothendieck’s Period Conjecture applied to $M$ will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind. address: 'Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, Italy' author: - Cristiana Bertolin title: 'Third kind elliptic integrals and 1-motives' --- Introduction {#introduction .unnumbered} ============ Let ${\mathcal{E}}$ be an elliptic curve defined over ${\mathbb{C}}$ with Weierstrass coordinate functions $x$ and $y$. On ${\mathcal{E}}$ we have the differential of the first kind $\omega = \frac{dx}{y},$ which is holomorphic, the differential of the second kind $ \eta = -\frac{xdx}{y},$ which has a double pole with residue zero at each point of the lattice ${\mathrm{H}}_1({\mathcal{E}}({\mathbb{C}}),{\mathbb{Z}})$ and no other pole, and the differential of the third kind $$\xi_Q = \frac{1}{2} \frac{y-y(Q)}{x - x(Q)} \frac{dx}{y},$$ for any point $Q $ of $ {\mathcal{E}}({\mathbb{C}}), Q \not=0,$ whose residue divisor is $D=-(0)+(-Q).$ Let $\gamma_1, \gamma_2$ be two closed paths on ${\mathcal{E}}({\mathbb{C}})$ which build a basis for the lattice ${\mathrm{H}}_1({\mathcal{E}}({\mathbb{C}}),{\mathbb{Z}})$. In his Peccot lecture at the Collège de France in 1977, M. Waldschmidt observed that the periods of the Weierstrass $\wp$-function (\[eq:periods-wp\]) are the elliptic integrals of the first kind $ \int_{\gamma_i} \omega = \omega_i$ $(i=1,2)$, the quasi-periods of the Weierstrass $\zeta$-function (\[eq:periods-zeta\]) are the elliptic integrals of the second kind $ \int_{\gamma_i} \eta = \eta_i$ $(i=1,2)$, but *there is no function whose quasi-quasi-periods are elliptic integrals of the third kind*. J.-P. Serre answered this question furnishing the function $$f_q(z)= \frac{\sigma(z+q)}{\sigma(z) \sigma(q)} e^{-\zeta(q) z } \qquad \mathrm{with}\; q \in {\mathbb{C}}\setminus \Lambda$$ whose *quasi-quasi periods* (\[eq:periods-fq\]) are *the exponentials of the elliptic integrals of the third kind* $ \int_{\gamma_i} \xi_Q = \eta_i q - \omega_i \zeta(q)$ $(i=1,2),$ where $q$ is an elliptic logarithm of the point $Q$. Consider now an extension $G$ of ${\mathcal{E}}$ by ${\mathbb{G}}_m$ parameterized by the divisor $D=(-Q)-(0)$ of $\mathrm{Pic}^0({\mathcal{E}}) \cong {\mathcal{E}}^* = \underline{{\mathrm{Ext}}}^1({\mathcal{E}},{\mathbb{G}}_m)$. Since the three differentials $\{\omega, \eta,\xi_Q\}$ build a basis of the De Rham cohomology ${\mathrm{H}}^1_{{\mathrm{dR}}}(G)$ of the extension $G$, elliptic integrals of the third kind play a role in the Generalized Grothendieck’s Period Conjecture (\[eq:GCP\]). The aim of this paper is to understand this role applying the Generalized Grothendieck’s Period Conjecture to 1-motives whose underlying semi-abelian variety is a *non trivial extension* of a product of elliptic curves by a torus. At the end of this paper the reader can find - an appendix by M. Waldschmidt in which he quotes transcendence results concerning elliptic integrals of the third kind; - a letter of Y. André containing an overview of Grothendieck’s Period Conjecture and its generalization. A 1-motive $M=[u:X \rightarrow G]$ over a sub-field $K$ of ${\mathbb{C}}$ consists of a finitely generated free ${\mathbb{Z}}$-module $X$, an extension $G$ of an abelian variety by a torus, and a homomorphism $u:X \to G(K)$. Denote by $M_{\mathbb{C}}$ the 1-motive defined over ${\mathbb{C}}$ obtained from $M$ extending the scalars from $K$ to ${\mathbb{C}}$. In [@D75] Deligne associates to the 1-motive $M$ - its De Rham realization ${\mathrm{T}}_{{\mathrm{dR}}}(M)$: it is the finite dimensional $K$-vector space ${\mathrm{Lie}}(G^\natural)$, with $ M^\natural =[u:X \rightarrow G^\natural]$ the universal extension of $M$ by the vector group ${\mathrm{Hom}}({\mathrm{Ext}}^1(M,{\mathbb{G}}_a),{\mathbb{G}}_a)$, - its Hodge realization ${\mathrm{T}}_{{\mathbb{Q}}}(M_{\mathbb{C}})$: it is the finite dimensional ${\mathbb{Q}}$-vector space ${\mathrm{T}}_{{\mathbb{Z}}}(M_{\mathbb{C}}) \otimes_{\mathbb{Z}}{\mathbb{Q}}$, with ${\mathrm{T}}_{{\mathbb{Z}}}(M_{\mathbb{C}})$ the fibered product of ${\mathrm{Lie}}(G)$ and $X$ over $G$ via the exponential map $ \exp : {\mathrm{Lie}}(G) \to G$ and the homomorphism $u:X \to G.$ The ${\mathbb{Z}}$-module ${\mathrm{T}}_{{\mathbb{Z}}}(M_{\mathbb{C}})$ is in fact endowed with a structure of ${\mathbb{Z}}$-mixed Hodge structure, without torsion, of level $\leq 1$, and of type $\{(0,0),(0,-1),(-1,0), (-1,-1)\}.$ Since the Hodge realizations attached to 1-motives are mixed Hodge structures, 1-motives are mixed motives. In particular they are the mixed motives coming geometrically from varieties of dimension $\leq 1$. In [@D75 (10.1.8)], Deligne shows that the De Rham and the Hodge realizations of $M$ are isomorphic $$\label{eq:betaM} \beta_M: {\mathrm{T}}_{{\mathrm{dR}}}(M) \otimes_K {\mathbb{C}}\longrightarrow {\mathrm{T}}_{{\mathbb{Q}}}(M_{\mathbb{C}})\otimes_K {\mathbb{C}}.$$ The *periods of M* are the coefficients of the matrix which represents this isomorphism with respect to $K$-bases. By Nori’s and Ayoub’s works (see [@Ay14] and [@N00]), it is possible to endow the category of 1-motives with a tannakian structure with rational coefficients, and therefore to define the motivic Galois group $${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$$ of a 1-motive $M$ as the fundamental group of the tannakian sub-category $< M>^\otimes$ generated by $M$ (see [@D89 Def 6.1] or [@D90 Def 8.13]). Applying the Generalized Grothendieck’s Period Conjecture proposed by André (see conjecture (?!) of André’s letter) to 1-motives we get Let $M$ be a 1-motive defined over a sub-field $K$ of ${\mathbb{C}}$, then $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, K (\mathrm{periods}(M)) \geq \dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \label{eq:GCP}$$ where $K (\mathrm{periods}(M))$ is the field generated over $K$ by the periods of $M$. In [@B02] we showed that the conjecture (\[eq:GCP\]) applied to a 1-motive of type $$M=[ u:{\mathbb{Z}}^{r} \, \longrightarrow \,\Pi^n_{j=1} {{\mathcal{E}}}_j \times {{\mathbb{G}}}_m^s]$$ is equivalent to the elliptico-toric conjecture (see [@B02 1.1]) which involves elliptic integrals of the first and second kind and logarithms of complex numbers. Consider now the 1-motive $$\label{eq:M} M=[ u:{\mathbb{Z}}^{r} \, \longrightarrow \, G]$$ where $G$ is a *non trivial* extension of a product $\Pi^n_{j=1} {\mathcal{E}}_j $ of pairwise not isogenous elliptic curves by the torus ${\mathbb{G}}_m^s.$ In this paper we introduce *the 1-motivic elliptic conjecture* (§\[conjecture\]) which involves elliptic integrals of the first, second and third kind. Our main Theorem is that this 1-motivic elliptic conjecture is equivalent to the Generalized Grothendieck’s Period Conjecture applied to the 1-motive (\[eq:M\]) (Theorem \[thmMain\]). The presence of elliptic integrals of the third kind in the 1-motivic elliptic conjecture corresponds to the fact that the extension $G$ underlying $M$ is not trivial. If in the 1-motivic elliptic conjecture we assume that the points defining the extension $G$ are trivial, then this conjecture coincides with the elliptico-toric conjecture stated in[@B02 1.1] (see Remarks \[Rk1\]). Observe that the 1-motivic elliptic conjecture contains also the Schanuel conjecture (see Remarks \[Rk2\]). In Section \[EllipticIntegral\] we recall basic facts about differential forms on elliptic curves. In Section \[periods\] we study the short exact sequences which “dévissent” the Hodge and De Rham realizations of 1-motives and which are induced by the weight filtration of 1-motives. In Lemma \[lem:decomposition\] we prove that instead of working with the 1-motive (\[eq:M\]) we can work with a direct sum of 1-motives having $r=n=s=1$. Using Deligne’s construction of a 1-motive starting from an open singular curve, in [@Ber08 §2] D. Bertrand has computed the periods of the 1-motive (\[eq:M\]) with $r=n=s=1.$ Putting together Lemma \[lem:decomposition\] and Bertrand’s calculation of the periods in the case $r=n=s=1$, we compute explicitly the periods of the 1-motive (\[eq:M\]) (see Proposition \[proof-periods\]). In section \[motivicGaloisgroup\], which is the most technical one, we study the motivic Galois group of 1-motives. We will follow neither Nori and Ayoub’s theories nor Grothendieck’s theory involving mixed realizations, but we will work in a completely geometrical setting using *algebraic geometry on tannakian categories*. In Theorem \[eq:dimUR\] we compute explicitly the dimension of the unipotent radical of the motivic Galois group of an arbitrary 1-motive over $K$. Then, as a corollary, we calculate explicitly the dimension of the motivic Galois group of the 1-motive (\[eq:M\]) (see Corollary \[eq:dimGalMot\]). For this last result, we restrict to work with a 1-motive whose underlying extension $G$ involves a product of elliptic curves, because only in this case we know explicitly the dimension of the reductive part of the motivic Galois group (in general, the dimension of the motivic Galois group of an abelian variety is not known). In section \[conjecture\] we state the 1-motivic elliptic conjecture and we prove our main Theorem \[thmMain\]. In section \[lowDim\] we compute explicitly the Generalized Grothendieck’s Period Conjecture in the low dimensional case, that is assuming $r=n=s=1$ in (\[eq:M\]). In particular we investigate the cases where $\mathrm{End}({\mathcal{E}}) \otimes_{\mathbb{Z}}{\mathbb{Q}}$-linear dependence and torsion properties affect the dimension of the unipotent radical of ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$. Acknowledgements {#acknowledgements .unnumbered} ================ I want to express my gratitude to M. Waldschmidt for pointing out to me the study of third kind elliptic integrals and for his appendix. I am very grateful to Y. André for his letter and for the discussions we had about the motivic Galois group. I also thank D. Bertrand and P. Philippon for their comments on an earlier version of this paper. This paper was written during a 2 months stay at the IHES. The author thanks the Institute for the wonderful work conditions. Notation {#notation .unnumbered} ======== Let $K$ be a sub-field of ${\mathbb{C}}$ and denote by $\overline{K}$ its algebraic closure. A 1-motive $M=[u:X \rightarrow G]$ over $K$ consists of a group scheme $X$ which is locally for the étale topology a constant group scheme defined by a finitely generated free ${\mathbb{Z}}\,$-module, an extension $G$ of an abelian variety $A$ by a torus $T$, and a homomorphism $u:X \to G(K)$. In this paper we will consider above all 1-motives in which $X= {\mathbb{Z}}^r$ and $G$ is an extension of a finite product $\Pi^n_{j=1} {\mathcal{E}}_j $ of elliptic curves by the torus ${\mathbb{G}}_m^s$ (here $r,n$ and $s$ are integers bigger or equal to 0). There is a more symmetrical definition of 1-motives. In fact to have the 1-motive $M=[u:{\mathbb{Z}}^r \rightarrow G]$ is equivalent to have the 7-tuple $({\mathbb{Z}}^r,{\mathbb{Z}}^s, \Pi^n_{j=1} {\mathcal{E}}_j ,\Pi^n_{j=1} {\mathcal{E}}_j^*, v ,v^*,\psi)$ where - ${\mathbb{Z}}^s$ is the character group of the torus ${\mathbb{G}}_m^s$ underlying the 1-motive $M$. - $v:{\mathbb{Z}}^r \rightarrow \Pi^n_{j=1} {\mathcal{E}}_j$ and $v^*:{\mathbb{Z}}^s \rightarrow \Pi^n_{j=1} {\mathcal{E}}_j^*$ are two morphisms of $K$-group varieties (here ${\mathcal{E}}_j^* := \underline{{\mathrm{Ext}}}^1({\mathcal{E}}_j,{\mathbb{G}}_m)$ is the Cartier dual of the elliptic curve ${\mathcal{E}}_j$). To have the morphism $v$ is equivalent to have $r$ points $P_k=(P_{1k}, \ldots, P_{nk})$ of $ \Pi^n_{j=1} {\mathcal{E}}_j(K)$ with $k=1, \ldots, r$, whereas to have the morphism $v^*$ is equivalent to have $s$ points $Q_i=(Q_{1i}, \ldots, Q_{ni})$ of $ \Pi^n_{j=1} {\mathcal{E}}_j^*(K)$ with $i=1, \ldots, s.$ Via the isomorphism $\underline{{\mathrm{Ext}}}^1(\Pi^n_{j=1}{\mathcal{E}}_j,{\mathbb{G}}_m^s) \cong (\Pi_{j=1}^n {\mathcal{E}}_j^*)^s ,$ to have the $s$ points $Q_i=(Q_{1i}, \ldots, Q_{ni})$ is equivalent to have the extension $G$ of $\Pi^n_{j=1} {\mathcal{E}}_j$ by ${\mathbb{G}}_m^s$. - $\psi$ is a trivialization of the pull-back $(v,v^*)^*\mathcal{P}$ via $(v,v^*)$ of the Poincar' e biextension $\mathcal{P}$ of $(\Pi^n_{j=1} {\mathcal{E}}_j,\Pi^n_{j=1} {\mathcal{E}}_j^*)$ by ${\mathbb{G}}_m$. To have this trivialization $\psi$ is equivalent to have $r$ points $R_k \in G(K)$ with $k=1, \ldots, r$ such that the image of $R_k$ via the projection $G \to \Pi^n_{j=1} {\mathcal{E}}_j$ is $P_k=(P_{1k}, \ldots, P_{nk})$, and so finally to have the morphism $u:{\mathbb{Z}}^r \rightarrow G. $ The index $k$, $0 \leq k \leq r,$ is related to the lattice ${\mathbb{Z}}^r$, the index $j$, $0 \leq j \leq n,$ is related to the elliptic curves, and the index $i$, $0 \leq i \leq s,$ is related to the torus ${\mathbb{G}}_m^s$. For $j=1, \ldots, n$, we index with a $j$ all the data related to the elliptic curve ${\mathcal{E}}_j$: for example we denote by $\wp_j(z)$ the Weierstrass $\wp$-function of ${\mathcal{E}}_j$, by $\omega_{j1}, \omega_{j2}$ its periods, ... On any 1-motive $M=[u:X \rightarrow G] $ it is defined an increasing filtration ${\mathrm{W}}_{\bullet}$, called the *weight filtration* of $M$: ${\mathrm{W}}_{0}(M)=M, {\mathrm{W}}_{-1}(M)=[0 \to G], {\mathrm{W}}_{-2}(M)=[0 \to T].$ If we set ${{\mathrm{Gr}}}_{n}^{{\mathrm{W}}} := {\mathrm{W}}_{n} / {\mathrm{W}}_{n-1},$ we have ${{\mathrm{Gr}}}_{0}^{{\mathrm{W}}}(M)= [ X \to 0], {{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}(M)= [0 \to A]$ and $ {{\mathrm{Gr}}}_{-2}^{{\mathrm{W}}}(M)= [0 \to T].$ Two 1-motives $M_i=[u_i:X_i \rightarrow G_i]$ over $K$ (for $i=1,2$) are isogeneous is there exists a morphism of complexes $(f_X,f_G):M_1 \to M_2$ such that $f_X:X_1 \to X_2$ is injective with finite cokernel, and $f_G:G_1 \to G_2$ is surjective with finite kernel. Since [@D75 Thm (10.1.3)] is true modulo isogenies, two isogeneous 1-motives have the same periods. Moreover, two isogeneous 1-motives build the same tannakian category and so they have the same motivic Galois group. Hence in this paper *we can work modulo isogenies*. In particular the elliptic curves ${\mathcal{E}}_1, \dots, {\mathcal{E}}_n$ will be pairwise not isogenous. Elliptic integrals of third kind {#EllipticIntegral} ================================ Let ${\mathcal{E}}$ be an elliptic curve defined over ${\mathbb{C}}$ with Weierstrass coordinate functions $x$ and $y$. Set $\Lambda := {\mathrm{H}}_1({\mathcal{E}}({\mathbb{C}}),{\mathbb{Z}}). $ Let $\wp(z)$ be the Weierstrass $\wp$-function relative to the lattice $\Lambda$: it is a meromorphic function on ${\mathbb{C}}$ having a double pole with residue zero at each point of $\Lambda$ and no other poles. Consider the elliptic exponential $$\begin{aligned} \nonumber \exp_{{\mathcal{E}}}: {\mathbb{C}}& \longrightarrow {\mathcal{E}}({\mathbb{C}}) \subseteq {\mathbb{P}}^2({\mathbb{C}})\\ \nonumber z & \longmapsto \exp_{{\mathcal{E}}}(z)=[\wp(z),\wp(z)',1]\end{aligned}$$ whose kernel is the lattice $\Lambda.$ In particular the map $\exp_{{\mathcal{E}}}$ induces a complex analytic isomorphism between the quotient ${\mathbb{C}}/ \Lambda$ and the ${\mathbb{C}}$-valuated points of the elliptic curve ${\mathcal{E}}$. In this paper, we will use small letters for elliptic logarithms of points on elliptic curves which are written with capital letters, that is $\exp_{{\mathcal{E}}}(p)=P \in {\mathcal{E}}({\mathbb{C}})$ for any $p \in {\mathbb{C}}$. Let $ \sigma(z)$ be the Weierstrass $\sigma$-function relative to the lattice $\Lambda$: it is a holomorphic function on all of ${\mathbb{C}}$ and it has simple zeros at each point of $\Lambda$ and no other zeros. Finally let $\zeta (z)$ be the Weierstrass $\zeta$-function relative to the lattice $\Lambda$: it is a meromorphic function on ${\mathbb{C}}$ with simple poles at each point of $\Lambda$ and no other poles. We have the well-known equalities $$\frac{d}{dz} \log \sigma(z)= \zeta(z) \quad \mathrm{and} \quad \frac{d}{dz} \zeta(z)= -\wp(z).$$ Recall that a meromorphic differential 1-form is of the *first kind* if it is holomorphic everywhere, of the *second kind* if the residue at any pole vanishes, and of the *third kind* in general. On the elliptic curve ${\mathcal{E}}$ we have the following differential 1-forms: 1. the differential of the first kind $$\label{eq:diffFirstk} \omega = \frac{dx}{y},$$ which has neither zeros nor poles and which is invariant under translation. We have that $\exp_{{\mathcal{E}}}^{*}(\omega) = dz.$ 2. the differential of the second kind $$\label{eq:diffSecondk} \eta = -\frac{xdx}{y}.$$ In particular $\exp_{{\mathcal{E}}}^{*}(\eta) = -\wp(z) dz$ which has a double pole with residue zero at each point of $\Lambda$ and no other poles. 3. the differential of the third kind $$\label{eq:diffThirdk} \xi_Q = \frac{1}{2} \frac{y-y(Q)}{x - x(Q)} \frac{dx}{y}$$ for any point $Q $ of $ {\mathcal{E}}({\mathbb{C}}), Q \not=0.$ The residue divisor of $\xi_Q$ is $-(0)+(-Q).$ If we denote $q \in {\mathbb{C}}$ an elliptic logarithm of the point $Q$, that is $\exp_{{\mathcal{E}}}(q)=Q$, we have that $$\exp_{{\mathcal{E}}}^{*}(\xi_Q) = \frac{1}{2} \frac{\wp'(z)- \wp'(q)}{\wp(z) - \wp(q)} dz,$$ which has residue -1 at each point of $\Lambda$. The 1-dimensional ${\mathbb{C}}$-vector space of differentials of the first kind is ${\mathrm{H}}^0({\mathcal{E}}, \Omega^1_{\mathcal{E}}).$ The 1-dimensional ${\mathbb{C}}$-vector space of differentials of the second kind modulo holomorphic differentials and exact differentials is ${\mathrm{H}}^1({\mathcal{E}}, \mathcal{O}_{\mathcal{E}}).$ In particular the first De Rham cohomology group ${\mathrm{H}}^1_{\mathrm{dR}}({\mathcal{E}})$ of the elliptic curve ${\mathcal{E}}$ is the direct sum ${\mathrm{H}}^0({\mathcal{E}}, \Omega^1_{\mathcal{E}}) \oplus {\mathrm{H}}^1({\mathcal{E}}, \mathcal{O}_{\mathcal{E}})$ of these two spaces and it has dimension 2. The ${\mathbb{C}}$-vector space of differentials of the third kind is infinite dimensional. The inverse map of the complex analytic isomorphism ${\mathbb{C}}/ \Lambda \to {\mathcal{E}}({\mathbb{C}})$ induced by the elliptic exponential is given by the integration ${\mathcal{E}}({\mathbb{C}}) \to {\mathbb{C}}/ \Lambda, P \to \int^{P}_{O} \omega \quad \mathrm{mod} \Lambda$, where O is the neutral element for the group law of the elliptic curve. Let $\gamma_1, \gamma_2$ be two closed paths on ${\mathcal{E}}({\mathbb{C}})$ which build a basis of ${\mathrm{H}}_1({\mathcal{E}}_{\mathbb{C}},{\mathbb{Q}})$. Then *the elliptic integrals of the first kind* $ \int_{\gamma_i} \omega = \omega_i$ $(i=1,2)$ are *the periods of the Weierstrass $\wp$-function*: $$\label{eq:periods-wp} \wp(z+\omega_i)= \wp(z) \quad \quad \mathrm{for} \; i=1,2.$$ Moreover *the elliptic integrals of the second kind* $ \int_{\gamma_i} \eta = \eta_i$ $(i=1,2)$ are *the quasi-periods of the Weierstrass $\zeta$-function*: $$\label{eq:periods-zeta} \zeta(z+\omega_i)= \zeta(z) + \eta_i \quad \quad \mathrm{for} \; i=1,2.$$ Consider Serre’s function $$\label{eq:def-fq} f_q(z)= \frac{\sigma(z+q)}{\sigma(z) \sigma(q)} e^{-\zeta(q) z } \qquad \mathrm{with}\; q \in {\mathbb{C}}\setminus \Lambda$$ whose logarithmic differential is $$\label{eq:expEXiq} \frac{f_q'(z)}{f_q(z)} dz = \frac{1}{2} \frac{\wp'(z)- \wp'(q)}{\wp(z) - \wp(q)} dz =\exp_{{\mathcal{E}}}^{*}(\xi_Q)$$ (see [@W84] and [@Ber08 §2]). *The exponentials of the elliptic integrals of the third kind* $ \int_{\gamma_i} \xi_Q = \eta_i q - \omega_i \zeta(q)$ $(i=1,2)$ are *the quasi-quasi periods* of the function $f_q(z):$ $$\label{eq:periods-fq} f_q(z+ \omega_i)= f_q(z) e^{\eta_i q - \omega_i \zeta(q)} \quad \quad \mathrm{for} \; i=1,2.$$ As observed in [@W84], we have that $$\frac{f_q(z_1+ z_2)}{f_q(z_1)f_q( z_2)}= \frac{\sigma(q+z_1+z_2)\sigma(q) \sigma(z_1)\sigma(z_2)}{\sigma(q+z_1)\sigma(z_1+z_2)\sigma(q+z_2)}. \label{eq:fq-sigma}$$ Consider now an extension $G$ of our elliptic curve ${\mathcal{E}}$ by ${\mathbb{G}}_m, $ which is defined over ${\mathbb{C}}$. Via the isomorphism $\mathrm{Pic}^0({\mathcal{E}}) \cong {\mathcal{E}}^* = \underline{{\mathrm{Ext}}}^1({\mathcal{E}},{\mathbb{G}}_m)$, to have the extension $G$ is equivalent to have a divisor $D=(-Q)-(0)$ of $\mathrm{Pic}^0({\mathcal{E}}) $ or a point $-Q$ of $ {\mathcal{E}}^*({\mathbb{C}})$. In this paper we identify ${\mathcal{E}}$ with ${\mathcal{E}}^*$. A basis of the first De Rham cohomology group ${\mathrm{H}}^1_{\mathrm{dR}}(G)$ of the extension $G$ is given by $\{\omega, \eta, \xi_Q \}$. Consider the semi-abelian exponential $$\label{eq:semiablog} \exp_{G}: {\mathbb{C}}^2 \longrightarrow G({\mathbb{C}}) \subseteq {\mathbb{P}}^4({\mathbb{C}})$$ $$(w,z) \longmapsto \exp_{G}(w,z)=\sigma(z)^3 \Big[\wp(z),\wp(z)',1, e^{w} f_q(z), e^{w} f_q(z) \Big( \wp(z) + \frac{\wp'(z)- \wp'(q)}{\wp(z)- \wp(q)} \Big) \Big]$$ whose kernel is ${\mathrm{H}}_1(G({\mathbb{C}}),{\mathbb{Z}})$. A basis of the Hodge realization ${\mathrm{H}}_1(G({\mathbb{C}}),{\mathbb{Q}})$ of the extension $G$ is given by a closed path $\delta_{Q}$ around $Q$ on $G({\mathbb{C}})$ and two closed paths $\tilde{\gamma}_1, \tilde{\gamma}_2$ on $G({\mathbb{C}})$ which lift a basis $\{\gamma_1, \gamma_2\}$ of ${\mathrm{H}}_1({\mathcal{E}}_{\mathbb{C}},{\mathbb{Q}})$ via the surjection $ {\mathrm{H}}_1(G_{\mathbb{C}},{\mathbb{Q}}) \rightarrow {\mathrm{H}}_1({\mathcal{E}}_{\mathbb{C}},{\mathbb{Q}}).$ We have that $$\label{eq:expGXiq} \exp_{G}^{*}(\xi_Q) = dw + \frac{f_q'(z)}{f_q(z)} dz.$$ Periods of 1-motives involving elleptic curves {#periods} ============================================== Let $M=[u:X \to G]$ be a 1-motive over $K$ with $G$ an extension of an abelian variety $A$ by a torus $T$. As recalled in the introduction, to the 1-motive $M_{{\mathbb{C}}}$ obtained from $M$ extending the scalars from $K$ to ${\mathbb{C}}$, we can associate its Hodge realization ${{\mathrm{T}}}_{{\mathbb{Q}}}(M_{\mathbb{C}})= ({\mathrm{Lie}}(G_{\mathbb{C}})\times_G X) \otimes_{\mathbb{Z}}{\mathbb{Q}}$ which is endowed with the weight filtration (defined over the integers) ${{\mathrm{W}}}_{0}{{\mathrm{T}}}_{{\mathbb{Z}}}(M_{\mathbb{C}}) ={\mathrm{Lie}}(G_{\mathbb{C}})\times_G X, {{\mathrm{W}}}_{-1}{{\mathrm{T}}}_{{\mathbb{Z}}}(M_{\mathbb{C}}) = {{\mathrm{H}}}_1(G_{\mathbb{C}},{\mathbb{Z}}), {{\mathrm{W}}}_{-2}{{\mathrm{T}}}_{{\mathbb{Z}}}(M_{\mathbb{C}}) = {{\mathrm{H}}}_1( T_{\mathbb{C}},{\mathbb{Z}}).$ In particular we have that ${{\mathrm{Gr}}}_0^{{\mathrm{W}}}{{\mathrm{T}}}_{{\mathbb{Z}}}(M_{\mathbb{C}})\cong X, {{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}{{\mathrm{T}}}_{{\mathbb{Z}}}(M_{\mathbb{C}}) \cong {{\mathrm{H}}}_{1}(A_{\mathbb{C}},{\mathbb{Z}})$ and $ {{\mathrm{Gr}}}_{-2}^{{\mathrm{W}}}{{\mathrm{T}}}_{{\mathbb{Z}}}(M_{\mathbb{C}}) \cong {{\mathrm{H}}}_{1}(T_{\mathbb{C}},{\mathbb{Z}}).$ Moreover to $M$ we can associate its De Rham realization ${{\mathrm{T}}}_{{\mathrm{dR}}}(M) = {\mathrm{Lie}}(G^\natural)$, where $M^\natural=[X \rightarrow G^\natural]$ is the universal vectorial extension of $M$, which is endowed with the Hodge filtration ${{\mathrm{F}}}^0{{\mathrm{T}}}_{{\mathrm{dR}}}(M)= \ker \big( {\mathrm{Lie}}( G^\natural) \rightarrow {\mathrm{Lie}}( G) \big).$ The weight filtration induces for the Hodge realization the short exact sequence $$\label{eq:Hodge} 0 \longrightarrow {\mathrm{H}}_1(G_{\mathbb{C}},{\mathbb{Z}}) \longrightarrow {\mathrm{T}}_{{\mathbb{Z}}} ( M_{\mathbb{C}}) \longrightarrow {\mathrm{T}}_{{\mathbb{Z}}} (X) \longrightarrow 0$$ which is not split in general. On the other hand, for the De Rham realization we have that The short exact sequence, induced by the weight filtration, $$\label{eq:DRham0} 0 \longrightarrow {{\mathrm{T}}}_{{\mathrm{dR}}}(G) \longrightarrow {{\mathrm{T}}}_{{\mathrm{dR}}}(M) \longrightarrow {{\mathrm{T}}}_{{\mathrm{dR}}}(X) \longrightarrow 0$$ is canonically split. Consider the short exact sequence $0 \to G \to M \to X[1] \to 0$. Applying ${\mathrm{Hom}}(-,{\mathbb{G}}_a)$ we get the short exact sequence of finitely dimensional $K$-vector spaces $$0 \longrightarrow {\mathrm{Hom}}(X,{\mathbb{G}}_a) \longrightarrow {\mathrm{Ext}}^1(M,{\mathbb{G}}_a) \to{\mathrm{Ext}}^1(G,{\mathbb{G}}_a) \longrightarrow 0$$ Taking the dual we obtain the short exact sequence $$0 \longrightarrow {\mathrm{Hom}}({\mathrm{Ext}}^1(G,{\mathbb{G}}_a),{\mathbb{G}}_a) \longrightarrow {\mathrm{Hom}}({\mathrm{Ext}}^1(M,{\mathbb{G}}_a),{\mathbb{G}}_a) \longrightarrow X \to 0$$ which is split since ${\mathrm{Ext}}^1(X, {\mathbb{G}}_a)=0$. Now consider the composite of the section $X \to {\mathrm{Hom}}({\mathrm{Ext}}^1(M,{\mathbb{G}}_a),{\mathbb{G}}_a)$ with the inclusion ${\mathrm{Hom}}({\mathrm{Ext}}^1(M,{\mathbb{G}}_a),{\mathbb{G}}_a) \to G^\natural$. Recalling that ${{\mathrm{F}}}^0{{\mathrm{T}}}_{{\mathrm{dR}}}(M) \cong {\mathrm{Hom}}({\mathrm{Ext}}^1(M,{\mathbb{G}}_a),{\mathbb{G}}_a)$, if we take Lie algebras we get the arrow ${\mathrm{T}}_{{\mathrm{dR}}} (X) = X \otimes K \to {{\mathrm{F}}}^0{{\mathrm{T}}}_{{\mathrm{dR}}}(M) \to {{\mathrm{T}}}_{{\mathrm{dR}}}(M) = {\mathrm{Lie}}(G^\natural)$ which is a section of the exact sequence (\[eq:DRham0\]). Denote by ${\mathrm{H}}_{{\mathrm{dR}}}(M) $ the dual $K$-vector space of ${{\mathrm{T}}}_{{\mathrm{dR}}}(M)$. By the above Lemma we have that $$\label{eq:DRham} {\mathrm{H}}_{{\mathrm{dR}}}(M) = {\mathrm{H}}_{{\mathrm{dR}}}^1(G) \oplus {\mathrm{H}}_{{\mathrm{dR}}}^1(X) .$$ Consider now a 1-motive $M=[u:{\mathbb{Z}}^r \rightarrow G]$ defined over $K$, where $G$ is an extension of a finite product $\Pi^n_{j=1} {\mathcal{E}}_j $ of elliptic curves by the torus ${\mathbb{G}}_m^s$. Let $\{ z_k \}_{k=1, \dots, r}$ be a basis of ${\mathbb{Z}}^r$ and let $\{ t_i \}_{i=1, \dots, s}$ be a basis of the character group ${\mathbb{Z}}^s$ of ${\mathbb{G}}_m^s$. For the moment, in order to simplify notation, denote by $A$ the product of elliptic curves $\Pi^n_{j=1} {\mathcal{E}}_j$. Denote by $G_i$ the push-out of G by $t_i: {\mathbb{G}}_m^s \to {\mathbb{G}}_m$, which is the extension of $A$ by ${\mathbb{G}}_m$ parameterized by the point $v^*(t_i)=Q_i=(Q_{1i}, \dots, Q_{ni})$, and by $R_{ik}$ the $K$-rational point of $G_i$ above $v(z_k)=P_k=(P_{1k}, \dots, P_{nk})$. Consider the 1-motive defined over $K$ $$M_{ik}= [u_{ik}:z_k {\mathbb{Z}}\rightarrow G_i]$$ with $u_{ik}(z_k)= R_{ik} $ for $i=1, \dots, s$ and $k=1, \dots, r$. In [@B02-2 Thm 1.7] we have proved geometrically that the 1-motives $M=[u:{\mathbb{Z}}^r \rightarrow G]$ and $\oplus_{i=1}^s \oplus_{k=1}^r M_{ik}$ generate the same tannakian category. Via the isomorphism $\underline{{\mathrm{Ext}}}^1(\Pi^n_{j=1}{\mathcal{E}}_j,{\mathbb{G}}_m) \cong \Pi_{j=1}^n \underline{{\mathrm{Ext}}}^1({\mathcal{E}}_j,{\mathbb{G}}_m) ,$ the extension $G_i$ of $A$ by ${\mathbb{G}}_m$ parametrized by the point $v^*(t_i)=Q_i=(Q_{1i}, \dots, Q_{ni})$ corresponds to the product of extensions $G_{1i} \times G_{2i} \times \dots \times G_{ni}$ where $ G_{ji}$ is an extension of ${\mathcal{E}}_j$ by ${\mathbb{G}}_m$ parametrized by the point $Q_{ji}$, and the $K$-rational point $R_{ik}$ of $G_i$ living above $P_k=(P_{1k}, \dots, P_{nk})$ corresponds to the $K$-rational points $(R_{1ik}, \dots, R_{nik})$ with $R_{jik} \in G_{ji}(K) $ living above $P_{jk} \in {\mathcal{E}}_j (K).$ Consider the 1-motive defined over $K$ $$\label{eq:jik} M_{jik}= [u_{jik}:z_k {\mathbb{Z}}\rightarrow G_{ji}]$$ with $u_{jik}(z_k)= R_{jik} $ for $i=1, \dots, s$, $k=1, \dots, r$ and $j=1, \dots, n.$ Let $(l_{jik},p_{jk}) \in {\mathbb{C}}^2$ be a semi-abelian logarithm (\[eq:semiablog\]) of $R_{jik},$ that is $$\label{eq:l} \exp_{G_{ji}} (l_{jik},p_{jk}) = R_{jik}.$$ \[lem:decomposition\] The 1-motives $M$ and $\oplus_{i=1}^s \oplus_{k=1}^r \oplus_{j=1}^n M_{jik}$ generate the same tannakian category. As in [@B02-2 Thm 1.7] we will work geometrically and because of loc. cit. it is enough to show that the 1-motives $\oplus_{i=1}^s \oplus_{k=1}^r M_{ik}$ and $\oplus_{i=1}^s \oplus_{k=1}^r \oplus_{j=1}^n M_{jik}$ generate the same tannakian category. Clearly $$\oplus_{j=1}^n \Big( \oplus_{i=1}^s \oplus_{k=1}^r M_{ik} \big/ [0 \to \Pi_{1 \leqslant l \leqslant n \atop l \not= j} G_{li} ] \Big) = \oplus_{i=1}^s \oplus_{k=1}^r \oplus_{j=1}^n M_{jik}$$ and so $< \oplus_{i=1}^s \oplus_{k=1}^r \oplus_{j=1}^n M_{jik}>^\otimes \; \; \subset \; \; < \oplus_{i=1}^s \oplus_{k=1}^r M_{ik}>^\otimes.$ On the other hand, if $\mathrm{d}_{\mathbb{Z}}: {\mathbb{Z}}\to {\mathbb{Z}}^n$ is the diagonal morphism, for fixed $i$ and $k$ we have that $$\oplus_{j=1}^n M_{jik} \big/ [ {\mathbb{Z}}^n / \mathrm{d}_{\mathbb{Z}}({\mathbb{Z}}) \to 0] = [\small{\Pi}_{j} u_{jik} : \mathrm{d}_{\mathbb{Z}}({\mathbb{Z}}) \longrightarrow G_{1i} \times G_{2i} \times \dots \times G_{ni}] = [u_{ik} :{\mathbb{Z}}\longrightarrow G_i] =M_{ik}$$ and so $$\oplus_{i=1}^s \oplus_{k=1}^r \Big( \oplus_{j=1}^n M_{jik} \big/ [ {\mathbb{Z}}^n / \mathrm{d}_{\mathbb{Z}}({\mathbb{Z}}) \to 0] \Big) = \oplus_{i=1}^s \oplus_{k=1}^r M_{ik}$$ that is $ < \oplus_{i=1}^s \oplus_{k=1}^r M_{ik}>^\otimes \; \; \subset \; \; < \oplus_{i=1}^s \oplus_{k=1}^r \oplus_{j=1}^n M_{jik}>^\otimes .$ The matrix which represents the isomorphism (\[eq:betaM\]) for the 1-motive $M=[u:{{\mathbb{Z}}}^r \to G]$, where $G$ is an extension of $\Pi^n_{j=1} {\mathcal{E}}_j $ by ${\mathbb{G}}_m^s$, is a huge matrix difficult to write down. The above Lemma implies that, instead of studying this huge matrix, it is enough to study the $rsn$ matrices which represent the isomorphism (\[eq:betaM\]) for the $rsn$ 1-motives $M_{jik}= [u_{jik}:z_k {\mathbb{Z}}\rightarrow G_{ji}].$ Following [@Ber08 §2], now we compute explicitly the periods of the 1-motive $M=[u:{\mathbb{Z}}\to G]$, where $G$ is an extension of one elliptic curve $ {\mathcal{E}}$ by the torus ${\mathbb{G}}_m.$ We need Deligne’s construction of $M$ starting from an open singular curve (see [@D75 (10.3.1)-(10.3.2)-(10.3.3]) that we recall briefly. Via the isomorphism $\mathrm{Pic}^0({\mathcal{E}}) \cong {\mathcal{E}}^* = \underline{{\mathrm{Ext}}}^1({\mathcal{E}},{\mathbb{G}}_m)$, to have the extension $G$ of ${\mathcal{E}}$ by ${\mathbb{G}}_m$ underlying the 1-motive $M$ is equivalent to have the divisor $D=(-Q)-(0)$ of $\mathrm{Pic}^0({\mathcal{E}})$ or the point $-Q$ of $ \cong {\mathcal{E}}^* $. We assume $Q$ to be a non torsion point. According to [@M74 page 227], to have the point $u(1)=R \in G(K)$ is equivalent to have a couple $$(P,g_R) \in {\mathcal{E}}(K) \times K({\mathcal{E}})^*$$ where $\pi(R)=P \in {\mathcal{E}}(K)$ (here $\pi: G \to {\mathcal{E}}$ is the surjective morphism of group varieties underlying the extension $G$), and where $g_R: {\mathcal{E}}\to {\mathbb{G}}_m, x \mapsto R+ \rho(x) -\rho(x+P)$ (here $\rho: {\mathcal{E}}\to G$ is a section of $\pi$), is a rational function on ${\mathcal{E}}$ whose divisor is $T^{*}_{P}D-D=(-Q+P)-(P)-(-Q)+(0)$ (here $T_P: {\mathcal{E}}\to {\mathcal{E}}$ is the translation by the point $P$). We assume also $R$ to be a non torsion point. Now pinch the elliptic curve ${\mathcal{E}}$ at the two points $-Q$ and $O$ and puncture it at two $K$-rational points $P_2$ and $P_1$ whose difference (according to the group law of ${\mathcal{E}}$) is $P$, that is $P=P_2-P_1.$ The motivic ${\mathrm{H}}^1$ of the open singular curve obtained in this way from ${\mathcal{E}}$ is the 1-motive $M=[u:{\mathbb{Z}}\rightarrow G]$, with $u(1)=R$. We will apply Deligne’s construction to each 1-motive $M_{jik}= [u_{jik}:z_k {\mathbb{Z}}\rightarrow G_{ji}]$ with $u_{jik}(z_k)= R_{jik} .$ \[proof-periods\] Choose the following basis of the ${\mathbb{Q}}$-vector space ${\mathrm{T}}_{{\mathbb{Q}}}(M_{jik \; {\mathbb{C}}}):$ - two closed paths $\tilde{\gamma}_{j1}, \tilde{\gamma}_{j2}$ on $G_{ji}({\mathbb{C}})$ which lift the basis $\{\gamma_{j1}, \gamma_{j2}\}$ of ${\mathrm{H}}_1({\mathcal{E}}_{j \;{\mathbb{C}}},{\mathbb{Q}})$ via the surjection $ {\mathrm{H}}_1(G_{ji \; {\mathbb{C}}},{\mathbb{Q}}) \rightarrow {\mathrm{H}}_1({\mathcal{E}}_{j \;{\mathbb{C}}},{\mathbb{Q}})$; - a closed path $\delta_{Q_{ji} }$ around $-Q_{ji}$ on $G_{ji}({\mathbb{C}})$ (here we identify $G_{ji}$ with the pinched elliptic curve ${\mathcal{E}}_j$); and - a closed path $\beta_{R_{jik}}$, which lifts the basis $\{z_k\}$ of ${\mathrm{T}}_{{\mathbb{Q}}}(z_k {\mathbb{Z}})$ via the surjection $ {\mathrm{T}}_{{\mathbb{Q}}} ( M_{jik \; {\mathbb{C}}}) \rightarrow {\mathrm{T}}_{{\mathbb{Q}}} (z_k {\mathbb{Z}}) $, and whose restriction to ${\mathrm{H}}_1(G_{ji \; {\mathbb{C}}},{\mathbb{Q}})$ is a closed path $\beta_{R_{jik}|G_{ji}}$ on $G_{ji}({\mathbb{C}})$ having the following properties: $\beta_{R_{jik}|G_{ji}}$ lifts a path $\beta_{P^1_{jk}P^2_{jk}}$ on ${\mathcal{E}}_{j }({\mathbb{C}}) $ from $P^1_{jk}$ to $P^2_{jk}$ (with $P^2_{jk}-P^1_{jk}=P_{jk}$) via the surjection $ {\mathrm{H}}_1(G_{ji \; {\mathbb{C}}},{\mathbb{Q}}) \rightarrow {\mathrm{H}}_1({\mathcal{E}}_{j \;{\mathbb{C}}},{\mathbb{Q}})$, and its restriction to ${\mathrm{H}}_1({\mathbb{G}}_m,{\mathbb{Q}})$ is a path $\beta_{jik}$ on ${\mathbb{G}}_m({\mathbb{C}})={\mathbb{C}}^* $ from $1$ to $l_{jik} (\ref{eq:l});$ and the following basis of the $K$-vector space ${\mathrm{H}}_{{\mathrm{dR}}}(M_{jik}):$ - the differentials of the first kind $\omega_j=\frac{dx_j}{y_j}$ (\[eq:diffFirstk\]) and of the second kind $\eta_j=-\frac{x_jdx_j}{y_j}$ (\[eq:diffSecondk\]) of ${\mathcal{E}}_j$; - the differential of the third kind $\xi_{Q_{ji} }= \frac{1}{2} \frac{y_j-y_j(Q_{ji})}{x_j - x_j(Q_{ji})} \frac{dx_i}{y_j}$ (\[eq:diffThirdk\]) of ${\mathcal{E}}_j$, whose residue divisor is $D=(-Q_{ji})-(0)$ and which lifts the basis $\{\frac{dt_i}{t_i}\}$ of ${\mathrm{H}}_{{\mathrm{dR}}}^1({\mathbb{G}}_m)$ via the surjection ${\mathrm{H}}_{{\mathrm{dR}}}^1(G_{ji}) \rightarrow {\mathrm{H}}_{{\mathrm{dR}}}^1({\mathbb{G}}_m)$; - the differential $df_j$ of a rational function $f_j$ on ${\mathcal{E}}_j$ such that $f_j(P^2_{jk})$ differs from $f_j(P^1_{jk})$ by 1. These periods of the 1-motive $M=[u:{{\mathbb{Z}}}^r \to G]$, where $G$ is an extension of $\Pi^n_{j=1} {\mathcal{E}}_j $ by ${\mathbb{G}}_m^s$, are then $$1, \omega_{j1},\omega_{j2},\eta_{j1}, \eta_{j2}, p_{jk},\zeta_j(p_{jk}), \eta_{j1} q_{ji} - \omega_{j1} \zeta_j(q_{ji}),\eta_{j2} q_{ji} - \omega_{j2} \zeta_j(q_{ji}) , \log f_{q_{ji}}(p_{jk}) + l_{jik}, 2i \pi$$ with $e^{l_{jik}} \in K^*,$ for $j=1, \ldots, n, k=1, \ldots, r$ and $i=1, \ldots,s.$ By Lemma \[lem:decomposition\], the 1-motives $M=[u:{\mathbb{Z}}^r \rightarrow G]$ and $\oplus_{i=1}^s \oplus_{k=1}^r \oplus_{j=1}^n [u_{jik}:z_k {\mathbb{Z}}\rightarrow G_{ji}]$ have the same periods and therefore we are reduced to prove the case $r=n=s=1$. Consider the 1-motive $M=[u: z{\mathbb{Z}}\to G],$ where $G$ is an extension of an elliptic curve ${\mathcal{E}}$ by ${\mathbb{G}}_m$ parameterized by $v^*(t)=-Q \in {\mathcal{E}}(K)$, and $u(z)=R$ is a point of $G(K)$ living over $v(z)=P \in {\mathcal{E}}(K).$ Let $(l,p) \in {\mathbb{C}}^2$ be a semi-abelian logarithm of $R,$ that is $$\exp_G (l,p) = R.$$ Let $P_2$ and $P_1$ be two $K$-rational points whose difference is $P$. Because of the weight filtration of $M$, we have the non-split short exact sequence $$0 \longrightarrow {\mathrm{H}}_{{\mathrm{dR}}}^1({\mathcal{E}}) \longrightarrow {\mathrm{H}}_{{\mathrm{dR}}}^1(G) \longrightarrow {\mathrm{H}}_{{\mathrm{dR}}}^1({\mathbb{G}}_m) \longrightarrow 0$$ As $K$-basis of ${\mathrm{H}}_{{\mathrm{dR}}}^1(G)$ we choose the differentials of the first kind $\omega $ and of the second kind $\eta$ of ${\mathcal{E}},$ and the differential of the third kind $\xi_Q$, which lifts the only element $\frac{dt}{t}$ of the basis of ${\mathrm{H}}_{{\mathrm{dR}}}^1({\mathbb{G}}_m)$. Because of the decomposition (\[eq:DRham\]), we complete the basis of ${\mathrm{H}}_{{\mathrm{dR}}}(M)$ with the differential $df$ of a rational function $f$ on ${\mathcal{E}}$ such that $f(P_2)$ differs from $f(P_1)$ by 1. Always because of the weight filtration of $M$, we have the non-split short exact sequence $$0 \longrightarrow {\mathrm{H}}_1({\mathbb{G}}_m,{\mathbb{Z}}) \longrightarrow {\mathrm{H}}_1(G_{\mathbb{C}},{\mathbb{Z}}) \longrightarrow {\mathrm{H}}_1({\mathcal{E}}_{\mathbb{C}},{\mathbb{Z}}) \longrightarrow 0$$ As ${\mathbb{Q}}$-basis of $ {\mathrm{H}}_1(G_{\mathbb{C}},{\mathbb{Q}})$ we choose two closed paths $\tilde{\gamma}_1, \tilde{\gamma}_2$ which lift the basis $\gamma_1, \gamma_2$ of ${\mathrm{H}}_1({\mathcal{E}}_{\mathbb{C}},{\mathbb{Q}})$ and a closed path $\delta_{Q}$ around $-Q$. Because of the non-split exact sequence (\[eq:Hodge\]), we complete the basis of ${\mathrm{T}}_{{\mathbb{Q}}}(M)$ with a closed path $\beta_R$, which lifts the only element $z$ of the basis of ${\mathrm{T}}_{{\mathbb{Q}}}(z {\mathbb{Z}}) = {\mathbb{Z}}\otimes {\mathbb{Q}}$ via the surjection $ {\mathrm{T}}_{{\mathbb{Q}}} ( M_{ {\mathbb{C}}}) \rightarrow {\mathrm{T}}_{{\mathbb{Q}}} (z {\mathbb{Z}}) $, and whose restriction to ${\mathrm{H}}_1(G_{ {\mathbb{C}}},{\mathbb{Q}})$ is a closed path $\beta_{R|G}$ on $G({\mathbb{C}})$ having the following properties: $\beta_{R|G}$ lifts a path $\beta_{P_1P_2}$ on ${\mathcal{E}}({\mathbb{C}}) $ from $P_1$ to $P_2$, and its restriction to ${\mathrm{H}}_1({\mathbb{G}}_m,{\mathbb{Q}})$ is a path $\beta_l$ on ${\mathbb{G}}_m({\mathbb{C}})={\mathbb{C}}^* $ from $1$ to $l.$ With respect to these bases of ${\mathrm{T}}_{{\mathbb{Q}}}(M)$ and ${\mathrm{H}}_{{\mathrm{dR}}}(M)$, the matrix which represents the isomorphism (\[eq:betaM\]) for the 1-motive $M=[u: z {\mathbb{Z}}\to G]$ is $$\label{eq:matrix-integrales} \left( {\begin{array}{cccc} \int_{\beta_R} df &\int_{\beta_{P_1P_2}} \omega & \int_{\beta_{P_1P_2}} \eta &\int_{\beta_{R|G}} \xi_Q \\ \int_{\tilde{\gamma}_1}df &\int_{\gamma_1} \omega & \int_{\gamma_1} \eta &\int_{\tilde{\gamma}_1} \xi_Q \\ \int_{\tilde{\gamma}_2}df &\int_{\gamma_2} \omega & \int_{\gamma_2} \eta &\int_{\tilde{\gamma}_2} \xi_Q \\ \int_{\delta_{Q}}df &\int_{\delta_{Q}}\omega & \int_{\delta_{Q}}\eta & \int_{\delta_{Q}} \xi_Q \\ \end{array} } \right)$$ Recalling that $\exp_{{\mathcal{E}}}^{*}(\omega) = dz, \exp_{{\mathcal{E}}}^{*}(\eta) = d \zeta(z)$, (\[eq:expEXiq\]) and (\[eq:expGXiq\]) we can now compute explicitly all these integrals: - $\int_{\beta_R} df= f(P_2)-f(P_1)=1,$ - $\int_{\tilde{\gamma}_1}df=\int_{\tilde{\gamma}_2}df = \int_{\delta_{Q}}df =0$ because of the decomposition (\[eq:DRham\]), - $\int_{\beta_{P_1P_2}} \omega= \int_{p_1}^{p_2} dz= p_2 - p_1 =p,$ - $\int_{\gamma_i} \omega= \int_{0}^{\omega_i} dz= \omega_i$ for $i=1,2,$ - $\int_{\delta_{Q}}\omega = \int_{\delta_{Q}}\eta= 0$ since the image of $\delta_{Q}$ via ${\mathrm{H}}_1(G_{\mathbb{C}},{\mathbb{Q}}) \to {\mathrm{H}}_1({\mathcal{E}}_{\mathbb{C}},{\mathbb{Q}})$ is zero, - $\int_{\gamma_i} \eta = \int_{0}^{\omega_i} d \zeta= \zeta(\omega_i) - \zeta(0) =\eta_i$ for $i=1,2,$ - $\int_{\beta_{P_1P_2}} \eta= \int_{p_1}^{p_2} d \zeta(z) = \zeta(p_2) - \zeta(p_1). $ By the pseudo addition formula for the Weierstrass $\zeta$-function (see [@WW Example 2, p 451]), $\zeta(z+y) - \zeta(z)- \zeta(y) = \frac{1}{2} \frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)} \in K({\mathcal{E}})$, and so it exists a rational function $g$ on ${\mathcal{E}}$ such that $g(p_2)-g(p_1)= - \zeta(p + p_1) + \zeta(p)+ \zeta(p_1).$ Since the differential of the second kind $\eta$ lives in the quotient space ${\mathrm{H}}^1({\mathcal{E}}, \mathcal{O}_{\mathcal{E}}),$ we can add to the class of $\eta $ the exact differential $dg$, getting - $\int_{\beta_{P_1P_2}}( \eta + dg) = \int_{p_1}^{p_2}( d \zeta(z) + dg) = \zeta(p_2) - \zeta(p_1) +g(p_2) -g(p_1) = \zeta(p),$ - $\int_{\beta_{R|G}} \xi_Q= \int_0^l dw +\int_{p_1}^{p_2} \frac{f_q'(z)}{f_q(z)} dz = l+ \int_{p_1}^{p_2} d \log f_q(z) = l + \log \frac{f_q(p_2)}{f_q(p_1)} .$ Since by [@WW 20-53] the quotient of $\sigma$-functions is a rational function on ${\mathcal{E}}$, from the equality (\[eq:fq-sigma\]) it exists a rational function $g_q(z)$ on ${\mathcal{E}}$ such that $ \frac{g_q(p_2)}{g_q(p_1)} = (\frac{f_q(p+p_1)}{f_q(p)f_q(p_1)})^{-1}$, and therefore we get - $\int_{\beta_{R|G}} ( \xi_Q + d \log g_q(z)) = \int_0^l dw+ \int_{p_1}^{p_2}( d \log f_q(z) + d \log g_q(z)) = l+ \log \big( \frac{f_q(p_2)}{f_q(p_1)} \frac{g_q(p_2)}{g_q(p_1)} \big) = \\ l+ \log \big( \frac{f_q(p_2)}{f_q(p_1)} \frac{f_q(p)f_q(p_1)}{f_q(p_2)} \big) = l+ \log( f_q(p) ),$ with $e^l \in K^*$, - $\int_{\tilde{\gamma}_i} \xi_Q = \int_{0}^{\omega_i} \frac{f_q'(z)}{f_q(z)} dz = \int_{0}^{\omega_i} d \log f_q(z) = \log \frac{f_q(\omega_i)}{f_q(0)} =\eta_i q - \omega_i \zeta(q) $ by (\[eq:periods-fq\]) for $i=1,2,$ - $\int_{\delta_{-Q}} \xi_Q = 2i \pi \mathrm{Res}_{-Q} \xi_Q = 2 i \pi.$ The addition of the differential $d \log g_q(z)$ to the differential of the third kind $\xi_Q$ will modify the last two integrals by an integral multiple of $2 i \pi$ (see [@S Thm 10-7]) and this is irrelevant for the computation of the field generated by the periods of $M.$ Explicitly the matrix (\[eq:matrix-integrales\]) becomes $$\label{eq:matrix-periods} \left( {\begin{array}{cccc} 1 &p & \zeta(p) &\log f_q(p) + l \\ 0 & \omega_1 & \eta_1 &\eta_1 q - \omega_1 \zeta(q) \\ 0 & \omega_2 & \eta_2 &\eta_2 q - \omega_2 \zeta(q) \\ 0 &0 & 0 & 2 i \pi\\ \end{array} } \right),$$ with $e^l \in K^*,$ and so the periods of the 1-motive $M=[u:z {\mathbb{Z}}\to G], u(z)=R,$ are $1, \omega_1,\omega_2, \eta_1, \eta_2, p, \zeta(p) , \log f_q(p) + l , \eta_1 q - \omega_1 \zeta(q), \eta_2 q - \omega_2 \zeta(q), 2 i \pi.$ The determinations of the complex and elliptic logarithms, which appear in the first line of the matrix (\[eq:matrix-periods\]), are not well-defined since they depend on the lifting $\beta_{P_1P_2}$ of the basis of ${\mathrm{T}}_{{\mathbb{Q}}}(z{\mathbb{Z}})$ (recall that the short exact sequence (\[eq:Hodge\]) is not split). Nevertheless, the field $K (\mathrm{periodes}(M))$, which is involved in the Generalized Grothendieck’s Period Conjecture, is totally independent of these choices since it contains $2i \pi$, the periods of the Weierstrass $\wp$-function, the quasi-periods of the Weierstrass $\zeta$-function, and finally the quasi-quasi-periods of Serre’s function $f_q(z)$ (\[eq:def-fq\]). We finish this section with an example: Consider the 1-motive $M=[u:{{\mathbb{Z}}}^2 \to G]$, where $G$ is an extension of ${\mathcal{E}}_1 \times {\mathcal{E}}_2 $ by ${\mathbb{G}}_m^3$ parameterized by the $K$-rational points $Q_1=(Q_{11},Q_{21}), Q_2=(Q_{12},Q_{22}), Q_3=(Q_{13},Q_{23})$ of ${\mathcal{E}}_1^* \times {\mathcal{E}}_2^* $, and the morphism $u$ corresponds to two $K$-rational points $R_1,R_2$ of $G$ leaving over two points $P_1=(P_{11},P_{21}), P_2=(P_{12},P_{22})$ of ${\mathcal{E}}_1 \times {\mathcal{E}}_2. $ The more compact way to write down the matrix which represents the isomorphism (\[eq:betaM\]) for our 1-motive $M=[u:{{\mathbb{Z}}}^2 \to G]$ is to consider the 1-motive $$M'= M/ [0 \longrightarrow {\mathcal{E}}_1] \oplus M/ [0 \longrightarrow {\mathcal{E}}_2],$$ that is, with the above notation $M'=[u_1={{\mathbb{Z}}}^2 \to \Pi_{i=1}^3 G_{1i} ] \oplus [u_2={{\mathbb{Z}}}^2 \to \Pi_{i=1}^3 G_{2i} ] $ with $u_1$ corresponding to two $K$-rational points $(R_{111},R_{121},R_{131} )$ and $(R_{112},R_{122},R_{132})$ of $\Pi_{i=1}^3 G_{1i}$ living over $P_{11}$ and $P_{12}$, and $u_2$ corresponding to two $K$-rational points $(R_{211},R_{221},R_{231} )$ and $(R_{212},R_{222},R_{232})$ of $\Pi_{i=1}^3 G_{2i}$ living over $P_{21}$ and $P_{22}$. The 1-motives $M$ and $M'$ generate the same tannakian category: in fact, it is clear that $ <M'>^\otimes \; \; \subset \; \; < M>^\otimes $ and in the other hand $M= M' / [ {\mathbb{Z}}^2 / \mathrm{d}_{\mathbb{Z}}({\mathbb{Z}}) \to 0]$. The matrix representing the isomorphism (\[eq:betaM\]) for the 1-motive $M'$ with respect to the $K$-bases chosen in the above Corollary is $$\left(\begin{matrix} & &\scriptstyle{ p_{11} } &\scriptstyle{ \zeta_1(p_{11})} & & 0 & 0 & \scriptstyle{\log f_{q_{11}}(p_{11})+l_{111}}& \scriptstyle{\log f_{q_{12}}(p_{11})+l_{121}} & \scriptstyle{\log f_{q_{13}}(p_{11})+l_{131}} \cr \scriptstyle{ {\rm Id}_{4 \times 4} } & & \scriptstyle{ p_{12} } & \scriptstyle{ \zeta_1(p_{12})} & & 0 &0 & \scriptstyle{\log f_{q_{11}}(p_{12})+l_{112}} & \scriptstyle{\log f_{q_{12}}(p_{12})+l_{121}} &\scriptstyle{\log f_{q_{13}}(p_{12})+l_{131}} \cr & &0 &0 & &\scriptstyle{ p_{21} } & \scriptstyle{ \zeta_2(p_{21})} & \scriptstyle{\log f_{q_{21}}(p_{21})+l_{211}}& \scriptstyle{\log f_{q_{22}}(p_{21})+l_{221}} & \scriptstyle{\log f_{q_{23}}(p_{21})+l_{231}} \cr & & 0 & 0 & & \scriptstyle{ p_{22} } &\scriptstyle{ \zeta_2(p_{22})} & \scriptstyle{\log f_{q_{21}}(p_{22})+l_{212}} & \scriptstyle{\log f_{q_{22}}(p_{22})+l_{222}} &\scriptstyle{\log f_{q_{23}}(p_{22})+l_{232}} \cr & & \scriptstyle{ { \omega}_{11}} &\scriptstyle{ { \eta}_{11}}& & & & \scriptstyle{ { \eta}_{11} q_{11}-{ \omega}_{11} \zeta_1(q_{11}) }& \scriptstyle{ { \eta}_{11} q_{12}-{ \omega}_{11} \zeta_1(q_{12}) }& \scriptstyle{ { \eta}_{11} q_{13}-{ \omega}_{11} \zeta_1(q_{13}) }\cr & & \scriptstyle{ { \omega}_{12}} &\scriptstyle{ { \eta}_{12}}& & & & \scriptstyle{ { \eta}_{12} q_{11}-{ \omega}_{12} \zeta_1(q_{11})}& \scriptstyle{ { \eta}_{12} q_{12}-{ \omega}_{12} \zeta_1(q_{12})}& \scriptstyle{ { \eta}_{12} q_{13}-{ \omega}_{12} \zeta_1(q_{13})} \cr & & & & &\scriptstyle{ { \omega}_{21}} & \scriptstyle{ { \eta}_{21} } & \scriptstyle{ { \eta}_{21} q_{21}-{ \omega}_{21} \zeta_2(q_{21})}& \scriptstyle{ { \eta}_{21} q_{22}-{ \omega}_{21} \zeta_2(q_{22})}& \scriptstyle{ { \eta}_{21} q_{23}-{ \omega}_{21} \zeta_2(q_{23})} \cr & & & & &\scriptstyle{ { \omega}_{22}} & \scriptstyle{ { \eta}_{22}}& \scriptstyle{ { \eta}_{22} q_{21}-{ \omega}_{22} \zeta_2(q_{21})} & \scriptstyle{ { \eta}_{22} q_{22}-{ \omega}_{22} \zeta_2(q_{22})} & \scriptstyle{ { \eta}_{22} q_{23}-{ \omega}_{22} \zeta_2(q_{23})} \cr & & & & & & & & \scriptstyle{ 2i \pi {\rm Id}_{3 \times 3} }& \end{matrix} \right).$$ In general, for a 1-motive of the kind $M=[u:{\mathbb{Z}}^r \rightarrow G]$, where $G$ is an extension of a finite product $\Pi^n_{j=1} {\mathcal{E}}_j $ of elliptic curves by the torus ${\mathbb{G}}_m^s$, we will consider the 1-motive $$M'= \oplus_{j=1}^n \big( M/ [0 \longrightarrow \Pi_{1 \leq l \leq n \atop l \not =j }{\mathcal{E}}_j] \big)$$ whose matrix representing the isomorphism (\[eq:betaM\]) with respect to the $K$-bases chosen in the above Corollary is $$\left(\begin{matrix} A&B&C \cr 0 & D&E \cr 0 &0 & F \end{matrix} \right)$$ with $A= {\rm Id}_{rn \times rn},B$ the $rn \times 2n$ matrix involving the periods coming from the morphism $v: {\mathbb{Z}}^r \to \Pi^n_{j=1} {\mathcal{E}}_j $ , $C$ the $rn \times s$ matrix involving the periods coming from the trivialization $\Psi$ of the pull-back via $(v,v^*)$ of the Poincaré biextension $\mathcal{P}$ of $(\Pi^n_{j=1}{\mathcal{E}}_j, \Pi^n_{j=1}{\mathcal{E}}_j^*)$ by ${\mathbb{G}}_m$ , $D$ the $2n \times 2n$ matrix having in the diagonal the period matrix of each elliptic curves ${\mathcal{E}}_j$, $ E$ the $2n \times s$ matrix involving the periods coming from the morphism $v^*: {\mathbb{Z}}^s \to \Pi^n_{j=1} {\mathcal{E}}_j^* $, and finally $ F = 2i \pi{\rm Id}_{s \times s}$ the period matrix of ${\mathbb{G}}_m^s$. Dimension of the unipotent radical of the motivic Galois group of a 1-motive {#motivicGaloisgroup} ============================================================================ Denote by $\mathcal{MM}_{\leq 1}(K)$ the category of 1-motives defined over $K$. Using Nori’s and Ayoub’s works (see [@Ay14] and [@N00]), it is possible to endow the category of 1-motives with a *tannakian structure with rational coefficients* (roughly speaking a tannakian category $\mathcal{T}$ with rational coefficients is an abelian category with a functor $\otimes :\mathcal{T} \times \mathcal{T} \to \mathcal{T}$ defining the tensor product of two objects of $\mathcal{T} $, and with a fibre functor over $\mathrm{Spec}({\mathbb{Q}})$ - see [@D90 2.1, 1.9, 2.8] for details). We work in a completely geometrical setting using algebraic geometry on tannakian category and defining as one goes along the objects, the morphisms and the tensor products that we will need (essentially we tensorize motives with pure motives of weight 0, and as morphisms we use projections and biextensions). The unit object of the tannakian category $\mathcal{MM}_{\leq 1}(K)$ is the 1-motive ${\mathbb{Z}}(0)= [ {\mathbb{Z}}\to 0]$. In this section we use the notation $Y(1)$ for the torus whose cocharacter group is $Y$. In particular ${\mathbb{Z}}(1)= [ 0 \to {\mathbb{G}}_m]$. If $M$ is a 1-motive, we denote by $M^\vee \cong {\underline{\mathrm{Hom}}}(M, {\mathbb{Z}}(0))$ its dual and by $ev_M : M \otimes M^\vee \to {\mathbb{Z}}(0), \delta_M: {\mathbb{Z}}(0) \to M^\vee \otimes M$ the arrows of $\mathcal{MM}_{\leq 1}(K)$ characterizing this dual. The Cartier dual of $M$ is $M^*= M^\vee \otimes {\mathbb{Z}}(1)$. If $M_1,M_2$ are two 1-motives, we set $$\label{eq:BiextHom} {\mathrm{Hom}}_{\mathcal{MM}_{\leq 1}(K)}(M_1 \otimes M_2, M_3):= \mathrm{Biext}^1 (M_1,M_2; M_3)$$ where $\mathrm{Biext}^1 ((M_1,M_2;M_3)$ is the abelian group of isomorphism classes of biextensions of $(M_1,M_2)$ by $M_3$. In particular the isomorphism class of the Poincaré biextension $\mathcal{P}$ of $(A,A^*)$ by ${\mathbb{G}}_m$ is the Weil pairing $P_\mathcal{P} : A \otimes A^* \to {\mathbb{Z}}(1)$ of $A.$ The tannakian sub-category $<M>^\otimes$ generated by the 1-motive $M$ is the full sub-category of $\mathcal{MM}_{\leq 1}(K)$ whose objects are sub-quotients of direct sums of $M^{\otimes \; n} \otimes M^{\vee \; \otimes \; m}$, and whose fibre functor is the restriction of the fibre functor of $\mathcal{MM}_{\leq 1}(K)$ to $<M>^\otimes$. Because of the tensor product of $<M>^\otimes$, we have the notion of commutative Hopf algebra in the category ${{\mathrm{Ind}}\,}<M>^\otimes$ of Ind-objects of $<M>^\otimes$, and this allows us to define the category of affine $<M>^\otimes$-group schemes, just called *motivic affine group schemes*, as the opposite of the category of commutative Hopf algebras in ${{\mathrm{Ind}}\,}<M>^\otimes.$ The Lie algebra of a motivic affine group scheme is a pro-object $\rm L$ of $\langle M \rangle^\otimes$ endowed with a Lie algebra structure, i.e. $\rm L$ is endowed with an anti-symmetric application $[\, , \,]: {\rm L} \otimes {\rm L} \to {\rm L}$ satisfying the Jacobi identity. The *motivic Galois group* ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$ of $M$ is the fundamental group of the tannakian category $< M >^\otimes$ generated by $M$, i.e. the motivic affine group scheme ${\rm Sp}( \Lambda),$ where $ \Lambda$ is the commutative Hopf algebra of $<M>^\otimes$ which is universal for the following property: for any object $X$ of $<M>^\otimes,$ it exists a morphism $$\label{eq:lambdaX} \lambda_X: X^{\vee} \otimes X \longrightarrow \Lambda$$ functorial on $X$, i.e. such that for any morphism $f: X \to Y$ in $<M>^\otimes$ the diagram $$\begin{matrix} Y^{\vee} \otimes X&{\buildrel f^t \otimes 1 \over \longrightarrow}& X^{\vee} \otimes X \cr {\scriptstyle 1 \otimes f}\downarrow \quad \quad & & \quad \quad \downarrow {\scriptstyle \lambda_X}\cr Y^{\vee} \otimes Y & {\buildrel \lambda_Y \over \longrightarrow} & \Lambda \end{matrix}$$ is commutative. The universal property of $\Lambda$ is that for any object $U$ of $<M>^\otimes$, the map $$\begin{aligned} \nonumber {{\mathrm{Hom}}}(\Lambda, U) & \longrightarrow \big\{ u_X: X^{\vee} \otimes X \to U, ~~ {\rm {functorial~ on~}} X \big\} \\ \nonumber f & \longmapsto f \circ \lambda_X\end{aligned}$$ is bijective. The morphisms (\[eq:lambdaX\]), which can be rewritten as $ X \to X \otimes \Lambda$, define the action of the motivic Galois group ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$ on each object $X$ of $<M>^\otimes$. If $\omega_{\mathbb{Q}}$ is the fibre functor Hodge realization of the tannakian category $<M>^\otimes$, $\omega_{\mathbb{Q}}(\Lambda)$ is the Hopf algebra whose spectrum ${\rm Spec} (\omega (\Lambda))$ is the ${\mathbb{Q}}$-group scheme $ {\underline {\rm Aut}}^{\otimes}_{\mathbb{Q}}(\omega_{\mathbb{Q}})$, i.e. the Mumford-Tate group $\mathrm{MT}(M)$ of $M$. In other words, the motivic Galois group of $M$ is *the geometric interpretation* of the Mumford-Tate group of $M$. By [@A19 Thm 1.2.1] these two group schemes coincides, and in particular they have the same dimension $$\label{dimGalMT} \dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) = \dim \mathrm{MT}(M).$$ Let $M=[u:X \to G]$ be a 1-motive defined over $K$, with $G$ an extension of an abelian variety $A$ by a torus $T$. The weight filtration ${\mathrm{W}}_{\bullet}$ of $M$ induces a filtration on its motivic Galois group ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$ ([@S72 Chp IV §2]): 0.3 true cm $ {\mathrm{W}}_{0}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))={{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) $ 0.3 true cm $ {\mathrm{W}}_{-1}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))= \big\{ g \in {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \, \, \vert \, \, (g - id)M \subseteq {\mathrm{W}}_{-1}(M) ,(g - id) {\mathrm{W}}_{-1}(M) \subseteq {\mathrm{W}}_{-2}(M),$ $ (g - id) {\mathrm{W}}_{-2}(M)=0 \big\} , $ 0.3 true cm $ {\mathrm{W}}_{-2}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))=\big\{ g \in {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \, \, \vert \, \, (g - id) M \subseteq {\mathrm{W}}_{-2}(M), (g - id) {\mathrm{W}}_{-1}(M) =0 \big\}, $ 0.3 true cm $ {\mathrm{W}}_{-3}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))=0.$ 0.3 true cm Clearly $ {\mathrm{W}}_{-1}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ is unipotent. Denote by ${\mathrm{UR}}(M)$ the unipotent radical of ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$. Consider the graduated 1-motive $$\widetilde{M}= {\mathrm{Gr}}_*^{\mathrm{W}}(M) = X+A+T$$ associated to $M$ and let $<\widetilde{M}>^\otimes$ be the tannakian sub-category of $<M>^\otimes$ generated by $\widetilde{M}$. The functor “take the graduated” ${\mathrm{Gr}}_*^{\mathrm{W}}: <M>^\otimes \twoheadrightarrow <\widetilde{M}>^\otimes$, which is a projection, induces the inclusion of motivic affine group schemes $$\label{eq:Gr_0} {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}) \hookrightarrow {\mathrm{Gr}}_*^{\mathrm{W}}{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M).$$ \[eq:dimGr0\] Let $M=[u:X \to G]$ be a 1-motive defined over $K$, with $G$ an extension of an abelian variety $A$ by a torus $T$. The quotient ${{\mathrm{Gr}}}_{0}^{{\mathrm{W}}}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ is reductive and the inclusion of motivic group schemes (\[eq:Gr\_0\]) identifies ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}) $ with this quotient. Moreover, if $X= {\mathbb{Z}}^r$ and $T= {\mathbb{G}}_m^s$ $$\dim {{\mathrm{Gr}}}_{0}^{{\mathrm{W}}}\big({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)\big)= \dim{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}) = \begin{cases} \dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(A) & \mbox{if } A \not= 0,\\ 1 & \mbox{if } A=0, T \not=0 ,\\ 0 & \mbox{if } A=T=0. \end{cases}$$ By a motivic analogue of [@By83 §2.2], ${{\mathrm{Gr}}}_{0}^{{\mathrm{W}}}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ acts via ${{\mathrm{Gal}}({\overline K}/K)}$ on ${{\mathrm{Gr}}}_{0}^{W}(M)$, by homotheties on ${{\mathrm{Gr}}}_{-2}^{{\mathrm{W}}}(M)$, and its image in the group of authomorphisms of ${{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}(M)$ is the motivic Galois group ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(A)$ of the abelian variety $A $ underlying $M$. Therefore ${{\mathrm{Gr}}}_{0}^{{\mathrm{W}}}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ is reductive, and via the inclusion (\[eq:Gr\_0\]) it coincides with ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}) .$ To conclude, observe that ${\mathrm{Lie}}\, {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}({\mathbb{G}}_m)= {\mathbb{G}}_m$ which has dimension 1, and ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}({\mathbb{Z}})= \mathrm{Sp}({\mathbb{Z}}(0))$ which has dimension 0. The inclusion $<\widetilde{M}>^\otimes \hookrightarrow <M>^\otimes $ of tannakian categories induces the following surjection of motivic affine group schemes $$\label{eq:RestrictionGr_0} {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \twoheadrightarrow {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M})$$ which is the restriction $g \mapsto g_{|\widetilde{M} }.$ As an immediate consequence of the above Lemma we have \[eq:DecomDim\] Let $M=[u:X \to G]$ be a 1-motive defined over $K$. Then $${\mathrm{W}}_{-1}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))=\ker \big[{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \twoheadrightarrow {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}) \big].$$ In particular, ${\mathrm{W}}_{-1}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ is the unipotent radical $ {\mathrm{UR}}(M)$ of ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$ and $$\dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) = \dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}) + \dim {\mathrm{UR}}(M).$$ Observe that we can prove the equality $ {\mathrm{W}}_{-1}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))=\ker \big[{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \twoheadrightarrow {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}) \big]$ directly using the definition of the weight filtration: $$\begin{aligned} g \in {\mathrm{W}}_{-1}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)) & \Longleftrightarrow (g - id) {{\mathrm{Gr}}}_{0}^{{\mathrm{W}}}(M) =0, (g - id) {{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}(M) =0,(g - id) {{\mathrm{Gr}}}_{-2}^{{\mathrm{W}}}(M) =0 \\ & \Longleftrightarrow g_{| {{\mathrm{Gr}}}_{*}^{{\mathrm{W}}}(M) } = {\mathrm{id}}, \; \; \mathrm{i.e.} \; \; g= {\mathrm{id}}\;\; \mathrm{in}\; \; {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(\widetilde{M}). \end{aligned}$$ The inclusion $<M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee) >^\otimes \hookrightarrow <M>^\otimes $ of tannakian categories induces the following surjection of motivic affine group schemes $$\label{eq:Gr_1} {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \twoheadrightarrow {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}\big(M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee)\big)$$ which is the restriction $g \mapsto g_{|M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee) }.$ \[eq:DecomRU\] Let $M=[u:X \to G]$ be a 1-motive defined over $K$. Then $${\mathrm{W}}_{-2}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))=\ker \big[{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) \twoheadrightarrow {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee)) \big].$$ In particular, the quotient ${{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ of the unipotent radical $ {\mathrm{UR}}(M)$ is the unipotent radical ${\mathrm{W}}_{-1} \big( {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee))\big)$ of ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}\big( M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee)\big)$. Using the definition of the weight filtration, we have: $$\begin{aligned} g \in {\mathrm{W}}_{-2}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)) & \Longleftrightarrow (g - id) M/{\mathrm{W}}_{-2}(M) =0,\; (g - id){\mathrm{W}}_{-1}(M) =0 \\ & \Longleftrightarrow g_{| M/{\mathrm{W}}_{-2}(M) } = {\mathrm{id}}, \; g_{| M^\vee/{\mathrm{W}}_{-2}(M^\vee) } = {\mathrm{id}}\\ & \Longleftrightarrow g= {\mathrm{id}}\; \; \mathrm{in}\;\; {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee)). \end{aligned}$$ Since the surjection of motivic affine group schemes (\[eq:Gr\_1\]) respects the weight filtration, ${\mathrm{W}}_{-2}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ is in fact the kernel of ${\mathrm{W}}_{-1} ({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)) \twoheadrightarrow {\mathrm{W}}_{-1} ({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M + M^\vee /{\mathrm{W}}_{-2} (M + M^\vee))) .$ Hence we get the second statement. From the definition of weight filtration, we observe that $${\mathrm{W}}_{-2}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)) \subseteq {\underline{\mathrm{Hom}}}(X,Y(1)) \cong X^\vee \otimes Y (1).$$ By the above Lemma, we have that $${{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)) \subseteq {\underline{\mathrm{Hom}}}(X+ Y^\vee,A+A^*) \cong X^\vee \otimes A+Y \otimes A^*.$$ In order to compute the dimension of the unipotent radical ${\mathrm{UR}}(M) $ of ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M )$ we use notations of [@B03 §3] that we recall briefly. Let $(X,Y^\vee, A,A^*, v:X \to A, v^*:Y^\vee \to A^*, \psi:X \otimes Y^\vee \to (v \times v^*)^* \mathcal{P})$ be the 7-tuple defining the 1-motive $M=[u:X \to G]$ over $K$, where $G$ an extension of $A$ by the torus $Y(1)$. Let $$E={\mathrm{W}}_{-1}( {\underline {{\mathrm{End}}}}(\widetilde{M})).$$ It is the direct sum of the pure motives $E_{-1}= X^\vee \otimes A + A^\vee \otimes Y(1)$ and $E_{-2}= X^\vee \otimes Y(1)$ of weight -1 and -2. As observed in [@B03 §3], the composition of endomorphisms furnishes a ring structure to $E$ given by the arrow $P: E \otimes E \to E$ of $\langle M \rangle^\otimes$ whose only non trivial component is $$E_{-1} \otimes E_{-1} \longrightarrow (X^\vee \otimes A) \otimes (A^* \otimes Y) \longrightarrow {{\mathbb{Z}}}(1) \otimes X^\vee \otimes Y = E_{-2},$$ where the first arrow is the projection from $ E_{-1} \otimes E_{-1}$ to $(X^\vee \otimes A) \otimes (A^* \otimes Y) $ and the second arrow is the Weil pairing $P_{\mathcal{P}}: A \otimes A^* \to {\mathbb{Z}}(1)$ of $A.$ Because of the definition (\[eq:BiextHom\]) the product $P:E_{-1} \otimes E_{-1} \to E_{-2}$ defines a biextension $\mathcal{B}$ of $(E_{-1},E_{-1})$ by $E_{-2} $, whose pull-back $ d^* \mathcal{B}$ via the diagonal morphism $d: E_{-1} \to E_{-1} \times E_{-1}$ is a $\Sigma - X^\vee \otimes Y (1)$-torsor over $E_{-1}$. By [@B03 Lem 3.3] this $\Sigma - X^\vee \otimes Y (1)$-torsor $ d^* \mathcal{B}$ induces a Lie bracket $[\, ,\,]: E \otimes E \to E$ on $E$ which becomes therefore a Lie algebra. The action of $E=W_{-1}( {\underline {{\mathrm{End}}}}(\widetilde{M}))$ on $\widetilde{M}$ is given by the arrow $ E \otimes \widetilde{M} \to \widetilde{M}$ of $\langle M \rangle^\otimes$ whose only non trivial components are $$\begin{aligned} \label{eq:alpha1} \alpha_1:& (X^\vee \otimes A) \otimes X \longrightarrow A \\ \nonumber \alpha_2:& (A^* \otimes Y) \otimes A \longrightarrow Y(1) \\ \nonumber \gamma :& (X^\vee \otimes Y(1)) \otimes X \longrightarrow Y(1),\end{aligned}$$ where the first and the last arrows are induced by $ev_{X^\vee}: X^\vee \otimes X \to {\mathbb{Z}}(0)$, while the second one is ${\mathrm{rk}}(Y)$-copies of the Weil pairing $P_{\mathcal{P}}: A \otimes A^* \to {\mathbb{Z}}(1)$ of $A$. By [@B03 Lem 3.3], via the arrow $(\alpha_1, \alpha_2, \gamma ): E \otimes \widetilde{M} \to \widetilde{M}$, the 1-motive $\widetilde{M}$ is in fact a $(E,[,])$-Lie module. As observed in [@B03 Rem 3.4 (3)] $E$ acts also on the Cartier dual $\widetilde{M}^*= Y^\vee + A^* + X^\vee(1)$ of $\widetilde{M}$ and this action is given by the arrows $$\begin{aligned} \label{eq:alpha2*} \alpha_2^*:& (A^* \otimes Y) \otimes Y^\vee \longrightarrow A^* \\ \nonumber\alpha_1^*:& (X^\vee \otimes A) \otimes A^* \longrightarrow X^\vee(1) \\ \nonumber \gamma^* :& (X^\vee \otimes Y(1)) \otimes Y^\vee \longrightarrow X^\vee(1),\end{aligned}$$ where $\alpha_2^*$ et $\gamma^*$ are projections, while $\alpha_1^*$ is ${\mathrm{rk}}(X^\vee)$-copies of the Weil pairing $P_{\mathcal{P}}: A \otimes A^* \to {\mathbb{Z}}(1)$ of $A$. Via the arrows $\delta_{ X^\vee}: {\mathbb{Z}}(0) \to X \otimes X^\vee$ et $\delta_{ Y}: {\mathbb{Z}}(0) \to Y^\vee \otimes Y$, to have the morphisms $v: X \to A$ and $v^*: Y^\vee \to A^*$ underlying the 1-motive $M$ is the same thing as to have the morphisms $V: {\mathbb{Z}}(0) \to A \otimes X^\vee $ and $V^*: {\mathbb{Z}}(0) \to A^* \otimes Y,$ i.e. to have a point $$b=(b_1,b_2) \in E_{-1}(K)= A \otimes X^\vee(K)+A^* \otimes Y(K).$$ Fix now an element $(x,y^\vee)$ in the character group $X \otimes Y^\vee$ of the torus $X^\vee \otimes Y(1)$. By construction of the point $b$, it exists an element $(s,t) \in X \otimes Y^\vee (K)$ such that $$\begin{aligned} \nonumber v(x) &= \alpha_1(b_1,s) \in A(K) \\ \nonumber v^*(y^\vee)& =\alpha_2^*(b_2,t)\in A^*(K).\end{aligned}$$ Let $i^*_{x,y^\vee} d^*\mathcal{B}$ be the pull-back of $d^* \mathcal{B}$ via the inclusion $i_{x,y^\vee}: \{ (v(x),v^*(y^\vee) )\} \hookrightarrow E_{-1}$ in $E_{-1}$ of the abelian sub-variety generated by the point $ (v(x),v^*(y^\vee) )$. The push-down $(x,y^\vee)_*i^*_{x,y^\vee} d^*\mathcal{B}$ of $i^*_{x,y^\vee} d^*\mathcal{B}$ via the character $(x,y^\vee):X^\vee \otimes Y(1)\to {\mathbb{Z}}(1)$ is a $\Sigma-{\mathbb{Z}}(1)$-torsor over $ \{ (v(x),v^*(y^\vee))\}: $ $$\begin{matrix} (x,y^\vee)_*i^*_{x,y^\vee} d^*\mathcal{B}& \longleftarrow & i^*_{x,y^\vee} d^*\mathcal{B} & \longrightarrow & d^* \mathcal{B} \\ \downarrow & &\downarrow & & \downarrow \\ \{ (v(x),v^*(y^\vee)) \} & = & \{ (v(x),v^*(y^\vee)) \} & {\buildrel i_{x,y^\vee} \over \longrightarrow} & E_{-1} \end{matrix}$$ To have the point $\psi(x,y^\vee)$ is equivalent to have a point $(\widetilde{b})_{x,y^\vee}$ of $(x,y^\vee)_*i^*_{x,y^\vee} d^*\mathcal{B}$ over $ (v(x),v^*(y^\vee))$, and so to have the trivialization $\psi$ is equivalent to have a point $$\widetilde{b} \in (d^*\mathcal{B})_{b}$$ in the fibre of $d^*\mathcal{B}$ over $b=(b_1,b_2).$ Consider now the following pure motives: 1. Let $B$ be the *smallest* abelian sub-variety (modulo isogenies) of $X^\vee \otimes A+A^* \otimes Y$ which contains the point $b=(b_1,b_2) \in X^\vee \otimes A (K) + A^* \otimes Y (K) $. The pull-back $i^*d^* \mathcal{B}$ of $d^* \mathcal{B}$ via the inclusion $i: B \hookrightarrow E_{-1}$ of $B$ on $E_{-1}$, is a $\Sigma-X^\vee \otimes Y(1)$-torsor over $B$. 2. Let $Z_1$ be the *smallest* ${{\mathrm{Gal}}({\overline K}/K)}$-sub-module of $X^\vee \otimes Y$ such that the torus $Z_1(1)$ contains the image of the Lie bracket $[\, ,\,]: B \otimes B \to X^\vee \otimes Y(1)$. The push-down $p_*i^*d^* \mathcal{B}$ of the $\Sigma-X^\vee \otimes Y(1)$-torsor $i^*d^* \mathcal{B}$ via the projection $p:X^\vee \otimes Y(1) \twoheadrightarrow (X^\vee \otimes Y/ Z_1)(1)$ is the trivial $\Sigma-(X^\vee \otimes Y/ Z_1)(1)$-torsor over $B$, i.e. $$p_*i^*d^* \mathcal{B}= B \times (X^\vee \otimes Y/ Z_1)(1).$$ Note by $\pi: p_*i^*d^* \mathcal{B} \twoheadrightarrow (X^\vee \otimes Y/ Z_1)(1)$ the canonical projection. We still note $\widetilde{b}$ the points of $i^*d^* \mathcal{B}$ and of $p_*i^*d^* \mathcal{B}$ living over $b \in B$. 3. Let $Z$ be the *smallest* ${{\mathrm{Gal}}({\overline K}/K)}$-sub-module of $X^\vee \otimes Y$ containing $Z_1$ and such that the sub-torus $(Z/ Z_1)(1)$ of $(X^\vee \otimes Y/ Z_1)(1)$ contains $\pi (\widetilde{b}) $. Let $A_{\mathbb{C}}$ be the abelian variety defined over ${\mathbb{C}}$ obtained from $A$ extending the scalars from $K$ to the complexes. Denote by $g$ the dimension of $A$. Consider the abelian exponential $$\exp_{A}: {\mathrm{Lie}}A_{\mathbb{C}}\longrightarrow A_{\mathbb{C}}$$ whose kernel is the lattice ${\mathrm{H}}_1(A_{\mathbb{C}}({\mathbb{C}}),{\mathbb{Z}}),$ and denote by $\log_A$ an abelian logarithm of $A$, that is a choice of an inverse map of $\exp_{A}$. Consider the composite $$P_\mathcal{P} \circ (v \times v^*): X \otimes Y^\vee \longrightarrow {\mathbb{Z}}(1)$$ where $P_\mathcal{P}: A \otimes A^* \to {\mathbb{Z}}(1)$ is the Weil pairing of $A.$ Since we work modulo isogenies, we identify the abelian variety $A$ with its Cartier dual $A^*$. Let $\omega_1, \dots , \omega_g $ be differentials of the first kind which build a basis of the $K$-vector space ${\mathrm{H}}^0(A, \Omega^1_A)$ of holomorphic differentials, and let $\eta_1, \dots , \eta_g $ be differentials of the second kind which build a basis of the $K$-vector space ${\mathrm{H}}^1(A, \mathcal{O}_A)$ of differentials of the second kind modulo holomorphic differentials and exact differentials. As in the case of elliptic curves, the first De Rham cohomology group ${\mathrm{H}}^1_{\mathrm{dR}}(A)$ of the abelian variety $A$ is the direct sum ${\mathrm{H}}^0(A, \Omega^1_A) \oplus {\mathrm{H}}^1(A, \mathcal{O}_A)$ of these two vector spaces and it has dimension $2g$. Let $\gamma_1, \dots, \gamma_{2g}$ be closed paths which build a basis of the ${\mathbb{Q}}$-vector space ${\mathrm{H}}_1(A_{\mathbb{C}},{\mathbb{Q}})$. For $n=1, \dots, g$ and $m=1,\dots, 2g$, the abelian integrals of the first kind $ \int_{\gamma_m} \omega_n = \omega_{nm}$ are the *periods* of the abelian variety $A$, and the abelian integrals of the second kind $ \int_{\gamma_m} \eta_n = \eta_{nm}$ are the *quasi-periods* of $A$. \[eq:dimUR\] Let $M=[u:X \to G]$ be a 1-motive defined over $K$, with $G$ an extension of an abelain variety $A$ by a torus $Y(1)$. Denote by $ F={\mathrm{End}}( A) \otimes_{\mathbb{Z}}{\mathbb{Q}}$ the field of endomorphisms of the abelian variety $A.$ Let $x_1, \dots, x_{\mathrm{rk}(X)}$ be generators of the character group $X$ and let $y^\vee_1, \dots , y^\vee_{\mathrm{rk}(Y^\vee)}$ be generators of the character group $Y^\vee.$ Then $$\dim_{{\mathbb{Q}}} {\mathrm{UR}}( M)=$$ $$2 \dim_{F} \mathcal{A}b \mathcal{L}og \; {\mathrm{Im}}(v,v^*) + \dim_{{\mathbb{Q}}} \mathcal{L}og \;{\mathrm{Im}}(P_\mathcal{P} \circ (v \times v^*)) + \dim_{{\mathbb{Q}}} \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))})$$ where - $ \mathcal{A}b\mathcal{L}og \; {\mathrm{Im}}(v,v^*)$ is the $F$-sub-vector space of ${\mathbb{C}}/ (\sum_{n=1, \dots, g \atop m=1, \dots, 2g}F \, \omega_{nm}) $ generated by the abelian logarithms $\{ \log_A v(x_k), \log_A v^*(y^\vee_i) \}_{ k=1, \ldots, \mathrm{rk}(X) \atop i=1, \ldots, \mathrm{rk}(Y^\vee)}$ ; - $ \mathcal{L}og \; {\mathrm{Im}}( P_\mathcal{P} \circ (v \times v^*))$ is the ${\mathbb{Q}}\,$-sub-vector space of ${\mathbb{C}}/ 2 i \pi {\mathbb{Q}}$ generated by the logarithms $\{ \log P_\mathcal{P}(v(x_k),v^*(y^\vee_i) ) \}_{ k=1, \ldots, \mathrm{rk}(X) \atop i=1, \ldots, \mathrm{rk}(Y^\vee)}$; - $ \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))})$ is the ${\mathbb{Q}}\,$-sub-vector space of ${\mathbb{C}}/ 2 i \pi {\mathbb{Q}}$ generated by the logarithms $\{ \log \psi(x_{k'},y^\vee_{i'} ) \}_{(x_{k'},y^\vee_{i'} ) \in \ker (P_\mathcal{P} \circ (v \times v^*)) \atop 1\leq {k'} \leq \mathrm{rk}(X), \; 1\leq {i'} \leq \mathrm{rk}(Y^\vee) } .$ By the main theorem of [@B03 Thm 0.1], the unipotent radical $W_{-1} ({\mathrm{Lie}}{{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}}(M))$ is the semi-abelian variety extension of $B$ by $Z(1)$ defined by the adjoint action of the Lie algebra $(B,Z(1), [\, , \,])$ over $B+Z(1).$ Since the tannakian category $<M>^\otimes$ has rational coefficients, we have that $ \dim_{{\mathbb{Q}}} W_{-1} ( {{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}}(M)) = 2 \dim B + \dim Z(1) $. Concerning the abelian part $$\dim B= \dim_F \mathcal{A}b\mathcal{L}og \; {\mathrm{Im}}(v,v^*).$$ On the other hand, for the toric part $\dim Z(1) = \dim (Z/ Z_1)(1) + Z_1(1)$ by construction. Because of the explicit description of the Lie bracket $[\, ,\,]: B \otimes B \to X^\vee \otimes Y(1)$ given in [@B03 (2.8.4)], $$\dim Z_1(1) = \dim_{{\mathbb{Q}}} \mathcal{L}og \;{\mathrm{Im}}(P_\mathcal{P} \circ (v \times v^*)).$$ Finally by construction $$\dim(Z/ Z_1)(1) = \dim_{{\mathbb{Q}}} \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))}) .$$ The dimension of the quotient ${{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))$ of the unipotent radical $ {\mathrm{UR}}( M)$ is twice the dimension of the abelian sub-variety $B$ of $X^\vee \otimes A+A^* \otimes Y,$ that is $$\dim_{\mathbb{Q}}{{\mathrm{Gr}}}_{-1}^{{\mathrm{W}}}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)) = 2 \dim_F \mathcal{A}b\mathcal{L}og \; {\mathrm{Im}}(v,v^*).$$ The dimension of $ W_{-2} ( {{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}}(M))$ is the dimension of the sub-torus $Z(1)$ of $X^\vee \otimes Y(1)$, that is $$\dim_{{\mathbb{Q}}} W_{-2} ( {{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}}(M)) = \dim_{{\mathbb{Q}}} \mathcal{L}og \;{\mathrm{Im}}(P_\mathcal{P} \circ (v \times v^*)) + \dim_{{\mathbb{Q}}} \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))})$$ A 1-motive $M=[u:X \to G]$ defined over $K$ is said to be *deficient* if $ {\mathrm{W}}_{-2}({{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M))=0.$ In [@JR] Jacquinot and Ribet construct such a 1-motive in the case $\mathrm{rk}(X)=\mathrm{rk}(Y^\vee)=1$. By the above Theorem we have that $M$ is deficient if and only if for any $(x,y^{\vee}) \in X \otimes Y^{\vee}$, $$P_\mathcal{P} (v(x), v^*(y^{\vee})) =1 \quad \mathrm{and} \quad \psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))}(x,y^{\vee}) =1,$$ that is if and only if the two arrows $P_\mathcal{P} \circ (v \times v^*): X \otimes Y^{\vee} \to {\mathbb{Z}}(1)$ and $\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))} : X \otimes Y^{\vee} \to {\mathbb{Z}}(1)$ are the trivial arrow. Now let $M=[u:{{\mathbb{Z}}}^r \to G]$ be a 1-motive defined over $K$, with $G$ an extension of a product $\Pi^n_{j=1} {\mathcal{E}}_j $ of pairwise not isogenous elliptic curves by the torus ${\mathbb{G}}_m^s.$ We go back to the notation used in Section \[periods\]. Denote by ${\mathrm{pr}}_h:\Pi^n_{j=1} {\mathcal{E}}_j \to {\mathcal{E}}_h$ and ${\mathrm{pr}}_h^*:\Pi^n_{j=1} {\mathcal{E}}_j^* \to {\mathcal{E}}_h^*$ the projections into the $h$-th elliptic curve and consider the composites $ v_h= {\mathrm{pr}}_h \circ v:{\mathbb{Z}}^r \rightarrow {\mathcal{E}}_h$ and $v^*_h={\mathrm{pr}}_h^* \circ v^* :{\mathbb{Z}}^s \rightarrow {\mathcal{E}}_h^*.$ Let $\mathcal{P}$ be the Poincaré biextension of $(\Pi^n_{j=1} {\mathcal{E}}_j, \Pi^n_{j=1} {\mathcal{E}}_j^*)$ by ${\mathbb{G}}_m$ and let $\mathcal{P}_j$ be the Poincaré biextension of $( {\mathcal{E}}_j, {\mathcal{E}}_j^*)$ by ${\mathbb{G}}_m$. The category of biextensions is additive in each variable, and so we have that $P_\mathcal{P} = \Pi^n_{j=1}P_{\mathcal{P}_j}$, where $P_{\mathcal{P}_j}: {\mathcal{E}}_j \otimes {\mathcal{E}}_j^*\to {\mathbb{Z}}(1)$ is the Weil pairing of the elliptic curve ${\mathcal{E}}_j$. \[eq:dimGalMot\] Let $M=[u:{{\mathbb{Z}}}^r \to G]$ be a 1-motive defined over $K$, with $G$ an extension of a product $\Pi^n_{j=1} {\mathcal{E}}_j $ of pairwise not isogenous elliptic curves by the torus ${\mathbb{G}}_m^s.$ Denote by $ k_j={\mathrm{End}}( {\mathcal{E}}_j) \otimes_{\mathbb{Z}}{\mathbb{Q}}$ the field of endomorphisms of the elliptic curve ${\mathcal{E}}_j$ for $j=1, \dots, n.$ Let $x_1, \dots, x_r$ be generators of the character group ${{\mathbb{Z}}}^r$ and let $y^\vee_1, \dots , y^\vee_{s}$ be generators of the character group ${{\mathbb{Z}}}^{s}.$ Then $$\dim_{\mathbb{Q}}{{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) = 4 \sum_{j=1}^n (\dim_{{\mathbb{Q}}} k_j)^{-1}-n+1 + \sum_{j=1}^n 2 \dim_{k_j} \mathcal{A}b \mathcal{L}og \; {\mathrm{Im}}(v_j,v^*_j) +$$ $$\dim_{{\mathbb{Q}}} \mathcal{L}og \;{\mathrm{Im}}(P_\mathcal{P} \circ (v \times v^*)) + \dim_{{\mathbb{Q}}} \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))})$$ - $\mathcal{A}b \mathcal{L}og \; {\mathrm{Im}}(v_j,v^*_j)$ is the $k_j$-sub-vector space of ${\mathbb{C}}/ k_j \, \omega_{j1}+ k_j \, \omega_{j2}$ generated by the elliptic logarithms $\{ p_{jk}, q_{ji} \}_{ k=1, \ldots, r \atop i=1, \ldots, s}$ of the points $\{ P_{jk}, Q_{ji} \}_{ k=1, \ldots, r \atop i=1, \ldots, s}$ for $j=1, \ldots, n;$ - $ \mathcal{L}og \; {\mathrm{Im}}( P_\mathcal{P} \circ (v \times v^*))$ is the ${\mathbb{Q}}\,$-sub-vector space of ${\mathbb{C}}/ 2 i \pi {\mathbb{Q}}$ generated by the logarithms $\{ \log P_{\mathcal{P}_j}(P_{jk}, Q_{ji} ) \}_{ k=1, \ldots,r, \; \; i=1, \ldots, s \atop j=1, \ldots, n}$; - $ \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))})$ is the ${\mathbb{Q}}\,$-sub-vector space of ${\mathbb{C}}/ 2 i \pi {\mathbb{Q}}$ generated by the logarithms $\{ \log \psi(x_{k'} ,y^\vee_{i'} ) \}_{(x_{k'} ,y^\vee_{i'} ) \in \ker (P_{\mathcal{P}_j} \circ (v_j\times v^*_j)) \atop 1\leq {k'} \leq r, \; 1\leq {i'} \leq s, \; j=1, \ldots, n } .$ Since the elliptic curves are pairwise not isogenous, by [@Moonen §2] and (\[dimGalMT\]) we have that $$\dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}\big(\Pi_{j=1}^{n} {\mathcal{E}}_j \big)=4 \; \sum_{j=1}^n (\dim_{\mathbb{Q}}k_j)^{-1}-n+1.$$ Therefore putting together Corollary \[eq:DecomDim\], Lemma \[eq:dimGr0\] and Theorem \[eq:dimUR\] we can conclude. We can express the dimension of the motivic Galois group of a product of elliptic curves also as $ 3 n_1 +n_2 +1,$ where $n_1$ is the number of elliptic curves without complex multiplication and $n_2$ is the number of elliptic curves with complex multiplication. Therefore $$\dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) =\dim{\mathrm{UR}}(M) + 3 n_1 +n_2 +1$$ The 1-motivic elliptic conjecture {#conjecture} ================================= **The 1-motivic elliptic conjecture** Consider - $n$ elliptic curves ${\mathcal{E}}_1, \dots, {\mathcal{E}}_n$ pairwise not isogenous. For $j=1, \ldots, n,$ denote by $ k_j={\mathrm{End}}( {\mathcal{E}}_j) \otimes_{\mathbb{Z}}{\mathbb{Q}}$ the field of endomorphisms of ${\mathcal{E}}_j$ and let $g_{2j}=60 \; \mathrm{G}_{4j}$ and $g_{3j}=140 \; \mathrm{G}_{6j}$, where $\mathrm{G}_{4j}$ and $\mathrm{G}_{6j}$ are the Eisenstein series relative to the lattice $ {\mathrm{H}}_1({\mathcal{E}}_j({\mathbb{C}}),{\mathbb{Z}})$ of weight 4 and 6 respectively; - $s$ points $Q_i= (Q_{1i},\dots,Q_{ni})$ of $\Pi^n_{j=1} {\mathcal{E}}_j^*({\mathbb{C}})$ for $i=1,\ldots, s$. These points determine an extension $G$ of $\Pi^n_{j=1} {\mathcal{E}}_j $ by $ {{\mathbb{G}}}_m^s$; - $r$ points $R_1,\dots,R_r $ of $G({\mathbb{C}})$. Denote by $(P_{1k},\dots,P_{nk}) \in \Pi^n_{j=1} {\mathcal{E}}_j({\mathbb{C}})$ the projection of the point $R_k$ on $\Pi^n_{j=1} {\mathcal{E}}_j$ for $k= \ldots, r.$ Then $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big(2i \pi, g_{2j},g_{3j}, Q_{ji},R_{k},\omega_{j1},\omega_{j2},\eta_{j1}, \eta_{j2}, p_{jk},\zeta_j(p_{jk}),$$ $$\eta_{j1} q_{ji} - \omega_{j1} \zeta_j(q_{ji}),\eta_{j2} q_{ji} - \omega_{j2} \zeta_j(q_{ji}) , \log f_{q_{ji}}(p_{jk}) +l_{jik} {\Big)}_{j=1,\ldots,n \;\; i=1,\ldots,s \atop k=1,\dots,r } \geq$$ $$4 \sum_{j=1}^n (\dim_{{\mathbb{Q}}} k_j)^{-1}-n+1 + \sum_{j=1}^n 2 \dim_{k_j} \mathcal{A}b \mathcal{L}og \; {\mathrm{Im}}(v_j,v^*_j) +$$ $$\dim_{{\mathbb{Q}}} \mathcal{L}og \;{\mathrm{Im}}(P_\mathcal{P} \circ (v \times v^*)) + \dim_{{\mathbb{Q}}} \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))})$$ where - $\mathcal{A}b \mathcal{L}og \; {\mathrm{Im}}(v_j,v^*_j)$ is the $k_j$-sub-vector space of ${\mathbb{C}}/ k_j \, \omega_{j1}+ k_j \, \omega_{j2}$ generated by the elliptic logarithms $\{ p_{jk}, q_{ji} \}_{ k=1, \ldots, r \atop i=1, \ldots,s}$ of the points $\{ P_{jk}, Q_{ji} \}_{ k=1, \ldots, r \atop i=1, \ldots, s}$ for $j=1, \ldots, n;$ - $ \mathcal{L}og \; {\mathrm{Im}}( P_\mathcal{P} \circ (v \times v^*))$ is the ${\mathbb{Q}}\,$-sub-vector space of ${\mathbb{C}}/ 2 i \pi {\mathbb{Q}}$ generated by the logarithms $\{ \log P_{\mathcal{P}_j}(P_{jk}, Q_{ji} ) \}_{ k=1, \ldots, r, \; \; i=1, \ldots, s \atop j=1, \ldots, n}$; - $ \mathcal{L}og \; {\mathrm{Im}}(\psi_{| \ker (P_\mathcal{P} \circ (v \times v^*))})$ is the ${\mathbb{Q}}\,$-sub-vector space of ${\mathbb{C}}/ 2 i \pi {\mathbb{Q}}$ generated by the logarithms $\{ \log \psi(x,y^\vee ) \}_{(x,y^\vee ) \in \ker (P_{\mathcal{P}_j} \circ (v_j\times v^*_j)) \atop (x,y^\vee ) \in {\mathbb{Z}}^r \otimes {\mathbb{Z}}^s} .$ Because of Proposition \[proof-periods\] and Corollary \[eq:dimGalMot\], we can conclude that \[thmMain\] Let $M=[u:{{\mathbb{Z}}}^r \to G]$ be a 1-motive defined over $K$, with $G$ an extension of a product $\Pi^n_{j=1} {\mathcal{E}}_j $ of pairwise not isogenous elliptic curves by the torus ${\mathbb{G}}_m^s.$ Then the Generalized Grothendieck’s Period Conjecture applied to $M$ is equivalent to the 1-motivic elliptic conjecture. \[Rk1\] If $Q_{ji}=0$ for $j=1, \dots,n$ and $i=1, \dots,s$, the above conjecture is the elliptic-toric conjecture stated in [@B02 1.1], which is equivalent to the Generalized Grothendieck’s Period Conjecture applied to the 1-motive $M=[u: \Pi_{k=1}^r z_k {{\mathbb{Z}}} \to {\mathbb{G}}_m^s \times \Pi^n_{j=1} {\mathcal{E}}_j]$ with $u(z_k) = (R_{1k}, \dots,R_{sk} , P_{1k}, \dots , P_{nk}) \in {\mathbb{G}}_m^s (K) \times \Pi^n_{j=1} {\mathcal{E}}_j(K).$ \[Rk2\] If $Q_{ji}=P_{ij}={\mathcal{E}}_j=0$ for $j=1, \dots,n$ and $i=1, \dots,s$, the above conjecture is equivalent to the Generalized Grothendieck’s Period Conjecture applied to the 1-motive $M=[u: \Pi_{k=1}^r z_k {{\mathbb{Z}}} \to {\mathbb{G}}_m^s]$ with $u(z_k) = (R_{1k}, \dots,R_{sk}) \in {\mathbb{G}}_m^s (K) $, which in turn is equivalent to the Schanuel conjecture (see [@B02 Cor 1.3 and §3]). Low dimensional case: $r=n=s=1$ {#lowDim} =============================== In this section we work with a 1-motive $M=[ u:{\mathbb{Z}}\rightarrow G], u(1)=R,$ defined over $K$ in which $G$ is an extension of just one elliptic curve ${\mathcal{E}}$ by the torus ${\mathbb{G}}_m$, i.e. $r=s=n=1$. Let $g_2=60 \; \mathrm{G}_4$ and $g_3=140 \; \mathrm{G}_6$ with $\mathrm{G}_4$ and $\mathrm{G}_6$ the Eisenstein series relative to the lattice $\Lambda := {\mathrm{H}}_1({\mathcal{E}}({\mathbb{C}}),{\mathbb{Z}})$ of weight 4 and 6 respectively. The field of definition $K$ of the 1-motive $M=[u:{\mathbb{Z}}\rightarrow G], u(1)=R$ is $${\mathbb{Q}}\big( g_2, g_3, Q,R \big).$$ By Proposition \[proof-periods\], the field $K (\mathrm{periods}(M))$ generated over $K$ by the periods of $M$, which are the coefficients of the matrix (\[eq:matrix-periods\]), is $${\mathbb{Q}}\Big(g_2, g_3, Q, R, 2 i \pi, \omega_1,\omega_2,\eta_1,\eta_2, p,\zeta(p), \eta_1 q - \omega_1 \zeta(q),\eta_2 q - \omega_2 \zeta(q) , \log f_q(p) +l \Big).$$ ${\mathrm{End}}({\mathcal{E}}) \otimes_{\mathbb{Z}}{\mathbb{Q}}$-linear dependence between the points $P$ and $Q$ and torsion properties of the points $P, Q,R$ affect the dimension of the unipotent radical of ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$. By Corollary \[eq:dimGalMot\] we have the following table concerning the dimension of the motivic Galois group ${{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M)$ of $M$: ----------------------------------------------------------------------------- ------------------------- -------------------------------------------------------- -------------------------------------------------------- ------------------------------------------------------------------------ $\dim {\mathrm{UR}}(M)$ $\dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) $ $\dim {{\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}}(M) $ $M$ ${\mathcal{E}}$ CM ${\mathcal{E}}$ not CM Q, R torsion 0 2 4 $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}\times {\mathbb{G}}_m]$ ($\Rightarrow$ P torsion) $ u(1)=(0,1)$ P,Q torsion 1 3 5 $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}\times {\mathbb{G}}_m] $ (R not torsion) $u(1)=(0,R) $ R torsion 2 4 6 $M=[u:{\mathbb{Z}}\rightarrow G]$ ($\Rightarrow$ P torsion) $ u(1)=0$ Q torsion 3 5 7 $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}\times {\mathbb{G}}_m]$ (P and R not torsion) $u(1)=(P,R) $ P torsion 3 5 7 $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}^* \times {\mathbb{G}}_m] $ (R and Q not torsion) $u(1)=(Q,R) $ P,Q 5 7 9 $M=[u:{\mathbb{Z}}\rightarrow G]$ ${\mathrm{End}}( {\mathcal{E}}) \otimes_{\mathbb{Z}}{\mathbb{Q}}$-lin indep $ u(1)=R$ ----------------------------------------------------------------------------- ------------------------- -------------------------------------------------------- -------------------------------------------------------- ------------------------------------------------------------------------ We can now state explicitly the Generalized Grothendieck’s Period Conjecture (\[eq:GCP\]) for the 1-motives involved on the above table: - $R$ and $Q$ are torsion: We work with the 1-motive $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}\times {\mathbb{G}}_m], u(1)=(0,1)$ or $M=[0 \to {\mathcal{E}}].$ If ${\mathcal{E}}$ is not CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2 \Big)\geq 4$$ that is 4 at least of the 6 numbers $ g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2$ are algebraically independent over ${\mathbb{Q}}$. If ${\mathcal{E}}$ is CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\eta_1 \Big)\geq 2$$ that is 2 at least of the 4 numbers $g_2,g_3,\omega_1,\eta_1$ are algebraically independent over ${\mathbb{Q}}.$ If we assume $g_2,g_3 \in \overline{{\mathbb{Q}}},$ we get the Chudnovsky Theorem: $\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}(\omega_1,\eta_1 )=2.$ - $P$ and $Q$ are torsion: We work with the 1-motive $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}\times {\mathbb{G}}_m], u(1)=(0,R)$ (this case was studied in the author’s Ph.D thesis, see [@B02]). If ${\mathcal{E}}$ is not CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\omega_2,\eta_1,\eta_2,R, \log(R) \Big)\geq 5$$ that is 5 at least of the 8 numbers $g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2,R, \log(R)$ are algebraically independent over ${\mathbb{Q}}$. If ${\mathcal{E}}$ is CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\eta_1,R, \log(R) \Big)\geq 3$$ that is 3 at least of the 6 numbers $g_2,g_3,\omega_1,\eta_1,R, \log(R)$ are algebraically independent over ${\mathbb{Q}}$. - $R$ is torsion: We work with the 1-motive $M=[u:{\mathbb{Z}}\rightarrow G], u(1)=0$ or $M=[v^*:{\mathbb{Z}}\to {\mathcal{E}}^*], v^*(1)=Q.$ If ${\mathcal{E}}$ is not CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2,Q,q, \zeta(q) \Big)\geq 6$$ that is 6 at least of the 9 numbers $g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2,Q,q, \zeta(q)$ are algebraically independent over ${\mathbb{Q}}$. If ${\mathcal{E}}$ is CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\eta_1,Q, q, \zeta(q) \Big)\geq 4$$ that is 4 at least of the 7 numbers $ g_2,g_3,\omega_1,\eta_1,Q, q, \zeta(q)$ are algebraically independent over ${\mathbb{Q}}$. - $Q$ is torsion: We work with the 1-motive $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}\times {\mathbb{G}}_m], u(1)=(P,R)$ (this case was considered in the author’s Ph.D thesis, see [@B02]). If ${\mathcal{E}}$ is not CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2, P,R,p,\zeta(p), \log(R) \Big)\geq 7$$ that is 7 at least of the 11 numbers $g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2, P,R,p,\zeta(p), \log(R) $ are algebraically independent over ${\mathbb{Q}}$. If ${\mathcal{E}}$ is CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big(g_2,g_3,\omega_1,\eta_1, P,R,p,\zeta(p), \log(R) \Big)\geq 5$$ that is 5 at least of the 9 numbers $g_2,g_3,\omega_1,\eta_1, P,R,p,\zeta(p), \log(R) $ are algebraically independent over ${\mathbb{Q}}$. - $P$ is torsion: We work with the 1-motive $M=[u:{\mathbb{Z}}\rightarrow G], u(1)=R \in {\mathbb{G}}_m(K)$ or $M=[u:{\mathbb{Z}}\rightarrow {\mathcal{E}}^* \times {\mathbb{G}}_m], u(1)=(Q,R).$ If ${\mathcal{E}}$ is not CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2,Q,R, q, \zeta(q),\log (R) \Big)\geq 7$$ that is 7 at least of the 11 numbers $ g_2,g_3,\omega_1,\omega_2,\eta_1, \eta_2,Q,R, q, \zeta(q),\log (R) $ are algebraically independent over ${\mathbb{Q}}$. If ${\mathcal{E}}$ is CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2,g_3,\omega_1,\eta_1,Q,R, q, \zeta(q),\log (R) \Big)\geq 5$$ that is 5 at least of the 9 numbers $ g_2,g_3,\omega_1,\eta_1,Q,R, q, \zeta(q),\log (R) $ are algebraically independent over ${\mathbb{Q}}$. - $P,Q,R$ are not torsion and $P,Q$ are ${\mathrm{End}}( {\mathcal{E}}) \otimes_{\mathbb{Z}}{\mathbb{Q}}$-linearly independent: We work with the 1-motive $M=[u:{\mathbb{Z}}\rightarrow G], u(1)=R \in G(K).$ If ${\mathcal{E}}$ is not CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2, g_3, Q, R, \omega_1,\omega_2,\eta_1,\eta_2, p,\zeta(p),q, \zeta(q), \eta_1 q - \omega_1 \zeta(q),\eta_2 q - \omega_2 \zeta(q) , \log f_q(p)+l \Big)\geq 9$$ that is 9 at least of the 15 numbers $ g_2, g_3, Q, R, \omega_1,\omega_2,\eta_1,\eta_2, p,\zeta(p),q, \zeta(q), \eta_1 q - \omega_1 \zeta(q),\eta_2 q - \omega_2 \zeta(q) , \log f_q(p) $ are algebraically independent over ${\mathbb{Q}}$. If ${\mathcal{E}}$ is CM, $$\mathrm{tran.deg}_{{\mathbb{Q}}}\, {\mathbb{Q}}\Big( g_2, g_3, Q, R, \omega_1,\eta_1, p,\zeta(p),q, \zeta(q), \eta_1 q - \omega_1 \zeta(q),\eta_2 q - \omega_2 \zeta(q) , \log f_q(p) +l \Big)\geq 7$$ that is 7 at least of the 13 numbers $ g_2, g_3, Q, R, \omega_1,\eta_1, p,\zeta(p),q, \zeta(q), \eta_1 q - \omega_1 \zeta(q),\eta_2 q - \omega_2 \zeta(q) , \log f_q(p) $ are algebraically independent over ${\mathbb{Q}}$. [10]{} Y. André. Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses \[Panoramas and Syntheses\], 17. Société Mathématique de France, Paris, 2004. Y. André. A note on 1-motives. to appear in Int. Math. Res. Not. (2019). J. Ayoub. L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle. J. Reine Angew. Math. 693 (2014), I: 1–149, II: 151–226. C. Bertolin. The Mumford-Tate group of 1-motives. Ann. Inst. Fourier (Grenoble) 52 (2002), no. 4, 1041–1059. C. Bertolin. Périodes de 1-motifs et transcendance. J. Number Theory 97 (2002), no. 2, 204–221. C. Bertolin. Le radical unipotent du groupe de Galois motivique d’un 1-motif. Math. Ann. 327 (2003), no. 3, 585–607. C. Bertolin. Motivic Galois theory for 1-motives Ann. Sci. Math. Québec 32 (2008), no. 2, 105–124. D. Bertrand. Schanuel’s conjecture for non-isoconstant elliptic curves over function fields. In *Model theory with applications to algebra and analysis*, Vol. 1, 41–62, London Math. Soc. Lecture Note Ser., 349, Cambridge Univ. Press, Cambridge, 2008. J.-L. Brylinski. 1-motifs et formes automorphes (théorie arithmétique des domaines des Siegel). Pub. Math. Univ. Paris VII 15 (1983). P. Deligne. Théorie de Hodge III. Inst. Hautes Études Sci. Publ. Math. No. 44 (1974), 5–77. P. Deligne. Le groupe fondamental de la droite projective moins trois points. Galois group over ${\mathbb{Q}}$, Math. Sci. Res. Inst. Pub. 16. 1989. P. Deligne. Catégories tannakiennes. The Grothendieck Festschrift II, Birkhäuser 87, 1990. A. Grothendieck. On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. No. 29, 1966, 95–103. O. Jacquinot and K. Ribet. Deficient points on extensions of abelian varieties by ${{\mathbb{G}}}_{m}$. J. Number Th. No. 25, 1987, 133–151. S. Lang. Introduction to transcendental numbers. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966. B. Mazur and W. Messing. Universal extensions and one dimensional crystalline cohomology. Lecture Notes in Mathematics, Vol. 370. Springer-Verlag, Berlin-New York, 1974. B. Moonen. Notes on Mumford-Tate groups. Notes de cours du centre Emile Borel, N. 19 (1999). D. Mumford. Abelian varieties. Oxford UP, 1974. M. Nori. Course at Tata Int. Fund. Res. Notes by N. Fakhruddin, Mumbai (2000). N. Saavedra Rivano. Catégories tannakiennes. Lecture Notes in Mathematics, Vol. 265. Springer-Verlag, Berlin-New York, 1972. ii+418 pp. G. Springer. Introduction to Riemann Surfaces. Addison-Wesley Publishing Company, Inc., Reading, Mass. 1957. viii+307 pp. M. Waldschmidt. Nombres transcendants et fonctions sigma de Weierstrass. C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 2, 111–114. M. Waldschmidt. Nombres transcendants et groupes algébriques. With appendices by Daniel Bertrand and Jean-Pierre Serre. Astérisque No. 69–70 (1987). E. T. Whittaker and G. N. Watson. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Reprint of the fourth (1927) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. vi+608 pp. Letter of Y. André {#letter-of-y.-andré .unnumbered} ================== , Following your query, I will try to summarize the formalism of Grothendieck’s period conjecture, present different variants, sketch their relations and give some historical hints and references. . Grothendieck’s period conjecture deals with transcendence properties of periods of algebraic varieties defined over a number field. In essence, it predicts that algebraic relations between periods come from geometry. Its first mention appears as a footnote in Grothendieck’s letter to Atiyah (Publ. IHES 29, 1966) where, after mentioning Schneider’s results on elliptic periods, he alludes to the existence of a general conjecture. A first published statement is contained in Lang’s book on transcendental number theory (Addison-Wesley 1966, chap. 4, historical note). The next published related statement, without mention of Grothendieck’s name/conjecture, is at the beginning of Deligne’s paper “Hodge cycles on abelian varieties" (Springer LN 900, 1982; see also the end of its announcement, Bull. SMF. 1980). The next published statement, and explanation of the relationship between the previous statements, is in chapter 9 of my book on G-functions (Vieweg 1989, recently reprinted by Springer), entitled “towards Grothendieck’s conjecture on periods of algebraic manifolds". A more complete exposition of the formalism, and its relation to a fullness conjecture of enriched realization of motives (parallel to the Hodge or Tate conjectures) is discussed in my SMF book on motives, denoted henceforth \[IM\] (2004); related material forms the whole third part of that book. In \[IM, 23.4-5\] and in other contemporary papers, I extended the period conjecture in two directions: $i)$ the idea of Galois theory of periods, $ii)$ the generalization of the period conjecture for motives defined over an arbitrary subfield of $\bf C$. A different but related thread came with Kontsevich’s period conjecture (preliminary version by the SMF, 1998; final version with Zagier: “periods", Springer 2001). Among other things, he conjectured that algebraic relations between periods come from the formal properties of $\int$, and indicated (relying on Nori’s work) that this conjecture is equivalent to Grothendieck’s period conjecture for all motives over number fields. . In order to express appropriately his intuition that algebraic relations between periods should come from geometry, Grothendieck uses his idea of motives and motivic Galois theory[^1]. At Grothendieck’s time, however, this theory was only a dream (with precise contours), so that the period conjecture was more a metaconjecture than a conjecture (in the sense that some terms were not well-defined); but some consequences of the conjecture could be formulated in well-defined terms, and all ensuing statements, however remote from the original intuition, were called “period conjectures”, creating many an ambiguity. Nowadays, there is an unconditional tannakian category of motives over any field $k$ of characteristic zero, which is of “geometric" nature (in the sense that the morphisms arise “somehow“ from algebraic correspondences), so that the conjecture can be neatly stated, and reflects the original intuition. In fact, three such theories have been constructed: the first (restricted to pure motives, i.e. motives coming from projective smooth $k$-varieties) was defined in my IHES paper (1996); the second by Nori (unpublished notes have circulated, and there is now the book by Huber and Müller-Stach, Springer 2017), the third by Ayoub (he defines a motivic Galois group using Voevodsky’s triangulated motives). These constructions look quite different, but turn out to be compatible (as shown by Arapura, Choudhury/Gallauer): in short, Nori’s and Ayoub’s absolute mixed motivic Galois groups over $k$ are ”the same", and my absolute pure motivic Galois group is just its pro-reductive quotient. In the sequel, I will thus speak about “the" motivic Galois group $G_{mot}(M)$ of a motive $M$ defined over $k\subset \bf C$, without being more explicit: this is a well-defined linear algebraic group defined over $\bf Q$ (a closed subgroup of the group of linear automorphisms of the Betti realization of $M$), which is reductive if $M$ is pure. The absolute motivic Galois group of $k$ is “their projective limit" for various $M$. These groups contain other previously defined groups: $$G_{mot}(M)\supset MTA(M) \supset MT(M),$$ where $MT(M)$ is the Mumford-Tate group attached to (the mixed Hodge structure of) $M_{\bf C}$, and $MTA(M)$ is the absolute Mumford-Tate group defined by Deligne (a.k.a. “absolute Hodge motivic Galois group"). The definition and computation of $MT(M)$ being easier than the others, it is interesting to know when these groups coincide. When $M$ is an abelian variety or a $1$-motive, defined over an algebraically closed $k \subset \bf C$, Deligne and Brylinsky proved that $MTA(M) \supset MT(M)$ is an equality, and I later proved the stronger statement that $G_{mot}(M) = MT(M)$ in those cases (Imrn 2019). . For any (pure or mixed) motive $M$ over $k \subset \bf C$, $G_{mot}(M)$ is defined as the group scheme of $\otimes$-automorphisms of the Betti realization of the tannakian category $\langle M\rangle$ generated by the motive $M$. One may also consider the algebraic De Rham realization, with values in $k$-vectors spaces. The scheme of $\otimes$-isomorphisms from De Rham to Betti$\otimes k$ is a torsor under $G_{mot}(M)_k$, the period torsor $\Pi(M)$. The name comes from the fact that integration gives rise to the “period isomorphism" $$\int: \,H_{dR}(M)\otimes_k {\bf C} \cong H_B(M)\otimes_{\bf Q}\bf C$$ (concretely, a matrix with entries the periods of $M$), and further to a canonical $\bf C$-point of $\Pi(M)$: $$\varpi: {\rm{Spec}}\,{\bf C}\to \Pi(M).$$ $(?) $ [*If $k\subset \bar{\bf Q}$, then $\varpi$ maps to the generic point of $\Pi(M)$.* ]{} In more heuristic words, [*the periods of $M$ generate the $k$-algebra of functions of the $k$-variety $\Pi(M)$*]{}, or else: [*the algebraic relations between periods of $M$ come from the morphisms in $\langle M\rangle$*]{} (which are of “geometric" nature). I insist that this should hold for any motive (pure or mixed) defined over any algebraic field $k$. The conjecture includes the subconjecture that $\Pi(M)$ is irreducible (=[connected]{}, since this is a torsor). In fact, it is equivalent to the connectedness of $\Pi(M)$, plus equality of dimensions: the dimension of the $k$-Zariski closure of the image of $\varpi$ is the dimension of $\Pi(M)$. By the relation between dimension and transcendence degree in commutative algebra, the former dimension is nothing but the transcendence degree over $k$ (or $\bf Q$) of the $k$-subalgebra of $\bf C$ generated by the periods of $M$, and the latter dimension is ${\rm{dim}}\, G_{mot}(M)$ since $\Pi(M)$ is a torsor under $G_{mot}(M)_k$. Therefore $(?)$ is equivalent to the [*connectedness of $\Pi(M)$*]{}, plus the equality: $$(??)\;\;\; {\rm{transc. deg}}_{\bf Q}\, k({\rm{periods}}(M)) = {\rm{dim}}\, G_{mot}(M).$$ This formulation is more congenial to transcendental number theorists (provided of course that one knows how to calculate or at least estimate the right hand side), while $(?)$ is more geometric. In some “applications" of the period conjecture, it may be necessary to take into account the geometry of the period torsor, and not just the numerical identity $(??)$, cf. e.g. \[IM 23.2\]. The formulation given in Lang’s book is the following: assume that $M$ is the motive of a projective smooth $k$-variety $X$, and note that any algebraic cycle on $X^n, n\in \bf N,$ has a De Rham class in $H_{dR}(X^n)=H_{dR}(X)^{\otimes n}$ and a Betti class in $H_{B}(X^n)=H_{B}(X)^{\otimes n}$, hence gives rise via $\int$ to polynomial relations of degree $n$ between periods of $M$; the conjecture predicts that $(???)\;\;\; $ [*these relations generate an ideal of definition for the period matrix of $M$.*]{} If one assumes Grothendieck’s standard conjecture[^2], $G_{mot}(M)$ is the group which fixes the classes of algebraic cycles in tensor powers of $H_B(M)$, and parallely, the previous period relations are equations for $\Pi(M)$. It is not difficult to deduce from there that [*in the pure case, $(???)$ is equivalent to $(?)$ plus the standard conjecture*]{}. The relation with Kontsevich’s period conjecture becomes apparent if one considers all motives together, and not just $\langle M\rangle $[^3]. A natural framework to deal with period problems is the tannakian category $Vec_{k, \bf Q}$ (appearing in \[IM, 7.5\]) consisting of a $k$-vector space, a $\bf Q$-vector space, and an isomorphism between their complexifications. De Rham and Betti realizations, together with $\varpi$, give rise to a $\otimes$-functor $dRB$ from $\langle M\rangle $ to $Vec_{k, \bf Q}$. The period conjecture $(?)$ implies that this functor is [*full*]{}. But fullness of $dRB$ is a much weaker conjecture[^4]. Let $G_{k, \bf Q}(M)$ be the tannakian group attached to $dRB(M)\in Vec_{k, \bf Q}$, a group defined purely in terms of the periods of $M$. One has $MTA(M)\supset G_{k, \bf Q}(M)$ , and the image of $\varpi$ lies in a $G_{k, \bf Q}(M)$-torsor contained in $\Pi(M)$. The fullness of $dRB$ follows if this torsor is $\Pi(M)$ itself (and conversely, if $G_{k, \bf Q}(M)$ is a so-called observable subgroup of $G_{mot}(M)$). . In $(??)$, inequality $\leq$ is unconditional. Moreover, it also holds with $G_{mot}(M)$ replaced by $MTA(M)$ or $G_{k, \bf Q}(M)$. It follows that $(?)$ implies $G_{mot}(M) = MTA(M)= G_{k, \bf Q}(M)$. In fact, $(??)$ splits into two equalities $$(??)'\;\;\; {\rm{transc. deg}}_{\bf Q}\, k({\rm{periods}}(M)) = {\rm{dim}}\, G_{k, \bf Q}(M) = {\rm{dim}}\, G_{mot}(M)$$ which can be studied separately. On the other hand, one can weaken inequality $\geq$ in $(??)$ on replacing $G_{mot}(M)$ by $MT(M)$ (but the reverse inequality becomes unclear; and in doing so, one looses the essence of the conjecture, which is that relations between periods should come from geometry[^5]). What is coined “motivic periods" or “formal periods" in the literature refers essentially to coordinates on $\Pi(M)$, or (less appropriately) on the corresponding torsors under $MTA(M)$ or $G_{k, \bf Q}(M)$, depending on the context; of course, under $(?)$, they may be “identified" with actual periods. The first published version of this conjecture (which I made around 1997) is \[IM, 23.4\]. In analogy with $(??)$, it predicts that for any $k\subset \bf C$, and any (pure or mixed) motive $M$ defined over $k$, $$(?!)\;\;\; {\rm{transc. deg}}_{\bf Q}\, k({\rm{periods}}(M)) \geq {\rm{dim}}\, G_{mot}(M).$$ Of course, since $k$ may contain the periods, one cannot hope for an equality. The first test that I made before stating it was the case of $1$-motives without abelian part, in which case one recovers Schanuel’s conjecture. Later, you studied many other cases in detail and gave evidence that this conjecture looks sharp and might be optimal. On the other hand, I am not aware of any “geometric" version of this conjecture in the style of $(?)$[^6]. To be complete, I ought to discuss the long story of the functional analog of Grothendieck’s period conjecture and Ayoub’s work which settles it (Ann. Maths 2015). But this letter is already too long, and as this lies beyond your query, I will content myself with the following indications. If $M$ is a motive over a function field $k$ in one variable over $\bf C$, one can define periods of $M$ as elements of the completion $\hat k$ at any place of good reduction for $M$. These periods are solutions of a Picard-Fuchs differential equation (a linear differential equation with coefficients in $k$). One can define a period torsor $\Pi(M)$ in this context, which is a torsor under the differential Galois group (= algebraic monodromy group, since the singularities are regular), as well as a canonical $\hat k$-point. By Kolchin’s theorem, the image of this point is Zariski-dense in $\Pi(M)$. It remains to relate the monodromy group to motives. This has been done by Ayoub[^7]: here, the algebraic mondromy group coincides with the [*relative motivic Galois group*]{}, i.e. the kernel of the map $G_{mot}(M)\to G_{mot, cst}(M)$ dual to the inclusion of the category of constant motives inside $\langle M\rangle$. My Bourbaki survey (1995) touches all the subjects of this letter in greater detail. With my best wishes, Yves. Appendix by M. Waldschmidt: Third kind elliptic integrals and transcendence {#appendix-by-m.-waldschmidt-third-kind-elliptic-integrals-and-transcendence .unnumbered} =========================================================================== This short appendix aims at giving references on papers related with transcendence results concerning elliptic integrals of the third kind. So far, results on transcendence and linear independence are known, but there are very few results on algebraic independence. In his book on transcendental numbers [@Schneider], Th. Schneider proposes eight open problems, the third of which is : [*Try to find transcendence results on elliptic integrals of the third kind.*]{} In [@Lang Historical Note of Chapter IV], S. Lang explains the connections between elliptic integrals of the second kind, Weierstrass zeta function and extensions of an elliptic curve by ${\mathbb{G}}_a$. He applies the so–called Schneider–Lang criterion to the Weierstrass elliptic and zeta functions and deduces the transcendence results due to Th. Schneider on elliptic integrals of the first and second kind. At that time, it was not known how to use this method for proving results on elliptic integrals of the third kind. The solution came from [@Serre], where J-P. Serre introduces the functions $f_q$ (with the notation of [@Bertolin]) related to elliptic integrals of the third kind, which satisfy the hypotheses of the Schneider-Lang criterion and are attached to extensions of an elliptic curve by ${\mathbb{G}}_m$. This is how the first transcendence results on these integrals were obtained [@MWAsterisque; @MWCanada]. In [@BertrandLaurent], D. Bertrand and M. Laurent give further applications of the Schneider-Lang criterion involving elliptic integrals of the third kind. Applications are given in [@Bertrand1983a; @Bertrand1983b; @Scholl1986], dealing with the Neron–Tate canonical height on an elliptic curve (including the $p$–adic height) and the arithmetic nature of Fourier coefficients of Eisenstein series. A first generalization to abelian integrals of the third kind is quoted in [@Bertrand1983b]. Transcendence measures are given in [@Reyssat1980AnnToulouse]. Properties of the smooth Serre compactification of a commutative algebraic group and of the exponential map, together with the links with integrals, are studied in [@FaltingsWustholz1984]. See also [@KnopLange1985]. In [@Masser2016 Chapter 20 – Elliptic functions] (see in particular Theorem 20.11 and exercises 20.104 and 20.105) more details are given on the functions associated with elliptic integrals of the third kind, the associated algebraic groups, which are extensions of an elliptic curve by ${\mathbb{G}}_m$, and the consequences of the Schneider-Lang criterion. The first results of linear independence of periods of elliptic integrals of the third kind are due to M. Laurent [@LaurentCrelle1980; @LaurentCrelle1982] (he announced his results in [@LaurentCRAS; @LaurentDPP]). The proof uses Baker’s method. More general results on linear independence are due to G. Wüstholz [@WustholzCrelle1984] (see also [@BakerWustholz2007 § 6.2]), including the following one, which answers a conjecture that M. Laurent stated in [@LaurentCrelle1982] where he proved special cases of it. Let $\wp$ be a Weierstrass elliptic function with algebraic invariants $g_2$, $g_3$. Let $\zeta$ be the corresponding Weierstrass zeta function, $\omega$ a nonzero period of $\wp$ and $\eta$ the corresponding quasi-period of $\zeta$. Let $u_1,\ldots,u_n$ be complex numbers which are not poles of $\wp$, which are ${\mathbb{Q}}$ linearly independent modulo ${\mathbb{Z}}\omega$ and such that $\wp(u_1),\dots,\wp(u_n)$ are algebraic. Define $$\lambda(u_i,\omega)=\omega\zeta(u_i)-\eta u_i.$$ Then the $n+3$ numbers $$1,\omega,\eta,\lambda(u_1),\ldots,\lambda(u_n)$$ are linearly independent over $\Qbar$. The question of the transcendence of the nonvanishing periods of a meromorphic differential form on an elliptic curve defined over the field of algebraic numbers is now solved [@BakerWustholz2007 Theorem 6.6]. See also [@HW2018], as well as [@Tretkoff2017 § 1.5] for abelian integrals of the first and second kind. A reference of historical interest to a letter from Leibniz to Huygens in 1691 is quoted in [@BakerWustholz2007 § 6.3] and [@Wustholz2012]. The only results on algebraic independence related with elliptic integrals of the third kind so far are those obtained by ƒ. Reyssat [@Reyssat1980Cras; @Reyssat1982] and by R. Tubbs [@Tubbs1987; @Tubbs1990]. We are very far from anything close to the conjectures in [@Bertolin]. For a survey (with an extensive bibliography including 254 entries), see [@MW2008]. The references below are listed by chronological order. [BeLau1981]{} Schneider, Theodor. Einführung in die transzendenten Zahlen. Springer-Verlag, Berlin-Gšttingen-Heidelberg, 1957. Lang, Serge. Introduction to transcendental numbers. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966. Serre, Jean–Pierre. Quelques propriétés des groupes algébriques commutatifs. Appendice II de [@MWAsterisque], 191 – 202. Waldschmidt, Michel. Nombres transcendants et groupes algŽbriques. With appendices by Daniel Bertrand and Jean-Pierre Serre. AstŽrisque No. 69-70 (1979), 218 pp.\ https://smf.emath.fr/publications/nombres-transcendants-et-groupes-algebriques-2e-edition Waldschmidt, Michel. Nombres transcendants et fonctions sigma de Weierstrass. C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 2, 111Ð114. Laurent, Michel. Transcendance de périodes d’intégrales elliptiques. C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 15, 699–701. Laurent, Michel. Transcendance de périodes d’intégrales elliptiques. Séminaire Delange-Pisot-Poitou, 20e année: 1978/1979. Théorie des nombres, Fasc. 1, Exp. No. 13, 4 pp. Laurent, Michel. Transcendance de périodes d’intégrales elliptiques. J. Reine Angew. Math. 316 (1980), 122–139. Reyssat, Éric. Approximation de nombres liés â la fonction sigma de Weierstrass. Ann. Fac. Sci. Toulouse Math. (5) 2 (1980), no. 1, 79–91. Reyssat, Éric. Fonctions de Weierstrass et indépendance algébrique. C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 10, A439–A441. Bertrand, Daniel & Laurent, Michel. Propriétés de transcendance de nombres liés aux fonctions thêta. C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 16, 747–749. Laurent, Michel. Transcendance de périodes d’intégrales elliptiques. II. J. Reine Angew. Math. 333 (1982), 144–161. Reyssat, Éric. Propriétés d’indépendance algébrique de nombres liés aux fonctions de Weierstrass. Acta Arith. 41 (1982), no. 3, 291–310. Bertrand, Daniel. Problémes de transcendance liés aux hauteurs sur les courbes elliptiques. Mathematics, pp. 55–63, CTHS: Bull. Sec. Sci., III, Bib. Nat., Paris, 1981. Bertrand, Daniel. Endomorphismes de groupes algébriques; applications arithmétiques. Diophantine approximations and transcendental numbers (Luminy, 1982), 1Ð45, Progr. Math., 31, Birkhäuser Boston, Boston, MA, 1983. Wüstholz, Gisbert. Transzendenzeigenschaften von Perioden elliptischer Integrale. J. Reine Angew. Math. 354 (1984), 164Ð174. Faltings, Gert & Wüstholz, Gisbert. Einbettungen kommutativer algebraischer Gruppen und einige ihrer Eigenschaften. J. Reine Angew. Math. 354 (1984), 175–205. Knop, Friedrich & Lange, Herbert. Some remarks on compactifications of commutative algebraic groups. Comment. Math. Helv. 60 (1985), no. 4, 497–507. Scholl, Antony. Fourier coefficients of Eisenstein series on noncongruence subgroups. Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 11–17. Tubbs, Robert. Algebraic groups and small transcendence degree. I. J. Number Theory 25 (1987), no. 3, 279–307. Tubbs, Robert. Algebraic groups and small transcendence degree. II. J. Number Theory 35 (1990), no. 2, 109–127. Baker, Alan & Wüstholz, Gisbert. Logarithmic forms and Diophantine geometry. New Mathematical Monographs, 9. Cambridge University Press, Cambridge, 2007. Waldschmidt, Michel. Elliptic functions and transcendence. Surveys in number theory, 143Ð188, Dev. Math., 17, Springer, New York, 2008. https://webusers.imj-prg.fr/ michel.waldschmidt/articles/pdf/SurveyTrdceEllipt2006.pdf Wüstholz, Gisbert. Leibniz’ conjecture, periods & motives. Colloquium De Giorgi 2009, 33Ð42, Colloquia, 3, Ed. Norm., Pisa, 2012. Masser, David. Auxiliary polynomials in number theory. Cambridge Tracts in Mathematics, 207. Cambridge University Press, Cambridge, 2016. Tretkoff, Paula. Periods and special functions in transcendence. Advanced Textbooks in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. Huber, Annette & Wstholz, Gisbert. Periods of 1-motives.\ arxiv.org/abs/1805.10104 Bertolin, Cristiana. Third kind elliptic integrals and 1-motives. [^1]: this statement is not just a conjecture in the history of mathematics: a decade ago in Montpellier, I had the priviledge to consult some unpublished notes by Grothendieck on motives (which by now may be online: (https://grothendieck.umontpellier.fr/archives-grothendieck/, cotes 10-19 - thanks to J. Fresan for the reference), and I saw that he really wrote the period conjecture essentially as formulated below. [^2]: usually, one writes: standard conjecture[*s*]{}, but in characteristic $0$ they amount to one single statement, cf. e.g. \[IM, chap. 5\]. [^3]: however, given your special interest in the case of $1$-motives, let me mention the recent work of Huber and Wüstholz, who manage to formulate a period conjecture in Kontsevich’s style just for $1$-motives. [^4]: for instance, if $M$ is the motive of an elliptic curve, fullness follows from known results in transcendental number theory, while $(?)$ is known only in the presence of complex multiplication. Another illustration of the difference arises if one considers all abelian varieties with complex multiplication by a cyclotomic field: fullness of $dRB$ (resp. period conjecture) is equivalent in this case to Rohrlich’s (resp. Lang’s) conjecture that all monomial (resp. algebraic) relations between special values of the gamma function come from the functional relations. There are several more recent results in the spirit of this fullness conjecture (Andreatta/Barbieri-Viale/Bertapelle, Huber/Wüstholz, Kahn, myself), but virtually nothing new about $(?)$. [^5]: but in the special case of $1$-motives, one actually looses nothing by my aformentioned results. [^6]: there are many other related open questions; e.g. recent work by Fresan and Jossen, following an intuition of Kontsevich, has shaped the contours of a theory of “exponential motives". A period conjecture in the style of $(??)$ may hold for them. Does it follow from $(?!)$? [^7]: there are unpublished similar works by Nori and by Jossen; and a related published result by Arapura, Adv. Math 233 (2013).

--- abstract: 'As semiconductor devices scale to new dimensions, the materials and designs become more dependent on atomic details. NEMO5 is a nanoelectronics modeling package designed for comprehending the critical multi-scale, multi-physics phenomena through efficient computational approaches and quantitatively modeling new generations of nanoelectronic devices as well as predicting novel device architectures and phenomena. This article seeks to provide updates on the current status of the tool and new functionality, including advances in quantum transport simulations and with materials such as metals, topological insulators, and piezoelectrics.' author: - 'J. E. Fonseca' - 'T. Kubis' - 'M. Povolotskyi' - 'B. Novakovic' - 'A. Ajoy' - 'G. Hegde' - 'H. Ilatikhameneh' - 'Z. Jiang' - 'P. Sengupta' - 'Y. Tan' - 'G. Klimeck' bibliography: - 'References\_N5BiTe.bib' - 'exportlistkeys.bib' - 'refs\_Unfolding.bib' - 'apssamp.bib' date: 'Received: date / Accepted: date' title: Efficient and realistic device modeling from atomic detail to the nanoscale --- Introduction {#intro} ============ Relentless downscaling of transistor size has continued according to Moore’s law for the past 40 years. Transistor size will continue to decrease in the next ten years, but foundational issues with currently unknown technology approaches must be pursued [@2012]. This downscaling has reached the range where the number of atoms in critical dimensions is countable, geometries are formed in three dimensions and new materials are being introduced. Under these conditions we argue that the overall geometry constitutes a new material that cannot be found as such in nature [@Fuechsle2012]. Quantum effects such as tunneling, state quantization, and atomistic disorder dominate the characteristics of these nano-scale devices. The interactions of electrons, photons, and phonons are now governed by these new material properties and long-range interactions such as strain and gate fields. The end-game of the transistor size down-scaling as we know it is now fundamentally in sight. The end-game transistor is expected to be about 5nm long and 1nm in its critical active region corresponding to about 5 atoms in width. The physical atomistic down-scaling limit will be reached in about 8-10 years. The overall agenda is to bridge *ab initio* materials science into TCAD simulations of realistically large scaled devices and get macroscopic quantities like current, voltages, absorption, etc., by mapping *ab initio* into basis sets of lower order and include them in a formalism that allows for transport. The NEMO5 nanoelectronics modeling software is aimed at comprehending the critical multi-scale, multi-physics phenomena and delivering results to engineers, scientists, and students through efficient computational approaches and quantitatively modelling new generations of nanoelectronic devices in industry, as well as predicting novel device architectures and phenomena. The basic functionality and history of the NEMO tool suite has been discussed previously [@Steiger2011; @Sellier2012]. NEMO5’s general software framework can easily include any kind of atomistic model and even semi-classical models if necessary. The scalable software implements Schrödinger’s equation and non-equilibrium Green’s function method (NEGF) in tight-binding formalism, for electronic structure and transport calculations, respectively. It also is able to take into account important effects such as atomistic strain, using valence force field (VFF) strain models. It then allows the calculation of electronic band structures, charge density, current and potential, eigen-energies and wave-functions, phonon spectra, etc., for a large variety of semiconductor materials and devices. This manuscript seeks to provide interested readers with an overview of the recent developments surrounding NEMO5. This paper discusses approaches a) to quantum transport solutions, b) newly-implemented approaches to achieve faster convergence in the self-consistent Poisson-transport solution, c) strain, d) phonons, e) semi-automated material parameterization, f) metals, g) piezoelectric materials such as SmSe, h) topological insulators and i) band structure unfolding. Transport ========= At the heart of NEMO5’s quantum transport approach is the non-equilibrium Green’s function method (NEGF) which is a computational approach to handle quantum transport in nanoelectronic devices [@Datta2005]. NEGF is numerically expensive when applied on atomistic tight-binding representations. NEGF requires storage, inversion and multiplication of matrices of the order of the number of electronic degrees of freedom. A well known method to ease the numerical burden is the recursive Green’s function method (RGF) that allows for limiting the calculation and storage of the retarded Green’s function to specific matrix blocks (such as only block diagonals and a single block column). Until recently, the RGF algorithm was limited to quasi 1D transport regimes, i.e. devices with 2 leads only. Generalizing work of Cauley et al., however, shows that RGF can be applied on virtually any transport problem, if the device Hamiltonian matrix is partitioned in a proper way [@Cauley2011]. NEMO5 allows partitioning the device ideally for 1D and quasi 1D transport problems according to the transport coordinate, but it also allows for the partition of complex, multi terminal devices and the application of RGF on them. Despite the RGF method, the computational burden in memory and CPU time is still limiting the maximum device size solvable with NEGF. To overcome this obstacle, NEMO5 offers incomplete spectral transformations of NEGF equations into a Hilbert space of smaller rank than the original tight-binding representation [@Zeng2013]. Special cases of this low rank approximation are known as CBR method (all ballistic NEGF) [@Mamaluy2003] and the mode space approach [@Wang2004]. This method allows approximating NEGF transport problems in electronic tight-binding representations within a fraction of the numerical load of exact NEGF solutions. The loss of the NEGF accuracy and predictive power is thereby negligible as shown in Fig.  \[figLRA\] This figure compares the conduction band electron density of a homogeneous 5x5nm Si nanowire in equilibrium calculated in an exact and a LRA-approximate NEGF calculation where the rank has been reduced down to 10% of the original problem size. Negligible discrepancies are magnified in the figure’s inset. Purely ballistic charge transport can be well described within the quantum transmitting boundary method (QTBM) [@Ting1992]. Since this method solves the quantum transport in the space of propagating lead modes, the numerical load is typically much smaller than in ballistic NEGF or RGF calculations which in general consider all modes. NEMO5 is able to solve the QTBM equations spatially distributed over large numbers of CPUs. For a given energy and transverse momentum, the boundary equations of the source and the drain are solved each on individual CPUs, whereas those sections of the device that are not in direct contact with the leads are solved on the remaining CPUs. ![ Comparison of the electron density of the exact NEGF calculation (circle) and of NEGF calculations with 10% of the original matrix rank.[]{data-label="figLRA"}](LRA.eps){width="48.00000%"} Self-consistent calculation =========================== The many-body problem is treated in the Hartree approximation by self-consistently solving the Poisson and transport equations (e.g. QTBM, as explained above). The self-consistent solution is a nonlinear problem and any efficient solution of this problem must take into account at least three components: the energy grid, the initial guess, and the self-consistent algorithm. The energy grid should resolve the features in the energy dependent device charge density, determined by the lead density of states and device transmission properties. A good energy grid should be inhomogeneous, so that it is able to resolve sharp features, yet have as few energy points as possible to facilitate efficient computations. Since the self-consistent process is necessarily iterative in nature, the initial guess is the first step in the solution. A good initial guess, close to the final solution, can prevent convergence problems. The self-consistent algorithm provides the next potential guess in each iteration. Ideally, the algorithm should prevent divergence and arrive at the solution with as few as possible iterations. Via PETSc, NEMO5 employs several kinds of Newton-Raphson algorithms [@Balay2013a], that rely on an efficient and approximate Jacobian implementation [@Trellakis1997; @Lake1997] and have protection against divergence by being able to control the potential update, or step size, between two iterations. The Newton-Broyden method and trust region methods [@Cowell1984] are also used. While robust, these methods do not always guarantee efficient solutions. We achieve the most efficient solutions by constructing an accurate and time-efficient initial guess, based on the semi-classical charge and locally constant Fermi level with the effective mass corrected for confinement effects, followed by the Newton-Raphson method with full step size. Results of one self-consistent simulation using the tight-binding formalism in NEMO5 are shown in Fig.  \[figSinanowire\]. The simulated device is n-type Si nanowire with 3x3 nm cross section (approximately 3 nm). The wire has 1 nm thick gate all-around and three doping regions: the channel under the gate is doped to 10$^{15}$ cm$^{-3}$, while the source and drain regions to 10$^{20}$ cm$^{-3}$. The length of the simulated device is 20 nm, of which 10 nm is the channel and 5 nm the source and drain regions each. The source and drain region length is chosen so that the potential becomes flat near the lead-device interface. Results are shown in Fig.  \[figSinanowire\]. The simulation is performed up to 0.6 V gate bias, to avoid unphysical effects at higher bias produced by ballistic transport in the absence of the transport barrier and subsequent Poisson/transport equation convergence issues. The convergence scheme consists of the Newton-Raphson method with full step size and the following initial guesses: for the first bias point the semi-classical initial guess is used; for the second bias point the previous solution is used as the initial guess; and for the third bias point upward the prediction/extrapolation based on the previous two solutions. This convergence scheme takes a total of 27 iterations for the 7 bias points simulated. The majority of inner bias points took only 3 iterations, while the first and the last bias point resulted in a slightly higher number of iterations. Even though the semi-classical initial guess with the effective mass corrected for confinement effects is very close to the final solution, the fact that the spatial effects of the confinement (i.e. quantum wave function) are not taken into account results in slightly more iterations. On the other hand, the quality of the semi-classical guess protects the simulation from divergence, as the full step size is used. The last bias point takes slightly more iterations, due to the fact that it is more difficult to achieve convergence for diminishing transport barrier at high gate bias. ![NEMO5 self-consistent simulation results for n-type all-around gate Si nanowire. The gate length is 10 nm, while the doping in the channel below the gate is 10$^{15}$ cm$^{-3}$. The source and drain regions are taken to be 5 nm with 10$^{20}$ cm$^{-3}$ doping. Panel a) shows the current-voltage characteristic, b) is the bulk band edge interpolated along the center of the nanowire, and c) is the same for electron density. The charge density is nonuniform in the cross section due to lateral quantum confinement and significantly larger than the converged average charge which equals the doping. The simulation had 7 voltage points and took in total 27 Poisson/transport equation iterations, thanks to an efficient convergence scheme.[]{data-label="figSinanowire"}](Sinanowire.eps){width="48.00000%"} Strain ====== In the last decade, strain was a major performance booster in ultra-scaled transistors [@Antoniadis2006] and it is of fundamental importance to consider the effect of strain on the band-structure and transport properties of novel devices. Heterostructures composed of lattice mismatched materials exhibit strain intrinsically. As shown in Fig. \[figstrain1\] NEMO5 is able to compute strain and relax the atomistic heterostructures using the Enhanced Valence Force Field (EVFF) [@Steiger2011a; @Paul2010; @Sui1993]. The energy functional contains not only Keating terms such as bond-stretching and bond-bending interactions, but also cross-stretching, stretch-bending, and second-nearest-neighbor angle-angle interactions. For polar materials, the long-range Coulomb interaction can be added in the case of 0-D (bulk) and 3-D (confined) simulations. NEMO5 contains two strategies for elastic energy minimization. One uses Jacobian and Hessian matrices and can be used only for small structures. The second method is approximate and is based only on the Jacobian. ![Strain simulation in a Nitride Hetero-structure Nanowire using NEMO5. (a) Physical structure and dimensions, (b) plot of strain component ezz which shows long range diffusion of strain.[]{data-label="figstrain1"}](strain1.eps){width="48.00000%"} Phonons ======= Nanowires show excellent thermo-electric properties which make them favorable for thermo-electric devices. For example silicon nanowires exhibit 100 times better ZT compared to bulk silicon and can achieve maximum ZT around 1 [@Boukai2008], creating a strong motivation for accurate phonon modeling in nanoscale devices. It is well known that the Keating model overstimates phonon energies of both optical and acoustic branches [@Sui1993]. NEMO5 is able to calculate phonon dispersion using the EVFF model which provides a reasonable match with experimental phonon dispersion (Fig. \[figphonon\]). The dynamical matrix has been calculated by the following: $$%D_{\lambda,\mu}^{i,j} = \begin{cases} \frac{1}{\sqrt{M_{i}M_{j}}} \frac{\partial^{2}U}{\partial r_{i}^{\lambda }\partial r_{j}^{\mu}}& \text{ if atom i or j is not periodic} \\ \frac{1}{\sqrt{M_{i}M_{j}}} \frac{\partial^{2}U}{\partial r_{i}^{\lambda }\partial r_{j}^{\mu}}e^{-i\vec{q}\cdot \vec{r_{ij}}} & \text{ if atom i or j is periodic } \end{cases} D_{\lambda,\mu}^{i,j} = \frac{1}{\sqrt{M_{i}M_{j}}} \frac{\partial^{2}U}{\partial r_{i}^{\lambda }\partial r_{j}^{\mu}}e^{-i\vec{q}\cdot \vec{r_{ij}}}$$ in which i and j are atom indexes, $\lambda$ and $\mu$ can be one of x, y or z directions, *${q}$*is the phonon wave vector, M$_{i}$ and M$_{j}$ are atom masses for atom i and j respectively and U is the total elastic energy of the system.** ![Phonon dispersion of Si calculated by EVFF model using NEMO5 vs. experiment.[]{data-label="figphonon"}](phonon.eps){width="48.00000%"} Metal insulator transition - SmSe ================================= With shrinking physical dimensions, the total transistor number in a single chip has been increasing exponentially for each generation. However, the scaling of the supply voltage in Silicon based MOSFET is limited by the 60mV/dec subthreshold swing (SS). The desire to reduce heat dissipation drives research for devices with different switching mechanisms. [@Theis2010] The Piezoelectronic Transistor (PET) [@Newns2012; @Newns2012a] is a promising approach to achieve a high ON/OFF ratio with very small voltage swing. In PET, the gate voltage is transduced to acoustic waves through a buffer layer made with piezoelectric (PE) materials. The channel layer of piezoresistive (PR) materials, e.g. Samarium monochalcogenides, is capable of modifying the conductance by several orders of magnitude subjected to moderate strain [@Mott1968] which is generated by deformation of PE. When the dimensions of PET are reduced to the nanometer scale, the device performance will be dominated by quantum effects. Quantum confinement will change band structure and minimum leakage is determined by tunneling. To simulate devices of realistic dimensions, computationally efficient models like Empirical Tight-Binding (ETB) are necessary. To obtain accurate parameterization, the SmSe band structure was first calculated in density functional theory (DFT) within the generalized gradient approximation with Hubbard-type U (GGA+U). A tight-binding band model including [*spdfs\**]{} orbitals is implemented based on analysis of the DFT angular momentum decomposition at the band minima [@Tan2013; @Jiang2013]. The inclusion of enhanced spin-orbit coupling for [*f*]{}-orbit is critical to account for the large [*4f$_{5/2}$-4f$_{7/2}$*]{} splitting due to a strong electron-electron interaction of localized [*f*]{} electrons. This model captures the band structure features and the variations of the bandgap in response to the strain predicted by DFT calculations (Fig. \[figSmSe\]). The obtained TB parameters are then used in quantum transport simulations with (NEGF). ![Band structure of SmSe calculated with ETB. (a) bulk band structure of SmSe. (b) modification of bandgap under hydrostatic and uniaxial strain.[]{data-label="figSmSe"}](SmSe.eps){width="48.00000%"} Material parameterization {#yaohua} ========================= The ETB method is widely used in atomistic device simulations. The reliability of such simulations depends very strongly on the choice of basis sets and the ETB parameters. The traditional way of obtaining the ETB parameters is by fitting to experiment data, or critical theoretical bandedges and symmetries rather than a foundational mapping. A further shortcoming of traditional ETB is the lack of an explicit basis. The mapping method is described here is shown in Fig. \[figflowchart\]. The first step is to perform ab-initio calculations of the band structure of a material. In general, any method that is capable to calculate electronic band structures and wave functions is suitable here. In the second step, the ETB basis functions for each type of atom are defined as ![The process of tight-binding (TB) parameters construction from DFT calculations.[]{data-label="figflowchart"}](flowchart_v5.eps){width="48.00000%"} $$\Psi _{n,l,m}\left( \mathbf{r}\right) \equiv \Psi _{n,l,m}\left( r,\theta ,\phi \right) =R_{n,l}\left( r\right) Y_{l,m}\left( \theta ,\phi \right) , \label{eq:definition_atomic_orbitals}$$ where the functions $Y_{l,m}$ are the complex spherical harmonics with angular quantum numbers $l$ and $m$; and the functions $R_{n,l}$ are exponentially damped plane waves$$R_{n,l}\left( r\right) =\sum_{i=1}^{N}\left[ a_{i}\sin \left( {\lambda _{i}r}% \right) {+b_{i}\cos }\left( {{\lambda _{i}r}}\right) \right] r^{n-1}\exp \left( -\alpha _{i}r\right) . \label{eq:definition_basis_Radialpart}$$The parameters $a_{i},b_{i},\alpha _{i},\lambda _{i}$ are the fitting parameters. With a given set of ETB basis functions $\Psi _{\text{TB} }^{\mathbf{k}}$, the DFT Hamiltonian is transformed to the tight-binding representation. Any non-zero off-diagonal element of the overlap matrix is neglected. The ETB Hamilton matrix elements are approximated by two center integrals according to the Slater-Koster table [@Slater_Tightbinding; @Podolskiy_TBElements]. ETB Hamilton matrix elements beyond either 1st or 2nd nearest neighbor coupling are neglected. In Step 4, the band edges, effective masses and eigen functions of the Hamiltonian at high symmetry points are calculated and compared to the corresponding DFT results. The overlaps of the ETB basis functions are also determined. In the fifth step, all fitting parameters are adjusted to improve the agreement of the ETB results with the DFT results and also to reduce the overlap matrix of the ETB basis functions to the unity matrix. Steps 2 - 5 are repeated until the convergence criterion is met. Step 6 requires to extract the converged ETB basis functions and the ETB two center integrals. ![Band structure and density of states of MgO by DFT and TB.[]{data-label="figfig_Ek_TE_MgO"}](MgO_TB_pub_apl_cmp_new_1.eps){width="48.00000%"} The DFT mapping method has been validated in NEMO5 using Si and GaAs[@Tan2013]. It is also applied successfully to Antimonides and new materials such as MgO[@Tan2013], SmSe[@Jiang2013]. MgO cyrstalizes in rock salt structure. Each oxygen atom has six magnesium atoms as 1st nearest neighbors and twelve oxygen atoms as 2nd nearest neighbors. MgO is parameterized for a 2nd nearest neighbor $sp3d5s^*$ ETB model. Within this model, the interaction between two oxygen atoms is required to produce the correct valence bands. It can be seen from Fig.\[figfig\_Ek\_TE\_MgO\] that the ETB band structure matches the DFT result well within the energies $-5$ to $15~\mathrm{eV}$. Tight-binding parameterization of metals {#metals} ======================================== Metals play a significant role in microprocessor operation. Besides connecting individual transistors, metal vias and interconnects deliver bias current, clock signals and metal-stacks are used for gate metallization. In the existing quantum mechanical atomistic device-modeling paradigm, metal contacts are used to set the Fermi levels in the source and drain of the transistor. Once this is done, the metals are essentially abstracted out of the usual Schrödinger-Poisson or NEGF-Poisson solution. Owing to decreasing device dimensions, the resistance drop across the metal-semiconductor contact is becoming an increasingly important issue. Additionally, from an overall power dissipation perspective, the increase in metal resistivity with decreasing via dimensionality is an extremely important unsolved problem[@2012]. Atomistic modeling of metal grain boundary interfaces, metal interconnect-liner interfaces and metal-semiconductor interfaces can provide significant guidance in the design of low-resistivity metal interconnects, liner materials and metal-semiconductor interfaces with low Schottky-barrier heights. With these objectives in mind, we have created accurate and computationally efficient Semi-Empirical Tight-Binding (SETB) models of Metals and Metal-Semiconductor interfaces suitable for studying electron transport in the aforementioned, technologically important systems. NEMO5 contains tight-binding models that have been formulated specifically to study the phenomenon of resistivity increase in metals with decreasing interconnect dimensions and electron transport across metal-semiconductor interfaces. As an example of the capabilities NEMO5 has in this regard, Fig. \[figmetalsa\] shows the bulk band structure of Cu obtained using an efficient 1st Nearest-Neighbor SETB representation of its FCC phase. This band structure is obtained by fitting to LCAO Density Functional Theory (DFT) band-structure for Cu using the exchange-correlation functional of Perdew and Zunger within the Generalized Gradient Approximation (GGA)[@Perdew1996]. It can be seen that the tight-binding model reproduces the DFT band structure accurately. In Fig. \[figmetalsb\], the transmission in bulk Cu along the \[001\] direction is computed using SETB and DFT and the results are compared to each other. It is evident that our SETB model reproduces the DFT results extremely accurately in the energy range of interest – a few kT’s below and above the Fermi level. ![Bulk band structure of Cu in FCC phase calculated using the SETB (dots) formalism and DFT (solid lines). Notice that our SETB model accurately captures DFT band structure features in all energies of interest in electronic transport. []{data-label="figmetalsa"}](figmetalsa.eps){width="48.00000%"} ![Transmission for a 1 nm cell along the \[001\] direction in Cu computed using SETB and DFT. []{data-label="figmetalsb"}](figmetalsb.eps){width="48.00000%"} Topological insulators - Bi$_{2}$Te$_{3}$ {#BiTe} ========================================= Topological Insulators (TIs) are a new state of matter with a bulk insulating gap and metal-like states on the surface or edge. The surface or edge states which are described by a linear Dirac Hamiltonian are robust and protected by time reversal symmetry [@hasanrmp; @fuprb07]. Topological insulators have acquired prominence because they offer a rich collection of fundamentally new phenomena along with a wide array of applications including optoelectronic THz detectors, spin-polarized contacts, ultra-fast switches, etc. [@Sengupta2013]. Several TI materials are known to exist at room temperature. Bi$_{2}$Te$_{3}$ and Bi$_{2}$Se$_{3}$ which possess bound surface states (Fig. \[figBiTespinprojection\]) are well-known examples. The unique properties of TIs are attributed to the linear dispersion of surface states that connect the conduction and valence band together. Further, these states have their spin locked perpendicular to momentum in-plane. NEMO5 offers the capability to compute the atomistic band structure of bulk and confined Bi$_{2}$Te$_{3}$ devices. The undoped Bi$_{2}$Te$_{3}$ is a narrow band-gap quintuple-layered semi-conductor with a rhombohedral crystal structure. The quintuple layer crystal structure is used in a twenty band tight-binding model. All parameters for these calculations were obtained from a orthogonal tight-binding model with sp$^{3}$d$^{5}$s\* orbitals, nearest-neighbor interactions, and spin-orbit coupling [@lee2006tight]. Additionally, the dispersion is spin-resolved and conforms exactly to experimentally observed spin-polarization (see Fig. \[figBiTespinprojection\]). ![The bandstructure of a $[100]$ grown Bi$_{2}$Te$_{3}$ quantum well. The surface states are shown within the boxed region. The conduction and valence bands are connected by a linear dispersion also known as a Dirac cone, depicted within the box. The color bar denotes the strength of spin-polarization.[]{data-label="figBiTespinprojection"}](BiTespinprojection.eps){width="48.00000%"} The corresponding Fermi-surface of the surface states exhibit a peculiar snow-flake structure. NEMO5 predicts this (Fig. \[figFermi\]), in agreement with experiments  [@souma2011direct]. ![The Fermi-surface of the surface states of Bi$_{2}$Te$_{3}$ with the distinctive snow-flake structure.[]{data-label="figFermi"}](figFermi.eps){width="48.00000%"} When conduction band and valence bands are connected (as in the case of TIs) conduction and valence bands cannot be unambiguously separated. Since an accurate prediction of device characteristics and material properties needs charge self-consistent calculations, it is imperative to devise electronic structure calculation models for materials where an explicit differentiation between electrons and holes is not possible. NEMO5 introduces the concept of a novel charge self-consistent full-band atomistic tight-binding method that avoids usage of holes. Hereby, the model of Andlauer and Vogl has been extended to atomistic tight-binding [@Andlauer2009]. Bandstructure unfolding {#unfolding} ======================= Semiconductor alloys do not possess translational symmetry, owing to a random distribution of atoms. For example, the cationic sites in Al$_{x}$Ga$_{1-x}$As can either accommodate an Al or a Ga atom. Thus semiconductor alloys cannot, in principle, have an associated bandstructure. Nevertheless, it is common to measure and use quantities associated with bandstructure (for example, energy bandgap and effective mass) to design and analyze devices in these materials. A compromise between the above two positions is to allow for an approximate bandstructure of alloys, where each energy band is broadened as a result of randomness. The supercell method [@Boykin_JPhysCondMatter_2007; @Popescu_PRB_2012] provides a computational framework to perform such a calculation. The essential idea is to construct a very large supercell which is randomly populated with atoms. A supercell of say Si$_{0.4}$Ge$_{0.6}$ would have roughly $40 \%$ of atoms being Si, while the rest being Ge. Periodic boundary conditions are imposed on this large supercell, and its energy spectrum determined (typically at a single $\vec{K}$ point). The supercell is viewed as being made up of fictitious primitive cells called *small-cells*. The supercell energy spectrum is finally unfolded onto the small-cell Brillouin zone and approximate small-cell energy bands determined. ![Energy bands of Si$_{0.5}$Ge$_{0.5}$ alloy obtained using the supercell method compared with those obtained with the VCA method.[]{data-label="fig_SiGe110"}](SiGe110.eps) In order to obtain adequate points along a particular direction $\vec{n}$ in the small-cell Brillouin zone (say $[100]$, $[110]$, $[111]$ etc.), it is convenient to work with specially chosen supercells. Reference [@Boykin_Physica_2009] describes special rectangular, non-primitive unitcells that are used as building blocks to construct such supercells in NEMO3D [@Klimeck_TED_2007A; @Klimeck_TED_2007B]. This approach has two drawbacks – (i) the non-primitive unit cell is itself made up of a number of small cells, requiring an additional unfolding step that is dependent on $\vec{n}$; (ii) it cannot be used for materials (like GaN) which do not have rectangular unit cells. NEMO5 implements a more general approach, based on [@Aravind_AJP_2006; @Ajoy_IWPSD_2012] where the supercell is built by cascading specially chosen primitive cells (which could be non-rectangular). Figure \[fig\_SiGe110\] shows the approximate energy bands of Si$_{0.5}$Ge$_{0.5}$ along the $[110]$ direction obtained by unfolding from a supercell containing 248 atoms. The atomic positions have been relaxed using a Keating model. Also shown are results of a virtual-crystal-approximation (VCA) [@Paul_EDL_2010], which computes energy bands using a primitive cell consisting of virtual atoms, whose properties are obtained by interpolating those of Si and Ge. It is interesting to note that the VCA approach provides a good estimate of the energy bands of bulk SiGe; nevertheless, the VCA approach has been known to be erroneous for SiGe wires [@Klimeck_TED_2007B]. Conclusion {#Conclusion} ========== An overview of the the NEMO5 nanoelectronics modeling tool has been given with updates regarding recent advances in physical models and associated code. With focus on efficient, scalable quantum transport algorithms, combined with flexibility to handle a wide variety of device structure and materials, NEMO5 seeks to be a cohesive package to provide accurate modeling of nanoscale devices. This work was partially supported by NSF PetaApps grant number OCI-0749140, NSF grant EEC-0228390 that funds the Network for Computational Nanotechnology, and SRC NEMO5 development: Semiconductor Research Corporation (SRC) (Task 2141), and Intel Corp. With kind permission from Springer Science+Business Media: Journal of Computational Electronics, Empirical tight binding parameters for GaAs and MgO with explicit basis through DFT mapping, volume 12, issue 1, 2013, pages 56-60, Yaohua Tan, Michael Povolotskyi, Tillmann Kubis, Yu He, Zhengping Jiang, Gerhard Klimeck, and Timothy B. Boykin, figures 6 and 7, ©Springer Science+Business Media New York 2013 DOI 10.1007/s10825-013-0436-0. The final publication the in the Journal of Computational Electronics is available at http://link.springer.com/article/10.1007%2Fs10825-013-0509-0

--- abstract: 'Initiatives by the IndIGO (Indian Initiative in Gravitational Wave Observations) Consortium during the past three years have materialized into concrete plans and project opportunities for instrumentation and research based on advanced interferometer detectors . With the LIGO-India opportunity, this initiative has a taken a promising path towards significant participation in gravitational wave (GW) astronomy and research, and in developing and nurturing precision fabrication and measurement technologies in India. The proposed LIGO-India detector will foster integrated development of frontier GW research in India and will provide opportunity for substantial contributions to global GW research and astronomy. Widespread interest and enthusiasm about these developments in premier research and educational institutions in India leads to the expectation that there will be a grand surge of activity in precision metrology, instrumentation, data handling and computation etc. in the context of LIGO-India. I discuss the scope of such research in the backdrop of the current status of the IndIGO action plan and the LIGO-India project.' author: - | C. S. Unnikrishnan[^1]\ Tata Institute of Fundamental Research, Mumbai 40005, India date: 'Based on a talk at the Fifth ASTROD Symposium, Raman Research Institute, Bangalore, July 2012. Close to the original published version LIGO document P1200166. See www.gw-indigo.org for updates on LIGO-India.' title: 'IndIGO and LIGO-India: Scope and Plans for Gravitational Wave Research and Precision Metrology in India' --- Preamble ======== The IndIGO Consortium for mobilizing and facilitating participation of Indian scientists and engineers in gravitational wave (GW) observations was set up in 2009 after a series of informal discussions during conferences and workshops related to gravitation because both the need and fresh opportunities for such participation was realized by the handful of people working in this area for over two decades. It was felt that with the plans for advanced GW detectors in the US and Europe, as well as with the possibility of new projects in Australia and Japan, there was new opportunity opening up, missed earlier due to limited resources during the era of initial GW interferometer detectors like LIGO and Virgo. This turned out to be true when the idea was strongly encouraged and supported by the global GW research community. For over two decades, there have been important and influential contributions to research in the theoretical aspects of the generation and detection of gravitational waves from India (S. V. Dhurandhar at IUCCA, Pune on data analysis strategies, methods and templates, and Bala Iyer at RRI, Bangalore on GW generation processes, source modeling and waveforms).  However, when the small community of researchers in gravitation experiments and gravitational waves discussed the possibility in the 1990s, the human and financial resources that could be mustered in India were not perhaps anywhere near the needs of the project and in any case no experimental activity in the field was taken up at that time. Meanwhile, the community of researchers in those theoretical aspects were steadily growing in India due to sustained research at IUCAA and RRI as well as due to the possibility of signatures of primordial gravitational waves in CMBR data attracting researchers in cosmology (like Tarun Souradeep at IUCAA, Pune) to the task of detection of gravitational waves. The possibility of an advanced large scale detector in India, after suitable smaller steps of research and implementation, was discussed enthusiastically again recently in the context of Advanced LIGO [@aLIGO] and Virgo [@Virgo], the upgraded GEO-HF [@GEO-hf] and the space-based LISA [@LISA], prompted and encouraged by several researchers (Bernard Schutz, Rana Adhikari and David Blair). The formation of the IndIGO consortium was followed up with a road-map for the Indian GW observational program that was charted out during a meeting I organized with Munawar Karim on experimental gravitation (ISEG2009) in Kochi in January 2009 [@iseg2009]. This document was prepared in association with ACIGA, the Australian consortium, represented in the meeting by D. Blair. It was natural to try to participate in the ACIGA plans for a new advanced detector (AIGO) in Perth, Australia [@AIGO] and it was envisaged that a limited yet active participation in hardware and human resources for an advanced interferometer detector was feasible and practical as a first serious step. Also, it was decided that a prototype detector in India was an essential element of this effort, for training and research. The overwhelming feeling during these steps was that the opportunity was ‘now, or never’. To assess the feasibility of this participation quantitatively, the IndIGO consortium members sought a grant for ‘Establishing Australia-India collaboration in GW Astronomy’ in 2009, mainly to facilitate meetings between AIGO and IndIGO, and this was immediately funded by the Australia-India Strategic Research Fund (AISRF) managed by the Department of Science & Technology (DST) in India. Research in gravitational waves was already recognized as a thrust area in the vision document of the DST. This was very important because it enabled timely participation of the IndIGO representatives in meetings in Shanghai (October 2009) and later in Perth and Gingin in Australia (February 2010) where the idea of LIGO-Australia was presented. LIGO-USA assessed the important scientific advantages of moving one of the 4-km arm-length Advanced LIGO (aLIGO) detectors from Hanford to a remote location outside the continent, and made the offer of contributing important hardware of the interferometer to the Australian detector project. Australian scientists had the responsibility for the infrastructure, assembly, commissioning and operation of the detector. IndIGO envisaged a 15% participation in hardware deliverables and human resources and prepared a project proposal to the DST seeking support. With LIGO-Australia [@LIGO-Aus-DPR] as the primary project for our participation, it was decided to attempt to construct a relatively small, yet advanced, prototype interferometer at the Tata Institute of Fundamental Research (TIFR), Mumbai where the availability of space is difficult, but facilities and support for fabrication and testing etc. are excellent. The proposal for the 3-m scale power recycled Michelson-Fabry-Perot interferometer with a conservative budget of about \$500,000 went through extensive reviews, being a new endeavour, and was successful in getting funded within a year, in 2011. Several aspects of Indian collaboration for LIGO-Australia was discussed in the India-Australia meeting under the AISRF grant, during February 2011 in Delhi. Based on a joint proposal with IUCAA, Pune and Caltech, USA as the node institutions, IndIGO obtained a grant for an Indo-US centre for gravitational wave physics and astronomy from the Indo-US Science & technology Forum (IUSSTF) in July 2011, to facilitate mutual visits and joint work. Another significant development was that the IndIGO consortium became a member of the Gravitational Wave International Committee (GWIC), in July 2011. However, there were serious difficulties for substantial funding in Australia for realizing the large scale LIGO-Australia detector within a reasonable schedule, set as about mid-2011. The flavour of the plans for GW research in India changed drastically after mid-2011 with the dawn of the possibility of a LIGO-India detector, to be constructed and operated in India with the hardware for the interferometer from LIGO-USA and the infrastructure to house it provided by India. The arrangement for LIGO-India is similar to what was envisaged for the LIGO-Australia detector, with the additional, and very important, task of selecting a suitable site in India for locating the detector. This ‘amazing opportunity’, as I call it, was facilitated due to both the mutual trust and confidence developed during several interactions between the US and Indian scientists in GW research as well as the support for the idea from key researchers in the field of gravity, like A. Ashtekar, K. Thorne, and B. Schutz. An important milestone in this fast-paced developments was the IndIGO consortium becoming a member of the LIGO Scientific Collaboration (LSC) in September 2011. The plan for LIGO-India was well in accordance with the goals of development of the field, especially for international collaboration and network operation, outlined in the road-map of the GWIC [@GWIC-rmap]. In October 2011, a panel of the National Science Foundation (NSF), USA reviewed the case LIGO-India and found its science case compelling enough to go forward, albeit with cautious evaluation criteria. Ever since, the LIGO-USA team has been working relentlessly to ensure with proper evaluation that the LIGO-India detector is feasible, by setting and examining several target criteria. The scientists, science mangers and the funding agencies in India were already sensitized to the importance of the national participation in GW research and astronomy, from the several meetings and discussions in the context of IndIGO and LIGO-Australia project, and this enabled IndIGO to prepare a detailed project proposal [@LIGO-India], which was submitted to the potential funding agencies - the Department of Science and Technology and Department of Atomic Energy, Government of India. This was discussed in a meeting in November 2011 and LIGO-India proposal received enthusiastic support and encouragement along with several other astronomy mega-projects. This paved way for the Planning commission of India discussing the project as a potential ‘12th plan’ mega-science project, to be initiated during the 12th 5-year plan of the government of India, during 2012-2017. When the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, and the two key technologically highly endowed institutes under the DAE – The Institute for Plasma Research (IPR), Gandhinagar and the Raja Ramanna Centre for Advanced Technology (RRCAT), Indore – agreed to take key responsibilities for the projects, things fell into sharp focus. Speedy and cautious response and support from the NSF, USA in the form of visits of key persons and reviews of the proposal by special committees resulted in quickly giving concrete form to the LIGO-India project. Four senior level visits from the LIGO-Laboratory to the LIGO-India lead-institutions for technical assessment and discussions were followed by three in depth reviews by a NSF panel. All this culminated in a review by the National Science Board, USA, in August 2012 and the following resolution: ‘Resolved, that the National Science Board authorize the Deputy Director at her discretion to approve the proposed Advanced LIGO Project change in scope, enabling plans for the relocation of an advanced detector to India’. The Department of Atomic Energy is putting together the papers for a note to the Cabinet seeking “in-principle” approval of the project, and permissions to sign the relevant MOUs and release of seed funding for the project. It is significant that LIGO-India is listed prominently in the communication from the US state department on U.S.- India Bilateral Cooperation on Science and Technology [@US-state-dept]. The LIGO-India project: Science case ==================================== The idea of LIGO-India (as well as the earlier LIGO-Australia) arose due to compelling science reasons when it was realized that a network of three advanced detectors, preferably of the same nature of design and sensitivity, forming a large triangle across the globe has significantly more advantages than the operation of the same three detectors in just two geographical sites, as was envisaged in the LIGO plan. In addition such a change of plan has the advantage of bringing in a new country and scientific community into the global GW research and astronomy effort. Of course, there is a change in the noise cancellation capability at one of the sites where the two detectors were supposed to operate simultaneously in the same vacuum enclosure (Hanford in this case). However, that was seen as a small price to pay for the great advantages LIGO-India would offer in terms of source localization, duty cycle, sensitivity and sky coverage [@Klimenko; @Fairhurst; @Fair-Sathya]. A study by B. Schutz, in which a general framework based on three new figures of merit was developed for studying the effectiveness of networks of interferometric gravitational wave detectors, showed that enlarging the existing LIGO–Virgo network with the planned detectors in India (LIGO-India)  and Japan (LCGT) brought major benefits, including much larger detection rates and more uniform antenna pattern and sky coverage [@Schutz2011]. I summarize here the main scientific advantages that are discussed in these papers. Source localization ------------------- The obvious advantage of locating a third aLIGO detector far away from the other two in the USA is the ability to locate a source in the sky with three identical detectors, with an accuracy of about a degree or so. In the pre-LIGO-India plans this was expected to be accomplished by timing measurements involving the aLIGO detectors in two locations and the Virgo detector at Cascina near Pisa. This of course is possible. However, there is significant improvement in the source localization ability when the LIGO-India detector is added to the network. The advantage is quantitatively similar with the third aLIGO detector in either Australia or India due to the bounded positioning possibilities on the spherical earth, with marginal advantage (10%) in the slightly larger baselines to Australia from the US sites (there is significant difference to the baseline to Virgo, however, of about 40%). Figs. 1 and 2, from ref. [@Fair-Sathya], indicate the locations of the detectors and the improvement in localization. (H - Hanford, USA, L - Livingston, USA, V - Virgo, Pisa, Italy and I - LIGO-India, assumed to be located near Bangalore for these estimates. The sensitivity of localization error to the exact position in India is mild). Since binary neutron star mergers are the most promising events for detection when the advanced detectors start their operation, the analysis in this paper is based on a population of binary neutron stars distributed over the sky at a luminosity distance of 200 Mpc, corresponding to the average distance reach of the advanced detectors. [fig1.eps]{} [fig2.eps]{} With a baseline of about 14000 km, the theoretical resolution for localization is about 10 sq. degree. Of course the details depend on the projection of this baseline on the sky, and hence varies with direction. With the four-detector network involving LIGO-India, HLVI, there is dramatic improvement, by an order of magnitude in the worst case of just the HHLV network. For 50% of the sources the average localization error improves from 30 sq. deg to 8 sq. deg. Apart from localizing the source on the two-dimensional sky, it is also possible to measure the distance to the source because binary neutron stars are standard candles, and it is estimated that four or three-detector networks involving LIGO-India can measure distances to better than 30%, the improvement resulting from the large baseline as well as the ability to sense the orbital orientation of the binary relative to the line of sight. These measurements will be important for the independent determination of the Hubble parameter. Sensitivity ----------- There will be a 10-fold improvement in the sensitivity of individual Advanced LIGO and Virgo detectors, compared to the earlier versions. In network operations, the overall sensitivity is determined by noise rejection capabilities achieved in coincidence detection. Therefore, the original HHL configuration had extra sensitivity coming from the fact that there are two Hanford detectors in the same UHV enclosure. Together with the Livingston detector, the sensitivity for detection along the normal to the US continent was almost twice or even 3 times of that in other direction rotated 90 degrees away (see left panel of Fig. 3). The sensitivity is smoothed out more uniformly in all directions in the HLVI configuration [@Klimenko]. Even though there is marginal reduction in the best sensitivity (about 15%), the worst sensitivity in some directions is only about half of that of the best sensitivity and significant region of the sky is visible with much better sensitivity (20%-30%) than possible with the HHLV configuration. [fig3.eps]{} Sky coverage ------------ Due to the fact that the new detector is significantly out of plane compared to the HLV configuration, the blind bands in the HHLV configuration are eliminated considerably in the HLVI configuration. Compared to a sky coverage of about 47% in the case HHLV, the HLVI network has 79% sky coverage. With the addition of the KAGRA detector in Japan (HLVIJ) this reaches 100%, compared to 74% of HHLVJ. The details are listed and discussed in ref. [@Fair-Sathya]. Duty cycle ---------- Each advanced detector is expected to have a duty cycle below 80%, due to the need for regular maintenance. Regular operation with good precision in localization requires a minimum of three detectors and therefore the duty cycle for a 3-detector operation is limited to about 52% in the HHLV configuration where the down time of one H detector has significant overlap with the down time for the other because they share the same UHV envelop, similar infrastructure etc. However, with HLVI, there are four 3-detector configurations available with total effective duty cycle of 41%, equal to the four-detector duty cycle itself, adding up to 82%. This is a significant improvement for the ‘on’ time for the network telescope, and this fact alone is worth relocating one detector to India, from the point of view of astronomy (Fig. 4). Of course, source localization is still considered as the most important single factor in the many scientific advantages of LIGO-India because it is a crucial factor in source identification with simultaneous observations with other types of telescopes operating in the electromagnetic spectrum. [fig4.eps]{} The LIGO-India project: Plan of execution ========================================= The LIGO-India project [@LIGO-India] is a joint endeavour between the consortium of research laboratories and universities funded by the Government of India and the LIGO Laboratory of the USA, funded by the National Science Foundation, USA. The LIGO Laboratory will provide the complete design and all the key detector components of the Advanced LIGO (aLIGO) detector. These include the vibration isolation platforms and systems, the pre-stabilized laser with amplifier, complete optics and suspension systems for mode cleaners and the interferometer, sensors, control systems and electronics, software, design and  assembly drawings and documents etc. India would provide the infrastructure and human resources to install the detector at a suitable site in India and would be responsible for commissioning and operating it. The infrastructure involves the appropriate site, suitably prepared, UHV enclosures and the 2x4km beam tubes, laboratories and clean rooms etc. The aim is to realize a third aLIGO detector, as close as possible in design and operating characteristics to the other two aLIGO detectors in the USA, in time to enable GW astronomical observations in a multi-detector network. The proposed observatory would be operated jointly by IndIGO and the LIGO Laboratory and would form a single network along with the LIGO detectors in USA and Virgo in Italy, the advanced detectors under assembly and commissioning, and possibly the KAGRA detector in Japan, which is in its beginning stage of construction. It will bring together scientists and engineers from different fields like optics, lasers, gravitational physics, astronomy and astrophysics, cosmology, computational science, mathematics and various branches of engineering. In order to fully realize the potential of multi-messenger astronomy, the LIGO-India project will join forces with several Indian astronomy projects. Potential collaborators include the space-based ‘Astrosat’ multi-wavelength astronomy project [@Astrosat], the high altitude gamma-ray observatory (HAGAR)[@hagar], the India-based Neutrino Observatory (INO) [@INO], the Giant Meter-wave Radio Telescope (GMRT) [@GMRT] and other optical/radio telescopes. The total estimated budget over 15 years is about \$250 million on the Indian side and the value of the interferometer hardware from LIGO-USA, including the research input, is estimated at about \$120 million. The arrangement drastically cuts down the time required, perhaps by 8-10 years, to fabricate the key components and modules to specifications, assemble and operate a new advanced detector at a third location with large baseline, and this is of course the greatest advantage from the point of view of Indian scientists. By using exactly the same hardware and software as the aLIGO detectors, the confidence in the quality and understanding of the data collected in the 3 similar detectors is ensured to large extent. The two aLIGO detectors are now operational in their first observational run (O1) with about 1/3 of their projected sensitivity. The assembly and testing of the LIGO-India detector is expected to start in 2019, and by then the two aLIGO detectors and the Virgo detector will be fully operational in the network and it is likely that the KAGRA detector will be ready as well. This staggered schedule, inevitable due to the large infrastructure to be prepared, especially the UHV enclosures and beam tubes, also helps in eventual speed up during 2018-22. Some of these details are discussed in the LIGO-India ‘detailed project report’ (DPR) public document, ref. [@LIGO-India]. The reasonable expectation from this tight schedule requiring systematic and sustained work is that India will operate an advanced gravitational wave detector with strain sensitivity similar to the aLIGO and advanced Virgo detectors by 2022. The LIGO-India project: National participation ============================================== A long term megascience project like the LIGO-India can be taken up and completed only with an inclusive national participation involving research institutes with different expertise and experience. In addition, strong industry participation is necessary while creating the infrastructure, especially the UHV enclosures and the clean room environments. Fortunately, IndIGO has been able to identify and mobilize several key research institutes in India to take up the challenge and contribute in various ways to the project. Most importantly, three major institutes volunteered to play the lead role, with crucial contributions of deliverables in their areas of expertise. These are the Institute for Plasma Research (IPR), Gandhinagar (near Ahmedabad), Raja Ramanna Centre for Advanced Technology (RRCAT), Indore and the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune. The IndIGO consortium that started out with several theoretical and data analysis scientists and two or three experimental physicists has now grown to a large membership of over 120, with half its strength coming from experimenters and engineers. The possibility of the mega-science detector project ‘at home’ with a global presence and collaboration is unprecedented and it is certain that a larger community will form as we progress through the detector construction towards operation. University participation ------------------------ In India, fundamental science projects that involve large scale infrastructure and collaboration of a large number of people have been pioneered and managed by some of the national laboratories, notably the Tata Institute of Fundamental Research, Mumbai. Indian universities, by and large, have kept away from large scale national projects due to constraints of funding and other administrative difficulties, even though small groups have been participating as partners in accelerator based particle physics research for some time now. The Indian Institutes of Technology (IIT) faculty have been very active in several projects of small and large scales with a technology flavour, but have been minor partners in large scale astronomy or physics projects. The Inter-University Centres were set up to increase the participation of university scientists in fundamental research through centralized facilities. The LIGO-India project has generated tremendous enthusiasm among both communities and the timing coincides with the new national initiative for high quality undergraduate education through several newly set up institutes (IISER) in several states of India. We expect that the LIGO-India project will bring together researchers and students from universities, IITs and IISERs in an unprecedented scale and level of integration. Industry participation ---------------------- From infrastructure development in the initial stages to the commissioning of the detector, strong industry participation is required for the success of LIGO-India project.` `The large scale sophisticated infrastructure required to house the GW detector, involving technologies for UHV, steel processing, robotic welding, clean rooms, hydraulics, computing clusters, power handling etc. can be realized only with the participation of the relevant industries. Particularly important is the creation of ultra-clean laboratory environment and ultra-clean ultra-high vacuum. Therefore, ensuring participation of some of these industries, especially those related to UHV technology, in some of the discussion meetings leading to the LIGO-India project has been a priority. Even though some of the fabrication aspects that are LIGO-specific are new for the industrial partners that we have started to identify, there is a sense of new national adventure in these circles that will help them to take up the challenges. This is possible only by adhering well to the schedule and budget expenditure plan. The lead institutes and the IndIGO consortium council are keenly following up the specific needs of the LIGO-India project in this context with the funding agencies and policy makers. We will be interacting strongly also with the electronics and computer industrial sources and a mutually beneficial long term partnership is envisaged. When the detector is operational a large amount of computing in the cloud environment is expected and it is natural that the project will contribute to the development in distributed computing in India. The detector components, contributed by LIGO-USA define several goals for technology achievements within the country for optical components, sensor technologies, feed-back control systems, mechanical fabrication etc. Our interaction with the industry in the context of ideas and hardware for next generation detectors will help to realize some of these goals with benefits of global exposure and market for relevant specialized industries. Outreach and higher education ----------------------------- There are very few internationally competitive physics and astronomy instruments operating within India. The lack of accessible and visible facilities for research that involves high technology and high finesse instrumentation directly affects the interest and motivation for choosing experimental physics and engineering physics as a career, especially in areas that requires a long term commitment and field work. The LIGO-India project will dramatically change this situation and a large number of tested remedial steps will built into the operation of the detector for continued engagement with undergraduate and post-graduate students in physics, astronomy and engineering. This will include introductory and advanced schools, hands-on laboratories at site as well as in associate centres at universities, IISERs and IITs, summer training programs at associated laboratories in India and abroad for selected motivated students, special LIGO-India fellowships for specialized training etc. The LIGO-India detector will be one of the very few research facilities in India of this scale, international relevance and technological innovation to which the general public and students can have access through an interface centre located not far from to the actual detector. It has the additional fascination as an instrument for astronomy of the neutron stars and black holes in distant galaxies. Creation and operation of a public outreach centre where key technologies and physical principle that make the detector will be on display, some of which for hands-on access, is an integral part of the project, as has been the practise in LIGO-USA, AIGO etc. as well. This will also serve, through direct interaction and through web-based services, as a centre for continuing education on these topics. Subsidiary centres of a similar nature will also be set up in the associate centres of the IndIGO consortium for wider reach, especially to school students. We will also tie up with national planetariums in different cities in India for programs on gravitational wave astronomy and teacher training in related areas. The home computer based ’Einstein@home’ program for data analysis for continuous wave sources that is currently running as part of the gravitational wave detection global activity will add to the public outreach program. It is not inconceivable that with the enthusiasm among students in India to get involved in such projects this program will touch a 1 million user mark and 1000 Tflops peak. Synergy of funding sources -------------------------- A national megaproject of this scale and spread of participation needs synergic funding and support from multiple national agencies. Indeed, the project proposal was considered by the two prominent national agencies that support fundamental research in physics and astronomy in India - the Department of Science and Technology (DST) and the Department of Atomic Energy (DAE). Projects requiring large financial support need to generate a consensus among scientists, science planners and public funding agencies. The IndIGO consortium prepared and submitted the LIGO-India proposal in time before the National Planning Commission finalized its allocations for the fresh 5-year plan for revenue spending in India (2012-17), after making several presentations in several forums, stressing the imminent detection of gravitational wave and the potential and importance of ensuing GW astronomy. In a joint meeting of representatives of the entire astronomy community in India, which discussed several large scale astronomy projects, LIGO-India received enthusiastic support, and the two agencies (DST and DAE) agreed to include the LIGO-India proposal in the list of Mega-Projects being considered by the planning commission of India. Prototype detector at TIFR, Mumbai ================================== It was realised from the outset that an advanced prototype interferometer detector that incorporates all essential features of the large scale detector is an important element in the road-map for GW research and astronomy in India. Therefore a detailed proposal for 3-m scale power-recycled interferometer detector was prepared in 2010 and submitted to the TIFR, Mumbai where associated expertise and facilities for taking up such a project existed. After extensive discussion and evaluation this project at an estimated cost of \$500,000 was approved. However, the need to construct an entirely new laboratory building and the delay in fund flow as well as in the approvals for LIGO-India have introduced some schedule uncertainty. It is expected that there will be an operational interferometer with a displacement sensitivity of about $10^{-17}m/\sqrt{Hz}$ by 2018, in the newly constructed laboratory. I now summarize the main design features and goals of this prototype detector (Fig. 5). [fig5.eps]{} The prototype detector is expected to serve as the research and training platform with all the features of the aLIGO-like detectors, scaled down to displacement sensitivity around $10^{-17}m/\sqrt{Hz}$ above 100 Hz. It will be the Indian research platform for features like signal recycling, DC read-out, and most importantly the use of squeezed light and noise reduction for precision metrology. It is envisaged that some parallel development on squeezed light based measurement technologies will be developed and this will be implemented in the prototype interferometer after 2018. It will also serve as a superb instrument for novel studies on short range gravity and QED force, especially a measurement of the Casimir force in the range 10 -100 microns where no previous measurements exist [@Raji-unni]. The main idea here is that even though the response of the suspended end mirror to a modulated force at frequency $\omega$ contains the attenuation factor $\left( \omega _{0}/\omega\right) ^{2},$ the already fine displacement sensitivity $10^{-17}m$ can be enhanced with integration to below $10^{-19}m$ over several hours. This makes the Casimir force measurable with good precision for separations larger than 10 microns. Possible coincidence operation with next generation cryo-mechanical detectors by optimizing the sensitivity at 2kHz+ by signal recycling and use of squeezed light (at strain sensitivity approaching $10^{-20}m/\sqrt{Hz}$) is also envisaged. Satellite projects and Precision metrology ========================================== A natural outcome of a project like LIGO-India is its catalyzing ability. With a large number of advanced technologies that have been stretched to their present limits in use in LIGO-India, the project is at once a model system for the use of similar technical strategies in other areas of precision metrology of both fundamental and practical nature, and a motivating platform for the next generation technologies. Only when the limits are visible, one is spurred into conceiving the next generation of technologies. In our laboratory, satellite projects to develop matter-wave interferometers based on both ultra-cold atoms and liquid helium will be taken up with a view to contribute to next generation gravitational experiments, potentially including GW detection. Cold atom interferometers are just an additional step or two from a  wealth of technologies that we already developed at TIFR, working with ultra-cold Rb and K atoms as well as Rb Bose-Einstein condensate. The fact that the gravitational coupling energy and therefore the gravitationally induced phase for an atom with a mass of 100 GeV is about $10^{11}$ times the gravitational energy of a 1 eV photon is the major advantage of a matter-wave interferometer exploring gravity. However, the size of matter-wave interferometers are tiny compared to optical interferometers and the possible configurations are limited. Yet, it is speculated that these might allow breakthroughs in several types of gravitational experiments, including the detection of gravitational waves [@Kasevich]. Matter wave interferometers are also important in addressing foundational questions on the quantum dynamics of particles in gravitational fields [@Unni-gill-PLA2012]. It is expected that a significant number of researchers in IISERs and IITs will take up small scale projects associated with LIGO-India aimed at technologies to be incorporated in the next generation detectors. Novel sensor modules and electronics, UHV compatible devices, compact and sensitive tilt meters, displacement and angle sensors, accelerometers and gyroscopes, computing strategies, integrated optics etc. are some examples. Due to the stringent requirements on the spatial placements of LIGO optical components, high precision survey has been an important aspect of the LIGO installation, and the methods used in defining the coordinates for LIGO-India precision installation are expected to contribute to geophysical measurements and survey strategies. The LIGO-India detector itself, equipped with its elaborate active vibration isolation system, is a sensitive instrument to monitor geophysical phenomena in the acoustic frequency range. LIGO-India and the Indian space science program =============================================== It is clear that the greatest potential for gravitational wave astronomy resides in the wavebands in the range $10^{-5}$ Hz - $1$ Hz$.$ Only a space-based detector with sufficient large arm length in that relatively noise-free environment can sense these signals. Even though the original LISA project is currently not realized, new thinking on this possibility will remain active due to its importance, and the inevitability from the point of view of the physicist and astronomer. Other possibilities are also being discussed, like the ASTROD-GW (China) [@Astrod], eLISA (ESA) [@eLISA] or DECIGO (Japan) [@Decigo]. In addition there are a few space-based gravitation experiments of high significance, like the GG [@GG], which can benefit from the involvement of an active space-based fundamental physics program in India. We expect that LIGO-India will motivate new initiatives within the highly successful Indian Space Research Organization itself towards new astronomy and fundamental physics. LIGO-India: Schedule and Progress ================================= Though many of the steps in bringing the proposal to the final stages of approvals have been taken, the final approval and funding allocation for the project from the cabinet committee of the government of India is still awaited, as of October 2015. Given this situation, there are constraints on going ahead with the initial tasks on site identification and selection, identifying potential partner institutions and industries, visits to LIGO-USA etc. However, the lead institutes have been active on the project and have now identified expert teams within, for several technical tasks of LIGO-India. Working visits from IndIGO members as well as training of researchers, engineers and post-doctoral fellows at the LIGO observatories at Hanford and Livingston in the several technical aspects of the aLIGO detectors is an intergral part of the early schedule. The EGO consortium in Europe, managing the Virgo detector, also has extended help in the matter of training and technology exposure. The IndIGO consortium members have been working on the initial tasks with support from the lead institutes. Most significantly, several potential sites in several states of India have been visited and preliminary measurements on seismicity and environmental noise factors have been conducted. The policy is not to compromise on the strict constraints laid out for a suitable site in terms of its long term isolation from seismic and man-made noise, while keeping in mind the accessibility for construction and operation on schedule. The ground noise models at the LIGO Hanford and Livingston sites are indicated, for reference, along with the requirement on residual noise after the pre-isolation stage, in figure 6. The active pre-isolation system for LIGO is already designed and fabricated to bring down the ground vibration noise at the two LIGO sites to the required level,  with isolation factor of about 1000 in the frequency range 1-30 Hz. India has several low seismicity areas, usually indicated as zone II in seismic activity maps in India. From short term seismic surveys conducted over two weeks each, it is now known that isolated areas in these zones are seismically quiet enough to install an advanced gravitational wave detector. Many potential sites that are more than 150 km from the ocean in such areas, spread around India in the Deccan plateau in Karnataka and Andhra Pradesh, and in Madhya Pradesh, Rajasthan and Chathisgarh, for example, have been visited for preliminary evaluations. Long term weather data from the meteorology department or from the repository at the nearest airport , as well as about 2 weeks of seismic noise data have been obtained. Typical ground noise measured in such sites during quietness is shown in the figure 7, obtained from IndIGO measurements spanning about 2 weeks with a 3-axis wideband Gurlap seismometer (more recent extended measurements confirm this trend). Since the typical ground noise at the sites we briefly explored in this frequency range seems comparable to that in the LIGO sites, the same isolators are expected to ensure their intended isolation levels in LIGO-India . [fig6.eps]{} [fig7.eps]{} As for development of other infrastructure, the review of technical drawings on the interferometer stations and the UHV infrastructure has progressed well enabling readiness for quick start after the cabinet approval. Capability for handling the pre-stabilized laser as well as for the fabrication and welding of the silica fiber suspensions of optics is now developed and the sense of technical readiness is strong. The era of LIGO-India operation and GW astronomy ================================================ India already has a strong community of young GW researchers involved in data analysis and source modeling aspects. Therefore, the user community for LIGO-data within India is substantial and will grow further. In fact, it was this aspect that was the initial driving force behind IndIGO and LIGO-India. One focus for theoretical developments will be strategies for handling independent noise from the different detectors, especially multi-detector coherent searches for sources where the data from different detectors in the network is combined and analyzed with the phase information (aperture synthesis), instead of just coincidences in time. The goal is efficient coherent search for GW signal from binary mergers using data from global detector network. The interface of cosmology (through CMBR) and GW is another key area of interest here. What is envisaged is an integrated environment for astronomy data storage and handling, more or less mirroring the evolving vision within the GW community that GW astronomy will need simultaneous and triggered observations by different types of telescopes and detectors, spanning the electromagnetic spectrum, as well as arrays of cosmic ray detectors. An advanced data centre with large computing power and storage is integral part of the LIGO-India plan and the first stage of the centre is being set up at IUCAA, Pune. This data centre with several hundred Terabytes storage and about 100 Tflops peak computing power is now operational and will eventually serve, with appropriate upgrades, as a Tier-2 data archival and computing centre for signals from the global GW detector network. Summary ======= I have sketched some key developments on the large canvas of the LIGO-India project, the extend of which is vast, transformational for Indian science, technology and higher education, and rejuvenating for Indian physics and astronomy. Acknowledgments =============== I thank my colleagues in the IndIGO consortium and its council who contributed much in the past 3 years to the developments described in this paper. IndIGO has grown ten-fold by 2015, with about 120 members, working together to realize LIGO-India and GW astronomy. Presentations on LIGO-India by Bala Iyer and Tarun Souradeep (IndIGO council) and Stan Whitcomb (LIGO-USA) on various occasions were of immense help in preparing this article. I thank Wei Tou Ni and Bala Iyer for the opportunity to publish this paper based on my talk at the Fifth ASTROD symposium (July 2012) at the Raman Research Institute, Bangalore, India. [99]{} G. M. Harry (for the LIGO Scientific Collaboration) Class. Quantum Grav. **27** (2010) 084006. Advanced Virgo, https://wwwcascina.virgo.infn.it/advirgo/docs.html B. Willke *et al*., Class. Quantum Grav. **23** (2006) S207. Laser Interferometer Space Antenna, see details at lisa.nasa.gov/ International Symposium on Experimental Gravitation, Kochi, India (2009): see details at www.tifr.res.in/iseg P. Barriga *et al*, Class. Quantum Grav. **27** (2010) 084005. LIGO-Australia - On the Crest of the Wave, The LIGO-Australia proposal, see www.aigo.org.au/aigo\_web\_docs/LIGO-AustraliaProposal.pdf The Gravitational Wave International Committee Roadmap, public document available at https://gwic.ligo.org/roadmap/ B. Iyer *et al*., LIGO-India Tech. rep. (2011), LIGO document M1100296-v2, at https://dcc.ligo.org. Also available from http://www.gw-indigo.org/tiki-index.php?page=LIGO-India. U.S.-India Bilateral Cooperation on Science and Technology’, fact sheet at http://www.state.gov/r/pa/prs/ps/2012/06/192271.htm S. Klimenko *et al*, Phys.Rev.D **83** (2011) 102001. S. Fairhurst, arXiv:1205.6611v2 (gr-qc). B. S. Sathyaprakash *et al.*, LIGO document T1200219-v1, at https://dcc.ligo.org. B. F. Schutz, Class. Quantum Grav. **28** (2011) 125023. B. Paul, ASTROSAT: Some Key Science Prospects, Int. J. Mod. Phys. D22 (2013) 1341009. (The satellite was launched on 28 September, 2015). HAGAR gamma ray laboratory at Hanle, Ladakh, India: www.tifr.res.in/hagar/experiment.html The India-based Neutrino Observatory: www.ino.tifr.res.in/ The Giant Meter-wave Radio Telescope, Pune, India: gmrt.ncra.tifr.res.in/ G. Rajalakshmi and C. S. Unnikrishnan, Class, Quantum Grav. **27** (2010) 215007. S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, Phys. Lett. B **678** (2009) 27. C. S. Unnikrishnan and G. T. Gillies, Phys. Lett. A 377 (2012) 60. G. Wang and W-T. Ni, Chinese Astron. Astrophys. **36** (2012) 211. P. Amaro-Seoane *et al*., arxiv:1201.3621. S Kawamura *et al.*, J. Phys.: Conf. Ser. **122** (2008) 012006 . A. M. Nobili *et al.*, Class. Quantum Grav. **29** (2012) 184011. R. Abbott *et al*., Class. Quantum Grav. **19** (2002) 1591. [^1]: On behalf of the LIGO Scientific Collaboration (LSC) and the IndIGO Consortium.  E-mail address: [email protected]

--- abstract: 'Any $pl$-stratified pseudomanifod can be normalized preserving its intersection homology. In this paper we extend this result for any topological stratified pseudomanifold and for a family of perversities which is larger than usual. Our construction is functorial. We also give a detailed description of the normalizer’s stratification in terms of the initial stratified pseudomanifold.' address: 'Universidad Central de Venezuela, Caracas' author: - 'G. Padilla' date: 'May 9/2002, revised on June 15/2002' title: On Normal Stratified Pseudomanifolds --- [^1] Foreword {#foreword .unnumbered} ======== [*For an entire version of this article the reader should go better to Extracta Math. Vol. 15 (3), 383-412.*]{} A stratified pseudomanifold is normal if its links are connected. A normalization of a stratified pseudomanifold $X$ is a normal stratified pseudomanifold $X^N$ together with a finite-to-one projection $\n:X^N\rightarrow X$ preserving the intersection homology. Recall that intersection homology is the suitable algebraic tool for the stratified point of view: it was first introduced by Goresky and MacPherson in the $pl$-category and later extended for any topological stratified pseudomanifold [@gm1], [@gm2]. Following Borel the map $\n$ is usually required to satisfy the following property: For each $x\in X$ there is a distinguished neighborhood $U$ such that the points of $\n^{-1}(x)$ are in correspondence with the connected components of the regular part of $U$. A normalization satisfying the above condition always exists for any $pl$-stratified pseudomanifold [@borel], [@maccrory]. In this article we study the main properties of the map $\n$. More precisely, we prove that $\n$ can be required to satisfy a stronger condition: it is a locally trivial stratified morphism preserving the conical structure transverse to the strata. We make an explicit construction of such a normalization for any topological stratified pseudomanifold. Our construction is functorial, thus unique. We exhibit the relationship between the stratifications of $X$ and $X^N$. Finally we prove that the normalization preserves the intersection homology with the family of perversities given in [@pervsheaf], see also [@king],[@illinois]. This family of perversities is larger than the usual one. With little adjust our procedure holds also in the $C^{\infty}$ category. [99]{} BOREL, A. et al. [*Intersection Cohomology.*]{} PM\#[**50**]{}(1984) Birkhäuser. BREDON, G. [*Topology and Geometry.*]{} GTM\#[**139**]{} (1993) Springer-Verlag. GORESKY, M. & R. MacPherson. [*Intersection Homology Theory.*]{} Topology \#[**19**]{} (1980), 135-162. GORESKY, M. & R. MacPherson. [*Intersection Homology II.*]{} Invent. Math. \#[**71**]{} (1983), 77-129. KING, H. [*Topology invariance of intersection homology without sheaves.*]{} Topoloy Appl. [**20**]{} (1984), 1091-1106. MACCRORY, C. [*Poincaré duality in spaces with singularities.*]{} Ph.D. Thesis. Brandeis University. (1972). MACPHERSON, R. [*Intersection homology and perverse sheaves.*]{} Colloquium Lectures, Annual Meeting of the AMS, San Francisco, June 1991. PFLAUM, M. [*Analytic and geometric study of Stratified Spaces.*]{} (2000) Springer-Verlag. SARALEGI, M. [*Homological Properties of Stratified Spaces.*]{} Illinois J. Math. \#[**38**]{} (1994), 47-70. [^1]: This paper was supported by the the ECOSNord project, Euskal Herria Unibertsitatea’s Math Department, the University of Artois’ LaboGA and the Universidad Central de Venezuela’s C.D.C.H. The author would like to thank M. Saralegi for encouraging us to look at this problem, and also R. Popper for some helpful conversations.

amstex =1 =5 true in =8.4 true in =.75 in \#1[2 \#1]{} cmbx12 = cmr12 =cmbx8 =cmti8 =cmr8 =cmr7 “”[AMS 1991 Subject Classification: Primary 37D50, 34D05.]{} “”[Key words and phrases: Non-uniformly hyperbolic dynamical systems, singularities, cylindric billiards, ergodicity, mixing, local ergodicity.]{} “”[This research was partially supported by the National Science Foundation, grant DMS-0098773.]{} 1 **Proving The Ergodic Hypothesis for Billiards With Disjoint Cylindric Scatterers** Nandor Simanyi University of Alabama at Birmingham Department of Mathematics Campbell Hall, Birmingham, AL 35294 U.S.A. [**1. Introduction**]{} Non-uniformly hyperbolic systems (possibly, with singularities) play a pivotal role in the ergodic theory of dynamical systems. Their systematic study started several decades ago, and it is not our goal here to provide the reader with a comprehensive review of the history of these investigations but, instead, we opt for presenting in nutshell a cross section of a few selected results. In 1939 G. A. Hedlund and E. Hopf \[He(1939)\], \[Ho(1939)\], proved the hyperbolic ergodicity of geodesic flows on closed, compact surfaces with constant negative curvature by inventing the famous method of “Hopf chains” constituted by local stable and unstable invariant manifolds. In 1963 Ya. G. Sinai \[Sin(1963)\] formulated a modern version of Boltzmann’s ergodic hypothesis, what we call now the “Boltzmann-Sinai ergodic hypothesis”: the billiard system of $N$ ($\ge2$) hard spheres of unit mass moving in the flat torus $\Bbb T^\nu=\Bbb R^\nu/\Bbb Z^\nu$ ($\nu\ge2$) is ergodic after we make the standard reductions by fixing the values of the trivial invariant quantities. It took seven years until he proved this conjecture for the case $N=2$, $\nu=2$ in \[Sin(1970)\]. Another 17 years later N. I. Chernov and Ya. G. Sinai \[S-Ch(1987)\] proved the hypothesis for the case $N=2$, $\nu\ge2$ by also proving a powerful and very useful theorem on local ergodicity. In the meantime, in 1977, Ya. Pesin \[P(1977)\] laid down the foundations of his theory on the ergodic properties of smooth, hyperbolic dynamical systems. Later on this theory (nowadays called Pesin theory) was significantly extended by A. Katok and J-M. Strelcyn \[K-S(1986)\] to hyperbolic systems with singularities. That theory is already applicable for billiard systems, too. Until the end of the seventies the phenomenon of hyperbolicity (exponential instability of the trajectories) was almost exclusively attributed to some direct geometric scattering effect, like negative curvature of space, or strict convexity of the scatterers. This explains the profound shock that was caused by the discovery of L. A. Bunimovich \[B(1979)\]: certain focusing billiard tables (like the celebrated stadium) can also produce complete hyperbolicity and, in that way, ergodicity. It was partly this result that led to Wojtkowski’s theory of invariant cone fields, \[W(1985)\], \[W(1986)\]. The big difference between the system of two spheres in $\Bbb T^\nu$ ($\nu\ge2$, \[S-Ch(1987)\]) and the system of $N$ ($\ge3$) spheres in $\Bbb T^\nu$ is that the latter one is merely a so called semi-dispersive billiard system (the scatterers are convex but not strictly convex sets, namely cylinders), while the former one is strictly dispersive (the scatterers are strictly convex sets). This fact makes the proof of ergodicity (mixing properties) much more complicated. In our series of papers jointly written with A. Krámli and D. Szász \[K-S-Sz(1990)\], \[K-S-Sz(1991)\], and \[K-S-Sz(1992)\], we managed to prove the (hyperbolic) ergodicity of three and four billiard spheres in the toroidal container $\Bbb T^\nu$. By inventing new topological methods and the Connecting Path Formula (CPF), in my two-part paper \[Sim(1992)\] I proved the (hyperbolic) ergodicity of $N$ hard spheres in $\Bbb T^\nu$, provided that $N\le\nu$. The common feature of hard sphere systems is — as D. Szász pointed this out first in \[Sz(1993)\] and \[Sz(1994)\] — that all of them belong to the family of so called cylindric billiards, the definition of which can be found later in this paragraph. However, the first appearance of a special, 3-D cylindric billiard system took place in \[K-S-Sz(1989)\], where we proved the ergodicity of a 3-D billiard flow with two orthogonal cylindric scatterers. Later D. Szász \[Sz(1994)\] presented a complete picture (as far as ergodicity is concerned) of cylindric billiards with cylinders whose generator subspaces are spanned by mutually orthogonal coordinate axes. The task of proving ergodicity for the first non-trivial, non-orthogonal cylindric billiard system was taken up in \[S-Sz(1994)\]. Finally, in our joint venture with D. Szász \[S-Sz(1999)\] we managed to prove the complete hyperbolicity of [*typical*]{} hard sphere systems. Consider the $d$-dimensional ($d\ge2$) flat torus $\Bbb T^d=\Bbb R^d/\Cal L$ supplied with the usual Riemannian inner product $\langle\, .\, ,\, .\, \rangle$ inherited from the standard inner product of the universal covering space $\Bbb R^d$. Here $\Cal L\subset\Bbb R^d$ is supposed to be a lattice, i. e. a discrete subgroup of the additive group $\Bbb R^d$ with $\text{rank}(\Cal L)=d$. The reason why we want to allow general lattices, other than just the integer lattice $\Bbb Z^d$, is that otherwise the hard sphere systems would not be covered. The geometry of the structure lattice $\Cal L$ in the case of a hard sphere system is significantly different from the geometry of the standard lattice $\Bbb Z^d$ in the standard Euclidean space $\Bbb R^d$, see subsection 2.4 of \[Sim(2002)\]. The configuration space of a cylindric billiard is $\bold Q=\Bbb T^d\setminus\left(C_1\cup\dots\cup C_k\right)$, where the cylindric scatterers $C_i$ ($i=1,\dots,k$) are defined as follows: Let $A_i\subset\Bbb R^d$ be a so called lattice subspace of $\Bbb R^d$, which means that $\text{rank}(A_i\cap\Cal L)=\text{dim}A_i$. In this case the factor $A_i/(A_i\cap\Cal L)$ is a subtorus in $\Bbb T^d=\Bbb R^d/\Cal L$ which will be taken as the generator of the cylinder $C_i\subset\Bbb T^d$, $i=1,\dots,k$. Denote by $L_i=A_i^\perp$ the orthocomplement of $A_i$ in $\Bbb R^d$. Throughout this paper we will always assume that $\text{dim}L_i\ge2$. Let, furthermore, the numbers $r_i>0$ (the radii of the spherical cylinders $C_i$) and some translation vectors $t_i\in\Bbb T^d=\Bbb R^d/\Cal L$ be given. The translation vectors $t_i$ play a crucial role in positioning the cylinders $C_i$ in the ambient torus $\Bbb T^d$. Set $$C_i=\left\{x\in\Bbb T^d:\; \text{dist}\left(x-t_i,A_i/(A_i\cap\Cal L) \right)<r_i \right\}.$$ In order to avoid further unnecessary complications, we always assume that the interior of the configuration space $\bold Q=\Bbb T^d\setminus\left(C_1\cup\dots\cup C_k\right)$ is connected. The phase space $\bold M$ of our cylindric billiard flow will be the unit tangent bundle of $\bold Q$ (modulo some natural identification at its boundary), i. e. $\bold M=\bold Q\times\Bbb S^{d-1}$. (Here $\Bbb S^{d-1}$ denotes the unit sphere of $\Bbb R^d$.) The dynamical system $\flow$, where $S^t$ ($t\in\Bbb R$) is the dynamics defined by uniform motion inside the domain $\bold Q$ and specular reflections at its boundary (at the scatterers), and $\mu$ is the Liouville measure, is called the cylindric billiard flow we want to investigate. (As to the notions and notations in connection with semi-dispersive billiards, the reader is kindly recommended to consult the work \[K-S-Sz(1990)\].) The main conjecture concerning the (hyperbolic) ergodicity of cylindric billiards is the “Erdőtarcsa conjecture” (named after the picturesque village in rural Hungary where it was originally formulated) that appeared as Conjecture 1 in Section 3 of \[S-Sz(2000)\]. Let $L_1,\dots,L_k\subset\Bbb R^d$ be subspaces, $\text{dim}L_i\ge2$, $A_i=L_i^\perp$, $i=1,\dots,k$. Set $$\Cal G_i=\left\{U\in\text{SO}(d):\, U|A_i=\text{Id}_{A_i}\right\},$$ and let $\Cal G=\left\langle\Cal G_1,\dots,\Cal G_k\right\rangle\subset\text{SO}(d)$ be the algebraic generate of the compact, connected Lie subgroups $\Cal G_i$ in $\text{SO}(d)$. The following notions appeared in Section 3 of \[S-Sz(2000)\]. We say that the system of base spaces $\{L_1,\dots,L_k\}$ (or, equivalently, the cylindric billiard system defined by them) is [*transitive*]{} if and only if the group $\Cal G$ acts transitively on the unit sphere $\Bbb S^{d-1}$ of $\Bbb R^d$. We say that the system of subspaces $\{L_1,\dots,L_k\}$ has the Orthogonal Non-splitting Property (ONSP) if there is no non-trivial orthogonal splitting $\Bbb R^d=B_1\oplus B_2$ of $\Bbb R^d$ with the property that for every index $i$ ($1\le i\le k$) $L_i\subset B_1$ or $L_i\subset B_2$. The next result can be found in Section 3 of \[S-Sz(2000)\] (see 3.1–3.6 thereof): For the system of subspaces $\{L_1,\dots,L_k\}$ the following three properties are equivalent: \(1) $\{L_1,\dots,L_k\}$ is transitive; \(2) the system of subspaces $\{L_1,\dots,L_k\}$ has the ONSP; \(3) the natural representation of $\Cal G$ in $\Bbb R^d$ is irreducible. A cylindric billiard flow is ergodic if and only if it is transitive. In that case the cylindric billiard system is actually a completely hyperbolic Bernoulli flow, see \[C-H(1996)\] and \[O-W(1998)\]. In order to avoid unnecessary complications, throughout the paper we always assume that $$\text{int}\bold Q\text{ is connected, and} \tag 1.1$$ $$\aligned \text{the }d\text{-dim spatial angle }\alpha(q)\text{ subtended by }\bold Q \\ \text{at any of its boundary points }q\in\partial\bold Q\text{ is positive.} \endaligned \tag 1.2$$ The Erdőtarcsa Conjecture has not been proved so far in full generality. Certain partial results, however, exist. Without pursuing the goal of achieving completeness, here we cite just two of such results: Almost every hard disk system (i. e. hard sphere system in a $2$-D torus) is hyperbolic and ergodic. (Here “almost every” is meant with respect to the outer geometric parameters $(r;\, m_1,\dots,m_N)$, where $r>0$ is the common radius of the disks, while $m_1,\dots,m_N$ are the masses.) Every hard sphere system is completely hyperbolic, i. e. all of its relevant Lyapunov exponents are nonzero almost everywhere. In this paper we are mainly interested in understanding the ergodic properties of cylindric billiard flows $\flow$ in which the closures $\bar C_i$ of the scattering cylinders $C_i$ are pairwise disjoint, i. e. $$\bar C_i\cap\bar C_j=\emptyset\text{ for }1\le i<j\le k. \tag 1.3$$ Elementary linear algebra shows that for such a (disjoint) cylindric billiard system it is true that $\text{span}\left\{A_i,A_j\right\}\ne\Bbb R^d$ for $1\le i,\, j\le k$ or, equivalently, $$L_i\cap L_j\ne\{0\}\text{ for }1\le i,\, j\le k. \tag 1.4$$ From now on we drop the disjointness condition (1.3) by only keeping the somewhat relaxed condition (1.4) above. The first, very simple, question that arises here is to characterize the transitivity of the $\Cal G$-action on $\Bbb S^{d-1}$ under the condition (1.4) above. The following proposition immediately follows from (1.4) and the characterization (2) of the transitivity above: A cylindric billiard system with the additional property (1.4) is transitive (that is, the $\Cal G$-action on the velocity sphere $\Bbb S^{d-1}$ is transitive) if and only if $\text{span}\left\{L_1,\dots,L_k\right\}=\Bbb R^d$ or, equivalently, $\bigcap_{i=1}^k A_i=\{0\}$. Now we are able to put forward the result of this paper: Assume that the cylindric billiard flow $\flow$ enjoys the geometric properties (1.1), (1.2), and (1.4) above. Then the transitivity condition $\bigcap_{i=1}^k A_i=\{0\}$ implies that the flow $\flow$ is completely hyperbolic and ergodic. In the case of a hard sphere system with masses $m_1,\,m_2,\dots,\,m_N$ the base space $L_{ij}$ of the cylinder $C_{ij}$ (describing the collision between the spheres labelled by $i$ and $j$) is obviously the set $$\left\{(\delta q_1,\dots,\delta q_N)\in\Cal T\bold Q\big|\; \delta q_k=0 \text{ for } k\not\in\{i,\,j\}\right\}.$$ (See also (4.4) in \[S-Sz(2000)\].) Therefore, in such systems the intersection of any two base spaces is zero. This shows that our present result is complementary to any possible past and future result about hard sphere systems. Section 2 contains the indispensable technical preparations, definitions, and notations. §3 is devoted to proving that if a non-singular orbit segment $S^{[a,b]}x$ of the flow $\flow$ has a combinatorially rich symbolic collision sequence (in a well defined sense) then $S^{[a,b]}x$ is sufficient (geometrically hyperbolic) modulo a codimension-2 algebraic subset of the phase space. Finally, the closing Section 4 contains the inductive proof of (H1) the so called “Chernov–Sinai Ansatz” for the flow $\flow$, i. e. that — informally speaking — for almost every singular phase point $x\in\bold M$ the forward semi-trajectory after the singularity is sufficient and (H2) outside of a slim subset $S\subset\bold M$ (for the notion of slimness, please see §2 below) it is true that \(i) $S^{(-\infty,\infty)}x$ has at most one singularity; \(ii) $S^{(-\infty,\infty)}x$ is sufficient. Section 4 concludes with putting together all the above results and applying the Theorem on Local Ergodicity for semi-dispersive billiards \[S-Ch(1987)\] to complete the proof of our Theorem. In order to simplify the notations, throughout the paper we will assume that the fundamental lattice $\Cal L\subset\Bbb R^d$ of the factorization $\Bbb T^d=\Bbb R^d/\Cal L$ is the standard integer lattice $\Bbb Z^d$. This is not a significant restriction of generality, for the entire proof of the theorem carries over to the general case easily by an almost word-by-word translation. [**2. Prerequisites**]{} We are going to briefly describe the discontinuity of the flow $\{S^t\}$ caused by a multiple collision at time $t_0$. Assume first that the pre–collision velocities of the particles are given. What can we say about the possible post–collision velocities? Let us perturb the pre–collision phase point (at time $t_0-0$) infinitesimally, so that the collisions at $\sim t_0$ occur at infinitesimally different moments. By applying the collision laws to the arising finite sequence of collisions, we see that the post-collision velocities are fully determined by the time– ordering of the considered collisions. Therefore, the collection of all possible time-orderings of these collisions gives rise to a finite family of continuations of the trajectory beyond $t_0$. They are called the [**trajectory branches**]{}. It is quite clear that similar statements can be said regarding the evolution of a trajectory through a multiple collision [**in reverse time**]{}. Furthermore, it is also obvious that for any given phase point $x_0\in\bold M$ there are two, $\omega$-high trees $\Cal T_+$ and $\Cal T_-$ such that $\Cal T_+$ ($\Cal T_-$) describes all the possible continuations of the positive (negative) trajectory $S^{[0,\infty)}x_0$ ($S^{(-\infty,0]}x_0$). (For the definitions of trees and for some of their applications to billiards, cf. the beginning of §5 in \[K-S-Sz(1992)\].) It is also clear that all possible continuations (branches) of the whole trajectory $S^{(-\infty,\infty)}x_0$ can be uniquely described by all possible pairs $(B_-,B_+)$ of $\omega$-high branches of the trees $\Cal T_-$ and $\Cal T_+$ ($B_-\subset\Cal T_-, B_+\subset \Cal T_+$). Finally, we note that the trajectory of the phase point $x_0$ has exactly two branches, provided that $S^tx_0$ hits a singularity for a single value $t=t_0$, and the phase point $S^{t_0}x_0$ does not lie on the intersection of more than one singularity manifolds. (In this case we say that the trajectory of $x_0$ has a “simple singularity”.) Consider a [**nonsingular**]{} trajectory segment $S^{[a,b]}x$. Suppose that $a$ and $b$ are [**not moments of collision**]{}. The neutral space $\Cal N_0(S^{[a,b]}x)$ of the trajectory segment $S^{[a,b]}x$ at time zero ($a<0<b$) is defined by the following formula: $$\aligned \Cal N_0(S^{[a,b]}x)=\big \{W\in\Cal Z\colon\;\exists (\delta>0) \; \text{s. t.} \; \forall \alpha \in (-\delta,\delta) \\ V\left(S^a\left(Q(x)+\alpha W,V(x)\right)\right)=V(S^ax)\text{ and } V\left(S^b\left(Q(x)+\alpha W,V(x)\right)\right)=V(S^bx)\big\}. \endaligned$$ ($\Cal Z$ is the common tangent space $\Cal T_q\bold Q$ of the parallelizable manifold $\bold Q$ at any of its points $q$, while $V(x)$ is the velocity component of the phase point $x=\left(Q(x),\,V(x)\right)$.) It is known (see (3) in §3 of \[S-Ch (1987)\]) that $\Cal N_0(S^{[a,b]}x)$ is a linear subspace of $\Cal Z$ indeed, and $V(x)\in \Cal N_0(S^{[a,b]}x)$. The neutral space $\Cal N_t(S^{[a,b]}x)$ of the segment $S^{[a,b]}x$ at time $t\in [a,b]$ is defined as follows: $$\Cal N_t(S^{[a,b]}x)=\Cal N_0\left(S^{[a-t,b-t]}(S^tx)\right).$$ It is clear that the neutral space $\Cal N_t(S^{[a,b]}x)$ can be canonically identified with $\Cal N_0(S^{[a,b]}x)$ by the usual identification of the tangent spaces of $\bold Q$ along the trajectory $S^{(-\infty,\infty)}x$ (see, for instance, §2 of \[K-S-Sz(1990)\]). Our next definition is that of the [**advance**]{}. Consider a non-singular orbit segment $S^{[a,b]}x$ with the symbolic collision sequence $\Sigma=(\sigma_1, \dots, \sigma_n)$ ($n\ge 1$), meaning that $S^{[a,b]}x$ has exactly $n$ collisions with $\partial\bold Q$, and the $i$-th collision ($1\le i\le n$) takes place at the boundary of the cylinder $C_{\sigma_i}$. For $x=(Q,V)\in\bold M$ and $W\in\Cal Z$, $\Vert W\Vert$ sufficiently small, denote $T_W(Q,V):=(Q+W,V)$. For any $1\le k\le n$ and $t\in[a,b]$, the advance $$\alpha(\sigma_k)\colon\;\Cal N_t(S^{[a,b]}x) \rightarrow \Bbb R$$ of the collision $\sigma_k$ is the unique linear extension of the linear functional $\alpha(\sigma_k)$ defined in a sufficiently small neighborhood of the origin of $\Cal N_t(S^{[a,b]}x)$ in the following way: $$\alpha(\sigma_k)(W):= t_k(x)-t_k(S^{-t}T_WS^tx).$$ Here $t_k=t_k(x)$ is the time moment of the $k$-th collision $\sigma_k$ on the trajectory of $x$ after time $t=a$. The above formula and the notion of the advance functional $$\alpha_k=\alpha(\sigma_k):\; \Cal N_t\left(S^{[a,b]}x\right)\to\Bbb R$$ has two important features: \(i) If the spatial translation $(Q,V)\mapsto(Q+W,V)$ is carried out at time $t$, then $t_k$ changes linearly in $W$, and it takes place just $\alpha_k(W)$ units of time earlier. (This is why it is called “advance”.) \(ii) If the considered reference time $t$ is somewhere between $t_{k-1}$ and $t_k$, then the neutrality of $W$ precisely means that $$W-\alpha_k(W)\cdot V(x)\in A_{\sigma_k},$$ i. e. a neutral (with respect to the collision $\sigma_k$) spatial translation $W$ with the advance $\alpha_k(W)=0$ means that the vector $W$ belongs to the generator space $A_{\sigma_k}$ of the cylinder $C_{\sigma_k}$. It is now time to bring up the basic notion of [**sufficiency**]{} (or, sometimes it is also called [**geometric hyperbolicity**]{}) of a trajectory (segment). This is the utmost important necessary condition for the proof of the fundamental theorem for semi-dispersive billiards, see Condition (ii) of Theorem 3.6 and Definition 2.12 in \[K-S-Sz(1990)\]. The nonsingular trajectory segment $S^{[a,b]}x$ ($a$ and $b$ are supposed not to be moments of collision) is said to be [**sufficient**]{} if and only if the dimension of $\Cal N_t(S^{[a,b]}x)$ ($t\in [a,b]$) is minimal, i.e. $\text{dim}\ \Cal N_t(S^{[a,b]}x)=1$. The trajectory segment $S^{[a,b]}x$ containing exactly one singularity (a so called “simple singularity”, see above) is said to be [**sufficient**]{} if and only if both branches of this trajectory segment are sufficient. The phase point $x\in\bold M$ with at most one singularity is said to be sufficient if and only if its whole trajectory $S^{(-\infty,\infty)}x$ is sufficient, which means, by definition, that some of its bounded segments $S^{[a,b]}x$ are sufficient. In the case of an orbit $S^{(-\infty,\infty)}x$ with a simple singularity, sufficiency means that both branches of $S^{(-\infty,\infty)}x$ are sufficient. By the results of Vaserstein \[V(1979)\], Galperin \[G(1981)\] and Burago-Ferleger-Kononenko \[B-F-K(1998)\], in a semi-dispersive billiard flow there can only be finitely many collisions in finite time intervals, see Theorem 1 in \[B-F-K(1998)\]. Thus, the dynamics is well defined as long as the trajectory does not hit more than one boundary components at the same time. We are going to summarize the basic properties of codimension-two subsets $A$ of a smooth manifold $M$. Since these subsets $A$ are just those negligible in our dynamical discussions, we shall call them [**slim**]{}. As to a broader exposition of the issues, see \[E(1978)\] or §2 of \[K-S-Sz(1991)\]. Note that the dimension $\dim A$ of a separable metric space $A$ is one of the three classical notions of topological dimension: the covering (Čech-Lebesgue), the small inductive (Menger-Urysohn), or the large inductive (Brouwer-Čech) dimension. As it is known from general general topology, all of them are the same for separable metric spaces. A subset $A$ of $M$ is called slim if and only if $A$ can be covered by a countable family of codimension-two (i. e. at least two) closed sets of $\mu$–measure zero, where $\mu$ is a smooth measure on $M$. (Cf. Definition 2.12 of \[K-S-Sz(1991)\].) The collection of all slim subsets of $M$ is a $\sigma$-ideal, that is, countable unions of slim sets and arbitrary subsets of slim sets are also slim. A subset $A\subset M$ is slim if and only if for every $x\in A$ there exists an open neighborhood $U$ of $x$ in $M$ such that $U\cap A$ is slim. (Cf. Lemma 2.14 of \[K-S-Sz(1991)\].) A closed subset $A\subset M$ is slim if and only if $\mu(A)=0$ and $\dim A\le\dim M-2$. If $A\subset M_1\times M_2$ is a closed subset of the product of two manifolds, and for every $x\in M_1$ the set $$A_x=\{ y\in M_2\colon\; (x,y)\in A\}$$ is slim in $M_2$, then $A$ is slim in $M_1\times M_2$. The following propositions characterize the codimension-one and codimension-two sets. For any closed subset $S\subset M$ the following three conditions are equivalent: “[(i)]{}” $\dim S\le\dim M-2$; “[(ii)]{}” $\text{int}S=\emptyset$ and for every open connected set $G\subset M$ the difference set $G\setminus S$ is also connected; “[(iii)]{}” $\text{int}S=\emptyset$ and for every point $x\in M$ and for any open neighborhood $V$ of $x$ in $M$ there exists a smaller open neighborhood $W\subset V$ of the point $x$ such that for every pair of points $y,z\in W\setminus S$ there is a continuous curve $\gamma$ in the set $V\setminus S$ connecting the points $y$ and $z$. (See Theorem 1.8.13 and Problem 1.8.E of \[E(1978)\].) For any subset $S\subset M$ the condition $\dim S\le\dim M-1$ is equivalent to $\text{int}S=\emptyset$. (See Theorem 1.8.10 of \[E(1978)\].) We recall an elementary, but important lemma (Lemma 4.15 of \[K-S-Sz(1991)\]). Let $R_2$ be the set of phase points $x\in\bold M\setminus\partial\bold M$ such that the trajectory $S^{(-\infty,\infty)}x$ has more than one singularities. The set $R_2$ is a countable union of codimension-two smooth sub-manifolds of $M$ and, being such, it is slim. The next lemma establishes the most important property of slim sets which gives us the fundamental geometric tool to connect the open ergodic components of billiard flows. If $M$ is connected, then the complement $M\setminus A$ of a slim set $A\subset M$ necessarily contains an arc-wise connected, $G_\delta$ set of full measure. (See Property 3 of §4.1 in \[K-S-Sz(1989)\]. The $G_\delta$ sets are, by definition, the countable intersections of open sets.) Denote by $\bold M^\#$ the set of all phase points $x\in\bold M$ for which the trajectory of $x$ encounters infinitely many non-tangential collisions in both time directions. The trajectories of the points $x\in\bold M\setminus\bold M^\#$ are lines: the motion is linear and uniform, see the appendix of \[Sz(1994)\]. It is proven in lemmas A.2.1 and A.2.2 of \[Sz(1994)\] that the closed set $\bold M\setminus\bold M^\#$ is a finite union of hyperplanes. It is also proven in \[Sz(1994)\] that, locally, the two sides of a hyperplanar component of $\bold M\setminus\bold M^\#$ can be connected by a positively measured beam of trajectories, hence, from the point of view of ergodicity, in this paper it is enough to show that the connected components of $\bold M^\#$ entirely belong to one ergodic component. This is what we are going to do in this paper. Denote by $\bold M^0$ the set of all phase points $x\in\bold M^\#$ the trajectory of which does not hit any singularity, and use the notation $\bold M^1$ for the set of all phase points $x\in\bold M^\#$ whose orbit contains exactly one, simple singularity. According to Proposition 2.12, the set $\bold M^\#\setminus(\bold M^0\cup\bold M^1)$ is a countable union of smooth, codimension-two ($\ge2$) submanifolds of $\bold M$, and, therefore, this set may be discarded in our study of ergodicity, please see also the properties of slim sets above. Thus, we will restrict our attention to the phase points $x\in\bold M^0\cup\bold M^1$. An essential precondition for the Theorem on Local Ergodicity by Bálint–Chernov–Szász–Tóth is the so called “Chernov-Sinai Ansatz” which we are going to formulate below. Denote by $\Cal S\Cal R^+\subset\partial\bold M$ the set of all phase points $x_0=(q_0,v_0)\in\partial\bold M$ corresponding to singular reflections (a tangential or a double collision at time zero) supplied with the post-collision (outgoing) velocity $v_0$. It is well known that $\Cal S\Cal R^+$ is a compact cell complex with dimension $2d-3=\text{dim}\bold M-2$. It is also known (see Lemma 4.1 in \[K-S-Sz(1990)\]) that for $\nu$-almost every phase point $x_0\in\Cal S\Cal R^+$ (Here $\nu$ is the Riemannian volume of $\Cal S\Cal R^+$ induced by the restriction of the natural Riemannian metric of $\bold M$.) the forward orbit $S^{(0,\infty)}x_0$ does not hit any further singularity. The Chernov-Sinai Ansatz postulates that for $\nu$-almost every $x_0\in\Cal S\Cal R^+$ the forward orbit $S^{(0,\infty)}x_0$ is sufficient (geometrically hyperbolic). The Theorem on Local Ergodicity by Chernov and Sinai (Theorem 5 of \[S-Ch(1987)\], see also Theorem 4.4 in \[B-Ch-Sz-T(2002)\]) claims the following: Let $\flow$ be a semi-dispersive billiard flow with the properties (1.1)–(1.2) and such that the smooth components of the boundary $\partial\bold Q$ of the configuration space are algebraic hypersurfaces. (The cylindric billiards with (1.1)–(1.2) automatically fulfill this algebraicity condition.) Assume – further – that the Chernov-Sinai Ansatz holds true, and a phase point $x_0\in\bold M\setminus\partial\bold M$ is given with the properties \(i) $S^{(-\infty,\infty)}x$ has at most one singularity, and \(ii) $S^{(-\infty,\infty)}x$ is sufficient. (In the case of a singular obit $S^{(-\infty,\infty)}x$ this means that both branches of $S^{(-\infty,\infty)}x$ are sufficient.) Then some open neighborhood $U_0\subset\bold M$ of $x_0$ belongs to a single ergodic component of the flow $\flow$. (Modulo the zero sets, of course.) [**3. Geometric Considerations**]{} Consider a non-singular trajectory segment $$S^{[a,b]}x_0=\left\{x_t=S^tx_0\big|\; a\le t\le b\right\}$$ of the cylindric billiard flow $\flow$ with the symbolic collision sequence $\symb$, meaning that there are time moments $a<t_1<t_2<\dots<t_n<b$ such that $S^tx\not\in\partial\bold M$ for $t\in[a,b]\setminus\{t_1,\dots,t_n\}$, and $Q\left(S^{t_i}x\right)\in\partial C_{\sigma_i}$, $i=1,\dots,n$. We assume that \(1) $\text{dim}\left(L_{\sigma_i}\cap L_{\sigma_j}\right)\ge2$ for $1\le i,\, j\le n$ (the so called “codimension-two condition” imposed on $\symb$), and \(2) $\text{span}\left\{L_{\sigma_1},\dots,L_{\sigma_n}\right\}=\Bbb R^d$, i. e. the system of cylinders $C_{\sigma_1},\dots,C_{\sigma_n}$ is transitive, see also §1. The first result of this section is Under the above conditions (1)–(2) the non-singular orbit segment $S^{[a,b]}x_0$ is hyperbolic (sufficient, cf. §2) modulo some codimension-two (i. e. at least two) submanifolds of the phase space. The proof is based upon the following, simple lemma: Let $n=2$, i. e. $\Sigma\left(S^{[a,b]}x_0\right)=(\sigma_1,\, \sigma_2)$. Then the advance functionals (cf. §2) $\alpha_1,\, \alpha_2:\; \Cal N_0\left(S^{[a,b]}x_0\right)\to\Bbb R$ corresponding to $\sigma_1$ and $\sigma_2$ are the same, unless the phase point $x_0$ belongs to some codimension-two ($\ge2$) submanifold. We may assume that the reference time $t=0$ is between the collisions $\sigma_1$ and $\sigma_2$, i. e. $t_1=t(\sigma_1)<0<t_2=t(\sigma_2)$. Consider an arbitrary neutral vector $\delta q\in\Cal N_0\left(S^{[a,b]}x_0\right)$. The neutrality of $\delta q$ with the advances $\alpha_i=\alpha_i(\delta q)$ ($i=1,2$) means that $$\delta q-\alpha_i v_0\in A_{\sigma_i}\quad (i=1,2), \tag 3.3$$ where $v_0=v(x_0)$, $x_0=(q_0,\, v_0)$. If $\alpha_1=\alpha_1(\delta q)$ happens to be different from $\alpha_2=\alpha_2(\delta q)$, then the equations in (3.3) yield that $(\alpha_1-\alpha_2)v_0\in\text{span}\left\{A_{\sigma_1}, \, A_{\sigma_2}\right\}$, i. e. $v_0\in\text{span}\left\{A_{\sigma_1},\, A_{\sigma_2}\right\}$. However, $$c:=\text{codim}\left(\text{span}\left\{A_{\sigma_1},\, A_{\sigma_2}\right\} \right)=\text{dim}\left(L_{\sigma_1}\cap L_{\sigma_2}\right)\ge2$$ (by our assumption (1)), and the event $v_0\in\text{span}\left\{A_{\sigma_1},\, A_{\sigma_2}\right\}$ is clearly described by a submanifold of codimension $c$. Finishing the proof of the proposition: According to the lemma, apart from a codimension-two ($\ge2$) exceptional set $E\subset\bold M$ it is true that all advance functionals $\alpha_i=\alpha_{\sigma_i}:\; \Cal N\left(S^{[a,b]}x_0\right)\to\Bbb R$ coincide. Assume that $x_0\not\in E$ and the reference time $t=0$ is chosen to be right before the first collision $\sigma_1$ of $\symb$. Consider an arbitrary neutral vector $w=\delta q\in\Cal N_0\left(S^{[a,b]}x_0\right)$. By replacing $w$ with $w-\alpha v_0$ ($x_0=(q_0,v_0)$, $\alpha=\alpha_i(w)$ is the common value of the advances $\alpha_i(w)$) we easily achieve that $\alpha_i(w)=0$ for $i=1,2,\dots,n$. The relation $\alpha_1(w)=0$ means that $w\in A_{\sigma_1}$ and $DS^{t^*_1}\left((w,0)\right)=(w,0)$, where $t(\sigma_1)<t^*_1<t(\sigma_2)$. Similarly, $\alpha_2(w)=0$ means that $w\in A_{\sigma_2}$ and $DS^{t^*_2}\left((w,0)\right)=(w,0)$, where $t(\sigma_2)<t^*_2<t(\sigma_3)$, etc. We get that $$w\in\bigcap_{i=1}^n A_{\sigma_i}=\text{span}\left\{L_{\sigma_i}\big|\; 1\le i\le n\right\}^\perp=\{0\},$$ thus $w=0$. This shows that the original neutral vector was indeed a scalar multiple of the velocity $v_0$, so $\text{dim}\Cal N_0\left(S^{[a,b]}x_0\right)=1$ whenever $x_0\not\in E$. It is clear from the above proof that without the assumption (1) of the proposition we obtain a codimension-one exceptional set $E\subset \bold M$ outside of which the statement holds true. Indeed, the overall assumption on the geometry of our cylindric billiard system is that $L_i\cap L_j\ne\{0\}$ for any pair of base spaces $L_i$ and $L_j$. Some Observations Concerning Codimension-one\ Exceptional Manifolds $J\subset\bold M$ The last remaining question of this section is this: In the original set-up (i. e. when only $\text{dim}\left(L_{\sigma_i}\cap L_{\sigma_j}\right)\ge1$ is assumed in (1) of Proposition 3.1) how an enhanced version of Proposition 3.1 excludes the existence of a codimension-one, smooth sub-manifold $J\subset\bold M$ separating different ergodic components of the flow $\flow$? Given a codimension-one, flow-invariant, smooth sub-manifold $J\subset\bold M$, consider a normal vector $n_0=(z,w)$ ($\ne 0$) of $J$ at the phase point $y\in J$, i. e. for any tangent vector $(\delta q,\, \delta v)\in\Cal T_y\bold M$ the relation $(\delta q,\, \delta v)\in\Cal T_yJ$ is true if and only if $\langle\delta q,z\rangle+\langle\delta v,w\rangle=0$. Here $\langle\, .\, ,\, .\, \rangle$ is the Euclidean inner product of the tangent space $\Bbb R^d$ of $\Bbb T^d$ at every point $q\in\Bbb T^d$. Let us determine first the time-evolution $n_0\longmapsto n_t$ ($t>0$) of this normal vector as time $t$ elapses. If there is no collision on the orbit segment $S^{[0,t]}y$, then the relationship between $(\delta q,\, \delta v)\in\Cal T_y\bold M$ and $(\delta q',\, \delta v')=\left(DS^t\right)(\delta q,\, \delta v)$ is obviously $$\aligned \delta v'&=\delta v, \\ \delta q'&=\delta q+t\delta v, \endaligned \tag 3.5$$ from which we obtain that $$\aligned (\delta q',\, \delta v')\in\Cal T_{y'}J&\Leftrightarrow\langle\delta q'-t \delta v',\, z\rangle+\langle\delta v',\, w\rangle=0 \\ &\Leftrightarrow \langle\delta q',\, z\rangle+\langle\delta v',\, w-tz\rangle=0. \endaligned$$ This means that $n_t=(z,\, w-tz)$. It is always very useful to consider the quadratic form $Q(n)=Q((z,w))=:\langle z,w\rangle$ associated with the normal vector $n=(z,w)\in\Cal T_y\bold M$ of $J$ at $y$. $Q(n)$ is the so called “infinitesimal Lyapunov function”, see \[K-B(1994)\] or part A.4 of the Appendix in \[Ch(1994)\]. For a detailed exposition of the relationship between the quadratic form $Q$, the relevant symplectic geometry and the dynamics, please see \[L-W(1995)\]. Since the normal vector $n=(z,w)$ of $J$ is only determined up to a nonzero scalar multiplier, the value $Q(n)$ is only determined up to a positive multiplier. However, this means that the sign of $Q(n)$ (which is the utmost important thing for us) is uniquely determined. This remark will gain a particular importance in the near future. From the above calculations we get that $$Q(n_t)=Q(n_0)-t||z||^2\le Q(n_0). \tag 3.6$$ The next question is how the normal vector $n$ of $J$ gets transformed $n^-\mapsto n^+$ through a collision (reflection) at time $t=0$? Elementary geometric considerations show (see Lemma 2 of \[Sin(1979)\], or formula (2) in §3 of \[S-Ch(1987)\]) that the linearization of the flow $$\left(DS^t\right)\Big|_{t=0}:\; (\delta q^-,\, \delta v^-)\longmapsto (\delta q^+,\, \delta v^+)$$ is given by the formulae $$\aligned \delta q^+&=R\delta q^-, \\ \delta v^+&=R\delta v^-+2\cos\phi RV^*KV\delta q^-, \endaligned \tag 3.7$$ where the operator $R:\; \Cal T_q\bold Q\to \Cal T_q\bold Q$ is the orthogonal reflection across the tangent hyperplane $\Cal T_q\partial\bold Q$ of $\partial\bold Q$ at $q\in \partial\bold Q$ ($y^-=(q,v^-)\in\partial\bold M$, $y^+=(q,v^+)\in\partial\bold M$), $V:\; (v^-)^\perp\to\Cal T_q\partial\bold Q$ is the $v^-$-parallel projection of the orthocomplement hyperplane $(v^-)^\perp$ onto $\Cal T_q\partial\bold Q$, $V^*:\; \Cal T_q\partial\bold Q\to (v^-)^\perp$ is the adjoint of $V$, i. e. it is the projection of $\Cal T_q\partial\bold Q$ onto $(v^-)^\perp$ being parallel to the normal vector $\nu(q)$ of $\partial\bold Q$ at $q\in\partial\bold Q$, $K:\; \Cal T_q\partial\bold Q\to \Cal T_q\partial\bold Q$ is the second fundamental form of $\partial\bold Q$ at $q$ and, finally, $\cos\phi=\langle\nu(q),\, v^+\rangle$ is the cosine of the angle $\phi$ subtended by $v^+$ and the normal vector $\nu(q)$. For the formula (3.7), please also see the last displayed formula of §1 in \[S-Ch(1982)\], or (i) and (ii) of Proposition 2.3 in \[K-S-Sz(1990)\]. We note that it is enough to deal with the tangent vectors $(\delta q^-,\, \delta v^-)\in(v^-)^\perp\times(v^-)^\perp$ ($(\delta q^+,\, \delta v^+)\in(v^+)^\perp\times(v^+)^\perp$), for the manifold $J$ under investigation is supposed to be flow-invariant, so any vector $(\delta q,\, \delta v)=(\alpha v,\, 0)$ ($\alpha\in\Bbb R$) is automatically inside $\Cal T_yJ$. The backward version (inverse) $$\left(DS^t\right)\Big|_{t=0}:\; (\delta q^+,\, \delta v^+)\mapsto (\delta q^-,\, \delta v^-)$$ can be deduced easily from (3.7): $$\aligned \delta q^-&=R\delta q^+, \\ \delta v^-&=R\delta v^+-2\cos\phi RV_1^*KV_1\delta q^+, \endaligned \tag 3.8$$ where $V_1:\; (v^+)^\perp\to\Cal T_q\partial\bold Q$ is the $v^+$-parallel projection of $(v^+)^\perp$ onto $\Cal T_q\partial\bold Q$. By using formula (3.8), one easily computes the time-evolution $n^-\longmapsto n^+$ of a normal vector $n^-=(z,w)\in\Cal T_{y^-}\bold M$ of $J$ if a collision $y^-\longmapsto y^+$ takes place at time $t=0$: $$\aligned (\delta q^+,\, \delta v^+)\in\Cal T_{y^+}J\Leftrightarrow\langle R\delta q^+, \, z\rangle+\langle R\delta v^+-2\cos\phi RV_1^*KV_1\delta q^+,\, w\rangle &=0 \\ \Leftrightarrow\langle\delta q^+,\, Rz-2\cos\phi V_1^*KV_1Rw\rangle+ \langle\delta v^+,\, Rw\rangle &=0. \endaligned$$ This means that $$n^+=\left(Rz-2\cos\phi V_1^*KV_1Rw,\, Rw\right) \tag 3.9$$ if $n^-=(z,\, w)$. It follows that $$\aligned Q(n^+)&=Q(n^-)-2\cos\phi\langle V_1^*KV_1Rw,\, Rw\rangle \\ &=Q(n^-)-2\cos\phi\langle KV_1Rw,\, V_1Rw\rangle\le Q(n^-). \endaligned \tag 3.10$$ Here we used the fact that the second fundamental form $K$ of $\partial\bold Q$ at $q$ is positive semi-definite, which just means that the billiard system is semi-dispersive. The last simple observation on the quadratic form $Q(n)$ regards the involution $I:\; \bold M\to\bold M$, $I(q,v)=(q,-v)$ corresponding to the time reversal. If $n=(z,w)$ is a normal vector of $J$ at $y$, then, obviously, $I(n)=(z,-w)$ is a normal vector of $I(J)$ at $I(y)$ and $$Q\left(I(n)\right)=-Q(n). \tag 3.11$$ By switching — if necessary — from the separating manifold $J$ to $I(J)$, and by taking a suitable remote image $S^t(J)$ ($t>>1$), in the spirit of (3.6), (3.10)–(3.11) we can assume that $$Q(n)<0 \tag 3.12$$ for every [*unit*]{} normal vector $n\in\Cal T_y\bold M$ of $J$ near a phase point $y\in J$. There could be, however, a little difficulty in achieving the inequality $Q(n)<0$, i. e. (3.12). Namely, it may happen that $Q(n_t)=0$ for every $t\in\Bbb R$. According to (3.6), the equation $Q(n_t)=0$ ($\forall\, t\in\Bbb R$) implies that $n_t=:(z_t,\, w_t)=(0,\, w_t)$ for all $t\in\Bbb R$ and, moreover, in the view of (3.9), $w_t^+=Rw_t^-$ is the transformation law at any collision $y_t=(q_t,\, v_t)\in\partial\bold M$. Furthermore, at every collision $y_t=(q_t,\, v_t)\in\partial\bold M$ the projected tangent vector $V_1Rw_t^-=V_1w_t^+$ lies in the null space of the operator $K$ (see also (3.9)), and this means that $w_0$ is a neutral vector for the entire trajectory $S^{\Bbb R}y$, i. e. $w_0\in\Cal N\left(S^{\Bbb R}y\right)$. (For the notion of neutral vectors and $\Cal N\left(S^{\Bbb R}y\right)$, cf. §2 above.) On the other hand, this is impossible for the following reason: Any tangent vector $(\delta q,\delta v)$ from the space $\Cal N\left(S^{\Bbb R}y\right)\times\Cal N\left(S^{\Bbb R}y\right)$ is automatically tangent to the separating manifold $J$, thus for any normal vector $n=(z,w)\in\Cal T_y\bold M$ of a separating manifold $J$ one has $$(z,\, w)\in\Cal N\left(S^{\Bbb R}y\right)^\perp\times\Cal N\left( S^{\Bbb R}y\right)^\perp. \tag 3.14$$ (As a direct inspection shows. We always tacitly assume that the exceptional manifold $J$ is locally defined by the equation $J=\left\{x\in U_0\big|\;\text{ dim}\Cal N\left(S^{[a,b]}x\right)>1\right\}$ with orbit segments $S^{[a,b]}x$ whose symbolic sequence is combinatorially rich, i. e. it typically provides sufficient phase points.) The membership in (3.14) is, however, impossible with a nonzero vector $w\in\Cal N\left(S^{\Bbb R}y\right)$. Consider a smooth, connected piece $\Cal S\subset\bold M$ of a singularity manifold corresponding to a singular (tangential or double) reflection [*in the past*]{}. Such a manifold $\Cal S$ is locally flow-invariant and has one codimension, so we can speak about its normal vectors $n$ and the uniquely determined sign of $Q(n)$ for $0\ne n\in\Cal T_y\bold M$, $y\in\Cal S$, $n\perp\Cal S$ (depending on the foot point, of course). Consider first a phase point $y^+\in\partial\bold M$ right after the singular reflection that is described by $\Cal S$. It follows from the proof of Lemma 4.1 of \[K-S-Sz(1990)\] and Sub-lemma 4.4 therein that at $y^+=(q,\, v^+)\in\partial\bold M$ any tangent vector $(0,\, \delta v)\in\Cal T_{y^+}\bold M$ lies actually in $\Cal T_{y^+}\Cal S$ and, consequently, the normal vector $n=(z,w)\in\Cal T_{y^+}\bold M$ of $\Cal S$ at $y^+$ necessarily has the form $n=(z,0)$, i. e. $w=0$. Thus $Q(n)=0$ for any normal vector $n\in\Cal T_{y^+}\bold M$ of $\Cal S$. According to the monotonicity inequalities (3.6) and (3.10) above, $$Q(n)<0 \tag 3.15$$ for any phase point $y\in\Cal S$ of a past singularity manifold $\Cal S$. The above observations lead to the following conclusion: Assume that the separating manifold $J\subset\bold M$ ($J$ is smooth, connected, $\text{codim}(J)=1$) is selected in such a way that $Q(n_y)<0$ for all normal vectors $0\ne n_y\in\Cal T_y\bold M$ of $J$ at any point $y\in J$, see above. Suppose further that the non-singular orbit segments $S^{[a,b]}y$ ($y\in B_0$, $B_0$ is a small open ball, $0<a<b$ fixed) have the common symbolic collision sequence $\symb$ with the relaxed properties (1)’ $\text{dim}\left(L_{\sigma_i}\cap L_{\sigma_j}\right)\ge1$ ($1\le i,\, j\le n$), and (2)’ (the same as (2)) $\text{span}\left\{L_{\sigma_1},\dots,L_{\sigma_n}\right\}=\Bbb R^d$. We claim that for almost every phase point $y\in J\cap B_0$ (i. e. apart from an algebraic variety $E'\subset J\cap B_0$ with $\text{dim}(E')<\text{dim}(J)$) the orbit segment $S^{[a,b]}y$ is hyperbolic (sufficient). Consider the algebraic variety $E\subset B_0$ of exceptional phase points characterized by the proof of Proposition 3.1, i. e. let $$E=\left\{y\in B_0\big|\; S^{[a,b]}y \text{ is not hyperbolic}\right\}.$$ The mentioned proof (in particular, the proof of Lemma 3.2) shows that the only way to have a codimension-one, smooth component in the variety $E$ is to have a submanifold defined by the relation $$v_t\in\text{span}\left\{A_{\sigma_i},\, A_{\sigma_{i+1}}\right\} \tag 3.17$$ for some $i\in\{1,2,\dots,n-1\}$, $t(\sigma_i)<t<t(\sigma_{i+1})$ ($v_t$ is the velocity at time $t$) with $\text{dim}\text{span}\left\{A_{\sigma_i},\, A_{\sigma_{i+1}}\right\}=d-1$. However, the normal vector $n_t=(z_t,w_t)\in\Cal T_{y_t}\bold M$ of the manifold defined by (3.17) at the point $y_t=S^ty=(q_t,v_t)$ obviously has the form $(z_t,w_t)=(0,w_t)$, thus $Q(n_t)=0$. By the assumption of this proposition (and by the monotone non-increasing property of $Q(n_t)$ in $t$, see the inequalities (3.6) and (3.10)) we get that any codimension-one, smooth component of $E$ is transversal to $J$, thus proving the proposition. In view of the inequality (3.15) (valid for past-singularity manifolds $\Cal S$), the exceptional manifold $J\subset\bold M$ featuring Proposition 3.16 may be replaced by any past-singularity manifold $\Cal S$ without hurting the proof of the proposition. Thus, we obtain Let $\Cal S$ be a smooth component of a past-singularity set, $y_0\in\Cal S$, $0<a<b$, and assume that the non-singular orbit segment $S^{[a,b]}y_0$ has the symbolic collision sequence $\Sigma=\symb$ fulfilling (1)’–(2)’ of Proposition 3.16. Then there is an open neighborhood $B_0$ of $y_0$ in $\bold M$ such that for almost every phase point $y\in\Cal S\cap B_0$ (with respect to the induced hypersurface measure of $\Cal S\cap B_0$) the symbolic collision sequence $S^{[a,b]}y$ is still the same $\Sigma=\symb$, and $S^{[a,b]}y$ is hyperbolic. (Note that in this result, as usual, the phrase “almost every” may be replaced by saying that “apart from a countable family of smooth, proper sub-manifolds”.) Yet Another Corollary of (3.6), (3.9), and (3.10) Assume that the orbit segment $S^{[a,b]}y_0$ is not singular, $\Sigma\left(S^{[a,b]}y_0\right)\allowmathbreak=\symb$ fulfills the relaxed conditions (1)’–(2)’ of Proposition 3.16 and, finally, there is a codimension-one, flow-invariant, smooth submanifold $E\ni y_0$ in $\bold M$ such that \(i) $\Sigma\left(S^{[a,b]}y\right)=\Sigma\left(S^{[a,b]}y_0\right)$ for all $y\in E$, and \(ii) $S^{[a,b]}y$ is not hyperbolic for all $y\in E$. Let $0\ne n_t=(z_t,w_t)$ be a normal vector of $S^t(E)$ at the point $y_t=(q_t,v_t)$. We claim that $Q(n_t)<0$ for all $t>b$. As we have seen before, the manifold $E$ is defined by the relation $v_t\in\text{span}\left\{A_{\sigma_i},\, A_{\sigma_{i+1}}\right\}$ with some $i\in\{1,2,\dots,n-1\}$, $\text{dim}\left(\text{span}\left\{A_{\sigma_i},\, A_{\sigma_{i+1}}\right\} \right)=d-1$, $t(\sigma_i)<t<t(\sigma_{i+1})$. Thus $n_t=(0,\bar w)$, $0\ne\bar w\perp\text{span}\left\{A_{\sigma_i},\, A_{\sigma_{i+1}}\right\}$, $t(\sigma_i)<t<t(\sigma_{i+1})$, meaning also that $Q(n_t)=0$. Assume, to the contrary of the assertion of this corollary, that $Q(n_\tau)=0$ for some $\tau>b$. Then, by the non-increasing property of $Q(n_\tau)$ in $\tau$, there is a small $\epsilon>0$ such that $Q(n_\tau)=0$ for all $\tau$, $t(\sigma_{i+1})<\tau<t(\sigma_{i+1})+\epsilon$. By (3.10) this means that the vector $V_1R\bar w=V\bar w$ is in the null space of the operator $K$, i. e. $V\bar w\in A_{\sigma_{i+1}}$. This means, in particular, that $\bar w\in\text{span}\left\{v_t,\, A_{\sigma_{i+1}}\right\}$ for $t(\sigma_i)<t<t(\sigma_{i+1})$. On the other hand, $\text{span}\left\{v_t,\, A_{\sigma_{i+1}}\right\}\subset\text{span} \left\{A_{\sigma_{i}},\, A_{\sigma_{i+1}}\right\}$, and $\bar w\perp\text{span}\left\{A_{\sigma_{i}},\, A_{\sigma_{i+1}}\right\}$, a contradiction. An almost immediate consequence of the above corollary and the proof of Proposition 3.16 is Assume that the non-singular orbit segments $S^{[a,b]}y$ ($y\in B_0$, $B_0$ is a small, open ball) have the common symbolic collision sequence $$\left(\Sigma^{(1)},\, \Sigma^{(2)}\right)=\left(\sigma_1^{(1)},\dots, \sigma_m^{(1)};\, \sigma_1^{(2)},\dots,\sigma_n^{(2)}\right)$$ such that both $\Sigma^{(j)}$ are [*combinatorially rich*]{}, i. e. $$\text{span}\left\{L_{\sigma_1^{(1)}},\, L_{\sigma_2^{(1)}},\, \dots,\, L_{\sigma_m^{(1)}}\right\}= \text{span}\left\{L_{\sigma_1^{(2)}},\, L_{\sigma_2^{(2)}},\, \dots,\, L_{\sigma_n^{(2)}}\right\}=\Bbb R^d.$$ Then the exceptional set $$E=\left\{y\in B_0\big|\; S^{[a,b]}y \text{ is not hyperbolic}\right\}$$ has codimension at least two. Let $E^{(j)}$ be a smooth, codimension-one exceptional manifold for $\Sigma^{(j)}$, $j=1,\, 2$. (The word “exceptional” refers to the fact that these manifolds consist of atypical phase points for which the corresponding $\Sigma^{(j)}$-part of the orbit is not hyperbolic, despite the assumed combinatorial richness of $\Sigma^{(j)}$.) Let $t(\sigma_m^{(1)})<t<t(\sigma_1^{(2)})$. By the previous corollary, the manifold $S^t\left(E^{(1)}\right)$ has a normal vector $n_t^{(1)}$ with $Q\left(n_t^{(1)}\right)<0$ at any point $y_t=S^ty$ ($y\in E^{(1)}$), while, by the same corollary again (applied in reverse time), at any phase point $y_t=S^ty$, $y\in E^{(2)}$, the manifold $S^t\left(E^{(2)}\right)$ has a normal vector $n_t^{(2)}$ with $Q\left(n_t^{(2)}\right)>0$, i. e. $E^{(1)}$ and $E^{(2)}$ are transversal at any point of their intersection. This finishes the proof of the corollary. [**4. Hyperbolicity Is Abundant The Inductive Proof**]{} Below we present the inductive proof of the Theorem of this paper. The induction will be performed with respect to the number of cylinders $k$. Beside the ergodicity (and, therefore, the Bernoulli property, see \[C-H(1996)\]) and \[O-W(1998)\]) we will prove (and use as the induction hypothesis!) a few technical properties listed below: (H1) The Chernov–Sinai Ansatz (see §2 above) holds true for the cylindric billiard flow $\flow$; (H2) There exists a slim subset $S\subset\bold M$ (see §2 for the concept of “slimness”) such that for all $x\in\bold M\setminus S$ 0.3truein (i) $S^{(-\infty,\infty)}x$ has at most one singularity and 0.3truein (ii) $S^{(-\infty,\infty)}x$ is hyperbolic (in the singular case both branches of $S^{(-\infty,\infty)}x$ are supposed to be hyperbolic, see §2 above). Consequently, according to the Fundamental Theorem for semi-dispersive billiards by Chernov and Sinai (Theorem 5 of \[S-Ch(1987)\], see also Theorem 4.4 in \[B-Ch-Sz-T(2002)\]) (H3) For every $x\in\bold M\setminus S$ the assertion of the Fundamental Theorem holds true in some open neighborhood $U_0$ of $x$ in $\bold M$, in particular, $x$ is a so called “zig-zag point”, see Definition 5.1 in \[Sz(2000)\]. Consequently, since the complementer set $\bold M\setminus S$ is known to contain a connected set of full measure (see §2) and the open neighborhood $U_0$ of $x$ belongs to a single ergodic component, we get that (H4) $\flow$ is ergodic, hence it is a Bernoulli flow by \[C-H(1996)\] and \[O-W(1998)\]. The above properties (H1)—(H2) will serve for us as the induction hypothesis. 1\. The base of the induction: $k=1$ In this case, necessarily, $L_1=\Bbb R^d$ and $A_1=\{0\}$, so the cylindric billiard system is actually a genuine, $d$-dimensional Sinai–billiard with a single spherical scatterer which has been well known to enjoy the properties (H1)—(H2) since the seminal work \[S-Ch(1987)\]. 2\. The induction step: $<k\longrightarrow k$, $k\ge2$. Let $k\ge2$, $\flow$ be a cylindric billiard flow fulfilling all the hypotheses of our Theorem, and suppose that the induction hypotheses have been successfully proven for every system (within the framework of the Theorem) with less than $k$ cylindric scatterers. First we prove (H1) for $\flow$. The upcoming proof of the Chernov-Sinai Ansatz is going to be a local argument by nature, for stating that “the forward orbit of almost every phase point $x$ on a past-singularity manifold is hyperbolic” is a local assertion. Let $\Cal S_0\subset\bold M^{\#}\setminus\partial\bold M$ (For the definition of $\bold M^{\#}$, please see §2.) be a small piece of a past-singularity manifold with the following properties: \(1) $\Cal S_0$ is smooth (analytic) and diffeomorphic to $\Bbb R^{2d-2}$; \(2) For every phase point $x\in\Cal S_0$ the last collision on the backward orbit $S^{(-\infty,0)}x$ is a singular collision taking place at time $\tau(x)<0$ so that the collision at $S^{\tau(x)}x$ is a simple singularity, see §2. Consequently, the type of this singularity (see §2) is the same for all $x\in\Cal S_0$. We will measure the size of the subsets $A\subset\Cal S_0$ by using the hypersurface measure $\nu$ induced on $\Cal S_0$. First of all, we restrict our attention to the subset $$A=\left\{x\in\Cal S_0\big|\; S^{(0,\infty)}x \text{ is non-singular}\right\} \tag 4.1$$ of $\Cal S_0$, being a $G_\delta$ subset of $\Cal S_0$ with full measure, $\nu(\Cal S_0\setminus A)=0$, for the set $\Cal S_0\setminus A$ is a countable union of smooth, proper submanifolds of $\Cal S_0$, see Lemma 4.1 in \[K-S-Sz(1990)\]. With each phase point $x\in A$ we associate the infinite symbolic collision sequence $$\Sigma(x)=\left(\sigma_1(x),\, \sigma_2(x),\dots\right)$$ of the forward orbit $S^{(0,\infty)}x$, the set of cylinders $$C(x)=\left\{\sigma_n(x)|\quad n=1,2,\dots\right\}, \tag 4.2$$ and the linear subspace $L(x)$ of $\Bbb R^d$ spanned by the base spaces of these cylinders: $$L(x)=\text{span}\left\{L_i\big|\; i\in C(x)\right\}. \tag 4.3$$ In view of Corollary 3.18, for $\nu$-almost every phase point $x$ of the (open) subset $$A_0=\left\{x\in A\big|\; L(x)=\Bbb R^d\right\} \tag 4.4$$ of $A$ the forward orbit $S^{(0,\infty)}x$ is hyperbolic (sufficient). Thus, in order to prove the Ansatz it is enough to show that $A_0=A$. We argue by contradiction. Assume that $$A_1=\left\{x\in A\big|\; C(x)=\Cal C_0\right\},\quad \nu(A_1)>0 \tag 4.5$$ for some $\Cal C_0\subset\{1,2,\dots,k\}$ with $$L^*=\text{span}\left\{L_i\big|\; i\in\Cal C_0\right\}\ne\Bbb R^d.$$ We will get a contradiction by applying (essentially) the invariant manifold construction ideas borrowed from the proof of Theorem 6.1 of \[Sim(1992-A)\]. Indeed, we pay attention to the forward orbits $S^{(0,\infty)}x$ of the points $x\in A_1$ governed solely by the sub-billiard dynamics defined by the cylinders $C_i$ with $i\in\Cal C_0$ in $\Bbb T^d=\Bbb R^d/\Bbb Z^d$. This sub-billiard dynamics obviously does not fulfill the transitivity condition, for it is invariant under all spatial translations by the elements of the subtorus $\tilde A=A^*/\left(A^*\cap\Bbb Z^d\right)$, where $A^*=(L^*)^\perp$ is the orthogonal complement of $L^*$ in $\Bbb R^d$. (Recall that the subspaces $L^*=\text{span}\left\{L_i\big|\; i\in\Cal C_0\right\}$ and $A^*$ are lattice subspaces, as elementary linear algebra shows.) The orthogonal direct sum $$\tilde L\oplus\tilde A=L^*/\left(L^*\cap\Bbb Z^d\right)\oplus A^*/\left(A^*\cap\Bbb Z^d\right)$$ of the sub-tori $\tilde L$ and $\tilde A$ provides a finite covering of $\Bbb T^d=\Bbb R^d/\Bbb Z^d$. Therefore, the sub-billiard dynamics $\left\{S_{\Cal C_0}^t\right\}$ defined by the cylinders with index in $\Cal C_0$ is finitely covered by the direct product flow $\left\{S^t_*\times T^t_*\right\}$, where $\left\{S^t_*\right\}$ is the (transitive) cylindric billiard flow in the torus $\tilde L=L^*/\left(L^*\cap\Bbb Z^d\right)$ defined by the intersections of the cylinders $C_i$ ($i\in\Cal C_0$) with the sub-torus $\tilde L$, while $\left\{T^t_*\right\}$ is the almost periodic (uniform) motion in $\tilde A=A^*/\left(A^*\cap\Bbb Z^d\right)$: More precisely, the phase point $x=(q,v)\in\Bbb T^d\times\Bbb R^d$ ($x\in A_1$) is decomposed locally as $q=q_1+q_2$, $q_1\in\tilde L$, $q_2\in\tilde A$, $v=v_1+v_2$, $v_1\in L^*$, $v_2\in A^*=(L^*)^\perp$, $S^t\left((q,v)\right)=(q(t),v(t))$, $q(t)=q_1(t)+q_2(t)$, $v(t)=v_1(t)+v_2(t)$, $\left(q_1(t),v_1(t)\right)=S^t_*(q_1,v_1)$, $v_2(t)=v_2(0)=v_2$, $q_2(t)=q_2+tv_2=q_2(0)+tv_2(0)$. We are about to construct the local, weakly stable manifolds $\gamma^{ws}(x)$ for $x=(q,v)=(q_1+q_2,\, v_1+v_2)\in A_1$ in the following way: $$\aligned \gamma^{ws}(x)=\bigg\{y=(q_1+\delta q_1+q_2+\delta q_2,\, v_1+\delta v_1+v_2) \bigg| \\ \text{dist}\left(S^t_*(q_1,v_1),\, S^t_*(q_1+\delta q_1,v_1+\delta v_1)\right)\to 0 \\ \text{exp. fast as }t\to\infty,\text{ and } ||\delta q_1||+||\delta v_1||+||\delta q_2||<\epsilon_0\bigg\}. \endaligned \tag 4.6$$ We see that $\gamma^{ws}(x)$ (if it exists as a manifold containing $x$ in its interior) is indeed the weakly stable manifold of the phase point $x$ corresponding to the artificially defined dynamics $S^t_*\times T^t_*$ for $t>0$. There are two important facts here: \(A) The weakly stable manifolds $\gamma^{ws}(x)$ (yet to be constructed for typical $x\in A_1$) are concave, local orthogonal sub-manifolds (see the “Invariant Manifolds” part of §2 in \[K-S-Sz(1990)\]) and, as such, they are uniformly transversal to the manifold $\Cal S_0$, see Sub-lemma 4.2 in \[K-S-Sz(1990)\]; \(B) The exponentially stable part $$\aligned \gamma^{es}(x)=\bigg\{y=(q_1+\delta q_1+q_2,\, v_1+\delta v_1+v_2)\bigg| \\ \text{dist}\left(S^t_*(q_1,v_1),\, S^t_*(q_1+\delta q_1,v_1+\delta v_1)\right)\to 0 \\ \text{exp. fast as }t\to\infty,\text{ and } ||\delta q_1||+||\delta v_1||<\epsilon_0\bigg\} \endaligned \tag 4.6/a$$ of $\gamma^{ws}(x)$ ($x=(q_1+q_2,v_1+v_2)\in A_1$) is to be constructed by using the statement of the Fundamental Theorem (Theorem 5 of \[S-Ch(1987)\]) for the $\Cal C_0$-sub-billiard system $\left\{S^t_*\right\}$. This statement can be used, for the $\nu$-typical phase points $x=(q,v)=(q_1+q_2,v_1+v_2)$ of $A_1$ have the property that the $S_*$-part $\left\{S^t_*(q_1,v_1)\right\}$ of their forward orbit is hyperbolic with respect to the sub-billiard system defined by the cylinders $C_i$, $i\in\Cal C_0$, see Corollary 3.18. According to the above points (A) and (B), there exists a measurable subset $A_2\subset A_1$ with $\nu(A_2)>0$ and a number $\delta_0>0$ such that for every $x\in A_2$ the manifold $\gamma^{ws}(x)$ exists and its boundary is at least at the distance $\delta_0$ from $x$ (these distances are now measured by using the induced Riemannian metric on $\gamma^{ws}(x)$). Then, by the absolute continuity of the foliation, see Theorem 4.1 in \[K-S(1986)\], the union $$B_2=\bigcup_{x\in A_2}\gamma^{ws}(x)\subset\bold M$$ has a positive $\mu$-measure in the phase space $\bold M$. Finally, the genuine forward orbits $S^{(0,\infty)}x$ of all points $x\in A_2$ avoid a fixed open ball $B_{r_0}$ of radius $r_0>0$. (For example: We may take any open ball $B_{r_0}$ inside the interior of any avoided cylinder $C_j$, $j\not\in\Cal C_0$.) Therefore, the forward orbit in the direct product dynamics $\left(S^t_*\times T^t_*\right)(y)$ of any point $y\in B_2$ ($y\in\gamma^{ws}(x)$, $x\in A_2$) avoids a slightly shrunk open ball $B_{r_0-\delta_0}$ of reduced radius $r_0-\delta_0$. However, this is clearly impossible, for the following reason: For $y=(q_1+q_2,\, v_1+v_2)$ ($q_1\in\tilde L$, $q_2\in\tilde A$, $v_1\in L^*$, $v_2\in A^*$) the $v_2$ component is left invariant by the product flow $S^t_*\times T^t_*$, and for almost every fixed value $v_2\in A^*$ (namely, for those vectors $v_2$ for which the orbit $tv_2/\left(A^*\cap\Bbb Z^d\right)$ ($t\in\Bbb R$) is dense in the torus $\tilde A=A^*/\left(A^*\cap\Bbb Z^d\right)$) the product flow $S^t_*\times T^t_*$ is ergodic on the corresponding level set, since it is the product of a mixing and an ergodic flow. (The flow $S^t_*$ is mixing by the induction hypothesis (H1)–(H4).) The obtained contradiction finishes the indirect proof of the Chernov-Sinai Ansatz, that is, (H1). The set $$NH(\Cal S_0)=\left\{x\in\Cal S_0\big|\; S^{(0,\infty)}x \text{ is not hyperbolic}\right\}$$ is a slim set. (In the case of a singular forward orbit non-hyperbolicity of $S^{(0,\infty)}x$ is meant that at least one branch of $S^{(0,\infty)}x$ is not hyperbolic, see §2.) Since the complement set $\Cal S_0\setminus A=\Cal S_0\setminus A_0$ is a countable union of smooth, proper sub-manifolds of $\Cal S_0$, the set $\Cal S_0\setminus A_0$ is slim. Therefore, it is enough to prove that the intersection $NH(\Cal S_0)\cap A_0$ is slim. However, according to Corollary 3.18, the forward orbit $S^{(0,\infty)}x$ of every $x\in A_0$ is hyperbolic, unless $x$ belongs to a countable union of smooth, proper sub-manifolds of $\Cal S_0$. Thus $NH(\Cal S_0)\cap A_0$ is slim. In view of Lemma 4.1 of \[K-S-Sz(1990)\], the set $R_2$ of phase points with more than one singularity on their orbit is slim, see also §2. Therefore, the final step in proving the remaining unproven induction hypothesis (i. e. (H2)) for our considered model $\flow$ with $k$ ($\ge2$) cylindric scatterers is to show that the set $$D=\left\{x\in\bold M^0\setminus\partial\bold M \big|\; S^{(-\infty,\infty)}x \text{ is not hyperbolic}\right\} \tag 4.8$$ is slim, i. e. it can be covered by a countable collection of closed subsets $F\subset\bold M$ with $\mu(F)=0$ and $\text{dim}F\le\text{dim}\bold M-2$. By the locality of slimness, see §2 above, it is enough to prove that for every element $x\in D$ the point $x$ has an open neighborhood $U$ (in $\bold M$) such that the set $U\cap D$ is slim. We want to classify the phase points $x\in D$. Consider, therefore, an arbitrary phase point $x=(q,v)\in D$. Denote the doubly infinite, symbolic collision sequence of $S^{(-\infty,\infty)}x$ by $\Sigma=\left(\dots,\sigma_{-2},\sigma_{-1},\,\sigma_{1},\sigma_{2},\dots\right)$ so that $\sigma_1$ is the first collision in positive time. (The index $0$ is not used.) We distinguish between two cases: $L^*=\text{span}\left\{L_{\sigma_i}|\; i\in\Bbb Z\setminus\{0\}\right\}\ne\Bbb R^d$. In this case, as we have seen before, the dynamics of $S^{(-\infty,\infty)}x$ is finitely covered by the direct product flow $\left\{S^t_*\times T^t_*\right\}$, where $\left\{S^t_*\right\}$ is the cylindric billiard flow in the sub-torus $\tilde L=L^*/\left(L^*\cap\Bbb Z^d\right)$ with the scatterers $C_{\sigma_i}\cap\tilde L$, while $\left\{T^t_*\right\}$ is the almost periodic (uniform) motion in the orthocomplement torus $\tilde A=A^*/\left(A^*\cap\Bbb Z^d\right)$, $A^*=(L^*)^\perp$. Now the point is that for the cylindric billiard flow $(\tilde L,\left\{S^t_*\right\},\mu_{\tilde L})$ both of the induction hypotheses (H1)–(H2) and, consequently, Theorem 5.2 of \[Sz(2000)\] apply. For the phase point $x\in D$ the direct product flow $\left(S^t_*\times T^t_*\right)(x)$ avoids an open ball, namely any open ball in the interior of any avoided cylinder $C_j$ with $$j\not\in\left\{\sigma_i|\; i\in\Bbb Z\setminus\{0\}\right\}.$$ Consequently, for each component $(q_2,v_2)\in\tilde A\times A^*$ of the canonical decomposition of $x=(q,v)=(q_1+q_2,v_1+v_2)$, $q_1\in\tilde L$, $v_1\in L^*$, $q_2\in\tilde A$, $v_2\in A^*$ it is true that the $\tilde L$-orbit $S^t_*(q_1,v_1)$ of $(q_1,v_1)$ avoids an open set $\emptyset\ne B\subset\tilde L$ on a doubly unbounded set $H$ of time moments, $\inf H=-\infty$, $\sup H=+\infty$. Therefore, in view of Theorem 5.2 of \[Sz(2000)\], the $(q_1,v_1)$-part of the phase point $x=(q_1+q_2,v_1+v_2)$ belongs to a slim subset $S_1$ of the phase space $\tilde L\times L^*$. According to the integrability property of closed slim sets (cf. Property 4 in §4.1 of \[K-S-Sz(1989)\]), even the closure $\bar D_1$ of the set $$D_1=\left\{x\in D\big|\; \text{span}\left\{L_{\sigma_i(x)}|\; i\in\Bbb Z \setminus\{0\}\right\}\ne\Bbb R^d\right\} \tag 4.9$$ (covered by Case I) is a slim subset of the phase space $\bold M$. We note that the set $\bar D_1$ is contained in the closed zero-set $$\aligned K=\big\{x\in\bold M^\#\big|\; x \text{ has a trajectory branch with a symbolic sequence } \\ (\dots,\, \sigma_{-1},\, \sigma_1,\, \dots) \text{ such that }\text{span}\left\{L_{\sigma_i}|\; i\in\Bbb Z\setminus\{0\} \right\}\ne\Bbb R^d\big\}, \endaligned$$ and the argument with “integrating up” the closed slim sets (by using Property 4 in §4.1 of \[K-S-Sz(1989)\]) is applied to the closed set $K$. $L^*=\text{span}\left\{L_{\sigma_i}|\; i\in\Bbb Z\setminus\{0\}\right\}=\Bbb R^d$. Select a vector $0\ne w\in\Cal N\left(S^{(-\infty,\infty)}x\right)$, $w\perp v$, from the neutral space $$\Cal N\left(S^{(-\infty,\infty)}x\right)=\Cal N(x)$$ of the considered phase point $x\in D$. For $i\in\Bbb Z\setminus\{0\}$ denote by $\alpha_i=\alpha_i(w)$ the “advance” of the collision $\sigma_i$ corresponding to the neutral vector $w$, see §2. Since $w$ is not parallel to $v$, at least two advances with neighboring indices are unequal; we may assume that $\alpha_{-1}\ne\alpha_1$. It follows from the proof of Lemma 3.2 that the event $\alpha_{-1}\ne\alpha_1$ can only occur if $$v=v_0\in\text{span}\left\{A_{\sigma_{-1}},\, A_{\sigma_{1}}\right\}. \tag 4.10$$ If the event $\alpha_k\ne\alpha_{k+1}$ ($k\ne-1,\, 0$) took place for another pair of neighboring advances as well, then, again by the proof of Lemma 3.2, we would have $$v_t\in\text{span}\left\{A_{\sigma_{k}},\, A_{\sigma_{k+1}}\right\} \quad (t_k<t<t_{k+1}). \tag 4.11$$ If at least one of the two subspaces on the right-hand-sides of (4.10) and (4.11) is of codimension higher than one, then the corresponding event alone ensures that the studied phase point $x\in D$ belongs to some codimension-two (i. e. at least two), smooth submanifold of the phase space, and such phase points obviously constitute a slim set, therefore they may be discarded. Thus, we may assume that $$\aligned \text{dim}\left(\text{span}\left\{A_{\sigma_{-1}},\, A_{\sigma_{1}} \right\}\right) \\ =\text{dim}\left(\text{span}\left\{A_{\sigma_{k}},\, A_{\sigma_{k+1}} \right\}\right)=d-1. \endaligned \tag 4.12$$ Denote by $n_\tau$ a (unit) normal vector of the $S^\tau$-image of the manifold $$v_0\in\text{span}\left\{A_{\sigma_{-1}},\, A_{\sigma_{1}}\right\}$$ at the phase point $S^\tau x=x_\tau$, and by $\tilde n_\tau$ a (unit) normal vector of the $S^\tau$-image of the manifold $v_t\in\text{span}\left\{A_{\sigma_{k}},\, A_{\sigma_{k+1}}\right\}$ ($t_k<t<t_{k+1}$) at the phase point $S^\tau x=x_\tau$. It follows from the proof of Corollary 3.19 that $Q(n_\tau)=0$ for $t_{-1}<\tau<t_1$, $Q(n_\tau)<0$ for $\tau>t_1$, $Q(n_\tau)>0$ for $\tau<t_{-1}$, $Q(\tilde n_\tau)=0$ for $t_k<\tau<t_{k+1}$, $Q(\tilde n_\tau)<0$ for $\tau>t_{k+1}$, and $Q(\tilde n_\tau)>0$ for $\tau<t_{k}$. Therefore, the two codimension-one sub-manifolds defined by (4.10) and (4.11) are transversal, so the simultaneous validity of (4.10)–(4.11) again results in an event for $x\in D$ showing that $x$ belongs to a slim subset of $\bold M$. Thus we may assume that $$\alpha_k=\alpha_{-1}\ne\alpha_1=\alpha_l$$ for all $k\le -1$, $l\ge1$. By adding a suitable, scalar multiple of the velocity $v=v_0$ to the neutral vector $w$, we can achieve that $$\alpha_k=\alpha_{-1}\ne0=\alpha_l \tag 4.13$$ for all $k\le -1$, $l\ge1$. The equalities $\alpha_l=0$ ($l\ge1$) mean that $$w\in\bigcap_{l>0} A_{\sigma_l}=\left(\text{span}\left\{L_{\sigma_l}\big|\; l>0\right\}\right)^\perp, \tag 4.14$$ see also the closing part of the proof of Proposition 3.1. An analogous argument shows that $$w-\alpha_{-1}v \in\bigcap_{k<0} A_{\sigma_k}=\left(\text{span}\left\{L_{\sigma_k}\big|\; k<0\right\}\right)^\perp. \tag 4.15$$ The equations (4.14)–(4.15) and $\alpha_{-1}\ne0$ imply that $$\aligned v\in\text{span}\left\{\bigcap_{k<0}A_{\sigma_k},\, \bigcap_{l>0}A_{\sigma_l}\right\} \\ =\bigcap_{k<0}A_{\sigma_k}+\bigcap_{l>0}A_{\sigma_l}:=H. \endaligned \tag 4.16$$ Recall that $\bigcap_{n\ne0}A_{\sigma_n}=\{0\}$ in the actual Case II, and $H\ne\Bbb R^d$, since $\text{span}\allowmathbreak\left\{A_{\sigma_k},\, A_{\sigma_l}\right\}\ne \Bbb R^d$ for $k<0<l$. We can assume that the linear direct sum on the right-hand-side of (4.16) is a subspace with codimension one, otherwise (just as many times in the past) the containment in (4.16) would be a codimension-two condition on the initial velocity $v=v_0$, and all such phase points $x=(q,v)$ can be discarded. Following the tradition, denote by $J$ the codimension-one sub-manifold of $\bold M$ defined by (4.16). The proof of the Ansatz in the case of (4.5) can now be repeated almost word-by-word. Indeed, the phase points $$\aligned x=(q,\, v)\in\bar D=\bar D(\Cal A,\Cal B)=\big\{x\in D\big|\; v\in H, \\ \left\{\sigma_k(x)|\; k<0\right\}=\Cal A,\; \left\{\sigma_l(x)|\; l>0\right\}=\Cal B\big\} \endaligned \tag 4.17$$ (with given $\Cal A,\, \Cal B\subset\{1,2,\dots,k\}$ such that $\text{span}\left\{L_j|\; j\in\Cal A\cup\Cal B\right\}=\Bbb R^d$) of the considered type again decompose as $(q,v)=(q_1+q_2,\, v_1+v_2)$, $v_1\in L^*=\text{span}\allowmathbreak\left\{L_{\sigma_l}|\; l>0\right\}$, $v_2\in A^*=\left(L^*\right)^\perp=\bigcap_{l>0}A_{\sigma_l}$, $q_1\in\tilde L=L^*/\left(L^*\cap\Bbb Z^d\right)$, $q_2\in\tilde A=A^*/\left(A^*\cap\Bbb Z^d\right)$, and the forward orbit $S^{(0,\infty)}x$ of our considered phase point $x\in D\cap J$ (fulfilling all of the mentioned assumptions) is essentially (up-to a finite covering) is governed by the product flow $\left(q_1(t),v_1(t)\right)=S^t_*(q_1,v_1)$, $\left(q_2(t),v_2(t)\right)=T^t_*(q_2,v_2)=(q_2+tv_2,v_2)$, where (as said before) $S^t_*$ is the sub-billiard flow in $\tilde L$ defined by the intersections of the cylinders $\left\{C_{\sigma_l}|\; l>0\right\}$ with the torus $\tilde L$. The exponentially stable component $\gamma^{es}(x)$ of $\gamma^{ws}(x)$ ($x\in D\cap J$) defined by (4.6/a) is transversal to the codimension-one manifold $J$ described by the membership in (4.16). Argue by contradiction. Assume that $\Cal T_x\gamma^{es}(x)\subset\Cal T_xJ$. The tangent space $\Cal T_xJ$ is obviously given by the simple formula $$\Cal T_xJ=\left\{(\delta q,\delta v)\in\Cal T_x\bold M\big|\; \delta v\in H \right\}. \tag 4.19$$ The second fundamental form $B\left(\gamma^{es}(x)\right)$ of $\gamma^{es}(x)$ at the phase point $x=(q_1+q_2,v_1+v_2)$ is known to be negative definite, so its range is the entire orthocomplement $(v_1)^\perp$ of $v_1$ in the space $$L^*=\text{span}\left\{L_{\sigma_l}|\; l>0\right\}= \left(\bigcap_{l>0}A_{\sigma_l}\right)^\perp.$$ On the other hand, since $$v=v_1+v_2\in H=\bigcap_{k<0}A_{\sigma_k}+\bigcap_{l>0}A_{\sigma_l}$$ and $v_2\in A^*=\bigcap_{l>0}A_{\sigma_l}\subset H$, from the assumed relation $\Cal T_x\gamma^{es}(x)\subset\Cal T_xJ$ and from $v_1\in H$ we get that $L^*\subset H$. Since $A^*=\left(L^*\right)^\perp\subset H$, this means that $H=\Bbb R^d$, contradicting $\text{dim}H=d-1$. This finishes the proof of the lemma. Finally, the slimness of the set $D$ in (4.8) will be proven in Case II as soon as we show that $\nu_J(\bar D)=0$, where $\bar D=\bar D(\Cal A,\Cal B)$ is defined in (4.17). This is, however, obtained the same way as the relation $\nu(A_2)=0$ at the end of the proof of the Ansatz. Indeed, in the case $\nu_J(\bar D)>0$ the union $$\tilde D:=\bigcup_{x\in\bar D}\gamma^{es}(x)$$ would have a positive $\mu$-measure in $\bold M$ (by the transversality proved above and by the absolute continuity of the $\gamma^{es}(\,.\,)$ foliation, see Theorem 4.1 in \[K-S(1986)\]), but this is impossible, for all forward orbits $S^{(0,\infty)}y$ of the points $y\in\tilde D$ would avoid a common open ball that can be obtained by slightly shrinking any open ball inside the interior of any avoided cylinder $C_j$ with $j\not\in\Cal B$, see also the closing part of the proof of $\nu(A_2)=0$ above. This finishes the proof of the fact that the set $D$ in (4.8) is indeed slim. 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--- abstract: 'For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}({{\mathcal{O}}}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the $3$-torsion subgroup of $K_{2n}({{\mathcal{O}}}_F)$ is as predicted by this conjecture.' address: - 'Department of Mathematics, Baruch College, The City University of New York, One Bernard Baruch Way, New York, NY 10010-5526, USA' - 'Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121-1969' - 'Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA' - 'Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA' - 'Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540-1966, USA' author: - 'Bruce W. Jordan' - Zev Klagsbrun - Bjorn Poonen - Christopher Skinner - Yevgeny Zaytman date: 'January 28, 2019' title: 'Statistics of $K$-groups modulo $p$ for the ring of integers of a varying quadratic number field' --- [^1] Introduction ============ The original Cohen–Lenstra heuristics [@Cohen-Lenstra1984] predicted, for each prime $p \ne 2$, the distribution of the $p$-primary part of $\operatorname{Cl}(F)$ as $F$ varied over quadratic fields of a given signature. More recent work developed heuristics for other families of groups, including class groups of higher degree number fields [@Cohen-Martinet1990], Picard groups of function fields [@Friedman-Washington1989], Tate–Shafarevich groups of elliptic curves [@Delaunay2001; @Delaunay2007; @Delaunay-Jouhet2014a], Selmer groups of elliptic curves [@Poonen-Rains2012-selmer; @Bhargava-Kane-Lenstra-Poonen-Rains2015], and Galois groups of nonabelian unramified extensions of number fields and function fields [@Boston-Bush-Hajir2017; @Boston-Wood2017]. Let ${\mathcal{F}}$ be a number field. Let ${{\mathcal{O}}}_{\mathcal{F}}$ be the ring of integers of ${\mathcal{F}}$. For $m \ge 0$, the $K$-group $K_m({{\mathcal{O}}}_{\mathcal{F}})$ is a finitely generated abelian group. It is finite when $m$ is even and positive: see [@Weibel2005]\*[Theorem 7]{}. Our goal is to study, for a fixed $m$ and odd prime $p$, the $p$-torsion subgroup $K_m({{\mathcal{O}}}_{\mathcal{F}})_p$ as ${\mathcal{F}}$ varies in a family of number fields, always ordered by absolute value of the discriminant. As described in Section \[sec:odd K groups\], $K_m({{\mathcal{O}}}_{\mathcal{F}})_p$ is well understood for odd $m$. Therefore we focus on the case $m=2n$. Now suppose that ${\mathcal{F}}$ is a quadratic field $F$. The action of $\operatorname{Gal}(F/{{\mathbb{Q}}})$ decomposes $K_{2n}({{\mathcal{O}}}_F)_p$ into $+$ and $-$ parts, and we will see that the $+$ part is $K_{2n}({{\mathbb{Z}}})_p$, independent of $F$. Therefore we focus on the variation of the $-$ part. A Cohen–Lenstra style heuristic will lead us to the following conjecture, involving the constants $$\alpha_{p,u,r} \colonequals \frac{\prod_{i = r+1}^\infty (1 - p^{-i})} {p^{r(u+r)} \prod_{i = 1}^{r+u} (1 - p^{-i})}$$ for nonnegative integers $u$ and $r$: \[C:K-theory conjecture for rank\] Fix $n \ge 1$ and an odd prime $p$ and $r \ge 0$. As $F$ ranges over all real resp. imaginary quadratic fields, $\operatorname{Prob}\left( \dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathcal{O}}}_F)^-_p = r \right)$ is given in the following table by the entry in the row determined by $n$ and column determined by the signature: To pass from the distribution of $\dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathcal{O}}}_F)^-_p$ to that of $\dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathcal{O}}}_F)_p$ itself, add the constant $\dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathbb{Z}}})_p$, which can be expressed in terms of a class group (see Section \[S:even K-groups of Z\]). Conjecture \[C:K-theory conjecture for rank\] implies an average order for $K_{2n}({{\mathcal{O}}}_F)_p$ as $F$ varies over all real or imaginary fields (see Conjecture \[C:average K-theory conjecture\]). We prove that this conjectured average order is correct for $p = 3$: \[T:average of K\_3\] Fix $n \ge 1$. The average order of $K_{2n}({{\mathcal{O}}}_F)_3$ as $F$ ranges over real resp. imaginary quadratic fields is as follows: $\textup{real}$ $\textup{imaginary}$ --------------------- ----------------- ---------------------- $\textup{$n$ even}$ $25/12$ $11/4$ $\textup{$n$ odd}$ $9/4$ $19/12$. By Theorem \[T:K-groups of Z\], $K_{2n}({{\mathbb{Z}}})_3 = 0$ for all $n$ since ${{\mathbb{Q}}}(\zeta_3)$ has class number $1$. Thus $K_{2n}({{\mathcal{O}}}_F)^-_3 = K_{2n}({{\mathcal{O}}}_F)_3$ for all $n$. Theorem \[T:average of K\_3\] is an analogue of the Davenport–Heilbronn theorem giving the average order of $\operatorname{Cl}(F)_3$ as $F$ varies over all real or imaginary quadratic fields [@Davenport-Heilbronn1971]\*[Theorem 3]{}. After this article was written, the second author proved an analogue of Theorem \[T:average of K\_3\] for $K_{2n}({{\mathcal{O}}}_{\mathcal{F}})_2$ as ${\mathcal{F}}$ varies over cubic fields [@Klagsbrun-preprint2]. Methods ------- The $p$-torsion subgroup $G_p$ of a finite abelian group $G$ has the same ${{\mathbb{F}}}_p$-dimension as $G/p \colonequals G/pG$; therefore we study $K_{2n}({{\mathcal{O}}}_F)/p$. The latter is isomorphic to an étale cohomology group $H^2_{{\textup{\'et}}}({{\mathcal{O}}}_F[1/p],\mu_p^{{\otimes}(n+1)})$, which we relate to isotypic components of the class group and Brauer group of ${{\mathcal{O}}}_E[1/p]$, where $E \colonequals F(\zeta_p)$. The Brauer group can be computed explicitly, and we develop heuristics for the class groups; combining these gives the conjectural distribution of $K_{2n}({{\mathcal{O}}}_F)_p$. In the case $p=3$, the isotypic components of $\operatorname{Cl}({{\mathcal{O}}}_E[1/p])$ are related to $\operatorname{Cl}({{\mathcal{O}}}_K[1/p])$ for quadratic fields $K$. The *average order* of the latter class groups can be computed unconditionally by using a strategy of Davenport and Heilbronn, which we refine using recent work of Bhargava, Shankar, and Tsimerman, to control averages in subfamilies with prescribed local behavior at $3$. This yields unconditional results on the average order of $K_{2n}({{\mathcal{O}}}_F)_3$. Prior work ---------- As far as we know, Cohen–Lenstra style conjectures have not been proposed for $K$-groups before, but some results on the distribution of $K_2({{\mathcal{O}}}_F)$ have been proved. Guo [@Guo2009] proved that $4$-ranks of $K_2({{\mathcal{O}}}_F)$ for quadratic fields $F$ follow a Cohen–Lenstra distribution, just as Fouvry and Klüners proved for $4$-ranks of $\operatorname{Cl}({{\mathcal{O}}}_F)$ [@Fouvry-Klueners2007]. Studying $4$-ranks is natural, since the $2$-rank of $K_2({{\mathcal{O}}}_F)$ for a quadratic field $F$ is determined by genus theory just as the $2$-rank of $\operatorname{Cl}({{\mathcal{O}}}_F)$ is (see [@Browkin-Schinzel1982], for example). Similar results on the $3$-ranks of $K_2({{\mathcal{O}}}_L)$ for cyclic cubic fields $L$ are due to Cheng, Guo, and Qin [@Cheng-Guo-Qin2014]. In addition, Browkin showed that Cohen–Martinet heuristics suggest a conjecture on $\operatorname{Prob}(3 \mid \#K_2({{\mathcal{O}}}_F))$ as $F$ ranges over quadratic fields of fixed signature [@Browkin2000]. Notation {#S:notation} -------- If $G$ is an abelian group and $n \ge 1$, let $G_n \colonequals \{g \in G:ng=0\}$ and $G/n \colonequals G/nG$. For any $k$-representation $V$ of a finite group $G$ such that $\operatorname{char}k \nmid \#G$, and for any irreducible $k$-representation $\chi$ of $G$, let $V^\chi$ be the $\chi$-isotypic component. Throughout the paper, $p$ is an odd prime, ${\mathcal{F}}$ is an arbitrary number field, $\overline{{\mathcal{F}}}$ is an algebraic closure of ${\mathcal{F}}$, the element $\zeta \in \overline{{\mathcal{F}}}$ is a primitive $p$th root of unity, and ${\mathcal{E}}\colonequals {\mathcal{F}}(\zeta_p)$. Later we specialize ${\mathcal{F}}$ to a quadratic field $F$ and define $E \colonequals F(\zeta_p)$. Let ${{\mathcal{O}}}_{\mathcal{F}}$ be the ring of integers of ${\mathcal{F}}$. If $S$ is a finite set of places of ${\mathcal{F}}$ containing all the archimedean places, define the ring of $S$-integers ${{\mathcal{O}}}_S := \{x \in {\mathcal{F}}: v(x) \ge 0 \textup{ for all $v \notin S$}\}$. Let $d_{\mathcal{F}}\in {{\mathbb{Z}}}$ be the discriminant of ${\mathcal{F}}$. Let $\mu({\mathcal{F}})$ be the group of roots of unity in ${\mathcal{F}}$. If ${{\mathcal{O}}}$ is a Dedekind ring, let $\operatorname{Cl}({{\mathcal{O}}})$ denote its class group, and let $\operatorname{Br}({{\mathcal{O}}})$ be its (cohomological) Brauer group, defined as $H^2_{{\textup{\'et}}}(\operatorname{Spec}{{\mathcal{O}}},{{\mathbb{G}}}_m)$ [@Poonen2017]\*[Definition 6.6.4]{}. From now on, all cohomology is étale cohomology, and we drop the subscript ${\textup{\'et}}$. From K-theory to class groups and Brauer groups {#S:from K-theory} =============================================== In this section, following Tate’s argument for $K_2$ [@Tate1976], we relate the even $K$-groups to more concrete groups: class groups and Brauer groups. \[T:K-theory and etale cohomology\] For any number field ${\mathcal{F}}$ and any $n \ge 1$, $$K_{2n}({{\mathcal{O}}}_{\mathcal{F}})/p {\simeq}H^2({{\mathcal{O}}}_{\mathcal{F}}[1/p],\mu_p^{{\otimes}(n+1)}).$$ \[L:H\^2 of mu\_p\] There is a canonical exact sequence $$0 {\longrightarrow}\operatorname{Cl}({{\mathcal{O}}}_{\mathcal{F}}[1/p])/p {\longrightarrow}H^2({{\mathcal{O}}}_{\mathcal{F}}[1/p],\mu_p) {\longrightarrow}\operatorname{Br}({{\mathcal{O}}}_{\mathcal{F}}[1/p])_p {\longrightarrow}0.$$ Consider the exact sequence $$1 \to \mu_p \to {{\mathbb{G}}}_m \stackrel{p}\to {{\mathbb{G}}}_m \to 1$$ of sheaves on $(\operatorname{Spec}{{\mathcal{O}}}_{\mathcal{F}}[1/p])_{{\textup{\'et}}}$. Take the associated long exact sequence of cohomology, and substitute $H^1({{\mathcal{O}}}_{\mathcal{F}}[1/p],{{\mathbb{G}}}_m) = \operatorname{Pic}({{\mathcal{O}}}_{\mathcal{F}}[1/p]) = \operatorname{Cl}({{\mathcal{O}}}_{\mathcal{F}}[1/p])$ and $H^2({{\mathcal{O}}}_{\mathcal{F}}[1/p],{{\mathbb{G}}}_m) = \operatorname{Br}({{\mathcal{O}}}_{\mathcal{F}}[1/p])$. \[L:Hochschild-Serre\] Let ${\mathcal{F}}'/{\mathcal{F}}$ be a finite Galois extension of degree prime to $p$. Let $i \ge 0$ and $r \in {{\mathbb{Z}}}$. Then $$H^i({{\mathcal{O}}}_{\mathcal{F}}[1/p],\mu_p^{{\otimes}r}) = H^i({{\mathcal{O}}}_{{\mathcal{F}}'}[1/p],\mu_p^{{\otimes}r})^{\operatorname{Gal}({\mathcal{F}}'/{\mathcal{F}})}.$$ In the Hochschild–Serre spectral sequence $$H^i(\operatorname{Gal}({\mathcal{F}}'/{\mathcal{F}}),H^j({{\mathcal{O}}}_{{\mathcal{F}}'}[1/p],\mu_p^{{\otimes}r})) \implies H^{i+j}({{\mathcal{O}}}_{\mathcal{F}}[1/p],\mu_p^{{\otimes}r}),$$ the groups $H^i(\operatorname{Gal}({\mathcal{F}}'/{\mathcal{F}}),H^j({{\mathcal{O}}}_{{\mathcal{F}}'}[1/p],\mu_p^{{\otimes}r}))$ for $i>0$ are $0$ because they are killed by both $\#\operatorname{Gal}({\mathcal{F}}'/{\mathcal{F}})$ and $p$. We now specialize ${\mathcal{F}}'$ to ${\mathcal{E}}\colonequals {\mathcal{F}}(\zeta_p)$. The action of $\operatorname{Gal}({\mathcal{E}}/{\mathcal{F}})$ on the $p$th roots of $1$ defines an injective homomorphism $\chi_1 \colon \operatorname{Gal}({\mathcal{E}}/{\mathcal{F}}) \to ({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times$. For $m \in {{\mathbb{Z}}}$, composing $\chi_1$ with the $m$th power map on $({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times$ yields another $1$-dimensional ${{\mathbb{F}}}_p$-representation of $\operatorname{Gal}({\mathcal{E}}/{\mathcal{F}})$; call it $\chi_m$. \[L:split exact sequence for H\^2\] There is a split exact sequence $$0 \to \left( \operatorname{Cl}({{\mathcal{O}}}_{\mathcal{E}}[1/p])/p \right)^{\chi_{-n}} \to H^2({{\mathcal{O}}}_{\mathcal{F}}[1/p],\mu_p^{{\otimes}(n+1)}) \to \left( \operatorname{Br}({{\mathcal{O}}}_{\mathcal{E}}[1/p])_p \right)^{\chi_{-n}} \to 0.$$ First, we have $$\begin{aligned} \nonumber H^2({{\mathcal{O}}}_{\mathcal{F}}[1/p],\mu_p^{{\otimes}(n+1)}) &= H^2({{\mathcal{O}}}_{\mathcal{E}}[1/p],\mu_p^{{\otimes}(n+1)})^{\operatorname{Gal}({\mathcal{E}}/{\mathcal{F}})} \quad\textup{(Lemma~\ref{L:Hochschild-Serre})} \\ \nonumber &= \left( H^2({{\mathcal{O}}}_{\mathcal{E}}[1/p],\mu_p) {\otimes}\mu_p^{{\otimes}n} \right)^{\operatorname{Gal}({\mathcal{E}}/{\mathcal{F}})} \quad\textup{(since $\mu_p \subset {\mathcal{E}}$)}\\ \label{E:H^2 for E and F} &= H^2({{\mathcal{O}}}_{\mathcal{E}}[1/p],\mu_p)^{\chi_{-n}} \quad\textup{(since $\mu_p^{{\otimes}n} {\simeq}\chi_n$).}\end{aligned}$$ On the other hand, Lemma \[L:H\^2 of mu\_p\] for ${\mathcal{E}}$ yields a sequence of $\operatorname{Gal}({\mathcal{E}}/{\mathcal{F}})$-representations $$0 {\longrightarrow}\operatorname{Cl}({{\mathcal{O}}}_{\mathcal{E}}[1/p])/p {\longrightarrow}H^2({{\mathcal{O}}}_{\mathcal{E}}[1/p],\mu_p) {\longrightarrow}\operatorname{Br}({{\mathcal{O}}}_{\mathcal{E}}[1/p])_p {\longrightarrow}0,$$ which splits by Maschke’s theorem. Take $\chi_{-n}$-isotypic components, and substitute in the middle. Substituting Theorem \[T:K-theory and etale cohomology\] into Lemma \[L:split exact sequence for H\^2\] yields the main result of this section: \[T:K\_2n in terms of Cl and Br\] For each $n \ge 1$, $$K_{2n}({{\mathcal{O}}}_{\mathcal{F}})/p {\simeq}\left( \operatorname{Cl}({{\mathcal{O}}}_{\mathcal{E}}[1/p])/p \right)^{\chi_{-n}} {\oplus}\left( \operatorname{Br}({{\mathcal{O}}}_{\mathcal{E}}[1/p])_p \right)^{\chi_{-n}}.$$ Even K-groups of the ring of integers of a quadratic field {#S:K-groups of quadratic field} ========================================================== Let $p^* = (-1)^{(p-1)/2} p$, so ${{\mathbb{Q}}}(\sqrt{p^*})$ is the degree $2$ subfield of ${{\mathbb{Q}}}(\zeta_p)$. From now on, $F$ is a degree $2$ extension of ${{\mathbb{Q}}}$ not equal to ${{\mathbb{Q}}}(\sqrt{p^*})$. Thus $F={{\mathbb{Q}}}(\sqrt{d})$ for some $d \in {{\mathbb{Q}}}^\times$ such that $d$ and $p^*$ are independent in ${{\mathbb{Q}}}^\times/{{\mathbb{Q}}}^{\times 2}$. Let $E=F(\zeta_p)$. Then $$\operatorname{Gal}(E/{{\mathbb{Q}}}) {\simeq}({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times \times \{\pm 1\}.$$ Let $\tau$ be the generator of $\operatorname{Gal}(E/{{\mathbb{Q}}}(\zeta_p)) = \{\pm 1\}$, so $\tau$ restricts to the generator of $\operatorname{Gal}(F/{{\mathbb{Q}}})$. The action of $\tau$ decomposes $K_{2n}({{\mathcal{O}}}_F)/p$ into $+$ and $-$ eigenspaces. Let $\chi_{-n,-1} \colon G \to {{\mathbb{F}}}_p^\times$ be such that $\chi_{-n,-1}|_{\operatorname{Gal}(E/F)} = \chi_{-n}$ and $\chi_{-n,-1}(\tau)=-1$. \[T:K\_2n for quadratic field\] For each $n \ge 1$, $$\begin{aligned} (K_{2n}({{\mathcal{O}}}_F)/p)^+ &{\simeq}K_{2n}({{\mathbb{Z}}})/p \\ (K_{2n}({{\mathcal{O}}}_F)/p)^- &{\simeq}\left( \operatorname{Cl}({{\mathcal{O}}}_E[1/p])/p \right)^{\chi_{-n,-1}} {\oplus}\left( \operatorname{Br}({{\mathcal{O}}}_E[1/p])_p \right)^{\chi_{-n,-1}}.\end{aligned}$$ To obtain the first statement, use Theorem \[T:K-theory and etale cohomology\] to rewrite each term as an étale cohomology group and apply Lemma \[L:Hochschild-Serre\] with ${\mathcal{F}}'/{\mathcal{F}}$ there being $F/{{\mathbb{Q}}}$. To obtain the second, take minus parts in Theorem \[T:K\_2n in terms of Cl and Br\]. Brauer groups {#S:Brauer groups} ============= The goal of this section is to determine the rightmost term in Theorem \[T:K\_2n for quadratic field\]. Let ${\mathcal{F}}$ and ${{\mathcal{O}}}_S$ be as in Section \[S:notation\]. Let $r_1$ be the number of real places of ${\mathcal{F}}$. Then there is an exact sequence $$0 {\longrightarrow}\operatorname{Br}{{\mathcal{O}}}_S {\longrightarrow}\left(\tfrac12 {{\mathbb{Z}}}/{{\mathbb{Z}}}\right)^{r_1} {\oplus}{\bigoplus}_{\textup{finite $v \in S$}} {{\mathbb{Q}}}/{{\mathbb{Z}}}\stackrel{\textup{sum}}{\longrightarrow}{{\mathbb{Q}}}/{{\mathbb{Z}}}.$$ \[C:p-torsion in Br\] We have $\operatorname{Br}({{\mathcal{O}}}_S)_p {\simeq}({{\mathbb{Z}}}/p{{\mathbb{Z}}})^{\{\textup{finite $v \in S$}\}}_{\textup{sum 0}}$, where the “sum $0$” subscript denotes the subgroup of elements whose sum is $0$. \[C:Br Z\[zeta\_p,1/p\]\] We have $\operatorname{Br}({{\mathbb{Z}}}[\zeta_p,1/p])_p = 0$. There is only one prime above $p$ in ${{\mathbb{Z}}}[\zeta_p]$. \[P:p-torsion in Br for quadratic fields\] Let $F$ and $E$ be as in Section \[S:K-groups of quadratic field\]. Then $$(\operatorname{Br}({{\mathcal{O}}}_E[1/p])_p)^{\chi_{-n,-1}} = \begin{cases} {{\mathbb{Z}}}/p{{\mathbb{Z}}}, & \textup{if $n \equiv 0 \!\!\!\pmod{p-1}$ and $d \in {{\mathbb{Q}}}_p^{\times 2}$;} \\ {{\mathbb{Z}}}/p{{\mathbb{Z}}}, & \textup{if $n \equiv \dfrac{p-1}{2} \!\!\!\pmod{p-1}$ and $p^* d \in {{\mathbb{Q}}}_p^{\times 2}$;} \\ 0, & \textup{in all other cases.} \end{cases}$$ The hypothesis implies that $F$ is not the quadratic subfield ${{\mathbb{Q}}}(\sqrt{p^*})$ of ${{\mathbb{Q}}}(\zeta_p)$. Thus $E$ is the compositum of linearly disjoint extensions ${{\mathbb{Q}}}(\zeta_p)$ and $F$ over ${{\mathbb{Q}}}$, and $\operatorname{Gal}(E/{{\mathbb{Q}}}) {\simeq}({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times \times \{\pm 1\}$. Let ${{\mathfrak{p}}}$ be a prime of $E$ lying above $p$. Let $D \le \operatorname{Gal}(E/{{\mathbb{Q}}})$ be the decomposition group of ${{\mathfrak{p}}}$. Since $p$ totally ramifies in ${{\mathbb{Q}}}(\zeta_p)/{{\mathbb{Q}}}$, we have $p-1 \mid \#D$. Let $S_p$ be the set of primes of $E$ lying above $p$, so $S_p {\simeq}\operatorname{Gal}(E/{{\mathbb{Q}}})/D$, which by the previous sentence is of size $1$ or $2$; it is $2$ if and only if $p$ splits in one of the quadratic subfields of $E$. These quadratic subfields are ${{\mathbb{Q}}}(\sqrt{p^*})$, $F$, and the field $F' = {{\mathbb{Q}}}(\sqrt{p^* d})$, but $p$ is ramified in ${{\mathbb{Q}}}(\sqrt{p^*})$. Thus by Corollary \[C:p-torsion in Br\], $$\operatorname{Br}({{\mathcal{O}}}_E[1/p])_p = ({{\mathbb{Z}}}/p{{\mathbb{Z}}})^{S_p}_{\textup{sum $0$}} = \begin{cases} {{\mathbb{Z}}}/p{{\mathbb{Z}}}, & \textup{if $p$ splits in $F$ or $F'$;}\\ 0, & \textup{otherwise} \end{cases}$$ as an abelian group, and it remains to determine in the first case which character it is isomorphic to. We will compute the action of $\operatorname{Gal}(E/F) {\simeq}({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times$ and $\operatorname{Gal}(E/{{\mathbb{Q}}}(\zeta_p)) {\simeq}\{\pm1\}$ separately. If $p$ splits in $F$ (that is, $d \in {{\mathbb{Q}}}_p^{\times 2}$), then $D = \operatorname{Gal}(E/F)$, which acts trivially on $S_p {\simeq}\operatorname{Gal}(E/{{\mathbb{Q}}})/D$, so $\operatorname{Gal}(E/F)$ acts on $\operatorname{Br}({{\mathcal{O}}}_E[1/p])_p$ as the trivial character $\chi_0$. If instead $p$ splits in $F'$ (that is, $p^* d \in {{\mathbb{Q}}}_p^{\times 2}$), then $D \ne \operatorname{Gal}(E/F)$, so $\operatorname{Gal}(E/F)$ acts nontrivially on the two-element set $S_p {\simeq}\operatorname{Gal}(E/{{\mathbb{Q}}})/D$, so $\operatorname{Gal}(E/F)$ acts on $\operatorname{Br}({{\mathcal{O}}}_E[1/p])_p$ as the character $\operatorname{Gal}(E/F) {\twoheadrightarrow}\{\pm 1\}$, which is $\chi_{(p-1)/2}$. Finally, consider the action of the generator $\tau$ of $\operatorname{Gal}(E/{{\mathbb{Q}}}(\zeta_p)) {\simeq}\{\pm1\}$ on $\operatorname{Br}({{\mathcal{O}}}_E[1/p])_p$. Lemma \[L:Hochschild-Serre\] shows that the $+$ eigenspace is $\operatorname{Br}({{\mathbb{Z}}}[\zeta_p,1/p])_p$, which is $0$ by Corollary \[C:Br Z\[zeta\_p,1/p\]\]. Thus $\operatorname{Br}({{\mathcal{O}}}_E[1/p])_p$ equals its $-$ eigenspace. \[R:splits\] Let $d$ be a fundamental discriminant. Then $d \in {{\mathbb{Q}}}_p^{\times 2}$ if and only if ${\genfrac{(}{)}{}{1}{d}{p}}=1$, and $p^*d \in {{\mathbb{Q}}}_p^{\times 2}$ if and only if $p|d$ and ${\genfrac{(}{)}{}{1}{-d/p}{p}}=1$. Even K-groups of Z {#S:even K-groups of Z} ================== \[T:K-groups of Z\] For each $n \ge 1$, $$K_{2n}({{\mathbb{Z}}})/p {\simeq}\left(\operatorname{Cl}({{\mathbb{Z}}}[\zeta_p])/p\right)^{\chi_{-n}}.$$ In Theorem \[T:K\_2n in terms of Cl and Br\] for ${\mathcal{F}}={{\mathbb{Q}}}$, the Brauer term is $0$ by Corollary \[C:Br Z\[zeta\_p,1/p\]\], and $\operatorname{Cl}({{\mathbb{Z}}}[\zeta_p,1/p]) = \operatorname{Cl}({{\mathbb{Z}}}[\zeta_p])$ since the unique prime ideal above $p$ in ${{\mathbb{Z}}}[\zeta_p]$ is principal. For $n \ge 1$, let $$\kappa_{2n,p} \colonequals \dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathbb{Z}}})/p = \dim_{{{\mathbb{F}}}_p} \left(\operatorname{Cl}({{\mathbb{Z}}}[\zeta_p])/p\right)^{\chi_{-n}}.$$ Assuming Vandiver’s conjecture that $p \nmid \#\operatorname{Cl}({{\mathbb{Z}}}[\zeta_p+\zeta_p^{-1}])$ for every prime $p$, the $K$-groups of ${{\mathbb{Z}}}$ are known; see [@Weibel2005]\*[Section 5.9]{}. To state the results for even $K$-groups, let $B_{2k} \in {{\mathbb{Q}}}$ be the $(2k)$th Bernoulli number, defined by $$\frac{t}{e^t-1}=1-\frac{t}{2}+\sum_{k=1}^\infty B_{2k} \frac{t^{2k}}{(2k)!} .$$ Let $c_k$ be the numerator of $|B_{2k}/(4k)|$. Then (see [@Weibel2005]\*[Corollary 107]{}) - Vandiver’s conjecture implies that $K_{4k}({{\mathbb{Z}}})=0$ for all $k \ge 1$. - For $k \ge 1$, the order of $K_{4k-2}({{\mathbb{Z}}})$ is $c_k$ if $k$ is even, and $2c_k$ if $k$ is odd; moreover, Vandiver’s conjecture implies that $K_{4k-2}({{\mathbb{Z}}})$ is cyclic. In fact, for each prime $p$, Vandiver’s conjecture for $p$ implies the conclusions above for the $p$-primary part of the $K$-groups. Thus Vandiver’s conjecture for an odd prime $p$ implies that for any $n \ge 1$, $$\kappa_{2n,p} = \begin{cases} 1 & \textup{ if $n=2k-1$ and $p|c_k$;} \\ 0 & \textup{ otherwise.} \end{cases}$$ Moreover, Vandiver’s conjecture is known for $p<163577856$ [@Buhler-Harvey2011]. The smallest odd prime $p$ for which there exists $n$ such that $p | \#K_{2n}({{\mathbb{Z}}})$ is the smallest irregular prime, $37$, which divides $\#K_{2n}({{\mathbb{Z}}})$ if and only if $n \equiv 31 \pmod{36}$; thus $\kappa_{2n,37}$ is $1$ if $n \equiv 31 \pmod{36}$, and $0$ otherwise. Assuming Vandiver’s conjecture, the smallest $n$ such that $\#K_{2n}({{\mathbb{Z}}})$ is divisible by an odd prime is $n=11$: we have $K_{22}({{\mathbb{Z}}}) {\simeq}{{\mathbb{Z}}}/691{{\mathbb{Z}}}$. See [@Weibel2005]\*[Example 96]{} for these and other examples. Odd K-groups {#sec:odd K groups} ============ For any number field ${\mathcal{F}}$, positive integer $i$, and odd prime $p$, we have $$K_{2i-1}({{\mathcal{O}}}_{\mathcal{F}})_p = \begin{cases} {{\mathbb{Z}}}/p{{\mathbb{Z}}}, & \textup{ if $[{\mathcal{F}}(\zeta_p):{\mathcal{F}}]$ divides $i$;} \\ 0, & \textup{ otherwise.} \end{cases}$$ Define the group $$\mu^{(i)}({\mathcal{F}}) \colonequals\{\zeta \in \mu(\overline{{\mathcal{F}}}) : \sigma^i \zeta = \zeta \textup{ for all $\sigma \in \operatorname{Gal}(\overline{{\mathcal{F}}}/{\mathcal{F}})$}\}.$$ For $n \ge 1$, let $\zeta_n \in \overline{{\mathcal{F}}}$ be a primitive $n$th root of $1$, and let $H_n$ be the image of the restriction homomorphism $\operatorname{Gal}({\mathcal{F}}(\zeta_n)/{\mathcal{F}}) {\hookrightarrow}\operatorname{Gal}({{\mathbb{Q}}}(\zeta_n)/{{\mathbb{Q}}}) {\simeq}({{\mathbb{Z}}}/n{{\mathbb{Z}}})^\times$, so $\#H_n=[{\mathcal{F}}(\zeta_n):{\mathcal{F}}]$. Then the following are equivalent: - $\zeta_n \in \mu^{(i)}({\mathcal{F}})$; - $\sigma^i \zeta_n \equiv \zeta_n \text{ for all } \sigma \in \operatorname{Gal}(\overline{{\mathcal{F}}}/{\mathcal{F}})$; - $a^i = 1$ for all $a \in H_n$. Now suppose that $n$ is a prime power $\ell^e$ for some prime $\ell$. Then $H_n$ contains a cyclic subgroup of index at most $2$ (we allow the case $\ell=2$). The last condition above implies $\#H_n | 2i$, which after multiplication by $[{\mathcal{F}}:{{\mathbb{Q}}}]$ becomes the statement that $[{\mathcal{F}}(\zeta_n):{{\mathbb{Q}}}]$ divides $2i[{\mathcal{F}}:{{\mathbb{Q}}}]$, which implies that the integer $\phi(n) \colonequals [{{\mathbb{Q}}}(\zeta_n):{{\mathbb{Q}}}]$ divides $2i[{\mathcal{F}}:{{\mathbb{Q}}}]$, which bounds $\phi(n)$ and hence $n$. Thus $\mu^{(i)}({\mathcal{F}})$ contains $\ell^e$th roots of $1$ for only finitely many prime powers $\ell^e$, so it is finite. Define $w^{(i)}({\mathcal{F}}) \colonequals \# \mu^{(i)}({\mathcal{F}})$. By Theorem 70 in [@Weibel2005], if $p$ is an odd prime, $K_{2i-1}({{\mathcal{O}}}_{\mathcal{F}})_p$ is ${{\mathbb{Z}}}/p{{\mathbb{Z}}}$ or $0$, according to whether $p$ divides $w^{(i)}({\mathcal{F}})$ or not. The previous paragraph shows that the latter condition is equivalent to $H_p$ being killed by $i$, and to $\#H_p | i$ since $H_p$ is cyclic (a subgroup of the cyclic group $({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times$). Finally, $\#H_p = [{\mathcal{F}}(\zeta_p):{\mathcal{F}}]$. Heuristics for class groups {#S:heuristics for class groups} =========================== Let $A$ be an abelian extension of ${{\mathbb{Q}}}$. Suppose that the Galois group $G \colonequals \operatorname{Gal}(A/{{\mathbb{Q}}})$ is of exponent dividing $p-1$. Let $I^p_A$ be the group of fractional ideals of ${{\mathcal{O}}}_A[1/p]$, that is, the free abelian group on the set of finite primes of $A$ not lying above $p$. We have the standard exact sequence of ${{\mathbb{Z}}}G$-modules $$\label{E:4-term} 1 \to {{\mathcal{O}}}_A[1/p]^\times \to A^\times \to I^p_A \to \operatorname{Cl}({{\mathcal{O}}}_A[1/p]) \to 0.$$ Let $S$ be a finite set of places of ${{\mathbb{Q}}}$ including $p$ and $\infty$. Let $S_A$ be the set of places of $A$ above $S$. We approximate by using only $S_A$-units and ideals supported on $S_A$. Let $S_A^p$ be the set of places of $A$ above $S-\{p\}$. Let ${{\mathcal{O}}}_{A,S}$ be the ring of $S_A$-integers in $A$. Let $I^p_{A,S}$ be the free abelian group on the nonarchimedean places in $S_A^p$. If $S$ is large enough that the finite primes in $S_A^p$ generate $\operatorname{Cl}({{\mathcal{O}}}_A[1/p])$, then we have an exact sequence of ${{\mathbb{Z}}}G$-modules $$\label{E:approximate 4-term} 1 \to {{\mathcal{O}}}_A[1/p]^\times \to {{\mathcal{O}}}_{A,S}^\times \to I^p_{A,S} \to \operatorname{Cl}({{\mathcal{O}}}_A[1/p]) \to 0.$$ Dropping the first term and tensoring with ${{\mathbb{F}}}_p$ yields an exact sequence of ${{\mathbb{F}}}_p G$-modules $${{\mathcal{O}}}_{A,S}^\times/p \to I^p_{A,S}/p \to \operatorname{Cl}({{\mathcal{O}}}_A[1/p])/p \to 0.$$ Let $\chi$ be an irreducible ${{\mathbb{F}}}_p$-representation of $G$; our assumption on $G$ guarantees that $\chi$ is $1$-dimensional. Taking $\chi$-isotypic components yields $$\label{E:class group as cokernel} \left({{\mathcal{O}}}_{A,S}^\times/p \right)^\chi \to \left( I^p_{A,S}/p \right)^\chi \to \left( \operatorname{Cl}({{\mathcal{O}}}_A[1/p])/p \right)^\chi \to 0.$$ Let $u = u(A,\chi) \colonequals \dim_{{{\mathbb{F}}}_p} \left({{\mathcal{O}}}_A[1/p]^\times/p \right)^\chi$. \[L:u-lemma\] Assume that $\mu_p(A)^\chi=0$. 1. \[E:u-formula\] Let $S_\infty$ resp. $S_p$ be the set of places of $A$ lying above $\infty$ resp. $p$. Then $$u = \dim_{{{\mathbb{F}}}_p} ({{\mathbb{F}}}_p^{S_\infty})^\chi + \dim_{{{\mathbb{F}}}_p} ({{\mathbb{F}}}_p^{S_p})^\chi - \begin{cases} 1, &\textup{if $\chi=1$;} \\ 0, &\textup{otherwise.} \end{cases}$$ 2. We have $$\dim_{{{\mathbb{F}}}_p} \left({{\mathcal{O}}}_{A,S}^\times/p \right)^\chi = \dim_{{{\mathbb{F}}}_p} \left(I^p_{A,S}/p \right)^\chi + u.$$ 3. The quantity $\dim_{{{\mathbb{F}}}_p} \left(I^p_{A,S}/p \right)^\chi$ can be made arbitrarily large by choosing $S$ appropriately. <!-- --> 1. The Dirichlet $S$-unit theorem implies that the abelian group ${{\mathcal{O}}}_A[1/p]^\times$ is finitely generated with torsion subgroup $\mu(A)$. Let $M$ be the ${{\mathbb{Z}}}G$-module ${{\mathcal{O}}}_A[1/p]^\times/\mu(A)$. Tensoring the exact sequence $$\mu(A) {\longrightarrow}{{\mathcal{O}}}_A[1/p]^\times {\longrightarrow}M {\longrightarrow}0$$ with ${{\mathbb{F}}}_p$ and taking $\chi$-isotypic components yields $$0 {\longrightarrow}\left( {{\mathcal{O}}}_A[1/p]^\times/p \right)^\chi {\longrightarrow}(M/p)^\chi {\longrightarrow}0,$$ so $u = \dim_{{{\mathbb{F}}}_p} (M/p)^\chi$. On the other hand, the proof of the Dirichlet $S$-unit theorem yields $$M {\otimes}{{\mathbb{R}}}{\simeq}{{\mathcal{O}}}_A[1/p]^\times {\otimes}{{\mathbb{R}}}{\simeq}\left({{\mathbb{R}}}^{S_\infty {\cup}S_p}\right)_{\textup{sum~$0$}}$$ as ${{\mathbb{R}}}G$-modules. A ${{\mathbb{Z}}}_{(p)} G$-module that is free of finite rank over ${{\mathbb{Z}}}_{(p)}$ is determined by its character, so $$M {\otimes}{{\mathbb{Z}}}_{(p)} {\simeq}\left( {{\mathbb{Z}}}_{(p)}^{S_\infty {\cup}S_p}\right)_{\textup{sum~$0$}}$$ as ${{\mathbb{Z}}}_{(p)} G$-modules. Both sides are free over ${{\mathbb{Z}}}_{(p)}$, so we may tensor with ${{\mathbb{F}}}_p$ to obtain $$M/p {\simeq}\left( {{\mathbb{F}}}_p^{S_\infty {\cup}S_p}\right)_{\textup{sum~$0$}}$$ as ${{\mathbb{F}}}_p G$-modules. In other words, there is an exact sequence $$0 {\longrightarrow}M/p {\longrightarrow}{{\mathbb{F}}}_p^{S_\infty} {\oplus}{{\mathbb{F}}}_p^{S_p} {\longrightarrow}{{\mathbb{F}}}_p {\longrightarrow}0.$$ Taking dimensions of the $\chi$-components yields the formula for $u$. 2. The composition $G \stackrel{\chi}\to ({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times {\hookrightarrow}{{\mathbb{Z}}}_p^\times$ lets us identify $\chi$ with a ${{\mathbb{Z}}}_p$-representation of $G$. Tensor with ${{\mathbb{Z}}}_p$, and take $\chi$-isotypic components: $$0 \to \left( {{\mathcal{O}}}_A[1/p]^\times {\otimes}{{\mathbb{Z}}}_p \right)^\chi \to \left( {{\mathcal{O}}}_{A,S}^\times {\otimes}{{\mathbb{Z}}}_p \right)^\chi \to \left( I^p_{A,S} {\otimes}{{\mathbb{Z}}}_p \right)^\chi \to \left( \operatorname{Cl}({{\mathcal{O}}}_A[1/p]) {\otimes}{{\mathbb{Z}}}_p \right)^\chi \to 0.$$ Since $\mu_p(A)^\chi=0$, the first three ${{\mathbb{Z}}}_p$-modules are free; on the other hand, the last is finite as a set. Take ${{\mathbb{Z}}}_p$-ranks. If $V$ is a ${{\mathbb{Z}}}_p G$-module such that $V^\chi$ is a free ${{\mathbb{Z}}}_p$-module of finite rank, then $\dim_{{{\mathbb{F}}}_p} (V/p)^\chi = \operatorname{rank}_{{{\mathbb{Z}}}_p} V^\chi$. This proves the formula. 3. If $S$ contains $m$ rational primes that split completely in $A$, then $I^p_{A,S}$ contains $({{\mathbb{Z}}}G)^m$, so $\dim_{{{\mathbb{F}}}_p} \left(I^p_{A,S}/p \right)^\chi \ge m$, and $m$ can be chosen arbitrarily large. Sequence  and Lemma \[L:u-lemma\](b,c) imply that $\left( \operatorname{Cl}({{\mathcal{O}}}_A[1/p])/p \right)^\chi$ is naturally the cokernel of a linear map ${{\mathbb{F}}}_p^{m+u} \to {{\mathbb{F}}}_p^m$ for arbitrarily large $m$. In Section \[S:heuristics for even K-groups\], we will vary $(A,\chi)$ in a family with constant $u$-value and conjecture that the distribution of $\left( \operatorname{Cl}({{\mathcal{O}}}_A[1/p])/p \right)^\chi$ equals the limit as $m \to \infty$ of the distribution of the cokernel of a *random* linear map ${{\mathbb{F}}}_p^{m+u} \to {{\mathbb{F}}}_p^m$; the precise statement is Conjecture \[C:class group heuristic\]. For now, we mention that this limiting distribution and the limiting expected size of the cokernel are known: \[P:alpha distribution\] Fix a prime $p$ and an integer $u \ge 0$. For $m \ge 0$, let $N$ be a linear map ${{\mathbb{F}}}_p^{m+u} \to {{\mathbb{F}}}_p^m$ chosen uniformly at random, and let ${\mathcal{N}}_{p,u,m}$ be the random variable $\dim_{{{\mathbb{F}}}_p} \operatorname{coker}(N)$. Then 1. \[I:alpha formula\] For each $r \ge 0$, $$\lim_{m \rightarrow \infty} \operatorname{Prob}({\mathcal{N}}_{p,u,m}=r) \; = \; \alpha_{p,u,r} \colonequals \frac{\prod_{i = r+1}^\infty (1 - p^{-i})} {p^{r(u+r)} \prod_{i = 1}^{r+u} (1 - p^{-i})}.$$ 2. \[I:alpha probability\] We have $\sum_{r = 0}^\infty \alpha_{p,u,r} = 1$. 3. \[I:alpha average\] We have $\sum_{r = 0}^\infty p^r \alpha_{p,u,r} = 1+p^{-u}$. <!-- --> 1. This is [@Kovalenko-Levitskaja1975a]\*[Theorem 1]{}. 2. This is the $q=1/p$ and $\alpha=0$ case of [@Cohen-Lenstra1984]\*[Corollary 6.7]{}. 3. This is the $q=1/p$ and $\alpha=1$ case of [@Cohen-Lenstra1984]\*[Corollary 6.7]{}. The constant $\alpha_{p,u,r}$ appeared also in [@Cohen-Lenstra1984]\*[Theorem 6.3]{}, as the $u$-probability that a random finite abelian $p$-group has $p$-rank $r$. The connection between $u$-probabilities and coranks of random matrices was made in [@Friedman-Washington1989]. Heuristics for class groups and even K-groups associated to quadratic fields {#S:heuristics for even K-groups} ============================================================================ Calculation of u ---------------- We now specialize Section \[S:heuristics for class groups\] to the setting of Section \[S:K-groups of quadratic field\]. Thus $F$ is ${{\mathbb{Q}}}(\sqrt{d})$ for some $d \in {{\mathbb{Q}}}^\times$ such that $d$ and $p^*$ are independent in ${{\mathbb{Q}}}^\times/{{\mathbb{Q}}}^{\times 2}$; by multiplying $d$ by a square, we may assume that $d$ is a fundamental discriminant: $d=d_F$. Also, $E \colonequals F(\zeta_p)$ and $\chi \colonequals \chi_{-n,-1}$. Define $u(E,\chi)$ as in the sentence before Lemma \[L:u-lemma\]. \[P:u-values for quadratic fields\] The value $u(E,\chi)$ is given by the following table: $d>0$ $d<0$ --------------------- ------------------------------------------------------------------------------ ------- ------- $\textup{$n$ even}$ $n \equiv 0 \pmod{p-1}$, $d \in {{\mathbb{Q}}}_p^{\times 2}$ $2$ $1$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^* d \in {{\mathbb{Q}}}_p^{\times 2}$ $2$ $1$ $\textup{all other cases}$ $1$ $0$ $\textup{$n$ odd}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^* d \in {{\mathbb{Q}}}_p^{\times 2}$ $1$ $2$ $\textup{all other cases}$ $0$ $1$ Because $\operatorname{Gal}(E/{{\mathbb{Q}}}(\zeta_p))$ acts on $\mu_p(E)$ as $+1$, we have $\mu_p(E)^\chi=0$, so Lemma \[L:u-lemma\] applies with $\chi = \chi_{-n,-1}$. The complex conjugation in $\operatorname{Gal}(E/{{\mathbb{Q}}}) {\simeq}({{\mathbb{Z}}}/p{{\mathbb{Z}}})^\times \times \{\pm1\}$ is $c \colonequals (-1,\pm 1)$, where the $\pm 1$ is $+1$ if $F$ is real, and $-1$ if $F$ is imaginary. The $G$-set $S_\infty$ is isomorphic to $G/\langle c \rangle$, so ${{\mathbb{F}}}_p^{S_\infty}$ is the $c$-invariant subrepresentation of the regular representation ${{\mathbb{F}}}_p G$. The multiplicity of $\chi$ in ${{\mathbb{F}}}_p G$ is $1$, so the multiplicity of $\chi$ in ${{\mathbb{F}}}_p^{S_\infty}$ is $1$ or $0$, according to whether $\chi(c)$ is $1$ or $-1$. By definition of $\chi_{-n,-1}$, we have $\chi(c)=(-1)^{-n} \operatorname{sgn}(d)$, so $$\dim_{{{\mathbb{F}}}_p} \left({{\mathbb{F}}}_p^{S_\infty}\right)^{\chi} = \begin{cases} 1, &\textup{if $(-1)^n d > 0$;} \\ 0, &\textup{if $(-1)^n d < 0$.} \\ \end{cases}$$ Next, Corollary \[C:p-torsion in Br\] implies $$\dim_{{{\mathbb{F}}}_p} \left({{\mathbb{F}}}_p^{S_p}\right)^\chi = \dim_{{{\mathbb{F}}}_p} (\operatorname{Br}({{\mathcal{O}}}_E[1/p])_p)^\chi,$$ which is given by Proposition \[P:p-torsion in Br for quadratic fields\]. The third term in Lemma \[L:u-lemma\] is $0$ since $\chi \ne 1$. Distribution ------------ Suppose that ${\mathscr{F}}$ is a family of quadratic fields. For $X>0$, let ${\mathscr{F}}_{<X}$ be the set of $F \in {\mathscr{F}}$ such that $|d_F| < X$. For any function $\gamma \colon {\mathscr{F}}\to {{\mathbb{Z}}}_{\ge 0}$, define the following notation: $$\begin{aligned} \operatorname{Prob}(\gamma(F) = r) &\colonequals \lim_{X \to \infty} \frac{\# \{F \in {\mathscr{F}}_{<X} : \gamma(F) = r \}} {\# {\mathscr{F}}_{<X}}, \\ \operatorname{Average}(\gamma) &\colonequals \lim_{X \to \infty} \frac{\sum_{F \in {\mathscr{F}}_{<X}} \gamma(F) } {\# {\mathscr{F}}_{<X}}.\end{aligned}$$ Our heuristic is formalized in the following statement. \[C:class group heuristic\] Fix one of the ten boxes below and right of the double lines in the table of Proposition \[P:u-values for quadratic fields\], and fix a corresponding $n \ge 1$. Let $F$ vary over the family ${\mathscr{F}}$ of quadratic fields with $d$ satisfying the conditions defining that box, and let $u$ be as calculated in Proposition \[P:u-values for quadratic fields\]. Then the distribution of $\left( \operatorname{Cl}({{\mathcal{O}}}_E[1/p])/p \right)^\chi$ equals the limit as $m \to \infty$ of the distribution of a cokernel of a random linear map ${{\mathbb{F}}}_p^{m+u} \to {{\mathbb{F}}}_p^m$; by this we mean, in the notation of Proposition \[P:alpha distribution\], that for each $r \in {{\mathbb{Z}}}_{\ge 0}$, $$\operatorname{Prob}\left( \dim_{{{\mathbb{F}}}_p} \left( \operatorname{Cl}({{\mathcal{O}}}_E[1/p])/p \right)^\chi = r \right) = \lim_{m \rightarrow \infty} \operatorname{Prob}({\mathcal{N}}_{p,u,m}=r),$$ which by Proposition \[P:alpha distribution\] equals $\alpha_{p,u,r}$. If Conjecture \[C:class group heuristic\] holds, then substituting it and Proposition \[P:p-torsion in Br for quadratic fields\] (with Remark \[R:splits\]) into Theorem \[T:K\_2n for quadratic field\] yields the following: \[C:K-theory conjecture\] Fix $n \ge 1$ and an odd prime $p$ and $r \ge 0$. As $F$ ranges over the quadratic fields ${{\mathbb{Q}}}(\sqrt{d})$ with $d$ satisfying the conditions defining a box below, $\operatorname{Prob}\left( \dim_{{{\mathbb{F}}}_p} (K_{2n}({{\mathcal{O}}}_F)/p)^- = r \right )$ is as follows: $d>0$ $d<0$ --------------------- ------------------------------------------------------------------------------ -------------------- -------------------- $\textup{$n$ even}$ $n \equiv 0 \pmod{p-1}$, $d \in {{\mathbb{Q}}}_p^{\times 2}$ $\alpha_{p,2,r-1}$ $\alpha_{p,1,r-1}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^*d \in {{\mathbb{Q}}}_p^{\times 2}$ $\alpha_{p,2,r-1}$ $\alpha_{p,1,r-1}$ $\textup{all other cases}$ $\alpha_{p,1,r}$ $\alpha_{p,0,r}$ $\textup{$n$ odd}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^* d \in {{\mathbb{Q}}}_p^{\times 2}$ $\alpha_{p,1,r-1}$ $\alpha_{p,2,r-1}$ $\textup{all other cases}$ $\alpha_{p,0,r}$ $\alpha_{p,1,r}$ The distribution of $\dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathcal{O}}}_F)/p$ is the distribution of $\dim_{{{\mathbb{F}}}_p} (K_{2n}({{\mathcal{O}}}_F)/p)^-$ shifted by the constant $\kappa_{2n,p} = \dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathbb{Z}}})_p$ of Section . In Conjecture \[C:K-theory conjecture\], the family of fields was defined by specifying both the sign of $d$ and a $p$-adic condition on $d$. To get the analogous probabilities for a larger family in which only the sign of $d$ is specified, we can take a weighted combination of probabilities from Conjecture \[C:K-theory conjecture\]. \[Ex:upper left\] Suppose that $n \equiv 0 \pmod{p-1}$ and ${\mathscr{F}}$ is the family of real quadratic fields $F$. By the first and third entries of the $d>0$ column of the table of Conjecture \[C:K-theory conjecture\], $\operatorname{Prob}\left( \dim_{{{\mathbb{F}}}_p} (K_{2n}({{\mathcal{O}}}_F)/p)^- = r \right)$ equals $$\label{E:weighted probability} \operatorname{Prob}(d \in {{\mathbb{Q}}}_p^{\times 2}) \; \alpha_{p,2,r-1} \; + \; \operatorname{Prob}(d \notin {{\mathbb{Q}}}_p^{\times 2}) \; \alpha_{p,1,r}.$$ Since $p^2 \nmid d$, we have $d \in {{\mathbb{Q}}}_p^{\times 2}$ if and only if $(d \bmod p) \in {{\mathbb{F}}}_p^{\times 2}$; this condition is satisfied by $d$ lying in $p \cdot (p-1)/2$ of the $p^2-1$ nonzero residue classes modulo $p^2$. The discriminant $d$ is equally likely to be in any of the $p^2-1$ nonzero residue classes modulo $p^2$, as follows, for example, from [@Prachar1958]\*[(1)]{}, so $$\operatorname{Prob}(d \in {{\mathbb{Q}}}_p^{\times 2}) = \frac{p(p-1)/2}{p^2-1} = \frac{p}{2p+2}.$$ Substituting this and the complementary probability into  yields $$\operatorname{Prob}\left( \dim_{{{\mathbb{F}}}_p} (K_{2n}({{\mathcal{O}}}_F)/p)^- = r \right) = \frac{p}{2p+2} \alpha_{p,2,r-1} + \frac{p+2}{2p+2} \alpha_{p,1,r}.$$ Similar calculations show that Conjecture \[C:K-theory conjecture\] implies all ten cases in Conjecture \[C:K-theory conjecture for rank again\] below. \[C:K-theory conjecture for rank again\] Fix $n \ge 1$ and an odd prime $p$ and $r \ge 0$. As $F$ ranges over all real resp. imaginary quadratic fields, $\operatorname{Prob}\left( \dim_{{{\mathbb{F}}}_p} (K_{2n}({{\mathcal{O}}}_F)/p)^- = r \right)$ is given in the following table by the entry in the row determined by $n$ and column determined by the signature: The distribution of $\dim_{{{\mathbb{F}}}_p} K_{2n}({{\mathcal{O}}}_F)/p$ is the distribution of $\dim_{{{\mathbb{F}}}_p} (K_{2n}({{\mathcal{O}}}_F)/p)^-$ shifted by $\kappa_{2n,p}$. Average order ------------- Proposition \[P:u-values for quadratic fields\] combined with the reasoning of Section \[S:heuristics for class groups\] (Proposition \[P:alpha distribution\], in particular) suggests the following statement: \[C:average class group heuristic\] Fix an odd prime $p$. The average order of $\left( \operatorname{Cl}({{\mathcal{O}}}_E[1/p])/p \right)^\chi$ for $F$ ranging over the quadratic fields ${{\mathbb{Q}}}(\sqrt{d})$ satisfying the conditions defining a box below is as follows: $d>0$ $d<0$ --------------------- ------------------------------------------------------------------------------ ------------ ------------ $\textup{$n$ even}$ $n \equiv 0 \pmod{p-1}$, $d \in {{\mathbb{Q}}}_p^{\times 2}$ $1+p^{-2}$ $1+p^{-1}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^*d \in {{\mathbb{Q}}}_p^{\times 2}$ $1+p^{-2}$ $1+p^{-1}$ $\textup{all other cases}$ $1+p^{-1}$ $2$ $\textup{$n$ odd}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^* d \in {{\mathbb{Q}}}_p^{\times 2}$ $1+p^{-1}$ $1+p^{-2}$ $\textup{all other cases}$ $2$ $1+p^{-1}$ Conjecture \[C:average class group heuristic\], in turn, would imply the following: \[C:average K-theory in residue classes conjecture\] Fix $n \ge 1$ and an odd prime $p$. The average order of $(K_{2n}({{\mathcal{O}}}_F)/p)^-$ for $F$ ranging over the quadratic fields ${{\mathbb{Q}}}(\sqrt{d})$ satisfying the conditions defining a box below is as follows: $d>0$ $d<0$ --------------------- ------------------------------------------------------------------------------ ------------ ------------ $\textup{$n$ even}$ $n \equiv 0 \pmod{p-1}$, $d \in {{\mathbb{Q}}}_p^{\times 2}$ $p+p^{-1}$ $p+1$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^*d \in {{\mathbb{Q}}}_p^{\times 2}$ $p+p^{-1}$ $p+1$ $\textup{all other cases}$ $1+p^{-1}$ $2$ $\textup{$n$ odd}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$, $p^* d \in {{\mathbb{Q}}}_p^{\times 2}$ $p+1$ $p+p^{-1}$ $\textup{all other cases}$ $2$ $1+p^{-1}$ To get the average order of $K_{2n}({{\mathcal{O}}}_F)/p$ itself, multiply each entry by $p^{\kappa_{2n,p}}$. Conjecture \[C:average K-theory in residue classes conjecture\] implies Conjecture \[C:average K-theory conjecture\] below for the family of all quadratic fields of given signature, by taking weighted combinations of averages. For example, for $n \equiv 0 \pmod{p-1}$ and real quadratic fields (cf. Example \[Ex:upper left\]), $$\begin{aligned} \operatorname{Average}\#(K_{2n}({{\mathcal{O}}}_F)/p)^- &= \operatorname{Prob}(d \in {{\mathbb{Q}}}_p^{\times 2}) \; (p+p^{-1}) + \operatorname{Prob}(d \notin {{\mathbb{Q}}}_p^{\times 2}) \; (1+p^{-1}) \\ &= \frac{p}{2p+2} (p+p^{-1}) + \frac{p+2}{2p+2} (1+p^{-1}) \\ &= \frac{p^3+p^2+4p+2}{2p^2+2p}.\end{aligned}$$ \[C:average K-theory conjecture\] Fix $n \ge 1$ and an odd prime $p$. The average order of $(K_{2n}({{\mathcal{O}}}_F)/p)^-$ for $F$ ranging over all real resp. imaginary quadratic fields is given in the following table by the entry in the row determined by $n$ and column determined by the signature: --------------------- ------------------------------------- -------------------------------- ----------------------------- $\textup{$n$ even}$ $n \equiv 0 \pmod{p-1}$ $\frac{p^3+p^2+4p+2}{2p^2+2p}$ $\frac{p^2+3p+4}{2p+2}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$ $\frac{3p^2+3p+2}{2p^2+2p}$ $\frac{5p+3}{2p+2}$ $\textup{all other cases}$ $\frac{p+1}{p}$ $2$ $\textup{$n$ odd}$ $n \equiv \frac{p-1}{2} \pmod{p-1}$ $\frac{5p+3}{2p+2}$ $\frac{3p^2+3p+2}{2p^2+2p}$ $\textup{all other cases}$ $2$ $\frac{p+1}{p}$ --------------------- ------------------------------------- -------------------------------- ----------------------------- To get the distribution of the order of $K_{2n}({{\mathcal{O}}}_F)/p$ itself, multiply each entry by $p^{\kappa_{2n,p}}$. It is not quite clear that Conjectures \[C:class group heuristic\], \[C:K-theory conjecture\], and \[C:K-theory conjecture for rank again\], imply Conjectures \[C:average class group heuristic\], \[C:average K-theory in residue classes conjecture\], and \[C:average K-theory conjecture\], respectively. For such implications, one would need to know that the contribution to each average from the rare cases of very large class groups or $K$-groups is negligible. The average order of even K-groups modulo 3 =========================================== In this section, we prove Conjecture \[C:average class group heuristic\] for $p=3$; then Conjectures \[C:average K-theory in residue classes conjecture\] and \[C:average K-theory conjecture\] for $p=3$ follow too; the latter becomes Theorem \[T:average of K\_3\]. For $p=3$, the table in Conjecture \[C:average class group heuristic\] to be verified simplifies to $d>0$ $d<0$ --------------------- ------------------------------------------ -------- -------- $\textup{$n$ even}$ $d \in {{\mathbb{Q}}}_3^{\times 2}$ $10/9$ $4/3$ $d \notin {{\mathbb{Q}}}_3^{\times 2}$ $4/3$ $2$ $\textup{$n$ odd}$ $-3d \in {{\mathbb{Q}}}_3^{\times 2}$ $4/3$ $10/9$ $-3d \notin {{\mathbb{Q}}}_3^{\times 2}$ $2$ $4/3$ We have $E=F(\zeta_3)$ and $G = \operatorname{Gal}(E/{{\mathbb{Q}}}) {\simeq}({{\mathbb{Z}}}/3{{\mathbb{Z}}})^\times \times \{\pm 1\}$. Let $H \colonequals \ker \chi$, which is generated by $(-1,(-1)^n)$. Let $K \colonequals E^H$, which is $F$ if $n$ is even, and $F'={{\mathbb{Q}}}(\sqrt{-3d})$ if $n$ is odd. For any ${{\mathbb{F}}}_3$-representation $V$ of $G$, we have $V^H = V^G {\oplus}V^\chi$. Apply this to $V = \operatorname{Cl}({{\mathcal{O}}}_E[1/3])/3$, and use Lemma \[L:Hochschild-Serre\] twice to obtain $$\operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3 {\simeq}\operatorname{Cl}({{\mathbb{Z}}}[1/3])/3 {\oplus}\left( \operatorname{Cl}({{\mathcal{O}}}_E[1/3])/3 \right)^\chi {\simeq}\left( \operatorname{Cl}({{\mathcal{O}}}_E[1/3])/3 \right)^\chi,$$ since $\operatorname{Cl}({{\mathbb{Z}}}[1/3])$ is a quotient of $\operatorname{Cl}({{\mathbb{Z}}})=0$. It remains to show that for each box, the average order of $\operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3$ is as given in the table above. For fixed $n$, as $d$ ranges over fundamental discriminants of fixed sign in a fixed coset of ${{\mathbb{Q}}}_3^{\times 2}$ up to some bound, the field $K$ ranges over quadratic fields with fundamental discriminant of fixed sign in a fixed coset of ${{\mathbb{Q}}}_3^{\times 2}$ up to some bound (the same sign and coset if $n$ is even, or sign and coset multiplied by $-3$ if $n$ is odd). Thus it suffices to prove that the average order of $\operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3$ for such $K$ is given by the table $d_K>0$ $d_K<0$ ------------------------------------------ --------- --------- $d_K \in {{\mathbb{Q}}}_3^{\times 2}$ $10/9$ $4/3$ $d_K \notin {{\mathbb{Q}}}_3^{\times 2}$ $4/3$ $2$ \[R:cosets\] There are four cosets of ${{\mathbb{Q}}}_3^{\times 2}$ in ${{\mathbb{Q}}}_3^\times$. Which coset contains a given fundamental discriminant $d_K$ is determined by whether $d_K$ is $1 \bmod 3$, $2 \bmod 3$, $3 \bmod 9$, or $6 \bmod 9$. In particular, $d_K \in {{\mathbb{Q}}}_3^{\times 2}$ if and only if $d_K \equiv 1 \pmod{3}$. The results in Sections \[S:class groups to cubic fields\] to \[S:counting cubic fields\] have been generalized by the second author to compute the average size of $\operatorname{Cl}({{\mathcal{O}}}_K[1/S])/3$ for an arbitrary finite set of primes $S$ using essentially identical methods [@Klagsbrun-preprint]. Similar methods have been used to compute the average size of $\operatorname{Cl}(O_K[1/p])/3$ when $p$ splits in $K$ [@Wood2018]. From class groups to cubic fields {#S:class groups to cubic fields} --------------------------------- To compute the average order of $\operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3$, we adapt the strategy of Davenport and Heilbronn [@Davenport-Heilbronn1971], which relies on results involving the following setup. Let $L$ be a degree $3$ extension of ${{\mathbb{Q}}}$ whose Galois closure $M$ has Galois group $S_3$. Let $K$ be the quadratic resolvent of $L$, that is, the degree $2$ subfield $M^{A_3}$. \[L:S\_3 extension\] The following are equivalent: 1. $M/K$ is unramified. 2. $d_L=d_K$. 3. $L/{{\mathbb{Q}}}$ is nowhere totally ramified. Let $\mathfrak{f}$ be the conductor of the abelian extension $M/K$. By [@Hasse1930]\*[Satz 3]{}, $d_L = N_{K/{{\mathbb{Q}}}}(\mathfrak{f}) d_K$; this proves (i)$\leftrightarrow$(ii). Also, $d_L$ is an integer square times $d_K$ (see [@Hasse1930]\*[(1)]{}); this proves (ii)$\leftrightarrow$(iii). Finally, [@Cohen2000]\*[Proposition 8.4.1]{} yields (ii)$\leftrightarrow$(iv). Throughout this section, we consider two cubic fields to be the same if they are abstractly isomorphic (so conjugate cubic fields are counted only as one). \[T:Delone-Faddeev\] Fix a quadratic field $K$. Then the following are naturally in bijection: 1. The set of index $3$ subgroups of $\operatorname{Cl}({{\mathcal{O}}}_K)$. 2. The set of unramified ${{\mathbb{Z}}}/3{{\mathbb{Z}}}$-extensions $M$ of $K$. 3. The set of cubic fields $L$ with $d_L=d_K$. Class field theory gives (i)$\leftrightarrow$(ii). The nontrivial element of $\operatorname{Gal}(K/{{\mathbb{Q}}})$ acts as $-1$ on $\operatorname{Cl}({{\mathcal{O}}}_K)$, so each $M$ in (ii) is an $S_3$-extension of ${{\mathbb{Q}}}$. The map (ii)$\to$(iii) sends $M$ to one of its cubic subfields $L$. The map (iii)$\to$(ii) sends $L$ to its Galois closure $M$. That these are bijections follows from Lemma \[L:S\_3 extension\](i)$\leftrightarrow$(ii). We are interested in $\operatorname{Cl}({{\mathcal{O}}}_K[1/3])$ instead of $\operatorname{Cl}({{\mathcal{O}}}_K)$, so we need the following variant. \[C:generalized Delone-Faddeev\] Fix a quadratic field $K$. Then the following are naturally in bijection: 1. The set of index $3$ subgroups of $\operatorname{Cl}({{\mathcal{O}}}_K)[1/3]$. 2. The set of unramified ${{\mathbb{Z}}}/3{{\mathbb{Z}}}$-extensions $M$ of $K$ such that the primes above $3$ in $K$ split completely in $M/K$. 3. The set of cubic fields $L$ with $d_L=d_K$ such that if $d_K \in {{\mathbb{Q}}}_3^{\times 2}$ then $3$ splits completely in $L/{{\mathbb{Q}}}$. The group $\operatorname{Cl}({{\mathcal{O}}}_K)[1/3]$ is the quotient of $\operatorname{Cl}({{\mathcal{O}}}_K)$ by the group generated by the classes of the primes ${{\mathfrak{p}}}| 3$ in $K$. Thus we need to restrict the bijections in Theorem \[T:Delone-Faddeev\] to the index $3$ subgroups $H$ containing these classes. Because the class field theory isomorphism $\operatorname{Cl}({{\mathcal{O}}}_K)/H {\simeq}\operatorname{Gal}(M/K)$ sends $[{{\mathfrak{p}}}]$ to $\operatorname{Frob}_{{\mathfrak{p}}}$, which is trivial if and only if ${{\mathfrak{p}}}$ splits in $M/K$, we obtain (i)$\leftrightarrow$(ii). If $3$ is inert or ramified in $K/{{\mathbb{Q}}}$, then the prime above $3$ is of order dividing $2$ in $\operatorname{Cl}({{\mathcal{O}}}_K)$, so to require it to be in the index $3$ subgroup is no condition. If $3$ splits in $K/{{\mathbb{Q}}}$ (that is, $d_K \in {{\mathbb{Q}}}_3^{\times 2}$), then the primes above $3$ in $K$ split in $M/K$ if and only if $3$ splits completely in $M/{{\mathbb{Q}}}$, which is if and only if $3$ splits completely in $L/{{\mathbb{Q}}}$. \[C:class groups to cubic fields\] Let $K$ be a quadratic field. 1. If $d_K \in {{\mathbb{Q}}}_3^{\times 2}$, then $$\# \operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3 = 2 \, \#\{\textup{cubic fields $L$ with $d_L=d_K$ in which $3$ splits}\} + 1.$$ 2. If $d_K \notin {{\mathbb{Q}}}_3^{\times 2}$, then $$\# \operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3 = 2 \, \#\{\textup{cubic fields $L$ with $d_L=d_K$}\} + 1.$$ For an elementary abelian $3$-group $V$, $$\#V = 2 \, \#\{\textup{index~$3$ subgroups of $V$}\} + 1.$$ Take $V = \operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3$, and apply Corollary \[C:generalized Delone-Faddeev\](i)$\leftrightarrow$(iii). Counting quadratic fields {#S:counting quadratic fields} ------------------------- Fix a sign and a coset of ${{\mathbb{Q}}}_3^{\times 2}$ in ${{\mathbb{Q}}}_3^\times$; by Remark \[R:cosets\], each of the four cosets is defined by a congruence condition mod $3$ or mod $9$. Let ${\mathscr{D}}$ be the set of fundamental discriminants having this sign and lying in this coset. For $X>0$, define ${\mathscr{D}}_{<X} \colonequals \{d \in {\mathscr{D}}: |d| < X\}$. To compute the average of the appropriate left hand side in Corollary \[C:class groups to cubic fields\] as $K$ ranges over quadratic fields with $d_K \in {\mathscr{D}}$, we compute the average number of cubic fields appearing in the corresponding right hand side. That is, we need the limit as $X \to \infty$ of $$\label{E:quadratics over cubics} \frac{ \sum_{K: d_K \in {\mathscr{D}}_{<X}} \#\{\textup{cubic fields $L$ with $d_L=d_K$ such that if $d_K \in {{\mathbb{Q}}}_3^{\times 2}$ then $3$ splits in $L$}\}} {\#\{\textup{quadratic fields $K$ such that $d_K \in {\mathscr{D}}_{<X}$}\}}.$$ We first compute the denominator. \[P:counting quadratic fields\] The number of quadratic fields $K$ satisfying $|d_K|<X$ and prescribed sign and $3$-adic congruence conditions is $\alpha_2 X/\zeta(2) + o(X)$, where $\alpha_2$ is given by the following table: $d_K>0$ $d_K<0$ ------------------------------------ --------- --------- $\textup{$d_K \equiv 1\pmod{3}$ }$ $3/16$ $3/16$ $\textup{$d_K \equiv 2\pmod{3}$ }$ $3/16$ $3/16$ $\textup{$d_K \equiv 3\pmod{9}$ }$ $1/16$ $1/16$ $\textup{$d_K \equiv 6\pmod{9}$ }$ $1/16$ $1/16$ Use an elementary squarefree sieve. Even though many entries in the table of Proposition \[P:counting quadratic fields\] coincide, it is stronger to give the asymptotics for the individual field families without merging them, and we need the stronger results. Counting cubic fields {#S:counting cubic fields} --------------------- By Lemma \[L:S\_3 extension\](ii)$\leftrightarrow$(iii), the numerator in  equals the number of nowhere totally ramified cubic fields $L$ with $d_L \in {\mathscr{D}}_{<X}$ such that if $d_L \in {{\mathbb{Q}}}_3^{\times 2}$ then $3$ splits completely in $L$. To compute this number, we follow the Davenport–Heilbronn approach, in the form of a refinement due to Bhargava, Shankar, and Tsimerman [@Bhargava-Shankar-Tsimerman2013]. For every prime $p$, let $\widehat{\Sigma}_p$ be the set of maximal cubic ${{\mathbb{Z}}}_p$-orders that are not totally ramified, up to isomorphism. For $R \in \widehat{\Sigma}_p$, let $\operatorname{Disc}_p(R)$ be the power of $p$ generating the discriminant ideal of $R$. \[thm:BST\] For each prime $p$ (including $2$), let $\Sigma_p \subseteq \widehat{\Sigma}_p$. Suppose that $\Sigma_p = \widehat{\Sigma}_p$ for all $p$ outside a finite set ${{\mathcal{P}}}$. Define $$c_p \colonequals \frac{p}{p+1} \sum_{R \in \Sigma_p} \frac{1}{\operatorname{Disc}_p(R) \; \lvert\operatorname{Aut}R\rvert}.$$ 1. The number of nowhere totally ramified totally real cubic fields $L$ up to isomorphism such that $|d_L| < X$ and ${{\mathcal{O}}}_L {\otimes}{{\mathbb{Z}}}_p \in \Sigma_p$ for all $p \in {{\mathcal{P}}}$ is $\frac{1}{12\zeta(2)}\left (\prod_{p\in {{\mathcal{P}}}} c_p \right) X + o(X)$. 2. The number of nowhere totally ramified complex cubic fields $L$ up to isomorphism such that $|d_L| < X$ and ${{\mathcal{O}}}_L {\otimes}{{\mathbb{Z}}}_p \in \Sigma_p$ for all $p \in {{\mathcal{P}}}$ is $\frac{1}{4\zeta(2)}\left (\prod_{p\in {{\mathcal{P}}}} c_p \right) X + o(X)$. The definition of $c_p$ yields $$\frac{p-1}{p} \sum_{R \in \Sigma_p} \frac{1}{\operatorname{Disc}_p(R) \; \lvert\operatorname{Aut}{R}\rvert} = \left( 1 - \frac{1}{p^2} \right) c_p.$$ For each $p \notin {{\mathcal{P}}}$, enumerating $\widehat{\Sigma}_p$ explicitly shows that $c_p=1$. Substituting this into [@Bhargava-Shankar-Tsimerman2013]\*[Theorem 8]{} yields the result. \[cor:d\_L1mod3\] The number of nowhere totally ramified cubic fields $L$ with $|d_L|<X$ satisfying prescribed sign and $3$-adic congruence conditions below such that if $d_L \equiv 1 \pmod{3}$ then $3$ splits completely in $L$ is $\alpha_3 X/\zeta(2) + o(X)$, where $\alpha_3$ is given by the following table: $d_L>0$ $d_L<0$ ------------------------------------ --------- --------- $\textup{$d_L \equiv 1\pmod{3}$ }$ $1/96$ $1/32$ $\textup{$d_L \equiv 2\pmod{3}$ }$ $1/32$ $3/32$ $\textup{$d_L \equiv 3\pmod{9}$ }$ $1/96$ $1/32$ $\textup{$d_L \equiv 6\pmod{9}$ }$ $1/96$ $1/32$ We apply Theorem \[thm:BST\] with ${{\mathcal{P}}}= \{ 3\}$ and with $\Sigma_3$ tailored to the row. For the first row, let $\Sigma_3 \colonequals \{{{\mathbb{Z}}}_3 \times {{\mathbb{Z}}}_3 \times {{\mathbb{Z}}}_3\}$, so $$c_3 = \frac{3}{4} \cdot \frac{1}{\operatorname{Disc}_3({{\mathbb{Z}}}_3 \times {{\mathbb{Z}}}_3 \times {{\mathbb{Z}}}_3) \; \lvert \operatorname{Aut}\left({{\mathbb{Z}}}_3 \times {{\mathbb{Z}}}_3 \times {{\mathbb{Z}}}_3 \right) \rvert} = \frac{3}{4} \cdot \frac{1}{6} = \frac{1}{8}.$$ For each other row, let $\Sigma_3 \colonequals \{{{\mathbb{Z}}}_3 \times {{\mathbb{Z}}}_3(\sqrt{d_L})\}$, and calculate $c_3$ similarly. End of proof ------------ Corollary \[cor:d\_L1mod3\] and Proposition \[P:counting quadratic fields\] give the asymptotic behavior of the numerator and denominator, respectively, in (see the first sentence of Section \[S:counting cubic fields\]). Thus, as $X \to \infty$, the ratio  tends to $\alpha_3/\alpha_2$. Following Corollary \[C:class groups to cubic fields\], we multiply by $2$ and add $1$ to obtain the average order of $\operatorname{Cl}({{\mathcal{O}}}_K[1/3])/3$ as $K$ varies over quadratic fields with $d_K \in {\mathscr{D}}$. For each signature, the answer is the same for each of the three nontrivial cosets of ${{\mathbb{Q}}}_3^{\times 2}$, so we combine them into a single entry in the table before Remark \[R:cosets\]. This completes the proof of Conjecture \[C:average class group heuristic\] for $p=3$, and hence also Conjectures \[C:average K-theory in residue classes conjecture\] and \[C:average K-theory conjecture\] for $p=3$ and Theorem \[T:average of K\_3\]. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the referee for taking the time to provide many helpful comments. [^1]: B.P. was supported in part by National Science Foundation grant DMS-1601946 and Simons Foundation grants \#402472 (to Bjorn Poonen) and \#550033. C.S. was supported in part by National Science Foundation grant DMS-1301842 and by the Simons Investigator grant \#376203 from the Simons Foundation.

--- abstract: 'A cosmology is considered driven by a stress-energy tensor consisting of a perfect fluid, an inhomogeneous pressure term (which we call a “tachyonic dust” for reasons which will become apparent) and a cosmological constant. The inflationary, radiation dominated and matter dominated eras are investigated in detail. In all three eras, the tachyonic pressure decreases with increasing radius of the universe and is thus minimal in the matter dominated era. The gravitational effects of the dust, however, may still strongly affect the universe at present time. In case the tachyonic pressure is positive, it enhances the overall matter [*density*]{} and is a candidate for dark matter. In the case where the tachyonic pressure is negative, the recent acceleration of the universe can be understood without the need for a cosmological constant. The ordinary matter, however, has positive energy density at all times. In a later section, the extension to a variable cosmological term is investigated and a specific model is put forward such that recent acceleration and future re-collapse is possible.' author: - | [A. Das [^1]]{}\ *[Department of Mathematics]{}\ *[Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6]{}\ *[Department of Physics]{}\ *[Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6]{}**** date: '[February 15, 2004]{}' title: Inhomogeneous cosmologies with tachyonic dust as dark matter --- PACS numbers: 95.35.+d, 98.80.-k\ Key words: inhomogeneous cosmology, tachyon\ Introduction ============ There are compelling reasons to study a cosmology which is not homogeneous. Inhomogeneous models were studied early on by Lemaître [@ref:lemaitre] and Tolman [@ref:tolman] and by many authors since. Misner [@ref:misner], for example, postulated a chaotic cosmology in which the universe began in a highly irregular state but which becomes regular at late times. The models presented here possess exactly this property, which will be realized in a later section. That is, in the matter phase, the deviation from FLRW spatial geometry is *minimal* and we show this by calculating the Gaussian curvature of two spheres in all phases. The curvature of a two sphere is the same for all values of $r$ in the matter domain yielding a three-dimensional space which is isometric to a sphere. Our present location may therefore be anywhere in this universe and there is no conflict with observational cosmology. The book by Krasiński also contains many inhomogeneous models which do not require us to be located at the center of symmetry [@ref:krasinskibook]. Some more recent studies dealing with inhomogeneous cosmologies include [@ref:fein], [@ref:iban1], [@ref:bark], [@ref:bark2], [@ref:iban2], [@ref:karasinskipaper] and [@ref:barm]. Inhomogeneous cosmological models are not at odds with astrophysical data. It is well known that inhomogeneities in the early universe will generate anisotropies in the cosmic microwave background radiation (CMB). Such effects have been studied by many groups ([@ref:arnau1], [@ref:saez], [@ref:arnau2], [@ref:fullana]) using density amplitudes and sizes of inhomogeneities corresponding to those of observed current objects (galactic clusters, the Great Attractor and voids). These studies, utilizing a range of reasonable parameters, have found that temperature fluctuations in the CMB, $\Delta T/\langle T\rangle$, ($\langle T\rangle$ being the mean temperature and $\Delta T$ the deviation from the mean) amount to no more than about $10^{-7} - 10^{-5}$, which is compatible with observation. Also, arguments to reconcile inhomogeneous solutions with cosmological observations may be found in [@ref:karasinskipaper]. The inhomogeneity referred to in this paper is a “radial” inhomogeneity compatible with spherical symmetry and therefore its effect on the CMB is potentially more difficult to detect than the (small) angular deviations. In general, at very high energies, our knowledge of the state of the universe is highly limited and special assumptions about the matter content and symmetry should be relaxed. It therefore seems reasonable to investigate solutions which, at least at early times, are less symmetric than the FLRW scenarios. A thorough exposition on various inhomogeneous cosmological models may be found in the book by Krasiński [@ref:krasinskibook]. In section 2 we consider a cosmology consisting of two fluids, a perfect fluid (motivated by the successful standard cosmology) and “tachyonic” dust. We use the term tachyonic due to the association of this source with space-like vectors in the stress-energy tensor. This terminology is also popular in string-theory motivated cosmologies commenced by the pioneering works of Mazumdar, Panda, Pérez-Lornezana [@ref:mazumdar] and Sen [@ref:sen1] and studied by many others (see, for example, [@ref:tachcos1], [@ref:tachcos2], [@ref:feins], [@ref:fks], [@ref:tachcos3], [@ref:matlock], [@ref:tachcos4], [@ref:tachcos6], [@ref:tachcos8], [@ref:tachcos5], [@ref:tachcos7] [@ref:tachcos9] and references therein). It should be pointed out that in *neither* the case presented here nor the string theory motivated case is the source acausal as will be pointed out below . The tachyonic dust is chosen as a dark matter candidate for several reasons. First, it provides one of the simplest extensions to the standard perfect fluid cosmology and it is hoped that this model will provide insight into more complex scenarios. Second, as will be seen below, the tachyonic dust is a source of pressure or tension without energy density and cosmological observations strongly imply that there exists a large pressure or tension component in our universe. This pressure also affects the overall effective mass of the universe. Multi-fluid models in the context of charged black holes in cosmology have been studied in [@ref:daskay] In section \[sec:varlam\] we consider an extension of the model to the case of variable cosmological term. We discuss in detail how making this term dynamical affects the fate of the universe. Finally, this paper utilizes a number of techniques for analyzing global properties of the manifold and it is hoped that this will provide a useful reference for the mathematical analysis of cosmological models. Tachyonic dust and perfect fluid universe ========================================= We consider here a model of the universe which contains both a perfect fluid and tachyonic dust. This source possesses the desirable properties mentioned in the introduction. Namely, the dust contribution is a source of pressure as is required for the recent accelerating phase of the universe. A tachyonic dust is the simplest model which contributes to pressure and it will be shown that this pressure also makes a contribution to the mass of the universe. This field is therefore also a potentially interesting candidate for dark matter. Aside from spherical symmetry, the sole assumption is that the eigenvalues of stress-energy tensor be real. We may therefore write $$T^{\mu}_{\;\nu}=\left[{\mu(t, r)}+{p(t, r)}\right]u^{\mu}u_{\nu}+{p(t, r)}\delta^{\mu}_{\;\nu}+ {\alpha(t, r)}w^{\mu}w_{\nu}, \label{eq:stresstens}$$ with $$u^{\beta}u_{\beta}=-1, \;\;\;\;\; w^{\beta}w_{\beta}=+1, \;\;\;\;\; u^{\beta}w_{\beta}=0. \nonumber$$ Here ${\mu(t, r)}$, ${p(t, r)}$ and ${\alpha(t, r)}$ are the fluid energy density, fluid pressure, and tachyonic [*pressure*]{} (or tension) respectively. By comparison of the ${\alpha(t, r)}$ term in (\[eq:stresstens\]) to the stress-energy tensor of regular dust, it can be seen why we choose the term “tachyonic dust” to describe this source. Notice that a dust associated with a space-like vector possesses the desirable property in that it yields solely a pressure. It will be shown that this tension may produce the observed acceleration of the universe at late times [@ref:riess], [@ref:perl]. The source is *not* acausal as the algebraic structure of (\[eq:stresstens\]) is exactly similar to that of an anisotropic fluid which is a causal source under minor restrictions and is often used in general relativity (see [@ref:makhark], [@ref:ivanov], [@ref:dev] and references therein). The time coordinate, $t$, may be chosen to be coincident with the proper time along a fluid streamline (the comoving condition). This gauge, along with spherical symmetry, allows a special class of metrics to be written as $$\begin{aligned} d\sigma^{2}:=&\left[\frac{d r^2}{1-\epsilon r^{2}+eg( r)} + r^2\,d\theta^{2} + r^2\sin^{2}\theta\,d\phi^{2}\right], \label{eq:submetric} \\ ds^{2}=& -dt^{2}+ {a^{2}(t)}\, d\sigma^{2}. \label{eq:metric}\end{aligned}$$ This form is particularly convenient as one may readily analyze differences between models presented here and the standard FLRW models (the $e\rightarrow 0$ limit). Therefore, $e$ may be interpreted as the tachyon coupling constant. It is easy to show that (\[eq:submetric\]-\[eq:metric\]) falls in the Tolman-Bondi class of metrics, used extensively in studies of inhomogeneous cosmologies. Using (\[eq:stresstens\]) and (\[eq:metric\]) in the Einstein equations with cosmological constant yields: $$\begin{aligned} {8\pi}{\mu(t, r)}+\Lambda=&3\left[\frac{{\dot{a}(t)}}{{a(t)}}\right]^{2}+ \frac{3\epsilon}{{a^{2}(t)}} -\frac{e}{ r^{2}{a^{2}(t)}}\left[ r{g( r)}\right]^{\prime}, \label{eq:einstone} \\ {8\pi}{p(t, r)}-\Lambda=&-2\frac{{\ddot{a}(t)}}{{a(t)}}-\left[\frac{{\dot{a}(t)}}{{a(t)}}\right]^{2} -\frac{\epsilon}{{a^{2}(t)}}+\frac{e{g^{\prime}( r)}}{2 r{a^{2}(t)}}, \label{eq:einsttwo} \\ {8\pi}{\alpha(t, r)}=&-\frac{e r}{2{a^{2}(t)}}\left[\frac{{g( r)}}{ r^{2}} \right]^{\prime}, \label{eq:einstthree}\end{aligned}$$ where dots represent partial derivatives with respect to $t$ and primes with respect to $ r$. Enforcing conservation on (\[eq:stresstens\]) yields two non-trivial equations: $$\begin{aligned} {\mu(t, r)}_{,t} +\frac{{\dot{a}(t)}}{{a(t)}}\left\{3\left[{\mu(t, r)}+ {p(t, r)}\right] +{\alpha(t, r)}\right\}&=0, \label{eq:cons1} \\ \left[{p(t, r)}+{\alpha(t, r)}\right]_{, r} +\frac{2{\alpha(t, r)}}{ r}&=0. \label{eq:cons2}\end{aligned}$$ Throughout this paper, restrictions ${a(t)}> 0$, ${\dot{a}(t)}\neq 0$ and $r>0$ are assumed in solving the differential equations. In case the tachyon parameter $e=0$, one gets back the standard FLRW cosmology. The orthonormal Riemann components will be useful: $$\begin{aligned} R_{\hat{t}\hat{ r}\hat{t}\hat{ r}}=& -\frac{{\dot{a}(t)}}{{a(t)}}= R_{\hat{t}\hat{\theta}\hat{t}\hat{\theta}}, \label{eq:rtrtr} \\ R_{\hat{ r}\hat{\theta}\hat{ r}\hat{\theta}}\equiv R_{\hat{ r}\hat{\phi}\hat{ r}\hat{\phi}}=& \frac{1}{{a^{2}(t)}r}\left[ r{\dot{a}^{2}(t)}+ \epsilon r -\frac{e{g^{\prime}( r)}}{2} \right], \label{eq:rrthrth} \\ R_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}}=& \left[\frac{{\dot{a}(t)}}{{a(t)}}\right]^{2}+\frac{2\epsilon r -e {g^{\prime}( r)}}{ r{a(t)}^2}, \label{eq:rthphthph}\end{aligned}$$ as well as those related by symmetry (hatted indices denote the orthonormal frame). The solutions, being local and valid in some domain, need not possess the neighbourhood near $r=0$. The singularity at $r=0$ will be addressed in a later section. In cosmology, two measurable parameters considered are the Hubble parameter $H(t)$ and the deceleration parameter, $q(t)$. These are: $$\begin{aligned} H(t):=&\frac{{\dot{a}(t)}}{{a(t)}}=-R_{\hat{t}\hat{ r}\hat{t}\hat{ r}}, \\ q(t):=& -\frac{{a(t)}{\ddot{a}(t)}}{{\dot{a}^{2}(t)}}.\end{aligned}$$ The field equations, (\[eq:einstone\]), (\[eq:einsttwo\]) and (\[eq:einstthree\]) yield $$\begin{aligned} \left[H(t)\right]^{2}=&\frac{{8\pi}}{3}{\mu(t, r)}+\frac{\Lambda}{3}- \frac{\epsilon}{{a^{2}(t)}} +\frac{e\left[ r{g( r)}\right]^{\prime}}{3 r^{2}{a^{2}(t)}}, \label{eq:modeinstone} \\ 6q(t)\left[H(t)\right]^{2}=&{8\pi}\left[{\mu(t, r)}+3{p(t, r)}\right] -2\Lambda -\frac{e r}{2{a^{2}(t)}} \left[\frac{{g( r)}}{ r^{2}}\right]^{\prime}, \label{eq:modeinsttwo} \\ \left[H(t)\right]^{2}\left[2q(t)-1\right] =&{8\pi}{p(t, r)}-\Lambda +\frac{1}{{a^{2}(t)}}\left[\epsilon -\frac{e{g^{\prime}( r)}}{2 r}\right] \label{eq:modeinstthree}.\end{aligned}$$ To study inhomogeneity, the orthonormal Riemann components of the three-dimensional sub-space (\[eq:submetric\]) are useful: $$\begin{aligned} \tilde{R}_{\hat{ r}\hat{\theta}\hat{ r}\hat{\theta}}\equiv \tilde{R}_{\hat{ r}\hat{\phi}\hat{ r}\hat{\phi}} =& \epsilon-\frac{e{g^{\prime}( r)}}{2 r}, \\ \tilde{R}_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}}=& \epsilon-\frac{e{g( r)}}{ r^{2}}.\end{aligned}$$ where the tilde is used to denote quantities calculated using the three dimensional subspace metric of $t=t_{0}$ spatial hyper-surfaces (\[eq:submetric\]). Finally, it is useful to define a measure of the inhomogeneity of the spatial universe via an inhomogeneity parameter: $$\begin{aligned} I( r):=&\frac{\tilde{R}_{\hat{ r}\hat{\theta}\hat{ r}\hat{\theta}}} {\tilde{R}_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}}} \\ =&\frac{ r\left[ 2\epsilon r- e{g^{\prime}( r)}\right]}{2\left[\epsilon r^{2}-e{g( r)}\right]}. \label{eq:inhomgen}\end{aligned}$$ A homogeneous space is characterized by $\frac{dI(r)}{dr}\equiv 0$. Specifically, for the FLRW ($e=0$) limit, $I( r) \equiv 1$. We next investigate the three major eras of cosmological evolution. Matter dominated era -------------------- In the universe’s recent history, the galaxies which constitute the bulk of the ordinary matter have negligible motion relative to the cosmic expansion. Therefore the pressure of ordinary matter is approximately zero. Reasonable physics also demands that ${\mu(t, r)}>0$. Setting the pressure equal to zero from the equation (\[eq:einsttwo\]) yields (assuming ${\dot{a}(t)}\neq0$) $$\frac{\left[{a(t)}{\dot{a}^{2}(t)}\right]^{\cdot}}{{\dot{a}(t)}} -\Lambda{a^{2}(t)}=- \left[\epsilon -\frac{e}{2 r}{g^{\prime}( r)}\right] = -C =\mbox{a constant}. \label{eq:matspeeqn}$$ Here, $C$ is the constant of separation. Solving this equation for ${g( r)}$ one obtains $$e{g( r)}=\left(\epsilon-C\right) r^{2}+eb, \label{eq:matg}$$ with $b$ a constant arising from integration. The equation for the expansion factor can be analyzed using techniques, many of which are well known in cosmology. We include details here for completion. The equation, after an integration, may be written as: $$\frac{1}{2}{\dot{a}^{2}(t)}-\frac{M_{0}}{{a(t)}}- \frac{\Lambda}{6}{a(t)}^{2}=-\frac{C}{2}. \label{eq:matenergy}$$ Here $M_{0}$ is a constant arising from the integration. In the standard cosmology this equation is often compared to total energy conservation and similar equations have been studied at least as early as Lemaitré and Eddington [@ref:lem] [@ref:edd]. The terms on the left hand side correspond to a kinetic energy, gravitational potential energy and vacuum energy respectively. The total energy being constant, $\left(-C/2\right)$. The constant $M_{0}$ may therefore be interpreted as an effective mass of the universe and it is of interest to investigate how the tachyon affects this constant. The equation (\[eq:matenergy\]) may be used in (\[eq:einstone\]) along with (\[eq:matg\]) to give the current effective mass of the universe: $$M_{0}={\mu(t, r)}\left[\frac{4}{3}\pi a^{3}(t)\right] +\frac{eb}{6 r^{2}}{a(t)}. \label{eq:matmass}$$ The second term in this equation gives the tachyonic contribution to the effective mass of the universe and therefore represents the present mass due to dark matter (which is independent of $\Lambda$ in this section). The fluid and tachyonic energy density and pressures are given by: $$\begin{aligned} 8\pi{\mu(t, r)}=& -\Lambda+3\left[\frac{{\dot{a}(t)}}{{a(t)}}\right]^{2} +\left(3C-\frac{eb}{ r^{2}}\right) \frac{1}{{a^{2}(t)}}, \\ 8\pi{p(t, r)}\equiv& 0, \\ 8\pi{\alpha(t, r)}=&\frac{eb}{r^{2}{a^{2}(t)}}. \label{eq:matalpha}\end{aligned}$$ Note that the tachyon pressure can be very small today (for large ${a(t)}$) although its effects through (\[eq:matmass\]) can be very large. Acceleration in the matter phase may be analyzed by studying the equations (\[eq:modeinsttwo\]) and (\[eq:modeinstthree\]): $$\begin{aligned} 2q -1=& \frac{1}{{\dot{a}^{2}(t)}} \left[C -\Lambda{a^{2}(t)}\right], \\ H^{2}(t)\left(2q-1\right)=&\frac{C}{{a(t)}}-\Lambda , \\ q=&{\mu(t, r)}\left[\frac{4}{3}\pi \frac{{a^{2}(t)}}{{\dot{a}^{2}(t)}}\right] +\frac{eb}{6 r^{2}{\dot{a}^{2}(t)}} -\frac{\Lambda}{3}\frac{{a^{2}(t)}}{{\dot{a}^{2}(t)}}. \label{eq:matacpar}\end{aligned}$$ Note that for positive ${\mu(t, r)}$, $q$ may be negative even with $\Lambda=0$. This result indicates that the tachyonic dust may drive the relatively recent acceleration phase indicated by supernova observations [@ref:riess], [@ref:perl], [@ref:riess2]. (Recall that positive acceleration corresponds to a negative deceleration parameter.) We will discuss in a later section the values the parameters in (\[eq:matacpar\]) must possess for this scenario. However, the emphasis in this paper will be on Lambda driven acceleration, the tachyon assuming the role of dark matter. Acceleration can also be studied by differentiating (\[eq:matenergy\]) to obtain $${\ddot{a}(t)}=-\frac{M_{0}}{{a^{2}(t)}}+\frac{\Lambda}{3}{a(t)}. \label{eq:matacceleq}$$ The above “force” equation nicely demonstrates the fact that positive $M_{0}$ tends to produce an attractive force whereas positive $\Lambda$ produces a negative or repulsive force. The tachyonic effect is inherent in $M_{0}$ via (\[eq:matmass\]). The fate of the universe is governed by the scale factor, ${a(t)}$. In general, the equation for ${a(t)}$ cannot be solved explicitly. Here we use effective potential techniques to study properties of ${a(t)}$. Figure \[fig:two\] shows plots of the effective potentials due to the matter fields (grey line indicating the function $-M_{0}/{a(t)}$) and the cosmological term (dashed line indicating the function $-\frac{\Lambda}{6{a^{2}(t)}}$) as well as the sum of the two (solid) for various signs of $M_{0}$ and $\Lambda$. ![[Effective potentials in the matter phase. Dashed lines denote cosmological potential, $-\frac{\Lambda}{6{a^{2}(t)}}$, grey lines denote matter potential, $-\frac{M_{0}}{{a(t)}}$, and solid black lines denote net effective potential. Re-collapse is possible for scenarios (a), (b) and (d). Parameters to produce the graphs are $|\Lambda|=0.1$ and $|M_{0}|=1$ although the qualitative picture remains unchanged for other values.]{}[]{data-label="fig:two"}](eff_pot3.eps) From the figure it can be seen that for $C > 0$ situations depicted in figures \[fig:two\] (a) and (d) allow solutions which re-collapse even for $\Lambda=0$. For $C < 0$, the configurations in figures \[fig:two\] (b) and (d) allow for re-collapse (there are no re-collapse solutions for $C<0$ if $\Lambda = 0$). In 2 (c) re-collapse is impossible. It is of interest to study the geometry of spatial sections generated by this solution. As mentioned in the introduction, the space-like hyper-surfaces are not surfaces of constant curvature. As well, if we consider the global picture, then the parameters discussed can also affect the topology of the universe. Spatial hyper-surfaces at $t=t_{0}$ possess the line element (equation \[eq:submetric\] except for a scale factor) $$d\sigma^{2}= \frac{d r^2}{1-C r^{2}+eb} + r^2\,d\theta^{2} + r^2\sin^{2}\theta\,d\phi^{2}. \label{eq:mat3dsub}$$ Although (\[eq:mat3dsub\]) bears a close resemblance to the standard FLRW line element, they are not equivalent. The orthonormal Riemann components for (\[eq:mat3dsub\]) yield: $$\begin{aligned} \tilde{R}_{\hat{ r}\hat{\theta}\hat{ r}\hat{\theta}}=&C, \label{eq:orthriemmat1}\\ \tilde{R}_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}}=& C -\frac{eb}{ r^{2}} \label{eq:orthriemmat2},\end{aligned}$$ and therefore, for $e\neq0$, the three dimensional hyper-surfaces are not of constant curvature. For small $e$ deviations are minimal and for $e=0$, the hyper-surface is of constant curvature $C$. The inhomogeneity parameter (\[eq:inhomgen\]) is calculated to be $$I( r)=\frac{{C r}^{2}}{{C r}^{2}-eb}. \label{eq:matinhom}$$ If we wish to treat the solutions as global, then the spatial topology may be studied. The two dimensional sub-manifold ($\theta=\pi/2$) of the three-metric (\[eq:mat3dsub\]) possesses line element $$d\sigma_{(2)}^{2}=\frac{d r^2}{1-C r^{2}+eb}+ r^{2}\,d\phi^{2}.$$ Transforming to the arc-length parameter, $l$, along a $ r$-coordinate curve, one can obtain $$d\sigma_{(2)}^{2}=\frac{\left[ r^{\prime}(l)\right]^{2}}{A-C r^{2}(l)}\, d l^2+ r^{2}(l)\,d\phi^{2} =dl^{2}+ r^{2}(l)\,d\phi^{2}, \label{eq:twoarc}$$ with $A:=1+eb\,$ and $\left[ r^{\prime}(l)\right]^{2}=A-C r^{2}(l)$. Integrating for $ r(l)>0$, the following solutions are derived: $$r(l)=\left\{ \begin{array}{lll} \sqrt{\frac{A}{2C}}\sin\left[\sqrt{C}(l-l_{0})\right] & \mbox{ for } & C > 0\; , \; A > 0 \\ \sqrt{\frac{A}{2|C|}} \sinh\left[\sqrt{|C|}\left(l-l_{0}\right) \right] & \mbox{ for }& C < 0 \; , \; A > 0 \\ \sqrt{\frac{A}{2}}\left(l-l_{0}\right) & \mbox{ for }& C = 0 \; , \; A > 0 \; , \end{array} \right.$$ ($l_{0}$ is a constant arising from integration). It is clear from (\[eq:twoarc\]) that $l$ is a geodesic coordinate. From the periodicity of the sine function, it may be seen that the two conjugate points on the radial geodesic congruences are given by $\, r(l_{0})= r(l_{0}+\pi/\sqrt{C}) =0$. Thus, one concludes that spatially closed universes correspond only to $C > 0$. Radiation dominated era ----------------------- Here we study the next major phase in the evolution of the universe. The radiation dominated phase is characterized by the relativistic fluid equation of state ${\mu(t, r)}=3{p(t, r)}$. Using this along with (\[eq:einstone\]) and (\[eq:einsttwo\]) yields: $$\frac{1}{2}\left[{a^{2}(t)}\right]^{\cdot\cdot} -\frac{2}{3}\Lambda{a^{2}(t)}= -\epsilon+\frac{5e}{12 r^{7/5}}\left[ r^{2/5}{g( r)}\right]^{\prime} =-C,$$ again $C$ is a separation constant. The equation for ${g( r)}$ is satisfied by $$e{g( r)}=\left(\epsilon-C\right) r^{2}+\frac{eb}{ r^{2/5}}. \label{eq:radg}$$ Solving for the scale factor, ${a(t)}$, one obtains: $${a^{2}(t)}=\left\{ \begin{array}{lll} -Ct^{2}+\kappa_{1}t+\kappa_{2} & \mbox{ for } & \Lambda = 0 \\ \kappa_{1}e^{2\sqrt{\frac{\Lambda}{3}}t} + \kappa_{2}e^{-2\sqrt{\frac{\Lambda}{3}}t} +\frac{3C}{2\Lambda}& \mbox{ for }& \Lambda > 0 \\ \kappa_{1}\sin\left[2\sqrt{\frac{-\Lambda}{3}}t\right] + \kappa_{2}\cos\left[2\sqrt{\frac{-\Lambda}{3}}t\right]+ \frac{3C}{2\Lambda} & \mbox{ for }& \Lambda < 0 \; . \end{array} \right.$$ Here, $\kappa_{1}$ and $\kappa_{2}$ are arbitrary constants of integration. However, the domain of $t$ and the signs of these constants must respect ${a^{2}(t)}>0$. The densities and pressures are given by $$\begin{aligned} 8\pi{\mu(t, r)}=&-\Lambda +3\left[\frac{{\dot{a}(t)}}{{a(t)}}\right]^{2} +\frac{3}{{a^{2}(t)}}\left[C-\frac{eb}{5 r^{12/5}} \right], \\ 8\pi{p(t, r)}=&\frac{8\pi}{3}{\mu(t, r)}, \\ 8\pi{\alpha(t, r)}=&\frac{6eb}{5 r^{12/5}{a^{2}(t)}}.\end{aligned}$$ The Hubble parameter is calculated to be: $$H(t)=\left\{ \begin{array}{lll} \frac{1}{2}\frac{\kappa_{1}-2Ct}{-Ct^{2}+\kappa_{1}t+\kappa_{2}} & \mbox{ for } & \Lambda = 0 \\ \sqrt{\frac{\Lambda}{3}}\left[ \frac{\kappa_{1}\exp\left(2\sqrt{\frac{\Lambda}{3}}t\right) - \kappa_{2}\exp\left(-2\sqrt{\frac{\Lambda}{3}}t\right)}{\kappa_{1}e^{2\sqrt{\frac{\Lambda}{3}}t} + \kappa_{2}e^{-2\sqrt{\frac{\Lambda}{3}}t}+3C/\Lambda} \right] & \mbox{ for }& \Lambda > 0 \\ \sqrt{\frac{-\Lambda}{3}} \left[\frac{\kappa_{1}\cos\left(2\sqrt{\frac{-\Lambda}{3}}t\right) - \kappa_{2}\sin\left(2\sqrt{\frac{-\Lambda}{3}}t\right)} {\kappa_{1}\sin\left(2\sqrt{\frac{-\Lambda}{3}}t\right) + \kappa_{2}\cos\left(2\sqrt{\frac{-\Lambda}{3}}t\right)+3C/\Lambda}\right] & \mbox{ for }& \Lambda < 0 \; . \end{array} \right.$$ The deceleration parameter is provided by: $$q(t)=\left\{ \begin{array}{lll} \frac{4C\kappa_{2}+\kappa_{1}^{2}}{\left(\kappa_{1}-2Ct\right)^{2}} & \mbox{ for } & \Lambda = 0 \nonumber \\ -\frac{6\kappa_{1}\kappa_{2}\Lambda + 3C\left[ \kappa_{1} \exp\left(2\sqrt{\frac{\Lambda}{3}}t\right) + \kappa_{2} \exp\left(-2\sqrt{\frac{\Lambda}{3}}t\right)\right] +\Lambda\kappa_{1}^{2}\exp\left(4\frac{\Lambda}{3}t\right) +\Lambda\kappa_{2}^{2}\exp\left(-4\frac{\Lambda}{3}t\right)}{\Lambda \left[\kappa_{1}e^{2\sqrt{\frac{\Lambda}{3}}t} - \kappa_{2}e^{-2\sqrt{\frac{\Lambda}{3}}t}\right]^{2}} & \mbox{ for }& \Lambda > 0 \nonumber \\ \frac{\kappa_{1}^{2}+\kappa_{2}^{2} + \left[ \kappa_{1}\sin\left(2\sqrt{\frac{-\Lambda}{3}}t\right) + \kappa_{2}\cos\left(2\sqrt{\frac{-\Lambda}{3}}t\right) \right]^{2} +\left(\frac{3C}{\Lambda}\right) \left[ \kappa_{1}\sin\left(2\sqrt{\frac{-\Lambda}{3}}t\right) + \kappa_{2}\cos\left(2\sqrt{\frac{-\Lambda}{3}}t\right) \right]} {\left[\kappa_{1}\sin\left(2\sqrt{\frac{-\Lambda}{3}}t\right) + \kappa_{2}\cos\left(2\sqrt{\frac{-\Lambda}{3}}t\right)\right]^{2}} & \mbox{ for }& \Lambda < 0 \; . \nonumber \end{array} \right. \nonumber$$ Finally, the inhomogeneity parameter of the spatial hyper-surfaces is given by: $$I( r)=\frac{\left( 5\,{ C r}^{{\frac {12}{5}}}+eb \right)}{5\left( {C r}^{{\frac {12}{5}}}-eb \right)}.$$ The presence of the tachyon affects the spatial geometry. Here spatial geometry is again studied via the arc-length parameter $l$. The geodesic equation along an $r$-coordinate curve yields $$\begin{aligned} \left[\frac{d r(l)}{dl}\right]^{2} + 2V( r(l)) =1 \label{eq:radspatgeod} \\ V( r):=\frac{1}{2}C r^{2} -\frac{eb}{2 r^{2/5}}.\end{aligned}$$ One may analyze (\[eq:radspatgeod\]) via similar “effective potential” techniques as in the dynamics. For positive $C$, $ r(l)$ is bounded regardless of the sign of the tachyonic potential, $eb$ (spatially closed universe). For negative $C$, all allowed solutions are unbounded (spatially open universe). For $C=0$, the spatial universe is also open. Inflationary era ---------------- We now investigate the inflationary phase. It is generally believed that the universe experienced tremendous expansion over a short period of time. There are many physical reasons for believing in this scenario and an excellent review may be found in [@ref:brandeninfl]. Some studies of the scalar tachyon’s relevance to inflation may be found in [@ref:fks], [@ref:tachcos3], [@ref:tachcos4], [@ref:tachcos6]. In the scenario presented here, the tachyon does not play the role of the inflaton. However, the inflationary phase provides one possible mechanism for the transition from high tachyon concentration to low concentration. Inflationary scenarios are generally supported by the equation of state ${\mu(t, r)}+{p(t, r)}=0$. This linear combination of (\[eq:einstone\]) and (\[eq:einsttwo\]) yields: $${a^{2}(t)}\left[\ln|{a(t)}|\right]^{\cdot\cdot}=\epsilon-\frac{e}{4 r^{3}} \left[ r^{2}{g( r)}\right]^{\prime} = C.$$ The solution for ${g( r)}$ is given by $$eg( r)= \left(\epsilon-C\right) r^{2} + \frac{eb}{ r^{2}}\,. \label{eq:infg}$$ As well, the following modes are found for ${a(t)}$: $$\begin{aligned} {a(t)}=&\beta_{0} e^{H t}& \mbox{for } C=0,\;\;\; \beta_{0}>0, \label{eq:infb1} \\ {a(t)}=&\sqrt{\frac{C}{\beta_{1}}}\cosh\left[\sqrt{\beta_{1}}(t-t_{0})\right] &\mbox{for } C>0,\;\;\; \beta_{1}>0, \label{eq:infb2} \\ {a(t)}=&\sqrt{\frac{|C|}{\beta_{2}}}\sinh\left[\sqrt{\beta_{2}}(t-t_{0})\right] &\mbox{for } C<0,\;\;\; \beta_{2}>0, \label{eq:infb5} \\ {a(t)}=&\sqrt{\frac{C}{\beta_{2}}} \sin\left[\sqrt{|\beta_{2}|}(t-t_{0})\right] & \mbox{for } C<0,\;\;\; \beta_{2}<0, \label{eq:infb4} \\ {a(t)}=&\sqrt{|C|}(t-t_{0}) &\mbox{for } C<0,\;\;\; \beta_{2}=0. \label{eq:infb3}\end{aligned}$$ Here $\beta_{0}$, $\beta_{1}$, $\beta_{2}$ and $H$ are constants of integration. The Hubble factor is given by $$H(t)=\left\{ \begin{array}{lllll} H=\mbox{constant} & \mbox{ for } & C = 0, \;\;\; \beta_{0}>0 \\ \sqrt{\beta_{1}}\tanh\left[\sqrt{\beta_{1}}(t-t_{0})\right] & \mbox{ for } & C > 0, \;\;\; \beta_{1}>0 \\ \sqrt{\beta_{2}}\coth\left[\sqrt{\beta_{2}}(t-t_{0})\right] & \mbox{ for } & C < 0, \;\;\; \beta_{2} >0 \\ \sqrt{|\beta_{2}|}\cot\left[\sqrt{|\beta_{2}|}(t-t_{0})\right] & \mbox{ for } & C < 0, \;\;\; \beta_{2} < 0 \\ (t-t_{0})^{-1} & \mbox{ for } & C < 0, \;\;\; \beta_{2}=0 . \end{array} \right.$$ The corresponding deceleration parameter $$q(t)=\left\{ \begin{array}{lllll} -1 & \mbox{ for } & C = 0, \;\;\; \beta_{0}>0 \\ -\coth^{2}\left[\sqrt{\beta_{1}}(t-t_{0})\right] & \mbox{ for } & C > 0, \;\;\; \beta_{1}>0 \\ -\tanh^{2}\left[\sqrt{\beta_{2}}(t-t_{0})\right] & \mbox{ for } & C < 0, \;\;\; \beta_{2} >0 \\ \tan^{2}\left[\sqrt{|\beta_{2}|}(t-t_{0})\right] & \mbox{ for } & C < 0, \;\;\; \beta_{2} < 0 \\ 0 & \mbox{ for } & C < 0, \;\;\; \beta_{2}=0\, . \end{array} \right.$$ The source terms are: $$\begin{aligned} 8\pi{\mu(t, r)}=-8\pi{p(t, r)}=&\frac{1}{{a^{2}(t)}}\left[3C+{\dot{a}^{2}(t)}+\frac{eb}{ r^{4}}\right] -\Lambda, \\ 8\pi{\alpha(t, r)}=&\frac{2eb}{ r^{4}{a^{2}(t)}}\end{aligned}$$ from which it can be seen that the tachyon is naturally diluted by the presence of a scale factor which increases rapidly. The fluid density and pressures, however, need not dilute as their expressions contain terms proportional to ${\dot{a}(t)}/{a(t)}$ which may tend to constant (as in (\[eq:infb1\]) and (\[eq:infb2\])) or increase (as in (\[eq:infb4\])). We demonstrate several scenarios next. In the figure \[fig:infex\]. The graphs on the left represent the scenario with $C>0$ (“closed inflation”) whereas the graphs on the right represent the $C = 0$ scenario (“flat inflation”) at some fixed value of $r$. Both scenarios are with $\Lambda=0$ so that ${\mu(t, r)}$ represents the energy density of all fields (dominated by the inflaton, with minor contributions from other fields) save for the tachyon, whose density is given by ${\alpha(t, r)}$ in graphs a) and b). The space-time coordinates possess units of $10^{-24}$ metres here. Note that for an acceptable interval of inflation (approx a few times $10^{-32}$s), we have, in the $C=0$ scenario, a dramatic decrease in the density of the tachyon field but not the necessarily the inflaton field. In this scenario, inflation must terminate by the standard phase transition of the inflaton field. At the end of inflation, the tachyon density is much smaller than the densities of the other matter which will be dominated by radiation leading to the radiation era. In the $C>0$ scenario, both ${\alpha(t, r)}$ *and* ${\mu(t, r)}$ vary with time although ${\mu(t, r)}$ (initially primarily the inflaton) approaches a constant value while ${\alpha(t, r)}$ decreases as $a^{-2}(t)$ (this is not obvious from graph c, however it can easily be seen, by examining the analytic expressions for ${\alpha(t, r)}$ and ${\mu(t, r)}$ with $C >0$, that ${\mu(t, r)}$ possesses a term which does not decay with time whereas ${\alpha(t, r)}$ does not possess such a term). It is a simple matter to show that parameters exist to produce an increase in the expansion factor by many orders of magnitude. The figures \[fig:infex\] show this although their time axes have been truncated to show the behaviour of $a(t)$ more clearly. ![[Inflationary scenarios: graphs on the left represent a $C > 0$ model whereas graphs on the right represent a $C=0$ model. Space-time coordinates are measured in $10^{-24}$m and densities are scaled accordingly. Graphs a) and b) represent the evolution of ${\alpha(t, r)}$, graphs c) and d) the evolution of ${\mu(t, r)}$ and graphs e) and f) the increase in the scale factor (see text for explanation).]{}[]{data-label="fig:infex"}](infpics2.eps) As inflation progresses, both models yield a tachyon density whose value decreases to a smaller value than ${\mu(t, r)}$. This value can be made small enough as not to intertfere with the physical processes that must have occured during the radiation dominated era. The spatial geometry is again studied using the arc-length parameter, $l$, as in the matter dominated era. In this case $$r(l)=\left\{ \begin{array}{lll} \frac{1}{\sqrt{2C}}\left\{\left(1+4Ceb\right)^{1/2} \sin\left[2\sqrt{C}\left(l-l_{0}\right) \right] +1\right\}^{1/2} & \mbox{ for } & C > 0 \\ \frac{1}{\sqrt{2|C|}} \left\{\left(4|C|eb-1\right)^{1/2} \sinh\left[2\sqrt{|C|}\left(l-l_{0}\right) \right] -1\right\}^{1/2} & \mbox{ for }& C < 0 \\ \sqrt{\left(l-l_{0}\right)^{2}-eb} & \mbox{ for }& C = 0 \; . \end{array} \right.$$ Here we see that, from periodicity of the sine function, $C>0$ again can yield the closed spatial universe. Finally, the inhomogeneity parameter in this phase is $$I( r)={\frac {C{ r}^{4}+eb}{C{ r}^{4}-eb}}.$$ An extension to variable Lambda cosmology {#sec:varlam} ========================================= Recent experiments suggest that the universe is presently in an accelerating phase. If one accepts that the net mass of the universe is positive, then the present acceleration can be explained by the figure 2a alone. Thus, the choice $\Lambda > 0$ must be made. In case $\Lambda >0$ is a constant, re-collapse is incompatible with acceleration. Therefore, we consider the generalization of the previous sections to the variable $\Lambda(t)$ case. This scenario has relevance in light of recent models (mainly based on supergravity considerations) which predict that the dark energy decreases and that the universe re-collapses within a time comparable to the present age of the universe (see [@ref:kalinde] and references therein). Time dependent fields with equation of state $p(t)\approx - \mu(t)$ have been employed in the literature to explain certain evolutionary periods requiring positive acceleration. There are also compelling reasons from particle physics for treating the cosmological term as a dynamic quantity (see [@ref:varlam1], [@ref:varlam2], [@ref:varlam3] [@ref:varlam4] and references therein). The field equations (\[eq:einstone\]), (\[eq:einsttwo\]), (\[eq:einstthree\]) formally remain the same with the exception that $\Lambda = \Lambda(t)$. However, the conservation equation (\[eq:cons1\]) needs to be augmented by an additional term. The definitions of the matter, radiation and inflationary phases are retained exactly as before. Therefore, the equations for ${g( r)}$ in all three phases remain intact. The solutions for ${g( r)}$ can be summarized as: $$e{g( r)}=(\epsilon-C) r^{2} +\frac{eb}{ r^{\nu}},$$ with $$\nu=\left\{ \begin{array}{lll} 0 & \mbox{ for the matter phase} \\ 2/5 & \mbox{ for the radiation phase } \\ 2 & \mbox{ for the inflationary phase }. \end{array} \right.$$ The three-geometries are specified as $$d\sigma^{2}=\frac{d r^{2}}{1-C r^{2}+\frac{eb}{ r^{\nu}}} + r^{2}\,d\theta^{2} + r^{2}\sin^{2}\theta \, d\phi^{2},$$ $$\begin{aligned} \tilde{R}_{\hat{ r}\hat{\theta}\hat{ r}\hat{\theta}} =&C +\frac{\nu eb}{2 r^{\nu+2}},\label{eq:genorthriem1} \\ \tilde{R}_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}} =&C -\frac{ eb}{2 r^{\nu+2}}, \label{eq:genorthriem2}\end{aligned}$$ from which one obtains $$I( r)=\frac{2C r^{\nu+2}+\nu eb}{2\left[C r^{\nu+2} -eb\right]} .$$ The field equations are $$\begin{aligned} 8\pi{\mu(t, r)}+ \Lambda(t)&=\frac{1}{{a^{2}(t)}}\left\{3 \left[{\dot{a}^{2}(t)}+C\right] + \frac{\left(\nu-1\right)eb}{ r^{\nu+2}}\right\}, \label{eq:varlam1} \\ 8\pi{p(t, r)}-\Lambda(t) &=-\frac{1}{{a^{2}(t)}} \left\{2{a(t)}{\ddot{a}(t)}+{\dot{a}^{2}(t)}+C +\frac{\nu eb}{2 r^{\nu+2}} \right\}. \label{eq:varlam2} \\ 8\pi{\alpha(t, r)}&=\frac{\left(\nu+2\right)eb}{2 r^{\nu+2}{a^{2}(t)}} \label{eq:varlam3},\end{aligned}$$ with conservation laws: $$\begin{aligned} {\mu(t, r)}_{,t}+\frac{\dot{\Lambda}(t)}{8\pi} +H(t) \left\{3\left[{\mu(t, r)}+{p(t, r)}\right] +{\alpha(t, r)}\right\}&=0, \\ \left[{p(t, r)}+{\alpha(t, r)}\right]_{,r} +\frac{\left(\nu+2\right)eb}{8\pi r^{\nu+3}{a^{2}(t)}}&=0.\end{aligned}$$ The dynamical quantities are given by $${\ddot{a}(t)}=-\left\{\frac{4\pi}{3} \left[{\mu(t, r)}+{\alpha(t, r)}+3{p(t, r)}\right] {a(t)}\right\} +\frac{1}{3}\Lambda(t){a(t)}. \label{eq:varlamadotdot}$$ The Hubble parameter and the deceleration parameter are furnished as $$\begin{aligned} H^{2}(t)=&\frac{8\pi}{3}{\mu(t, r)}+\frac{\Lambda(t)}{3} -\frac{C}{{a^{2}(t)}} -\frac{(\nu-1)eb}{3 r^{\nu+2}{a^{2}(t)}} , \\ H^{2}(t)\left[2q(t)-1\right] =&8\pi{p(t, r)}-\Lambda(t) +\frac{1}{{a^{2}(t)}}\left[C+ \frac{\nu eb}{2 r^{\nu+2}}\right].\end{aligned}$$ As far as experimental evidences are concerned, the matter domain is the most relevant. Therefore, the maximum information possible will be elicited from the field equations for that domain. Setting ${p(t, r)}= \nu = 0$ and integrating (\[eq:varlam2\]) yields the “energy” conservation equation: $$\frac{1}{2}{\dot{a}^{2}(t)}-\frac{1}{6{a(t)}}\int_{t_{2}}^{t} \Lambda(\tau) \,d[a^{3}(\tau)] -\frac{M_{0}}{{a(t)}} = -\frac{C}{2} \label{eq:varlamenergy}.$$ Again $M_{0}$ arises from the integration and represents the total effective mass of the universe. Furthermore, $t_{2}$ is another constant representing the beginning of the matter phase. In terms of the matter fields, the mass is: $$\begin{aligned} M_{0}=&{\mu(t, r)}\left[\frac{4}{3}\pi a^{3}(t)\right] +\frac{eb}{6 r^{2}}{a(t)}+\frac{a^{3}(t)}{6} \Lambda(t) -\frac{1}{6}\int_{t_{0}}^{t} \Lambda(\tau)\,d\left[a^{3}(\tau)\right], \label{eq:varlammass} \\ =&\left[\mu(t_{2},r)+\alpha(t_{2},r)\right] \left[\frac{4}{3}\pi a^{3}(t_{2})\right] +\frac{a^{3}(t_{2})}{6}\Lambda(t_{2}) .\end{aligned}$$ Here, a possible interpretation is that the first term represents the total mass of observed matter (“normal” matter), the second term the tachyonic contribution to the dark matter (non-baryonic mass for pressure, “dark energy” for tension) and the third gives rise to potential “dark energy” responsible for acceleration. The acceleration is provided by (\[eq:varlamadotdot\]) and (\[eq:varlamenergy\]) as: $$\begin{aligned} {\ddot{a}(t)}=&-\frac{M_{0}}{{a^{2}(t)}} +\frac{1}{2}\Lambda(t){a(t)}- \frac{1}{6{a^{2}(t)}}\int_{t_{2}}^{t}\Lambda(\tau) \,d\left[a^{3}(\tau)\right] \nonumber \\ =&-\left\{ \frac{\left[{\mu(t, r)}+{\alpha(t, r)}\right] \left[\frac{4}{3}\pi a^{3}(t) \right]}{{a^{2}(t)}}\right\} +\frac{1}{3}\Lambda(t){a(t)}.\end{aligned}$$ In case ${\mu(t, r)}> 0$, $eb > 0$ and $\Lambda(t) > 0$, the above terms on the right hand side produce a combination of both attractive and repulsive forces. The Hubble parameter in the matter domain is provided by: $$\begin{aligned} H^{2}(t)=&\frac{8}{3}\pi\left[{\mu(t, r)}+{\alpha(t, r)}\right] +\frac{\Lambda(t)}{3}-\frac{C}{{a^{2}(t)}} \label{eq:hsqvarlam} \\ =&\frac{2M_{0}}{a^{3}(t)}-\frac{C}{{a^{2}(t)}} +\frac{1}{3 a^{3}(t)} \int_{t_{2}}^{t}\Lambda(\tau)\, d\left[a^{3}(\tau)\right] .\end{aligned}$$ The deceleration parameter in this domain is $$H^{2}(t)\left[2q(t)-1\right]= \frac{C}{{a^{2}(t)}} -\Lambda(t). \label{eq:varlammatdec}$$ It is clear from (\[eq:varlammatdec\]) that $q$ can be positive, negative or zero depending on the values of $C$ and $\Lambda(t)$. A specific model will be proposed which accommodates a spatially closed, re-collapsing universe with an accelerating period in the matter domain. In the cosmology presented here, this may be realized by setting $\Lambda(t_{2})> 0$, $C >0$ and $eb >0$. Observations indicate that $C$ has a value very close to zero. A $C\leq 0$ universe has cubic divergent volume at all times save the origin when the volume is zero. If, however, $C$ is extremely small yet positive, one has finite large volume in the matter domain without contradicting observations. The time periods for inflation, radiation and matter dominated eras are $[\epsilon, t_{1}]$, $[t_{1},t_{2}]$ and $[t_{2},T/2]$ respectively. The time $T/2$ indicates the initiation of re-collapse and thus represents the half-life of the universe. Of course, the boundaries separating the domains are not sharp as we have indicated and therefore the above simply represents a rough guideline. A possible evolutionary scenario is depicted in figure \[fig:evolution\]. Here we plot both the scale factor ${a(t)}$ and cosmological term $\Lambda(t)$ as a function of cosmic time. The scale factor increases greatly during the inflationary phase (in the model presented in the figure, the inflation is driven by some matter field, not the cosmological term). This is followed by a decelerating phase and, near the present time, a period of acceleration follows. This scenario is based on the tachyonic positive pressure model and therefore this acceleration is $\Lambda$ driven. To allow for re-collapse, the cosmological term decays (starting at $t=t_{3}$) so that ${\ddot{a}(t)}$ becomes negative causing deceleration and eventual re-collapse. The figure is symmetric about $T/2$. Furthermore, one may have a cyclic universe where the scenario repeats after the “big crunch”. ![[A possible scenario for the evolution of the universe. The present time is denoted by $t=t_{0}$ and the half-life of the universe denoted by $t=T/2$. The solid line represents the qualitative evolution of the scale factor and the dashed line the cosmological term. ]{}[]{data-label="fig:evolution"}](evolve3.eps) A suitable $\Lambda(t)$ function may be defined by $$\Lambda(t)=\left\{ \begin{array}{llll} \Lambda_{0}\;\;\;\mbox{a positive constant} & \mbox{ for } & 0 \leq t < t_{1} \\ \Lambda_{0} & \mbox{ for }& t_{1} \leq t < t_{2} \\ \Lambda_{0} & \mbox{ for }& t_{2} \leq t < t_{3} \\ \Lambda_{0} -\epsilon_{2}(t-t_{3})^{2} + \epsilon_{4}(t-t_{4})^{4} & \mbox{ for }& t_{3} < t < T/2 \; , \end{array} \right.$$ with $\epsilon_{2}>0$ and $\epsilon_{4}>0$. The expansion factor for this case is given by: $$a(t)=\left\{ \begin{array}{lll} \sqrt{\frac{C}{\beta_{0}}} \cosh(\sqrt{\beta_{0}}t) \;\; & \mbox{ for } 0 \leq t < t_{1} \\ \sqrt{\frac{3C}{2\Lambda_{0}}+\kappa_{1} \exp\left[2\sqrt{\frac{\Lambda_{0}}{3}}t\right] + \kappa_{2} \exp\left[-2\sqrt{\frac{\Lambda_{0}}{3}}t\right]} \;\; & \mbox{ for } t_{1} \leq t < t_{2} \label{eq:avarlam} \\ f^{-1}(t-t_{2}) \;\;& \mbox{ for } t_{2} \leq t < t_{3} \; , \end{array} \right.$$ $f(a):= \int_{a(t_{2})}^{a(t)} \frac{dx} {\sqrt{\frac{\Lambda_{0}}{3}x^{2} +\left[2M_{0} -\frac{\Lambda_{0}}{3}a^{3}(t_{2})\right]x^{-1}-C}}$. There are enough arbitrary parameters in (\[eq:avarlam\]) so that ${a(t)}$ can be joined smoothly in the three phases if one wishes to enforce sharp boundaries between the phases. The function $a(t)$ satisfies the formidable integro-differential equation $$\left[{\dot{a}(t)}\right]^{2} -\frac{\Lambda_{0}}{3}{a^{2}(t)}-\frac{2M_{0}-\Lambda_{0} a^{3}(t)}{{a(t)}} +\frac{1}{3{a(t)}} \int_{t_{3}}^{t}\left(\epsilon_{2}\tau^{2}- \epsilon_{4} \tau^{4} \right)f\left[a^{3}(\tau)\right] =0$$ in the interval $t_{3} \leq t < T/2$. The above equation is too difficult to solve analytically at this stage. The spatial geometry for $C > 0$ is governed by $ r(l)$ as $$d\sigma^{2}=dl^{2}+ r^{2}(l) \left(d\theta^{2}+\sin^{2}\theta\, d\phi^{2} \right),$$ $$r(l)=\left\{ \begin{array}{lll} \frac{1}{\sqrt{2C}}\left\{\left(1+4Ceb\right)^{1/2} \sin\left[2\sqrt{C}\left(l-l_{0}\right) \right] +1\right\}^{1/2} & \mbox{ for inflation} \\ F^{-1}\left(l-l_{0}\right) & \mbox{ for radiation} \\ \sqrt{\frac{1+eb}{2C}} \sin\left[\sqrt{C}\left(l-l_{0}\right) \right] & \mbox{ for matter} \; . \end{array} \right.$$ Here, $F( r):=\int\frac{d r}{\sqrt{1-C r^{2}+eb/ r^{2/5}}}\:$. By previous discussions, in all phases the physical universes are closed. Moreover, the total volume corresponding to the three dimensional spatial sub-manifold in the matter phase is given by $$\frac{2\pi\left(1+eb\right)}{C} \left[a(t_{0})\right]^{3}\int_{l_{0}}^{l_{0}+\pi/\sqrt{C}} \sin^{2}\left[\sqrt{C}\left(l-l_{0}\right)\right]\, dl =\frac{\pi^{2}\left(1+eb\right)}{C^{3/2}}\left[a^{3}(t_{0})\right],$$ (Note that in the limit $C\rightarrow 0_{+}$, the above volume diverges). We now wish to address the singularity at $r=0$. The two dimensional geometries (restricted to $\theta=\pi/2$) yield: $$d\sigma_{2}^{2}=\frac{dr^{2}}{1-C r^{2}+\frac{eb}{r^{\nu}}} +r^{2}\,d\phi^{2}.$$ These two-dimensional surfaces embedded in a three-dimensional Euclidean space possess the following Gaussian curvatures: $$K(r,\phi)=\left\{ \begin{array}{lll} C+\frac{eb}{r^{4}} \;\; > 0 & \mbox{ for inflation} \\ C+ \frac{eb}{5r^{12/5}}\;\; > 0 & \mbox{ for radiation} \\ C \;\; > 0 & \mbox{ for matter} \; . \end{array} \right.$$ In the matter domain, the surface is *locally isometric to a spherical surface of radius* $1/\sqrt{C}$. However, the original three dimensional spaces in the equation (\[eq:genorthriem2\]) all exhibit a singularity at the limit $r\rightarrow 0_{+}$. Therefore, some possible global pictures for these three dimensional spaces are provided in figure \[fig:globals\]. ![[Possible global pictures for spatial sections in the inflationary (a, b), radiation (c, d) and matter (e, f) dominated eras. The diagrams on the left include the points corresponding to $r=0$. The diagrams on the right have a neighborhood about $r=0$ removed and the boundaries identified. Note that in the matter domain the surfaces are isometric to spheres yet singularities still exist at $r=0$ as indicated by the “hairs” in diagram e.]{}[]{data-label="fig:globals"}](globals.eps) In the figure \[fig:globals\], one of the angles is suppressed so that latitudinal lines represent two-spheres. Two possible scenarios exist; the figures on the left represent the spatial manifold for inflation, radiation dominated and matter dominated eras which include $r=0$ (the left and right points in each figure). The figures on the right have the domains in the neighborhood of $r=0$ excised. The left and right boundaries are therefore identified. Note that as the evolution progresses, the anisotropy of the spatial sections diminishes yielding a sphere in the matter domain. This is therefore quite compatible with observation. The poles of the sphere, however, are singular or must be excised. The singularity appears to be “soft” in that it is of the conical type. Also, the equations, being local, are valid in a domain $r_{1} < r < r_{2}$ which need not include $r=0$. It is likely that such a singularity would be absent in a quantum theory of gravity which would be manifest at high energies. Compatibility with current observations ======================================= Current observations indicate a universe which is approximately 5% baryonic matter, 20% non-baryonic matter and 75% “dark energy” which is responsible for the recent acceleration phase. The “directly” measurable quantities in cosmology are ${\mu(t, r)}$, $H$ and $q_{0}$. Roughly, in the present epoch (and, as we are dealing with an inhomogeneous universe, in our neighborhood of the universe) these quantities possess the following approximate values: $$\begin{aligned} \mu(t_{0}, r_{0}) \approx& 1.6\times 10^{-56}\, \mbox{m}^{-2}, \label{eq:nummu} \\ H^{2}:=&\left[\frac{{\dot{a}(t)}}{{a(t)}}\right]^{2} \approx 7.3\times 10^{-53}\, \mbox{m}^{-2}, \label{eq:nuH} \\ q_{0} \approx& -0.4\;\;\;. \label{eq:numq}\end{aligned}$$ Here $t_{0}$ and $ r_{0}$ are the current time and position respectively. The deceleration equation (\[eq:matacpar\]) provides a relationship (using the above parameters along with (\[eq:matalpha\])) between $\Lambda$ and ${\alpha(t, r)}$ (we assume that any time variation in $\Lambda$ can be ignored): $$\alpha(t_{0}, r_{0}) = \left( 8.0\times 10^{-2} \Lambda -6.9\times 10^{-54}\right) \mbox{m}^{-2}\;. \label{eq:alphalambda}$$ If there is no cosmological constant, then the second term in this equation indicates the approximate value the tachyon tension must possess in our region of the universe to drive the observed acceleration. If, on the other hand, the tachyon possesses positive pressure (contributing all or in part to the non-baryonic dark matter of the universe) then the acceleration is $\Lambda$ driven. In such a case $\alpha(t_{0}, r_{0})$ may take on the following values: $$0 \leq \alpha(t_{0}, r_{0}) \lesssim 4 \mu(t_{0}, r_{0}) \approx 6.4\times 10^{-56} \, \mbox{m}^{-2}.$$ The upper limit comes from noting the observational evidence that the dark matter contribution is approximately four times the baryonic contribution to the matter content. This sets a restriction on the cosmological constant to be of the order $$\Lambda = \mathcal{O}\left(10^{-52}\right)m^{-2} .$$ Alternately we may begin the analysis by using equation (\[eq:varlammatdec\]) and solving for $\Lambda$ (with the parameters quoted above) $$\Lambda_{0}=\frac{C}{{a^{2}(t)}}- H^{2}\left(2q-1\right) =\frac{C}{{a^{2}(t)}} +13.14\times 10^{-53}. \label{eq:lambdanot}$$ Also, by equation (\[eq:hsqvarlam\]), we may write $$\begin{aligned} 3H^{2}&=&8\pi\left[{\mu(t, r)}+{\alpha(t, r)}\right] +\Lambda_{0} -\frac{3C}{{a^{2}(t)}} \nonumber \\ 2.19\times10^{-52}&=&8\pi\left[{\mu(t, r)}+{\alpha(t, r)}\right] + 1.31\times 10^{-52}-2\frac{C}{{a^{2}(t)}} \label{eq:threehsq}\end{aligned}$$ (in the last equation (\[eq:lambdanot\]) has been used.) Isolating the ${\mu(t, r)}+{\alpha(t, r)}$ term and using in (\[eq:varlammass\]) yields $$\frac{M_{0}}{\frac{4\pi}{3}a^{3}(t_{0})} = 3.48\times10^{-54} +\frac{1}{8\pi} \left[\frac{C}{a^{2}(t_{0})}+1.3\times10^{-52} \right]\left[\frac{a(t_{2})}{a(t_{0})}\right]^{3} +\frac{C}{4\pi a^{2}(t_{0})}. \label{eq:univdens}$$ The $C/{a^{2}(t)}$ terms represent the present “radius”-squared of the universe. The left hand side of (\[eq:univdens\]) is an analogue of the present Newtonian density of the universe. The above equation is therefore useful in determining the radius of the universe given the density or vice-versa. Concluding remarks ================== This paper considers a simple cosmological model consisting of perfect fluid matter supplemented with a “tachyonic dust”. The perfect fluid, with positive mass density, makes up the ordinary matter as in the standard cosmology. The tachyonic dust term is a source of pressure which, interestingly, can increase the effective mass of the universe. In this case it could potentially be utilised as a source of dark matter although the clustering properties need to be studied. In case the tachyonic dust term is a source of tension, it may be responsible for the observed recent acceleration of the universe. This model provides the simplest pressure enhancing extension to the successful FLRW scenario. At late times, the solution generates a geometry compatible with FLRW. Acknowledgements {#acknowledgements .unnumbered} ================ The authors thank their home institutions for various support during the production of this work. Also, A. DeB. thanks the S.F.U. Mathematics department for kind hospitality. A. Das thanks Dr. S. Kloster for useful informal discussions. We thank the anonymous referees for helpful suggestions. [10]{} G. Lemaître, (1933) 51. Engl. trans. [*Gen. Rel. Grav.*]{} [**29**]{} (1997) 641. 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P. Matlock, R. C. Rashkov, K. S. Viswanathan and Y. Yang, (2002) 026004. M. Sami, P. Chingangbam and T. Qureshi, (2002) 043530. X. Li, D. Liu and J. Hao, . J. S. Bagla, H. K. Jassal and T. Padmanabhan, . D. Choudhury, D. Ghoshal, D. P. Jatkar and S. Panda, (2003) 231. C. Kim, H. B. Kim and Y. Kim, (2003) 111. G. W. Gibbons, . A. Das and D. Kay, (1988) 1031. A. G. Reiss, et al, (1998) 1009. S. Perlmutter, et al, (1999) 565. M. K. Mak and T. Harko, (2003) 393. B. V. Ivanov, (2002) 104011. K. Dev and M. Gleiser, (2002) 1793. A. G. Reiss, et al, (2001) 49. G. Lemaitré, (1927) 49. A. S. Eddington, (1930) 668. R. Brandenberger, Kish Island, Iran, (Kluwer, Dordrecht, 2000). R. Kallosh and A. Linde, (2003) 002. P. G. Bergmann, (1968) 25. R. V. Wagoner, (1970) 3209. A. D. Linde, (1974) 183. J. M. Overduin and F. I. Cooperstock, (1998) 043506. [^1]: e-mail: [email protected]

--- abstract: 'We prove that the set of maps which exhibit distributional chaos of type 1 (DC1) is $C^0$-dense in the space of continuous self-maps of given any compact topological manifold (possibly with boundary).' address: 'Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro, Tokyo 153-8914, Japan' author: - Noriaki Kawaguchi title: 'Distributionally chaotic maps are $C^0$-dense' --- Introduction ============ Throughout this paper, $X$ denotes a compact metric space endowed with a metric $d$. We denote by $C(X)$ the space of continuous self-maps of $X$, equipped with the metric $d_{C^0}$ defined by $$d_{C^0}(f,g)=\sup_{x\in X}d(f(x),g(x))$$ for $f,g\in C(X)$. The aim of this paper is to prove the following theorem. Let $M$ be a compact topological manifold (possibly with boundary). Then, the set of maps which exhibit distributional chaos of type 1 (DC1) is dense in $C(M)$. The notion of [*distributional chaos*]{} was introduced by Schweizer and Smítal [@SS]. For a continuous map $f:X\to X$, a pair of points $(x,y)\in X^2$ is said to be a [*DC1-pair*]{} for $f$ if $$\limsup_{n\to\infty}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x),f^i(y))<\delta\}|=1\quad\text{for every $\delta>0$},$$ and $$\limsup_{n\to\infty}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x),f^i(y))>\delta_0\}|=1\quad\text{for some $\delta_0>0$}.$$ Then, $f$ is said to exhibit [*distributional chaos of type 1*]{} (DC1) if there is an uncountable set $S\subset X$ such that for any $x,y\in S$ with $x\ne y$, $(x,y)$ is a DC1-pair for $f$. Indeed, the distributional chaos has two more versions: DC2 and DC3. They are numbered in the order of decreasing strength, so DC1 is the strongest, and DC2 is stronger than Li-Yorke chaos. Hereafter, $M$ denotes a compact topological manifold (possibly with boundary). For any $f\in C(X)$, we denote by $h_{top}(f)$ the topological entropy of $f$. In [@Y], it was proved by Yano that a generic $f\in C(M)$ satisfies $h_{top}(f)=\infty$. Downarowicz proved that for any $f\in C(X)$, $h_{top}(f)>0$ implies DC2 [@D]. These together imply that DC2 is generic in $C(M)$. Then, it is natural to ask whether DC1 is still generic in $C(M)$ or not. When $M$ is the unit interval (denoted by $I$), we know that for any $f\in C(I)$, $h_{top}(f)>0$ iff $f$ exhibits DC1 [@SS]; therefore, DC1 is generic in $C(I)$. Theorem 1.1 gives a partial answer to the question for a general $M$. It should be noticed that Pikuła showed that $h_{top}(f)>0$ does not necessarily imply DC1 for any $f\in C(X)$ [@Pik]. This indicates that some additional assumptions besides positive topological entropy are needed to ensure DC1 for general continuous maps. [*Shadowing*]{} is a natural candidate for such an assumption. Recently in [@LLT], Li [*et al.*]{} proved that for any $f\in C(X)$ with the shadowing property, $f$ exhibits DC1 if one of the following properties holds: (1) $f$ is non-periodic transitive and has a periodic point, or (2) $f$ is non-trivial weakly mixing. Here, note that we have $h_{top}(f)>0$ in both cases. According to Mazur and Oprocha [@MO], we know that the shadowing property is generic in $C(M)$. However, because of the additional assumption (1) or (2), it is not obvious that these results imply the genericity of DC1. The [*limit shadowing property*]{} introduced by Eirola [*et al.*]{} in [@ENP] is a variant of the shadowing property defined as follows. Given a continuous map $f:X\to X$, a sequence $(x_i)_{i\ge0}$ of points in $X$ is a [*limit pseudo orbit*]{} of $f$ if $\lim_{i\to\infty}d(f(x_i),x_{i+1})=0$. Then, $f$ is said to have the [*limit shadowing property*]{} if for any limit pseudo orbit $(x_i)_{i\ge0}$ of $f$, there is $x\in X$ such that $\lim_{i\to\infty}d(f^i(x),x_i)=0$, where such $x$ is called a [*limit shadowing point*]{} of $(x_i)_{i\ge0}$. The property provides a method of gaining information about the asymptotic behavior of the true orbits from pseudo-orbits of continuous maps. Then, we prove the following theorem. Suppose that a continuous map $f:X\to X$ satisfies the following properties. - The limit shadowing property. - $h_{top}(f)>0$. Then, $f$ exhibits distributional chaos of type 1 (DC1). Moreover, $E(X,f)\subset\overline{DC1(X,f)}$, where $E(X,f)$ (resp. $DC1(X,f)$) is the set of entropy pairs (resp. DC1-pairs) for $f$. This theorem guarantees DC1 for any $f\in C(X)$ with the limit shadowing property without any other assumption than the positive topological entropy. In [@MO], it was also proved that the set of maps with the [*s-limit shadowing property*]{} is dense in $C(M)$. We know that the s-limit shadowing property always implies the limit shadowing property [@BGO]. On the other hand, Yano proved that for any $K>0$, the set $$E_K(M)=\{f\in C(M):h_{top}(f)\ge K\}$$ contains an open and dense subset of $C(M)$ [@Y]. Thus, Theorem 1.1 is implied by Theorem 1.2. This paper consists of four sections. Some basic definitions and facts are briefly collected in Section 2. In Section 3, we introduce a relation on the chain recurrent set, and prove a few lemmas needed for the proof of Theorem 1.2. Then, Theorem 1.2 is proved in Section 4. Preliminaries ============= In this section, we collect some basic definitions and facts used in this paper. [*Chains, cycles, pseudo-orbits, and the shadowing property*]{} --------------------------------------------------------------- Given a continuous map $f:X\to X$, a finite sequence $(x_i)_{i=0}^{k}$ of points in $X$, where $k$ is a positive integer, is called a [*$\delta$-chain*]{} of $f$ if $d(f(x_i),x_{i+1})\le\delta$ for every $0\le i\le k-1$. A $\delta$-chain $(x_i)_{i=0}^{k}$ of $f$ is said to be a [*$\delta$-cycle*]{} of $f$ if $x_0=x_k$. For $\delta>0$, a sequence $(x_i)_{i\ge0}$ of points in $X$ is called a [*$\delta$-pseudo orbit*]{} of $f$ if $d(f(x_i),x_{i+1})\le\delta$ for all $i\ge0$. Then, for $\epsilon>0$, a $\delta$-pseudo orbit $(x_i)_{i\ge0}$ of $f$ is said to be [*$\epsilon$-shadowed*]{} by $x\in X$ if $d(f^i(x),x_i)\leq \epsilon$ for all $i\ge 0$. We say that $f$ has the [*shadowing property*]{} if for any $\epsilon>0$, there is $\delta>0$ such that every $\delta$-pseudo orbit of $f$ is $\epsilon$-shadowed by some point of $X$. A point $x\in X$ is said to be a [*chain recurrent point*]{} for $f$ if for any $\delta>0$, there is a $\delta$-cycle $(x_i)_{i=0}^{k}$ of $f$ such that $x_0=x_k=x$. The set of chain recurrent points for $f$ is denoted by $CR(f)$. [*Entropy pairs*]{} ------------------- Given a continuous map $f:X\to X$ and an open cover $\mathcal{U}$ of $X$, we denote by $h(f,\mathcal{U})$ the [*entropy of $f$ relative to $\mathcal{U}$*]{} (see [@W] for details). The notion of [*entropy pairs*]{} was introduced by Blanchard [@B]. A pair of points $(x,y)\in X^2$ with $x\ne y$ is said to be an [*entropy pair*]{} for $f$ if for any closed neighborhoods $A$ of $x$ and $B$ of $y$, we have $h(f,\{A^c,B^c\})>0$ whenever $A\cap B=\emptyset$. The set of entropy pairs for $f$ is denoted by $E(X,f)\:(\subset X^2)$. For two continuous maps $f:X\to X$ and $g:Y\to Y$, we say that $(Y,g)$ is a [*factor*]{} of $(X,f)$ if there is a surjective continuous map $\pi:X\to Y$ such that $\pi\circ f=g\circ\pi$. Such a map is called a [*factor map*]{}, and also denoted as $\pi:(X,f)\to(Y,g)$. The basic properties of entropy pairs is summarized in the following lemma. [@B] Given a factor map $\pi:(X,f)\to (Y,g)$, we have the following properties. - $h_{top}(f)>0$ if and only if $E(X,f)\ne\emptyset$. - For any $x,y\in X$, if $(x,y)\in E(X,f)$ and $\pi(x)\ne\pi(y)$, then $(\pi(x),\pi(y))\in E(Y,g)$. - For any $(z,w)\in E(Y,g)$, there is $(x,y)\in\pi^{-1}(z)\times\pi^{-1}(w)$ such that $(x,y)\in E(X,f)$. We say that a continuous map $f:X\to X$ has [*upe*]{} if for any $(x,y)\in X^2$ with $x\ne y$, $(x,y)\in E(X,f)$. It is known that if $f$ has upe, then $f$ is weakly mixing, and when $f$ has the shadowing property, the converse holds [@LO]. By this, especially, the shift map $\sigma:\{0,1\}^\mathbb{N}\to\{0,1\}^\mathbb{N}$ has upe. A chain relation and a few lemmas ================================= In this section, we prove a few lemmas needed for the proof of Theorem 1.2. First, for any continuous map $f:X\to X$, we define a chain relation $\sim$ on $CR(f)$ as follows. For any $x,y\in CR(f)$, $x\sim y$ if and only if for every $\delta>0$, there are integers $m=m(\delta)>0$ and $N=N(\delta)>0$ such that for any integer $n\ge N$, there are two $\delta$-chains $(x_i)_{i=0}^{mn}$, $(y_i)_{i=0}^{mn}\subset CR(f)$ of $f$ such that $x_0=y_{mn}=x$ and $x_{mn}=y_0=y$. Even if we replace ‘$\subset CR(f)$’ with ‘$\subset X$’ in the above definition, the relation $\sim$ does not change. This can be seen as follows. Given any $\epsilon>0$, there is $\delta=\delta(\epsilon)>0$ such that for any $\delta$-cycle $(z_i)_{i=0}^k$ of $f$, $d(z_i,CR(f))\le\epsilon$ holds for all $0\le i\le k$. Fix $x,y\in CR(f)$ and let $\alpha=(x_i)_{i=0}^l$, $\beta=(y_i)_{i=0}^l\subset X$ be $\delta$-chains of $f$ with $x_0=y_l=x$ and $x_l=y_0=y$. Then, since $\gamma=\alpha\beta$ is a $\delta$-cycle of $f$, the above property holds. Put $x'_0=x$, $x'_l=y$, and take $x'_i\in CR(f)$ with $d(x_i,x'_i)=d(x_i,CR(f))\le\epsilon$ for each $0<i<l$. Then, for any $\delta'>0$, if $\delta$ is sufficiently small, $\alpha'=(x'_i)_{i=0}^l\subset CR(f)$ gives a $\delta'$-chain of $f$, which has the same length and end points as $\alpha$. This argument also applies to $\beta$. We refer to the Robinson’s proof of $CR(f|_{CR(f)})=CR(f)$ in [@Ro] for a similar argument. Then, the basic properties of the relation $\sim$ are given in the following lemma. For any continuous map $f:X\to X$, the relation $\sim$ on $CR(f)$ is a closed $(f\times f)$-invariant equivalence relation. In other words, $\sim$ is an equivalence relation on $CR(f)$, $R=\{(x,y)\in CR(f)^2:x\sim y\}$ is a closed subset of $CR(f)^2$, and $(f\times f)(R)\subset R$. Actually, we have $(f\times f)(R)=R$ in the above notation. It is not difficult to give a direct proof of Lemma 3.1 based only on the definition of the relation $\sim$. However, we confirm it through an alternative description of $\sim$. In [@Ka1], it was shown that for any continuous map $f:X\to X$, $CR(f)$ admits the so-called [*$\delta$-cyclic decomposition*]{} for each $\delta>0$, that is, a family of disjoint subsets of $CR(f)$ $$\mathcal{D}(\delta)=\{D_{i,j}:1\le i\le K,\:0\le j\le m_i-1\}$$ with the following properties, where $J=\{(i,j):1\le i\le K,\:0\le j\le m_i-1\}$. - $CR(f)=\bigsqcup _{(i,j)\in J}D_{i,j}$, and every $D_{i,j}$, $(i,j)\in J$, is clopen in $CR(f)$. - Putting $D_{i,m_i}=D_{i,0}$, we have $f(D_{i,j})=D_{i,j+1}$ for every $(i,j)\in J$. - Given any $x,y\in D_{i,j}$ with $(i,j)\in J$, there is $N>0$ such that for any integer $n\ge N$, there is a $\delta$-chain $(x_\eta)_{\eta=0}^k\subset CR(f)$ of $f$ with $x_0=x$, $x_k=y$, and $k=m_in$. Then, Lemma 3.1 is an immediate consequence of the above properties (D1), (D2), and the following lemma. Let $f:X\to X$ be a continuous map. Then, for any $x,y\in CR(f)$, $x\sim y$ if and only if for every $\delta>0$, $x$ and $y$ are contained in the same component of $\mathcal{D}(\delta)$. Suppose $x\sim y$. For any given $\delta>0$, let $\mathcal{D}(\delta)=\{D_{i,j}:1\le i\le K,\:0\le j\le m_i-1\}$ be the $\delta$-cyclic decomposition of $CR(f)$. Without loss of generality, we may assume $x\in D_{1,0}$. Then, it must be shown that $y\in D_{1,0}$. For the purpose, we take $\delta'>0$ such that $d(A,B)=\inf\{d(a,b):a\in A, b\in B\}>\delta'$ for any $A,B\in\mathcal{D}(\delta)$ with $A\ne B$. For any integer $n=qm_1+r\ge0$, where $q\ge0$ and $0\le r\le m_1-1$, put $D_{1,n}=D_{1,r}$. Then, for every $\delta'$-chain $(x_n)_{n=0}^L\subset CR(f)$ of $f$ with $x_0=x$, by (D2) and the choice of $\delta'$, we have $x_n\in D_{1,n}$ for all $0\le n\le L$. Now, since $x\sim y$, there are $m=m(\delta')>0$ and $N=N(\delta')>0$ as in Definition 3.1. Put $M=mm_1N$ and note that $m_1N\ge N$. Thus, there is a $\delta'$-chain $(x_n)_{n=0}^M\subset CR(f)$ of $f$ such that $x_0=x$ and $x_M=y$, and this implies $y=x_{(mN)m_1}\in D_{1,(mN)m_1}=D_{1,0}$. The converse is a direct consequence of (D3). The following lemma relates the entropy pairs with the chain relation $\sim$. Let $f:X\to X$ be a continuous map. Then, for any $(x,y)\in E(X,f)$, $(x,y)\in CR(f)^2$ and $x\sim y$. Suppose $(x,y)\in E(X,f)$. Given any $\delta>0$, let $\mathcal{D}(\delta)$ be the $\delta$-cyclic decomposition of $CR(f)$. We define a relation $R\subset X^2$ by $$R=\{(a,a)\in X^2:a\in X\}\cup\{(a,b)\in CR(f)^2:\text{$\{a,b\}\subset A $ for some $A\in\mathcal{D}(\delta)$}\}.$$ Then, by (D1) and (D2), $R$ is a closed $(f\times f)$-invariant equivalence relation. Let $X_R$ be the quotient space, $\pi:X\to X_R$ be the quotient map, and $f_R:X_R\to X_R$ be the continuous map defined by $f_R\circ\pi=\pi\circ f$. $\pi$ is a factor map. We easily see that $\Omega(f_R)=\{\pi(A):A\in\mathcal{D}(\delta)\}$, a finite set (here $\Omega(\cdot)$ is the non-wandering set), and hence $h_{top}(f_R)=h_{top}(f_R|_{\Omega(f_R)})=0$. From Lemma 2.1, it follows that $\pi(x)=\pi(y)$ (or $(x,y)\in R$), which implies $\{x,y\}\subset A$ for some $A\in\mathcal{D}(\delta)$. Since $\delta>0$ is arbitrary, by Lemma 3.2, we conclude that $(x,y)\in CR(f)^2$ and $x\sim y$. For any $(x,y)\in CR(f)^2$, if $x\sim y$, then we easily see that the following holds: Given any $\delta>0$, there are an integer $a>0$ and $\delta$-chains $\gamma_{ij}=(y_{ij,\eta})_{\eta=0}^a$ of $f$ with $y_{ij,0}=i$ and $y_{ij,a}=j$ for all $i,j\in\{x,y\}$ (P). Conversely, it is obvious that if (P) holds for $(x,y)\in CR(f)^2$, then $x\sim y$. Keeping this equivalence in mind, Lemma 3.3 can also be proved as below by Kerr and Li characterization of the entropy pairs as the so-called (non-diagonal) [*IE-pairs*]{} (see [@KL] for details). Let $(x,y)\in E(X,f)$. Then, according to [@KL], $(x,y)$ is an IE-pair for $f$. It follows that for any $\epsilon>0$, $(B_\epsilon(x), B_\epsilon(y))$ has an [*independence set*]{} of positive density (here $B_\epsilon(\cdot)$ is the $\epsilon$-ball). In particular, this implies the existence of two integers $0\le m<n$ such that for all $i,j\in\{x,y\}$, there is $x_{ij}\in X$ with $f^m(x_{ij})\in B_\epsilon(i)$ and $f^n(x_{ij})\in B_\epsilon(j)$. Then, it is clear that the property (P) holds for $(x,y)$, thus $(x,y)\in CR(f)^2$ and $x\sim y$. Proof of Theorem 1.2 ==================== In this section, we prove Theorem 1.2. For the purpose, we first define a separation property for two chain recurrent points. Given a continuous map $f:X\to X$, we say that a pair of points $(x,y)\in CR(f)^2$ has property\* if there is $r>0$ such that for any $\delta>0$, there are two $\delta$-cycles $\gamma_1=(x_i)_{i=0}^k$, $\gamma_2=(y_i)_{i=0}^k\subset CR(f)$ of $f$ such that $x_0=x_k=x$, $y_0=y_k=y$, and $d(x_i,y_i)>r$ for each $0\le i\le k$. Let $f:X\to X$ be a continuous map. A pair of points $(x,y)\in X^2$ is said to be a [*proximal pair*]{} for $f$ if $\liminf_{n\to\infty}d(f^n(x),f^n(y))=0$. If $(x,y)\in X^2$ is not proximal, i.e. $\inf_{n\ge0}d(f^n(x),f^n(y))>0$, the pair is said to be [*distal*]{}. \(1) If $(x,y)\in CR(f)^2$ has property\*, then $(x,y)$ is a distal pair for $f$. $ $(2) If $(x,y)\in X^2$ is a distal pair for $f$, then any $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^2$ satisfies property\*. $ $(3) Assume that $(x,y)\in CR(f)^2$ with $x\sim y$ is a distal pair for $f$. Then, given any $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^2$, since $\sim$ is a closed $(f\times f)$-invariant relation (by Lemma 3.1), $z\sim w$. It also holds that $(z,w)$ has property\*. The following lemma gives a sufficient condition for DC1. Suppose that a continuous map $f:X\to X$ satisfies the following properties. - There is $(z,w)\in CR(f)^2$ which satisfies $z\sim w$ and property\*. - For any sequence $(x_i)_{i\ge0}$ of points in $CR(f)$, if $\lim_{i\to\infty}d(f(x_i),x_{i+1})=0$, then there is $x\in X$ such that $\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f^i(x),x_i)=0$. Then, $f$ exhibits distributional chaos of type 1 (DC1). By (1), there is $r>0$ with the following property: For any integer $n\ge1$, there are an integer $a_n>0$, two $n^{-1}$-cycles $\gamma_{0,n}=(p_i)_{i=0}^{a_n}$, $\gamma_{1,n}=(q_i)_{i=0}^{a_n}\subset CR(f)$ of $f$, and two $n^{-1}$-chains $\alpha_n=(r_i)_{i=0}^{a_n}$, $\beta_n=(s_i)_{i=0}^{a_n}\subset CR(f)$ of $f$ such that $p_0=p_{a_n}=z$, $q_0=q_{a_n}=w$, $r_0=s_{a_n}=z$, $r_{a_n}=s_0=w$, and $d(p_i,q_i)>r$ for every $0\le i\le a_n$. We take a sequence of integers $0<m_1<m_2<\cdots$ such that $m_1=2$, and, putting $b_n=\sum_{i=0}^{n-1}a_im_i$, we have $$\frac{a_n(m_n-2)+1}{b_n+a_nm_n+1}>1-n^{-1}$$ for every $n>1$. Put $c_{0,n}=\gamma_{0,n}^{m_n}$ and $c_{1,n}=\alpha_n\gamma_{1,n}^{m_n-2}\beta_n$ for each $n\ge1$. We have $l(c_{0,n})=l(c_{1,n})=a_nm_n$ for any $n\ge1$ (here $l(\cdot)$ denotes the length of the cycle). Then, for each $u=(u_n)_{n\in\mathbb{N}}\in\{0,1\}^\mathbb{N}$, define $\xi(u)=c_{u_1,1}c_{u_2,2}c_{u_3,3}\cdots\subset CR(f)$, a limit pseudo orbit of $f$, and take $x(u)$ as in the property (2). Let us fix an uncountable subset $S\subset\{0,1\}^\mathbb{N}$ such that for any $u,v\in S$ with $u\ne v$, both $\{n\in\mathbb{N}:u_n=v_n\}$ and $\{n\in\mathbb{N}:u_n\ne v_n\}$ are infinite sets. We shall prove that for any $u,v\in S$ with $u\ne v$, $(x(u),x(v))\in X^2$ is a DC1-pair for $f$. For the purpose, put $\xi(u)=(x_{0,i})_{i\ge0}$, $\xi(v)=(x_{1,i})_{i\ge0}$, and $(x(u),x(v))=(x_0,x_1)$. Note that, putting $\epsilon_{j,n}=\frac{1}{n}\sum_{i=0}^{n-1}d(f^i(x_j),x_{j,i})$, we have $\lim_{n\to\infty}\epsilon_{j,n}=0$ for each $j\in\{0,1\}$. Given any $\delta>0$, putting $c_n=b_n+a_nm_n+1$, we have $$|\{0\le i\le c_n-1: d(f^i(x_j),x_{j,i})<\frac{\delta}{2}\}|\ge c_n(1-2\delta^{-1}\epsilon_{j,c_n})$$ for any $n\ge1$ and $j\in\{0,1\}$. On the other hand, for any $n>1$, if $u_n=v_n$, then because $c_{u_n,n}=c_{v_n,n}$, $$\{b_n\le i\le b_n+a_nm_n\}\subset\{0\le i\le c_n-1:x_{0,i}=x_{1,i}\},$$ and so $$|\{0\le i\le c_n-1:x_{0,i}=x_{1,i}\}|\ge c_n\cdot\frac{a_nm_n+1}{c_n}>c_n(1-n^{-1}).$$ Hence, for any $n>1$ with $u_n=v_n$, we have $$|\{0\le i\le c_n-1:d(f^i(x_0),f^i(x_1))<\delta\}|>c_n(1-n^{-1}-2\delta^{-1}\epsilon_{0,c_n}-2\delta^{-1}\epsilon_{1,c_n}).$$ This together with $|\{n\in\mathbb{N}:u_n=v_n\}|=\infty$ yields $$\limsup_{n\to\infty}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x_0),f^i(x_1))<\delta\}|=1.$$ Note that $\delta>0$ is arbitrary. It only remains to prove the following: $$\limsup_{n\to\infty}\frac{1}{n}|\{0\le i\le n-1:d(f^i(x_0),f^i(x_1))>\frac{r}{3}\}|=1.$$ Similarly as above, we have $$|\{0\le i\le c_n-1: d(f^i(x_j),x_{j,i})<\frac{r}{3}\}|\ge c_n(1-3r^{-1}\epsilon_{j,c_n})$$ for any $n\ge1$ and $j\in\{0,1\}$. On the other hand, for any $n>1$, if $u_n\ne v_n$, then because $c_{u_n,n}\ne c_{v_n,n}$, $$\{b_n+a_n\le i\le b_n+a_n+a_n(m_n-2)\}\subset\{0\le i\le c_n-1:d(x_{0,i},x_{1,i})>r\},$$ and so $$|\{0\le i\le c_n-1:d(x_{0,i},x_{1,i})>r\}|\ge c_n\cdot\frac{a_n(m_n-2)+1}{c_n}>c_n(1-n^{-1}).$$ Thus, for any $n>1$ with $u_n\ne v_n$, we have $$|\{0\le i\le c_n-1:d(f^i(x_0),f^i(x_1))>\frac{r}{3}\}|>c_n(1-n^{-1}-3r^{-1}\epsilon_{0,c_n}-3r^{-1}\epsilon_{1,c_n}).$$ This together with $|\{n\in\mathbb{N}:u_n\ne v_n\}|=\infty$ implies the required equation. Now, It has been proved that for any $u,v\in S$ with $u\ne v$, $(x(u),x(v))$ is a DC1-pair for $f$. Since $S$ is an uncountable set, $f$ exhibits DC1. We need the following lemma from [@Ka2]. [@Ka2 Theorem 1.1] Let $f:X\to X$ be a continuous map with the limit shadowing property. Then, $f|_{CR(f)}:CR(f)\to CR(f)$ satisfies the shadowing property. It is known that the shadowing property with positive topological entropy allows us to obtain a subsystem of $(X,f^a)$ (for some $a>0$) which has $(\{0,1\}^\mathbb{N},\sigma)$ (here $\sigma$ is the shift map) as a factor (see, for example, [@LO; @MoO]). We use it and Lemma 3.3 to find a pair of chain recurrent points with the two properties assumed in Lemma 4.1. Since, especially, the second part of the proof of Theorem 1.2 relies on a specific construction of the subsystem through $\sim$, for completeness, we briefly describe it below. [*Construction*]{}: Suppose that $g=f|_{CR(f)}:CR(f)\to CR(f)$ has the shadowing property. Given $(x,y)\in CR(f)^2$ with $x\ne y$ and $x\sim y$, put $x_0=x$, $x_1=y$, and let $0<\epsilon<2^{-1}d(x,y)$. Take $\delta=\delta(\epsilon)>0$ such that every $\delta$-pseudo orbit of $g$ is $\epsilon$-shadowed by some point of $CR(f)$. Since $x\sim y$, there are an integer $a>0$ and $\delta$-chains $\gamma_{ij}=(y_{ij,\eta})_{\eta=0}^a\subset CR(f)$ of $g$ with $y_{ij,0}=x_i$ and $y_{ij,a}=x_j$ for each $i,j\in\{0,1\}$. For any $s=(s_i)_{i\in\mathbb{N}}\in\{0,1\}^\mathbb{N}$, consider the $\delta$-pseudo orbit $\xi(s)=\gamma_{s_1s_2}\gamma_{s_2s_3}\gamma_{s_3s_4}\cdots\subset CR(f)$ of $g$, which is $\epsilon$-shadowed by some $x(s)\in CR(f)$. Let $$Y=\{x\in CR(f): \text{$\xi(s)$ is $\epsilon$-shadowed by $x$ for some $s\in\{0,1\}^\mathbb{N}$}\}$$ and note that $Y$ is a compact $f^a$-invariant subset of $CR(f)$. Define a map $\pi:Y\to\{0,1\}^\mathbb{N}$ by the condition that $\xi(\pi(x))$ is $\epsilon$-shadowed by $x$ for each $x\in Y$. Then, it is easy to see that $\pi:(Y,f^a)\to(\{0,1\}^\mathbb{N},\sigma)$ is a factor map. By the above construction, we see that if $f|_{CR(f)}$ has the shadowing property, then for any $(x,y)\in CR(f)^2$ with $x\ne y$, $x\sim y$ implies $(x,y)\in E(X,f)$. Indeed, we have $x(s)\in\bigcap_{i\in\mathbb{N}}f^{-(i-1)a}(B_\epsilon(x_{s_i}))\ne\emptyset$ for every $s\in\{0,1\}^\mathbb{N}$ in the above notation, which implies $$h(f,\{B_\epsilon(x)^c,B_\epsilon(y)^c\})\ge a^{-1}\log2>0.$$ Since $0<\epsilon<2^{-1}d(x,y)$ is arbitrary, $(x,y)\in E(X,f)$. The fact that the shift map $\sigma:\{0,1\}^\mathbb{N}\to\{0,1\}^\mathbb{N}$ has upe, which is mentioned in Section 2, is a consequence of this remark. Combining this remark with Lemma 3.3 (and Lemma 2.1), we get the following corollary, which characterizes the set of entropy pairs by the relation $\sim$, under the assumption of the shadowing property. Let $f:X\to X$ be a continuous map. If $f|_{CR(f)}:CR(f)\to CR(f)$ has the shadowing property, then it holds that $$E(X,f)=\{(x,y)\in CR(f)^2:x\ne y\:\:\text{and}\:\:x\sim y\}.$$ In particular, $h_{top}(f)>0$ if only if $x\sim y$ for some $(x,y)\in CR(f)^2$ with $x\ne y$. Finally, let us prove Theorem 1.2. By (1) and Lemma 4.2, $f|_{CR(f)}:CR(f)\to CR(f)$ satisfies the shadowing property. Since $E(X,f)\ne\emptyset$ by (2) and Lemma 2.1, we can take $(x,y)\in E(X,f)$. Then, by Lemma 3.3, it holds that $(x,y)\in CR(f)^2$, $x\ne y$, and $x\sim y$; therefore, we have a factor map $\pi:(Y,f^a)\to(\{0,1\}^\mathbb{N},\sigma)$ as above. Since $\sigma$ has upe, especially, we have $(0^\infty,1^\infty)\in E(\{0,1\}^\mathbb{N},\sigma)$. According to Lemma 2.1, there is $(p,q)\in E(Y,f^a)$ such that $\pi(p)=0^\infty$ and $\pi(q)=1^\infty$. Because $(0^\infty,1^\infty)$ is a distal pair for $\sigma$, $(p,q)$ is a distal pair for $f^a$, so for $f$. On the other hand, since $$E(Y,f^a)\subset E(X,f^a)\subset E(X,f),$$ we have $(p,q)\in E(X,f)$, which together with Lemma 3.3 implies $(p,q)\in CR(f)^2$ and $p\sim q$. Fix $(z,w)\in\omega((p,q),f\times f)\subset CR(f)^2$. Then, as Remark 4.1, it satisfies $z\sim w$ and property\*. Thus, by (1) and Lemma 4.1, we conclude that $f$ exhibits DC1. Let us prove $E(X,f)\subset\overline{DC1(X,f)}$. Given any $(x,y)\in E(X,f)$, Lemma 3.3 implies $(x,y)\in CR(f)^2$, $x\ne y$, and $x\sim y$, so we have a factor map $\pi:(Y,f^a)\to(\{0,1\}^\mathbb{N},\sigma)$ as above (for any $0<\epsilon<2^{-1}d(x,y)$). Take $(p,q)\in E(Y,f^a)$ such that $\pi(p)=0^\infty$ and $\pi(q)=1^\infty$. Fix $(z,w)\in\omega((p,q),f^a\times f^a)$ and note that $$(z,w)\in\omega(p,f^a)\times\omega(q,f^a)\subset\pi^{-1}(0^\infty)\times\pi^{-1}(1^\infty)\subset B_\epsilon(x)\times B_\epsilon(y).$$ Since $\omega((p,q),f^a\times f^a)\subset\omega((p,q),f\times f)$, similarly as above, $(z,w)\in CR(f)^2$ satisfies $z\sim w$ and property\*. Then, we see that, because $f$ has the limit shadowing property, in the proof of Lemma 4.1, $x(u)$ can be taken as a limit shadowing point of $\xi(u)$ for any $u\in\{0,1\}^\mathbb{N}$. Also, in the proof of Lemma 4.1, it has been proved that for any $u,v\in\{0,1\}^\mathbb{N}$ with $|\{n\in\mathbb{N}:u_n=v_n\}|=\infty$ and $|\{n\in\mathbb{N}:u_n\ne v_n\}|=\infty$, $(x(u), x(v))$ is a DC1-pair for $f$. For concreteness, put $u=0^\infty$ and $v=(01)^\infty=0101\cdots$. Then, by the definition of $\xi(u)$ and $\xi(v)$, we easily see that there is a sequence of integers $0<n_1<n_2\cdots$ such that $$\lim_{j\to\infty}(f^{n_j}(x(u)),f^{n_j}(x(v)))=(z,w).$$ Since every $(f^{n_j}(x(u)),f^{n_j}(x(v)))$, $j\ge1$, is a DC1-pair for $f$ as $(x(u),x(v))$ is so, we have $(z,w)\in\overline{DC1(X,f)}$. Thus, by $(z,w)\in B_\epsilon(x)\times B_\epsilon(y)$, and since $0<\epsilon<2^{-1}d(x,y)$ is arbitrary, $(x,y)\in\overline{DC1(X,f)}$, completing the proof. In this appendix, as a complement to the proof of Theorem 1.2, we show that the existence of $(z,w)\in CR(f)^2$ with $z\sim w$ and property\* is a necessary condition for DC1. For the purpose, the next lemma is needed. Let $f:X\to X$ be a continuous map. If $(x,y)\in X^2$ is a proximal pair for $f$, then any $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^2$ satisfies $z\sim w$. Given any $\delta>0$, consider the $\delta$-cyclic decomposition $\mathcal{D}(\delta)$ of $CR(f)$. For any $p\in X$, note that $\lim_{n\to\infty}d(f^n(p),CR(f))=0$, and so by (D2), we have $\lim_{n\to\infty}$ $d(f^n(p),f^n(A_p))=0$ for some $A_p\in\mathcal{D}(\delta)$. It holds that $$\lim_{n\to\infty} d(f^n(x),f^n(A_x))=0\quad\text{and}\quad\lim_{n\to\infty} d(f^n(y),f^n(A_y))=0.$$ Since $(x,y)$ is proximal, again by (D2), $A_x$ and $A_y$ should be equal. Put $A=A_x=A_y$. Then, because $(z,w)\in\omega((x,y),f\times f)$, there are $B\in\mathcal{D}(\delta)$ and a sequence of integers $0<n_1<n_2<\cdots$ such that $f^{n_i}(A)=B$ for all $i\ge1$, and $\lim_{i\to\infty}(f^{n_i}(x),f^{n_i}(y))=(z,w)$. These properties give $\{z,w\}\subset B$, and since $\delta>0$ is arbitrary, by Lemma 3.2, we conclude $z\sim w$. Let $f:X\to X$ be a continuous map. If $(x,y)\in DC1(X,f)$, then there exists $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^2$ with $z\sim w$ and property\*. Let $(x,y)\in DC1(X,f)$. Then, $(x,y)$ is clearly a proximal pair for $f$. From the definition of DC1-pairs, it also follows that there is $\delta_0>0$ for which $$T=\{n\ge0:d(f^n(x),f^n(y))>\delta_0\}$$ is a so-called [*thick set*]{}, meaning that for any $n>0$, there is $j\ge0$ such that $\{j,j+1,\dots, j+n\}\subset T$. Let us choose two sequences of integers $0<n_1<n_2<\cdots$, $0<N_1<N_2<\cdots$, and $(p,q)\in X^2$ such that the following holds. - For any $j\ge1$ and any $n_j\le n\le n_j+N_j$, $d(f^n(x),f^n(y))>\delta_0$. - $\lim_{j\to\infty}(f^{n_j}(x),f^{n_j}(y))=(p,q)$. Then, $d(f^n(p),f^n(q))\ge\delta_0$ for all $n\ge0$; therefore, $(p,q)\in\omega((x,y),f\times f)$ is distal. By Lemma A.1, it also holds that $p\sim q$. Take $(z,w)\in\omega((p,q),f\times f)\subset\omega((x,y),f\times f)$. Then, as Remark 4.1, $z\sim w$ and property\* hold. We end with a simple application of Lemma A.1. A pair of points $(x,y)\in X^2$ is said to be a [*Li-Yorke pair*]{} for $f$ if $$\liminf_{n\to\infty}d(f^n(x),f^n(y))=0\quad\text{and}\quad\limsup_{n\to\infty}d(f^n(x),f^n(y))>0.$$ By Lemma A.1, a similar argument as the proof of Proposition A.1 shows that for any Li-Yorke pair $(x,y)$ for $f$, there is $(z,w)\in\omega((x,y),f\times f)\subset CR(f)^2$ such that $z\ne w$ and $z\sim w$. When $f|_{CR(f)}:CR(f)\to CR(f)$ has the shadowing property, due to Corollary 4.1, the existence of such a $(z,w)$ implies $h_{top}(f)>0$. Thus, by Lemma 4.2, the assumption (2) of Theorem 1.2 holds if $f$ has the limit shadowing property and has a Li-Yorke pair. Also, it follows that a continuous map $f$ with the limit shadowing property exhibits DC1 only if $h_{top}(f)>0$ because any DC1-pair is a Li-Yorke pair. [99]{} A.D. Barwell, C. Good, P. Oprocha, Shadowing and expansivity in subspaces. Fund. 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--- abstract: 'We prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant stationary product probability. We also give examples of a type ${\rm III}_{1}$ Bernoulli shift and a Markovian flow which are power weakly mixing and zero type.' author: - Zemer kosloff title: 'the zero-type property and mixing of Bernoulli shifts' --- [^1] Introduction {#introduction .unnumbered} ------------ This article deals with the concept of zero-type for invertible non-singular transformations $T$ of the standard probability space $\left(X,\cB,m\right)$. **** A non-singular transformation without an absolutely continuous invariant probability is called zero-type, sometimes also called mixing, if its Koopman operator is mixing, meaning that it’s maximal spectral type is a Rajchman measure. This generalises the notion of zero-type introduced by Hajian and Kakutani [@H-K]. **** ****$\ L_{0}^{2}\left(X,m\right)=L^{2}(X,m)\ominus\mathbb{C}$. In the interplay between non singular ergodic theory and infinitely divisible processes, the zero type transformations are associated with the classical notion of mixing of infinitely divisible processes, see [@RS] or [@Ro]. In the first part we prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant product probability. Thus the Hamachi shift [@Ham] and the type ${\rm III}_{1}$ shift given in [@Kos] are examples of conservative, ergodic, zero-type transformations which do not posses an $m$ equivalent $\sigma$-finite invariant measure, also known as type ${\rm III}$ and zero type transformations. For another construction of zero type, type ${\rm III}$ transformations see [@Da1 Theorem 0.3]. A non singular transformation $T$ is weakly mixing if $S\times T$ is ergodic for every ergodic probability preserving transformation $S$. It follows that if $T$ is a probability preserving weakly mixing transformation then $T\times\cdots\times T$ ($k-$times) is ergodic for every $k\in\mathbb{N}$. In general this is not true, as there exist weakly mixing transformations such that $T\times T$ is not ergodic. In an attempt to understand the notion of weak mixing for non singular transformations the concept of power weakly mixing transformations was introduced. $T$ is called *power weakly mixing* if for any $l_{1},l_{2},....,l_{k}\in\bbZ\backslash\{0\}$, $$\left(X^{k},P^{\otimes k},T^{l_{1}}\times T^{l_{2}}\times\cdots\times T^{l_{k}}\right)$$ is an ergodic automorphism where $$P^{\otimes k}=\underset{k-\mbox{times}}{\underbrace{P\otimes P\otimes\cdots\otimes P}}.$$ A weaker notion is that of infinite ergodic index which means that for every $k\in\mathbb{N}$, the $k$-fold product of $T$ is an ergodic automorphism. In [@AFS] it was shown that Chacon’s non singular type ${\rm III}_{\lambda}$ transformation, $0<\lambda\leq1$, is a power weakly mixing transformation. In [@Da2] a construction of a transformation which is of infinite ergodic index but not power weakly mixing was given. Later in [@DP] a flow was constructed such that all times are of infinite ergodic index. These constructions use the cutting and stacking method which usually doesn’t give a zero type transformation. We give a construction of a Bernoulli shift which is power weakly mixing and type ${\rm III}_{1}$. By the first part it is also zero type. Finally a continuous time flow $\left\{ \phi_{t}\right\} _{t\in\RR}$ is given such that for every $a_{1},a_{2},...,a_{n}\in\RR$, $$\phi_{a_{1}}\times\phi_{a_{2}}\times\cdots\times\phi_{a_{n}}$$ is ergodic. This flow is the time shifts of a continuous time Markov Chain. Preliminaries: {#preliminaries .unnumbered} -------------- Let $\left(X,\cB,P\right)$ be a probability space. An invertible measurable transformation $T:X\to X$ is said to be *non-singular* with respect to $P$ if it preserves the $P-$null sets. i.e for every $A\in\cB,$ $P(A)=0$ if and only if $$P\circ T(A):=P\left(TA\right)=0.$$ A measure $m$ on $X$ is said to be $T-$*invariant* if for every *$A\in\cB$, $$P\circ T\left(A\right)=P(A).$$* If $T$ is non-singular, then for every $n$, $P\circ T^{n}$ is absolutely continuous with respect to $P$. By the Radon-Nikodym theorem there exist measurable functions $\frac{dP\circ T^{n}}{dP}\in L_{1}(X,P)_{+}$ such that for every $A\in\mathcal{B}$ , $$P\circ T^{n}(A)=\int_{A}\left(\frac{dP\circ T^{n}}{dP}\right)\, dP.$$ Denote $\left(T^{n}\right)'(x):=\frac{dP\circ T^{n}}{dP}$. A set $W\in\cB$ is called *wandering* if $\left\{ T^{n}W\right\} _{n=-\infty}^{\infty}$ are disjoint. As in [@Aar p.7] denote by $\mathfrak{D}$ the measurable union of all wandering sets for $T$, this set is $T-$invariant. Its complement is denoted by $\mathfrak{C}$. We call $\left(X,\cB,P,T\right)$ *dissipative* if $\mathfrak{D}=X$. If $\mathfrak{C}=X$ then $\left(X,\cB,P,T\right)$ is said to be *conservative*. We will use the following version of Hopf’s decomposition theorem which says that the conservative and dissipative parts can be separated in the following way. \[ext: Hopf’s theorem\]For every non-singular transformation $T$ of the probability space $(X,\cB,P)$ there exists a decomposition $X=\mathfrak{D\cup C}$ such that $\left(X,\mathcal{B},P|_{\mathfrak{D}},T\right)$ is dissipative and $\left(X,\cB,P|_{\mathfrak{C}},T\right)$ is conservative. Furthermore $$\sum_{k=0}^{\infty}\left(T^{-n}\right)'(x)<\infty\ a.e.\ x\in\mathfrak{D}$$ and$$\sum_{k=0}^{\infty}\left(T^{-n}\right)'(x)=\infty\ a.e.\ x\in\mathfrak{C.}$$ The Hellinger Integral and definition of the Zero Type Property {#the-hellinger-integral-and-definition-of-the-zero-type-property .unnumbered} --------------------------------------------------------------- The Koopman operator $U:L_{2}\left(X,\cB,P\right)$ is then defined by$$Uf(x)=\sqrt{T'(x)}f\circ T(x).$$ It is a unitary operator and by the chain rule for the Radon-Nikodym derivatives for every $n\in\bbZ$,$$U^{n}f=\sqrt{\left(T^{n}\right)'}f\circ T^{n}.$$ A transformation is *Non-Singular zero-type* ( NS zero type) if the maximal spectral type $\sigma_{T}\in\mathcal{P}\left(\mathbb{T},\mathcal{B}\left(\mathbb{T}\right)\right)$ of $U_{T}$ is a Rajchman measure. That is its Fourier coefficients $\hat{\sigma}_{T}(n)$ tend to $0$ as $n\to\infty$. and so for every $f\in L_{2}(X,P)$,$$\int_{X}f\cdot U^{n}fdP\to0\ as\ n\to\infty.$$ By looking at the Koopman operators it is seen that the zero-type property depends only on the equivalence class of $P$. \[rem: Krengel mixing and H-K\]In the case when $\left(X,\cB,m,T\right)$ is a $\sigma$-finite measure preserving transformation Krengel and Sucheston [@KrS] have showed that mixing of the Koopman operator is equivalent to the Hajian and *Kakutani zero-type condition* [@H-K] which states that for every $A\in\cB$ with $m(A)<\infty$,$$\lim_{n\to\infty}m\left(A\cap T^{-n}A\right)=0.$$ Let $P,Q$ be two probability measures on $X$. The *Hellinger Integral* (see [@Hel] or [@Kak]) is then defined as $$\rho(P,Q)=\int_{X}\sqrt{\frac{dP}{dm}}\cdot\sqrt{\frac{dQ}{dm}}\, dm$$ where $m$ is any finite measure on $X$ such that $P\ll m\ ,\ Q\ll m$. In the special case where $P\ll Q$ we can take $m=Q$ and have $$\rho(P,Q)=\int_{X}\sqrt{\frac{dP}{dQ}}\: dQ.$$ The function $\rho(\cdot,\cdot)$ measures the amount of singularity of $P$ with respect to $Q$. This function satisfies that for every $P,Q\in\mathcal{P}(X)$ ,$0\leq\rho(P,Q)\leq1$. Also $\rho(P,Q)=0$ if and only if $P$ is singular with respect to $Q$. The proof of the following proposition is standard. \[pro:NS zero type and mixing. \]Let $T$ be a non-singular transformation of the probability space $\left(X,\cB,P\right)$. The following are equivalent. \(i) ${\displaystyle \lim_{n\to\infty}\rho\left(P,P\circ T^{n}\right)=0}$. \(ii) $\left(T^{n}\right)'\overset{P}{\longrightarrow0}.$ \(iii) $\sigma_{T}$ is a Rajchman measure. Bernoulli shifts are zero-type or mixing ======================================== Let $\bbX=\{0,1\}^{\bbZ}$ and $T$ be the left shift action on $\bbX$, that is $$\left(Tw\right)_{i}=w_{i+1}.$$ Denote the cylinder sets by $$\left[b\right]_{k}^{l}=\left\{ w\in\bbX:\ \forall i=k,...,l,\ w_{i}=b_{i}\right\} .$$ A measure $P={\displaystyle \prod_{k=-\infty}^{\infty}}P_{k}\in\mathcal{P}(\bbX)$ is called a product measure if for every $k<l$, and for every cylinder $[b]_{k}^{l},$ $$P\left(\left[b\right]_{k}^{l}\right)=\prod_{j=k}^{l}P_{j}\left(\left\{ b_{j}\right\} \right).$$ We will say that a product measure $P$ is *non-singular* if the shift is non-singular with respect to $P$. For two product probability measures $P,Q$ and $N\in\bbN\cup\{\infty\}$ define $$\begin{aligned} d_{N}\left(P,Q\right) & := & \sum_{k=-N}^{N}\left\{ \left(\sqrt{P_{k}\left(\left\{ 0\right\} \right)}-\sqrt{Q_{k}\left(\left\{ 0\right\} \right)}\right)^{2}\right..\\ & & \left.+\left(\sqrt{P_{k}\left(\left\{ 1\right\} \right)}-\sqrt{Q_{k}\left(\left\{ 1\right\} \right)}\right)^{2}\right\} \end{aligned}$$ Notice that $d_{N}(P,Q)\uparrow d_{\infty}(P,Q)$ as $N\to\infty$. Set $d(P,Q):=d_{\infty}(P,Q)$ The following lemma is a direct consequence of Kakutani’s Theorem on equivalence of product measures[@Kak]. \[cla: Zero Type con for shifts\]Let $P={\displaystyle \prod_{k=-\infty}^{\infty}P_{k}}$ be a product measure. Then \(1) For any two product measures $P,Q$, $$d(P,Q)\propto-\log\rho\left(P,Q\right).$$ \(2) $P$ is non-singular if and only if$$d(P,P\circ T)<\infty.\label{eq: condition for zero type 1.}$$ \(3) The shift $ $is NS zero-type if and only if$$\lim_{n\to\infty}d\left(P,P\circ T^{n}\right)=\infty.$$ \[thm:Let–be\]Let $P={\displaystyle \prod_{k=-\infty}^{\infty}P_{k}}$ be a non-singular product measure. Then either there exists a shift invariant $P$-equivalent probability or the shift $ $ $\left(\bbX,\cB(\bbX),P,T\right)$ is NS zero-type. Therefore a non-singular shift is either mixing in the probability preserving sense or mixing in the non-singular sense. Theorem \[pro:NS zero type and mixing. \] follows from lemmas \[pro:with limit\] and \[pro:no limit\]. \[pro:with limit\]Let $P$ be a non-singular product measure on $\{0,1\}^{\mathbb{Z}}$ such that $$\exists\lim_{k\to-\infty}P_{k}=(p,1-p):=\mu_{p}.$$ Denote by $Q={\displaystyle \prod_{k=-\infty}^{\infty}\mu_{p}}$. Then if $P\perp Q$ then $ $ $\left(\bbX,\cB(\bbX),P,T\right)$ is of NS-zero type. Else $Q$ is a $P$-equivalent shift-invariant probability measure. Assume that $P\perp Q$. Then by Kakutani’s theorem $$d\left(P,Q\right)=\infty.$$ By claim \[cla: Zero Type con for shifts\] its enough to show that ${\displaystyle \lim_{n\to\infty}d\left(P,P\circ T^{n}\right)=\infty}$. Let $M>0$. Since $d(P,Q)=\infty$ there exists a $N\in\mathbb{N}$ such that $$d_{N}(P,Q)>M.$$ For every $n\in\mathbb{N}$,$$\begin{aligned} d\left(P,P\circ T^{n}\right) & \geq & \sum_{k=-N}^{N}\left\{ \left(\sqrt{P_{k}\left(\left\{ 0\right\} \right)}-\sqrt{P_{k-n}\left(\left\{ 0\right\} \right)}\right)^{2}\right.\\ & & \left.+\left(\sqrt{P_{k}\left(\left\{ 1\right\} \right)}-\sqrt{P_{k-n}\left(\left\{ 1\right\} \right)}\right)^{2}\right\} \end{aligned}$$ Therefore since ${\displaystyle \lim_{j\to-\infty}}P_{j}=\mu_{p}$ then $$\begin{aligned} \liminf_{n\to\infty}d(P,P\circ T^{n}) & \geq & d_{N}(P,Q)\\ \geq & M.\end{aligned}$$ Since $M$ is arbitrary then$$\lim_{n\to\infty}d\left(P,P\circ T^{n}\right)=\infty.$$ \[pro:no limit\]Let $ $$P$ be a non-singular product measure on $\{0,1\}^{\mathbb{Z}}$ such that $$\liminf_{k\to-\infty}P_{k}\left(\left\{ 0\right\} \right)\neq\lim\sup_{k\to-\infty}P_{k}\left(\left\{ 0\right\} \right).$$ Then $\left(\bbX,\cB(\bbX),P,T\right)$ is NS zero-type. Write $q_{1}={\displaystyle \lim\inf_{k\to-\infty}P_{k}(\{0\})}$ and $q_{2}={\displaystyle \limsup_{k\to-\infty}P_{k}(\{0\})}$. Let $M>0$. Set $\alpha=\frac{q_{2}-q_{1}}{4}.$ Define $$A_{q_{i}}:=\left\{ n\in\mathbb{Z}:\ \left|P_{k}(\{0\})-q_{i}\right|<\alpha\right\} ,\ i=1,2.$$ Let $A_{q_{i}}^{N}=A_{q_{i}}\cap[-N,N]$. Choose $N$ large enough so that $$\left|A_{q}^{N}\right|\geq\frac{M}{\alpha}\ {\rm \ and\ }\left|A_{p}^{N}\right|\geq\frac{M}{\alpha}.$$ Since $d(P,P\circ T)<\infty$ then for every $j\in\mathbb{Z}\cap[-N,N]$,$$\lim_{n\to\infty}\left|P_{-n}(\{0\})-P_{-n+j}(\{0\})\right|=0.$$ Therefore for large enough $n\in\bbN$ either $$\left[-N-n,N-n\right]\cap A_{q_{1}}=\emptyset$$ or $$[-N-n,N-n]\cap A_{q_{2}}=\emptyset.$$ Therefore for large enough $n\in\mathbb{N}$, $$\begin{aligned} d\left(P,P\circ T^{n}\right) & \geq & d_{N}(P,Q)\\ & \geq & \sum_{k\in A_{q_{1}}^{N}}\left(\sqrt{P_{k}\left(\left\{ 0\right\} \right)}-\sqrt{P_{k-n}\left(\left\{ 0\right\} \right)}\right)^{2}\\ & & +\sum_{k\in A_{q_{2}}^{N}}\left(\sqrt{P_{k}\left(\left\{ 0\right\} \right)}-\sqrt{P_{k-n}\left(\left\{ 0\right\} \right)}\right)^{2}\\ & \geq & \min\left(\alpha\cdot\left|A_{q_{i}}^{N}\right|,\alpha\cdot\left|A_{q_{2}}^{N}\right|\right)\geq M.\end{aligned}$$ Therefore $$\lim_{n\to\infty}d\left(P,P\circ T^{n}\right)=\infty$$ and the shift is NS zero-type. A zero type and power weakly mixing Bernoulli shift =================================================== In this section we construct a Bernoulli shift which is zero type and power weak mixing. The construction is done by imposing a stronger growth condition on the shift constructed in [@Kos]. Construction of the product measure: The product measure will be $P={\displaystyle \prod_{k=-\infty}^{\infty}}P_{k}$ , where $$\forall i\geq0,\ P_{i}\left(0\right)=P_{i}\left(1\right)=\frac{1}{2}.\label{P on N+}$$ The definition of $P_{k}$ for negative $k$’s is more complicated as it involves an inductive procedure. The inductive definition of $P_{k}$ for negative $k's$. ------------------------------------------------------- We will need to define inductively 5 sequences $\left\{ \lambda_{t}\right\} _{t=1}^{\infty},\left\{ n_{t}\right\} _{t=1}^{\infty}$ , $\left\{ m_{t}\right\} _{t=1}^{\infty},\ \left\{ M_{t}\right\} _{t=0}^{\infty}$ and $\left\{ N_{t}\right\} _{t=1}^{\infty}$ . The sequence $\left\{ \lambda_{t}\right\} $ is of real numbers which decreases to $1$. The other four, $\left\{ n_{t}\right\} _{t=1}^{\infty}$ , $\left\{ m_{t}\right\} _{t=1}^{\infty},\ \left\{ M_{t}\right\} _{t=0}^{\infty}$ and $\left\{ N_{t}\right\} _{t=1}^{\infty}$ are increasing sequences of integers. First choose a positive summable sequence $\left\{ \epsilon_{t}\right\} _{t=1}^{\infty}$ and set $M_{0}=1$. : Set $\lambda_{1}=2$ , $n_{1}=2$ , $m_{1}=4$ . Set also $N_{1}=M_{0}+n_{1}=3$ and $M_{1}=N_{1}+m_{1}=7.$ $ $ Given $\left\{ \lambda_{u},n_{u},N_{u},m_{u},M_{u}\right\} _{u=1}^{t-1}$ , we will choose the next level $\left\{ \lambda_{t},n_{t},N_{t},m_{t},M_{t}\right\} $ in the following order. First we choose $\lambda_{t}$ depending on $M_{t-1}$ and $\epsilon_{t}$. Given $\lambda_{t}$ we will choose $n_{t}$ and then $N_{t}$ will be defined by $$N_{t}:=M_{t-1}+n_{t}.$$ Then given $N_{t}$ we will choose $m_{t}$ and finally set $$M_{t}:=N_{t}+m_{t}.$$ : Set $k_{t}:=\left\lfloor \log_{2}\left(\frac{M_{t-1}}{\epsilon_{t}}\right)\right\rfloor +1$ , where $\left\lfloor x\right\rfloor $ denotes the integral part of $x$. Then set $\lambda_{t}=e^{\frac{1}{2^{k_{t}}}}$. With this choice of $\lambda_{t}$ we have,$$\lambda_{t}^{M_{t-1}}<e^{\epsilon_{t}}.\label{Constraint on lambda(t)}$$ This choice of $\lambda_{t}$ has the property that for every $u<t$, $\lambda_{u}=\lambda_{t}^{2^{k_{t}-k_{u}}}.$ Define $$A_{t-1}:=\left\{ \prod_{u=1}^{t-1}\lambda_{u}^{x_{u}}:\ x_{u}\in\left[-n_{u},n_{u}\right]\right\} .$$ : Given $\{\lambda_{u},n_{u},m_{u}\}_{u=1}^{t-1}$ and $\lambda_{t}$, the set $A_{t-1}$ is a finite subset of $\lambda_{t}^{\bbZ}$. Choose $n_{t}$ which satisfies $$\lambda_{t}^{n_{t}/4}\geq\max\left\{ a^{2}:a\in A_{t-1}\right\} .$$ : Now that $n_{t}$ is chosen we set $N_{t}=M_{t-1}+n_{t}$ . Set $$tN_{t}\left(2+2^{tN_{t}}\right)=m_{t}.\label{constraint of m(t)}$$ Since $m_{t}$ satisfies \[constraint of m(t)\] then for every $k<t$ and $n\leq N_{t}$, $$\frac{m_{t}}{n}-N_{t}>2^{kN_{t}}\label{eq:The growth condition necessary for p.w.m}$$ Let $\left(X,\mu,T\right)$ be a non singular automorphism such that $\cB(X)\neq\left\{ \emptyset,X\right\} $ and let $\mathcal{F\subset B}$ be a factor algebra. Then: \(i) $\mathcal{F}$ is *exhaustive* if ${\displaystyle \vee_{n=0}^{\infty}T^{n}\mathcal{F=\cB}.}$ \(ii) $\mathcal{F}$ is *exact* if ${\displaystyle \cap_{n=0}^{\infty}T^{n}\mathcal{F=}\left\{ \emptyset,X\right\} .}$ \(iii) $T$ is *a K-automorphism* if it is conservative and admits a factor algebra $\mathcal{F\subset B}$ that is exhaustive and exact and such that $T'$ is $\mathcal{F}$ measurable. Krengel has shown in [@Kre p. 153-154] that all $K$-automorphisms are ergodic. See also [@S-T Proposition 4.8(a)]. The Bernoulli shift $\left(\bbX,\mathcal{B}\left(\bbX\right),P,T\right)$ is a non-singular, type ${\rm III}_{1}$, zero-type and power weakly mixing transformation. In [@Kos] it is shown that the shift is a type ${\rm III}_{1}$ transformation. Therefore by Theorem \[thm:Let–be\] the shift is a zero type transformation. It remains to show that the shift is power weak mixing. Let $l_{1},l_{2},...,l_{k}\in\bbZ\backslash\{0\}$ and denote by $S:=T^{l_{1}}\times T^{l_{2}}\times\cdots\times T^{l_{k}}$. Clearly $$\begin{aligned} S^{n'}\left(w_{1},w_{2},..,w_{k}\right) & = & \prod_{i=1}^{k}T^{\left(l_{i}n\right)'}\left(w_{i}\right)\end{aligned}$$ and $\left(\bbX^{k},P^{\times k},S\right)$ admits an exhaustive and exact factor. Therefore in order to prove the ergodicity of $S$ it is sufficient to show that $S$ is conservative. By [@Kos lemma 3] and a similar calculation for negative $n's$ there exists $t_{0}\in\bbN$ such that for every $t>t_{0}$ , $|n|\in\left[N_{t},m_{t}\right)$ and $w\in\bbX$$$T^{n'}(w)\geq\sqrt[k]{\frac{1}{2}}\prod_{u=1}^{t}\lambda_{u}^{\sum_{j=-N_{u}+1}^{-M_{u-1}}\left\{ w_{k+n}-w_{k}\right\} }\geq2^{-N_{t}-1/k}.$$ Here the last inequality follows from $\lambda_{u}<\lambda_{1}=2$. Let $L=\max\left\{ \left|l_{i}\right|:1\leq i\leq k\right\} .$Then for every $t>\max\left(t_{0},L\right)$ , $i\in\{1,..,k\}$ and $N_{t}\leq n\leq\frac{m_{t}}{L}$, $$T^{\left(l_{i}n\right)'}(w)\geq2^{-N_{t}-1/k}$$ and so $$S^{n'}\left(w_{1},w_{2},..,w_{k}\right)\geq2^{-kN_{t}-1}.$$ Which together with implies that every $\tilde{w}\in\bbX^{k}$ and $t>\max\left(t_{0},k,L\right)$ $$\sum_{n=N_{t}}^{m_{t}/L}S^{n'}(\tilde{w})\geq\left(\frac{m_{t}}{L}-N_{t}\right)2^{-kN_{t}-1}\geq\frac{1}{2}.$$ Therefore for every $ $$\tilde{w}\in\bbX^{k}$, $$\sum_{n=1}^{\infty}S^{n'}\left(\tilde{w}\right)\geq\sum_{t}\sum_{n=N_{t}}^{m_{t}/L}S^{n'}(\tilde{w})=\infty.$$ By Hopf’s theorem for non-singular transformations $S$ is conservative. The next example is a continuous time flow such that all the times are zero type and power weakly mixing. In this Markov Chain example the flow preserves an infinite measure. The Markov Chain Example. ------------------------- A Borel map $X\times\RR\ni(x,t)\mapsto\phi_{t}(x)$ such that $$\phi_{t}\phi_{s}=\phi_{t+s}$$ is called a *non-singular flow* on $(X,\cB)$. Given a measure $\mu$ on $X$, the semi-flow $\left\{ \phi_{t}\right\} _{t\in[0,\infty)}$ is called exact if $$\cap_{t\geq0}\phi_{t}^{-1}\cB=\left\{ \emptyset,X\right\} mod\mu.$$ $ $ A measure preserving flow $\left(X,\mathcal{B},\mu,\left\{ \phi_{t}\right\} \right)$ is a *K-flow* if it admits an exhaustive and exact factor. Clearly a natural extension of an exact semiflow is a K-flow. The flow is called *Power Weakly Mixing* if for every $t_{1},t_{2},..,t_{n}\in\RR$, $$\phi_{t_{1}}\times\phi_{t_{2}}\times\cdots\phi_{t_{n}}$$ is an ergodic transformation of $\left(\left(S^{\RR}\right)^{k},\cB^{\otimes k},\mu^{\otimes k}\right)$. A function $p:[0,\infty)\to[0,1]$ is a *Markovian Renewal Function* if there exists a countable state space, which will be denoted by $S$, Markov Chain $\left\{ X_{t}\right\} _{t\in[0,\infty)}$ and a state $a\in S$ such that $$P_{a,a}(t):=P\left(\left.X_{t}=a\right|X_{0}=a\right)=p(t).$$ We say that $p$ is *aperiodic* if $$\gcd\left\{ n\in\mathbb{N}:\ p(n)\neq0\right\} =1,$$ and *null recurrent* if $$\sum_{n=1}^{\infty}p(n)=\infty\ and\ p(n)\xrightarrow[n\to\infty]{}0.$$ Given a renewal function $p$ the sequence $\left\{ p(n)\right\} _{n\in\mathbb{N}}$ defines a renewal sequence for the discrete time Markov Chain $\left\{ X_{n}\right\} _{n=0}^{\infty}$. Thus if $p$ is aperiodic and null recurrent then the Markov chain $\left\{ X_{n}\right\} _{n=0}^{\infty}$ is aperiodic and null recurrent. Hence there exists a stationary ($\sigma$-finite) measure $\tilde{\mu}\in\mathcal{M}(S)$. It follows that the measure $$\mu(\{a\})=\int_{0}^{1}\left(\sum_{s\in S}P_{s,a}(t)\tilde{\mu}(\{s\})\right)dt$$ is a stationary measure for $\left(P_{t}\right)_{t\in\RR}=\left(\left\{ P_{a_{1},a_{2}}(t)\right\} _{a_{1},a_{2}\in S}\right)_{t\in\RR}$. Finally let $\nu=P^{\mu}$ be the measure on $S^{\RR}$ with finite dimensional distributions $$\nu\left[x_{t_{0}}=s_{0},x_{t_{1}}=s_{1},...,x_{t_{n}}=s_{n}\right]=\mu\left(\left\{ s_{0}\right\} \right)P_{s_{0},s_{1}}\left(t_{1}-t_{0}\right)\cdots P_{s_{n-1},s_{n}}\left(t_{n}-t_{n-1}\right),$$ for every $t_{0}<t_{1}<\cdots<t_{n}$ and $s_{0},s_{1},\ldots,s_{n}\in S$. The flow $\left\{ \phi_{t}\right\} _{t\in\mathbb{R}}$ on $S^{\RR}$ defined by$$\phi_{t}w(s)=w(s+t)$$ is $\nu$- measure preserving. It is the natural extension of the semiflow $\left\{ \phi_{t}\right\} _{t\in[0,\infty)}$. \[thm:Power weakly mixing flow\]Let $p:[0,\infty)\to[0,1]$ be an aperiodic and null recurrent Markov Renewal Function then the flow $\left(S^{\RR},\mathcal{B},\nu,\left\{ \phi_{t}\right\} _{t\in\RR}\right)$ is conservative, exact and zero-type. If in addition for every $t_{1},t_{2},..,t_{n}\in\RR_{+}$, $$\sum_{n=1}^{\infty}\prod_{j=1}^{k}p_{a,a}^{(n)}(t_{j})=\infty$$ then the flow is Power Weakly Mixing. Since $\left\{ X_{t}\right\} _{t\geq0}$ is a null recurrent Markov chain the flow is conservative and zero-type. First we show that the tail $\sigma$-field of $\left\{ X_{t}\right\} _{t\geq0}$ is trivial, hence the semiflow $\left\{ \phi_{t}\right\} _{t\in[0,\infty)}$ is exact. Let $h>0$ and observe that $\Upsilon=\left(S^{h\mathbb{N}},\cB_{S^{h\mathbb{N}}},\nu|_{h\mathbb{N}},\phi_{h}\right)$ is a factor of $ $$\mathfrak{X}=\left(S^{[0,\infty)},\mathcal{B},\nu,\phi_{h}\right)$. Since the discrete time chain $\left\{ X_{nh}\right\} _{n\in\mathbb{N}}$ is aperiodic and recurrent, it follows by [@BF] that its tail $\sigma$-algebra is trivial. Denote by $\mathcal{F}_{h}=\mathcal{B\cap}S^{[0,h]}.$ Then it follows from the Markov property that given $\Upsilon$, for every $\Lambda_{0},\Lambda_{1},...,\Lambda_{n}\in\mathcal{F}_{h}$, the sets $$\left[w\in\Lambda_{1}\right],\left[\phi_{h}w\in\Lambda_{2}\right],...,\left[\phi_{nh}w\in\Lambda_{n}\right]$$ are independent. By Kolmogorov’s zero-one law $\mathfrak{X}$ is an exact non-singular extension of $\Upsilon$ in the sense of [@AD]. Therefore, since $\Upsilon$ is exact, it follows by Proposition 4 in [@AD] that $\mathfrak{X}$ is exact. See also [@Ios Theorem 6]. Therefore since the flow $\left\{ \phi_{t}\right\} _{t\in\RR}$ is the natural extension of the semiflow, it is a K-flow. It follows that for every $t_{1},t_{2},..,t_{k}\in\RR$ the transformation $R=\phi_{t_{1}}\times\phi_{t_{2}}\times\cdots\phi_{t_{n}}$ is K and in order to prove ergodicity of $R$ it is enough to show conservativity which is a consequence of the fact that, $$\begin{aligned} \sum_{n=1}^{\infty}\prod_{j=1}^{k}p_{a,a}^{(n)}\left(\left|t_{j}\right|\right) & = & \infty.\end{aligned}$$ Example: Let $$p(t)=\frac{1}{\log(e+t)}.$$ Since $p(t)$ satisfies the conditions of [@Kin Theorem 6.6, p.144, see also p.41] there exists a continuous time Markov Chain $\left\{ X_{t}:t\in\mathbb{R}\right\} $ on a countable state space $S$ such that for some $a\in S$, $$p_{a,a}(t):=P\left(\left.X_{t}=a\right|X_{o}=a\right)=p(t).$$ Since $$p(n)=\frac{1}{\log(e+n)}\neq0,\ \sum_{n=1}^{\infty}p(n)=\infty$$ and for every $t_{1},t_{2},..,t_{n}>0$, $$\sum_{n=1}^{\infty}\prod_{j=1}^{k}p_{a,a}^{(n)}(t_{j})=\sum_{n=1}^{\infty}\prod_{j=1}^{k}p_{a,a}(n\cdot t_{j})=\sum_{n=1}^{\infty}\prod_{j=1}^{k}\frac{1}{\log\left(e+t_{j}\cdot n\right)}=\infty,$$ it satisfies the conditions of Theorem \[thm:Power weakly mixing flow\] and hence the Markov flow defined by the Markov Chain is conservative, zero type and power weakly mixing. ** [ST]{} J. 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Park, Rank-one flows of transformations with infinite ergodic index, Proc. Amer. Math. Soc. 139 (2011), 201-207. S. Day, B. Grivna, E. McCartney, and C.E. Silva, Power Weakly Mixing Infinite Transformations, New York J. of Math. 5 (1999), 17-24. A. Hajian and S. Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc. 110 (1964), pp 136151. MR 27:4904 [1.15]{} T. Hamachi, On a Bernoulli shift with non-identical factor measures, Ergod. Th. & Dynam. Sys. 1 (1981), pp 273283, MR 83i:28025, Zbl 597.28022. E. Hellinger, Neue Begr$\ddot{u}$ndung der Theorie quadratischer Formen von unendlich vielen Ver$\ddot{a}$nderlichen. Journal f$\ddot{u}$r die reine and angewandte Mathematik 36 (1909), pp 210271. Iosifescu, M. On finite tail $\sigma$-algebras. Z. Wahrscheinlichkeitstheorie 24 (1972), 159-166 Kingman, J. F. C. (1972). Regenerative Phenomena. John Wiley and Sons Ltd. ISBN 0-471-47905-5. [1.15]{} S. Kakutani, On equivalence of infinite product measures, Ann. Math. 49 (1948), pp 214224. Z.Kosloff, On a type ${\rm III}_{1}$ Bernoulli shift, preprint arXiv:0912.1094. U. Krengel, Transformations without finite invariant measures have finite strong generators, Contributions to Ergodic Theory and Probability (Ohio State Univ., Columbus, OH, 1970), Lecture Notes in Math., no. 160, Springer-verlag, Berlin, 1970, pp. 135157, MR 42 \#4703, Zbl 201.38303. U. Krengel and L. Sucheston, On mixing in infinite measure spaces. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 (1969), 150164. Rosinski, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoul li 2 365377. MR1440274 E. Roy. Ergodic properties of Poissonian ID processes. Ann. Probab., 35(2):551576, 2007. [1.15]{} C. E. Silva and P. Thieullen, A skew product entropy for nonsingular transformations, J. Lon. Math. Soc. (2) 52 (1995), 497516. [^1]: This research was was supported by THE ISRAEL SCIENCE FOUNDATION grant No. 1114/08.

--- author: - 'Roman Pflugfelder, ' bibliography: - 'IEEEabrv.bib' - 'survey-min.bib' title: 'Siamese Learning Visual Tracking: A Survey' --- learning as data-driven programming approach [@shalev-shwartz-BOOK2014] has steadily improved the robustness and accuracy of vision techniques for the last decades, even more, machine learning has now finally shifted our conceptual view on the design of vision algorithms since the advent of Nvidia’s CUDA platform in 2007 and since the availability of large manually labelled image datasets in 2010. State-of-the-art tells us to see vision algorithms as flexible reusable and for now neural learning processes, trained end-to-end on the application-specific data. The breakthrough in solving categorial object recognition (ILSVRC’10 [@russakovsky-IJCV2015]) has motivated the field to re-apply the connectionism paradigm [@mcculloch-1943; @rosenblatt-1958; @hinton-1981] to hard vision problems with in many cases outstanding success. For example, categorial object recognition has been pushed down on ILSVRC’10-14 from 28.2% classification error (SVM classifier [@lin-CVPR2011]) to 4.82% (CNN ensemble [@ioffe-ICML2015]) which proved experimentally deep learning as ample solution to the problem. Since connectionism [@nowlan-NIPS1994] grasped the visual tracking community after the AI winter again in 2013 [@wang-NIPS2013], results have been accordingly shown for the important problem of tracking. For example, Area under the Curve (AuC) performance on OTB-2013 and OPE has been improved by 31% from 49.9% (structured SVM [@hare-CVPR2011]) to 56.0% in 2014 (discriminative correlation filter[^1] [@danelljan-BMVC2014]) to 65.5% (Siamese network [@tao-CVPR2016]; see Sec. \[sec:sint\]) in 2016 with lots of room to improve. The remainder of this section introduces the state-of-the-art view on visual tracking and the significant step change of learning the tracking. Sec. \[sec:relwork\] overviews other significant surveys covering the tracking literature of the last 15 years and works out as well as summarises in Sec. \[sec:challenges\] the mentioned grand challenges of tracking. Sec. \[sec:siamese\] focuses on very recent works using Siamese networks and deep offline learning which promise some impact on tracking performance in the coming years, a discussion which concludes the paper in Sec. \[sec:conclude\]. Tracking Definition ------------------- Tracking [@challa-book2011] and in particular visual tracking [@maggio-book2011; @betke-book2016] is basically a sequential inference problem. As a result, tracking makes predictions of latent variables[^2] by the way of collecting sufficient evidence with visual sensory inferred from spatiotemporal stimuli in the scene. In most applications, the observations are made by RGB, infrared, depth i.e. LIDAR, ToF cameras, radar or some combination, but other cameras such as event cameras [@scaramuzza-ICRAW2017], hyperspectral cameras and fluorescence microscopy in microbiology [@skylaki-natbio2016] are becoming increasingly popular. Physical objects or phenomena in the scene cause usually these stimuli, for example natural objects such as humans, animals, cells or artificial objects like cars but also complex phenomena as the formation and movement of clouds [@chow-solar2015]. The latent variables originate from the mathematical description of scene, objects, camera and their relationships and the assumptions made by the designer. A description is a set of data structures, related by computational feature transforms that give the latent variables the necessary meaning to make inference useful. The image template obtained from the original image is a good example for a well-known simplest description with the four-parameter axis-aligned bounding box as latent variable. In many cases, simple descriptions of appearance, shape, location and motion are sufficient, although, some situations need more complex descriptions such as manoeuvring objects, tracking in clutter [@challa-book2011] and tracking multiple objects [@betke-book2016]. The latter, for example, needs mutual descriptions in order to infer identity. Tracker Design -------------- The paradigm shift from pure hand-crafted designs to connectionism in computer vision suggests to design trackers as multiple complementary and loosely coupled learning processes which are finally embedded and intertwined into some kind of management and control process [@winston-PHD1970; @sussman-PHD1973]. There is evidence in neuroscience and cognitive science that the visual brain and mind works in similar way [@grossberg-tcs2000; @kandel-book2000]. The motivation comes from the outstanding success of convolutional networks as solution to categorial object recognition and by the higher complexity of tracking compared to categorial object recognition. Each learning process is a hierarchical meaningful description of the visual input, with the aim to yield evidence of the tracker’s state and to finally make predictions of the particular latent variables given by the design. The tracker under this paradigm works basically in two operating states either it is learning from incoming visual input to adapt the parameters of the learning processes by end-to-end training or the tracker is inferring the latent variables of the tracker. The data flow of such a tracker’s design shows three fundamental pathways [@bazzani-ICML2011] which Fig. \[fig:pathways\] illustrates: Who? : Collects evidence of objects’ appearance and shape as well as their identities, similar to the ventral pathway in the human brain. Modules of this pathway are in literature known as object representation, appearance modelling, feature selection and data association. Where? : Collects evidence of objects’ location and motion (derivatives of location with respect to time) similar to the dorsal pathway in the human brain. Modules of this pathway are in literature known as object initialisation, object localisation and motion estimation. Why? : Formalises the objectives of the calculus. ![Data flow diagram of a tracker in three pathways: who, where and why. These interconnected pathways, constituted by the descriptions of scene, objects, camera and their relationships, process incoming observation data $Y_1, \ldots Y_T$ and predict finally the latent variables $X_1, \ldots, X_T$.[]{data-label="fig:pathways"}](tracker-pathways){width="\linewidth"} Trackers are inference machines, gathering evidence of who and where by collecting sequentially observations. Parametric descriptions are precisely formulated and initially given. Each prediction step completes the inference process by computing the latent variables given the evidence of who and where. The third why pathway formalises the objective function so that the tracker can come to a conclusion. For example, a popular objective is least-squares to given ground truth, e.g. a given image template of the object. Different interconnected modules capturing the various descriptions of the tracking problem constitute these pathways. The complexity of interconnection and the choice of the mathematical formulation has direct consequence to the robustness and accuracy of the tracking results as well as to the computational efficiency of the tracking algorithm. Popular designs are mode seeking, HMM’s, Kalman filter, variants and generalised SMC methods. Recently [@wang-ICCV2015] proposed five key building blocks of a tracker: motion model, feature extractor, observation model, model updater, ensemble post-processor. Learning Tracking ----------------- This network of pathways has a longtime been seen through the glass of dynamical systems theory [@bar-book2004; @challa-book2011]. Motivated by radar technology, a huge body of statistical estimators has been studied since the 1950s. Although learning has been considered in the form of system identification, i.e. the algorithms are able to identify and adapt the system parameters and may switch the underlying statistical model of the data, machine learning has not widely been recognised as tool to estimate the functional parts of the system itself whilst or before receiving the data. Radar tracking mostly aims for efficient statistical algorithms covering the identification of objects and the where and why pathways. This is opposite to visual tracking that mainly focuses on visual appearance - the who pathway. As images are complex data encoding a plethora of information, handcrafted algorithms are very constrained in their reliability, hence researchers started early in the 90s to apply machine learning to functions of feature selection with a huge body of literature since then. Learning rigorously the functional parts such as the camera model, the feature transform, the motion behaviour, the object identities individually as well as holistically is still in its infancy. There are results in human vision research, that learning tracking is an intentional, preconscious process [@jarvis-book2012], so there is a good chance for learning tracking from large weakly labeled data. There are also hints that humans simultaneous tracking is possible for a very low number of three to five objects, an incentive for artificial vision to beat human capabilities. Related Work {#sec:relwork} ============ Five significant surveys of tracking have been published recently [@yilmaz-ACM2006; @cannons-TR2008; @yang-Neurocomputing2011; @li-ACM2013; @smeulders-TPAMI2013]. Yilmaz et al. [@yilmaz-ACM2006] proposed a first taxonomy of tracking and proposed important challenges for trackers such as unconstrained video, context, efficient inference and application specific fine-tuning online either discriminative or un-/semisupervised by using machine learning. Cannons [@cannons-TR2008] gives a comprehensive overview of building blocks such as representation, initialisation, prediction, association and adaptation. The paper discusses pros and cons of features such as points, edges, contours, regions and their combinations. The work identifies as important challenges for future research feature selection i.e. the representation, the evolution of the representation over longer time periods, the combination of trackers and tracker evaluation. Yang et al. [@yang-Neurocomputing2011] review global and local features, the integration of object context and online generative versus discriminative learning. The work identifies ensemble learning to combine heterogeneous features into a coherent inference framework where particle filters are compared in detail. The work states the drift problem as important problem and proposes as future research consideration of adaptive priors, object context, pixel-wise segmentation, both generative and discriminative learning and SMC methods for inference. Li et al. [@li-ACM2013] survey solely representation with a summary of global and local features and a review on generative, discriminative and hybrid learning approaches. The survey concludes with the inevitable trade-off between robustness and accuracy by carefully balancing representation and prior information. The work proposes the need for a principled approach to reconstruct 3-D pose, the need for viewpoint and frame-rate invariance and the need for attentive mechanisms for initialisation similar to human vision. Finally, Smeulders et al. [@smeulders-TPAMI2013] compare 19 online trackers by their predicted bounding boxes on the ALOV benchmark. The paper concludes with the importance of the experimental setup which is crucial to assess performance and that tracking unknown objects in unknown scenarios is an open problem. The experiments show little conceptual consensus among the trackers and find none of the trackers superior. There is evidence that complex representations are inferior to simpler ones. Sparse and local features seem appropriate to handle rapid changes such as occlusion, clutter and illumination. Grand Challenges {#sec:challenges} ================ We summarise from this related work the following remaining challenges to solve tracking: Complexity ---------- A description of tracking limits the number of predictable latent variables. The higher the complexity, i.e. the information capacity of the description, the more information can be inferred by a set of latent variables. For example, the image template predicts either bounding boxes or pixelwise segmentations of the object but not 3-D pose which needs descriptions of depth. Higher complexity, however, comes with difficulty of collecting unambiguous evidence about the predictions of latent variables. Descriptions should be made simple to be informative but as complex to meet application specific demands. Uncertainty ----------- Trackers conform to a specification of quality parameters during a period of time such as the robustness needed for tolerating changes and for preventing failure. The description of tracking needs to be invariant to temporal changes of the scene, objects, camera and their relationships. For example, changes in object appearance need to be tolerated by the tracker. At this point, we have to distinguish vagueness and uncertainty. While vagueness refers to controllable risks of specified changes, uncertainty refers to risks of possible changes, i.e. a potential subset of changes is unspecified hence unknown to the tracker. An example is a tracker for arbitrary objects. Such trackers become important. Descriptions of arbitrary objects are therefore needed. Such descriptions should be sufficiently made time-invariant to meet application specific demands. Initialisation -------------- Application users or object detectors initialise trackers. Initialisation takes place in the first video frame and after full occlusions during tracking. For initialisation in the first video frame, the standard approach assumes either the human in the loop or the limitation to certain object categories. Online learning a detector of the individual object handles usually the problem of full occlusions during tracking. In case of uncertainty, a generalised initialisation is needed which is able to handle arbitrary objects. Such an attentive tracker detects and tracks salient objects occurring in the video. For example, a salient object might be an image region of atypical spatiotemporal characteristics compared to its neighbourhood. Attentive mechanisms based on saliency are also found in human vision. Trackers for arbitrary objects should be made attentive to fully exploit their potential. Computability ------------- Inference has to be as efficient and accurate as possible, being able to compute predictions of latent variables and satisfying quality parameters at the same time. Complexity of the description and the prediction of the latent variables go hand in hand, turning inference very quickly to computational intractability. A principled approach balances both, leading to an optimal graceful degradation of inference. Comparison ---------- Trackers need to be objectively compared by accepted experimental methodologies. This would foster scientific progress in the field. Disucssion ---------- The literature shows the enormous success of using machine learning to improve the robustness of tracking. While recent work and initiatives try to establish community platforms, evaluation protocols and allow new insights into tracking, only a few works consider the problem of initialisation. Vagueness, complexity and computability of tracking are strongly intertwined and suggest a common machine learning approach as principled solution. However, it is important to point out that although machine learning is very promising to control vagueness by fully exploiting video information, learning will fail in the most general case of uncertainty, as learning assume priors of the underlying random processes, a constraining assumption in case of real disjuncture between known random processes and new unknown processes with unknown statistics. [@amodei-CORR2016] emphasised recently this an other problems as robustness to distributional shift. Siamese Tracking {#sec:siamese} ================ Learning with a Siamese network [@baldi-JNC1993; @bromley-NIPS1993] is a promising approach to tackle some of these difficult tracking challenges. A Siamese network is a Y-shaped neural network that joins two network branches in final layers to produce a single output. The idea originated 1993 in fingerprint recognition [@baldi-JNC1993] and signature verification [@bromley-NIPS1993], where the task is to compare two imaged fingerprints or two hand-written signatures and infer identity. A Siamese network captures the comparison of the preprocessed input as a function[^3] of similarity with the advantageous ability to learn similarity and the features jointly and directly from the data [^4]. Despite their generality and usefulness in various applications, relatively less is known about statistical foundation and properties [@blackwell-SMSP1951; @nguyen-AOS2009]. Siamese networks have also been applied to face verification and recognition [@chopra-CVPR2005; @taigman-CVPR2014; @parkhi-BMVC2015; @schroff-CVPR2015], areal-to-ground image matching [@lin-CVPR2015], stereo matching [@zbontar-CVPR2015], patch matching [@han-CVPR2015; @simo-serra-ICCV2015; @zagoruyko-CVPR2015], optical flow [@dosovitskiy-ICCV2015], large-scale video classification [@karpathy-CVPR2014] and one-shot character recognition [@koch-ICMLW2015]. Overview -------- Motivated by these successful applications, some research groups studied very recently the Siamese networks for tracking [@chen-CORR2016; @tao-CVPR2016; @held-ECCV2016; @bertinetto-VOT2016; @valmadre-CVPR2017]. These proposed methods consider similarity as a priori given except for [@chen-CORR2016; @held-ECCV2016] who learn similarity and features jointly. Joint learning utilises the Y-shaped network architecture to its full extent, while assumptions, such as a given similarity, restrict parts of the network. Joint learning is currently little understood, while feature learning for the aim of compression has been extensively studied over decades by the signal processing community [@nguyen-AOS2009]. Learning similarity with given features as last case is rigorously studied in statistical decision theory and machine learning. All methods assume an initial given bounding box in the first frame and the presence of a search region in the next frame where the object template is matched. Object template and search region are input to the network except for [@tao-CVPR2016]. The attempt of all proposed methods is to learn a hierarchy of convolutional features of arbitrary training objects by ignoring categorisation and to train entirely offline, end-to-end the network by using back-propagation and infer at runtime the object’s bounding box either by regressing directly [@held-ECCV2016] or by estimating its centre position and scale in two subsequent steps [@chen-CORR2016; @bertinetto-VOT2016; @valmadre-CVPR2017] or by ranking proposed bounding box candidates given certain criteria to retrieve the best match [@tao-CVPR2016]. [@held-ECCV2016] showed that this approach learns very generic features which generalise to new objects and even new object categories not present in the training data. Siamese networks on one hand combine the expressive power of convolutional networks in the branches with real-time inference which is indispensable from an application point of view. On the other hand, the approach allows due to its simplicity a better understanding of the implications of learning jointly features and similarity. Proposed Methods ---------------- This section summarises the proposed methods concerning the technical details of network, training and inference and then compares and discusses the important differences to gain some insight into the approach of Siamese tracking. ### YCNN [@chen-CORR2016] propose as possibly first but unpublished attempt two identical branches similar to VGGNet [@simonyan-ICLR2015] with three conv and max-pooling layers, both linked to three fc layers. The conv layers share the same parameters. Each layer finishes with a ReLU except for the output which finishes with a sigmoid function. The network output is a 0-1 bounded prediction map with high values at pixels indicating object presence. The branches work as feature hierarchies aggregating fine-to-coarse spatial details, while the fc layers design spatiotemporal features as well as a general similarity function. Thus, YCNN learns discriminating features of growing complexity while simultaneously learning similarity between template and search region with corresponding prediction maps. Training is done in two stages on augmented images of objects from ImageNet and for fine-tuning with videos from VOT-15 [@kristan-ICCVW2015] and TB-100 [@wu-TPAMI2015]. Training minimises a weighted $L^2$ loss by using Adam [@Kingma-CORR2014], mini-batches and dropout. Weighting is important as nearly 95% of pixels in the prediction map have very low to zero values. During tracking, the feed forward pass infers then position as maximum in the prediction map. By averaging the prediction map over the five most confident maintained templates avoids drift. Repeating inference with scaled templates estimates additionally overall scale. ### SINT {#sec:sint} [@tao-CVPR2016] propose two identical query and search networks inherited from AlexNet [@krizhevsky-NIPS2012] and VGGNet with five conv and two max-pooling layers, three region pooling layers, an fc layer and a final $L^2$ normalisation layer. ReLUs follow each conv layer. Max-pooling is done after the first two conv layers. Both networks are unconnected but share the same weights. Instead of object and search region templates, the whole two subsequent frames are input, hence bounding boxes locating the object in the query frame and bounding boxes locating candidates in the search frame are additionally fed to the networks. The networks’ outputs are normalised features lying on the same manifold. Again, the networks work as feature hierarchies aggregating fine-to-coarse spatial details, however in this work similarity is a priori defined by the training loss function. So SINT learns discriminating solely features of growing complexity with bounding boxes in query and search frame and an additional binary variable indicating correct and incorrect pairs measured by the Jaccard index. Training is done on images of objects from ALOV [@smeulders-TPAMI2013]. Training minimises a margin contrastive loss and uses pre-training on ImageNet. During tracking, the query is fed with the initial bounding box in the first frame resulting in a query feature vector. Inference samples candidates at radial positions and different scales and feeds the search at once resulting in feature vectors for each candidate. An offline learned ridge regressor refines finally position and scale of the winning candidate with maximal inner product to the query. ### SiamFC [@bertinetto-VOT2016] propose two identical branches inherited from AlexNet with five conv layers, max-pooling following the first two conv layers and ReLUs after every conv layer except for conv5. A novel cross-correlation layer links the two conv5 layers. By waiving padding the whole network is fully-convolutional. The output is an unbounded correlation map with high values at pixels indicating object presence. As for YCNN and SINT, the branches can be seen as spatial description of increasing complexity which is embedded in a metric space where cross-correlation is used as similarity function. Like SINT, SiamFC learns discriminating solely the features with triplets of template, search region and corresponding prediction map. Values isotropically within a radius of the centre count correctly to the object’s position, hence are labeled positively whereas all other values are labeled negatively. Training is done on videos of objects from ImageNet [@russakovsky-IJCV2015]. Augmentation considers scale but not translation, because of the fully-convolutional network property. Training minimises a discriminative mean logistic loss by using SGD, mini-batches, Xavier initialisation and annealing of the learning rate. Tracking computes the position via the up-sampled prediction map for a given template. The tracker handles scale by searching over five different scale variations and updates scale by linear interpolation. ### CFNet [@valmadre-CVPR2017] adds a correlation filter and crop layers to the branch that concerns the template. These layers follow directly the convolutional network. The input is a larger region of the frame including the template, hence resolution of feature maps in the branches and prediction map is larger. Feature maps are further multiplied by a cosine window and cropped after correlation to remove the effect of circular boundaries. CFNet inherits the basic ability from SiamFC to discriminate spatial features with triplets of template region, search region and corresponding prediction map. Instead of unconstrained features, CFNet learns features that especially discriminate and solve the underlying ridge regression of the correlation layer by exploiting background samples in the surrounding region of the template. The learnt parameters of the correlation layer remain fixed during tracking, no online learning happens as shown by [@danelljan-ICCVW2015]. Training is done as with SiamFC by using the same algorithms on videos of objects from ImageNet. To make training end-to-end, emphasis has been on a differential correlation layer and on back-propagation of the parameters. Correlation is formulated in the Fourier space to preserve efficiency of computation. Tracking is similar simple as in SiamFC and computes position and scale by a single feed forward pass. The prediction map is multiplied by a spatial cosine window to penalise larger displacements. Instead of handling five different scale variations, scale is handled as by [@comaniciu-TPAMI2002]. To fully exploit the correlation filter, the initial template is updated in each frame by a moving average. ### GOTURN [@held-ECCV2016] proposes two convolutional branches inherited from AlexNet up to pool5. Both branches share the same parameters. These pool5 features of both branches are connected to a single vector and fed to three fc layers. ReLUs are used after each fc layer. The final fc layer links to an output layer with four nodes describing the bounding box. The output is scaled by a validated constant factor. GOTURN learns simultaneously the hierarchy of spatial features in the branches as well as spatiotemporal features and the similarity function in the fc layers to discriminate between template and search region with corresponding bounding boxes. Training is done in two stages on augmented images of objects from ImageNet and on videos from ALOV by using standard back-propagation of CaffeNet. Augmentation assumes linear translation and constant scale with parameters sampled from a Laplace distribution, hence small motion is assumed to occurs more frequently than larger displacements. Training minimises a $L^1$ loss between predicted and ground truth bounding box by using mini-batches, dropout and pre-training of the branches on ImageNet without fine-tuning these parameters to prevent overfitting. Tracking initialises the template in the first frame and updates the template with the predicted bounding box for each frame. Crops of the current and next frame yield template and search region. These crops are not exact but padded to add context. Discussion ---------- After understanding the methods’ details, this section compares the details concerning differences in branches, outputs and connections of branches. Differences in the networks’ training and inference are finally discussed. ![The proposed methods use network architectures of varying complexity: a) convolutional branches and fc connection layers (YCNN, GOTURN), b) convolutional branches and a single connection layer (SiamFC, CFNet), c) convolutional branches with equal final normalisation layers (SINT).[]{data-label="fig:net"}](siamese-networks){width="\linewidth"} ### Network Branches All proposed methods suggest convolutional branches inherited either from AlexNet or VGGNet with five conv layers except for YCNN that complements three layers by two fc layers and CFNet that studies one to five layers. The inheritance from AlexNet and VGGNet allows transfer learning from ImageNet and ALOV. The methods consider equal branches by effectively sharing the parameters which avoids during the fine-tuning overfitting to the small datasets currently available [@tao-CVPR2016]. The first two conv layers capture very local visual detail, for example edges, contributing to the accuracy of the tracker, while conv layers three to five aggregate this detail to an object specific description, for example category specific details, which is important for the robustness of the tracker [@danelljan-ICCVW2015]. Max-pooling as it is part of AlexNet and VGGNet introduces invariance to deformations of the object but it also reduces significantly image resolution and by that hinders improvements to the tracker’s accuracy. All authors except [@chen-CORR2016] recognise this limitation and use two max-pooling layers to trade-off accuracy and deformation invariance. [@valmadre-CVPR2017] also show important insights into the number of layers. They report saturation of tracking performance with increasing network depth and that more than five conv layers yield minor performance gains. CFNet implements the object specific description on higher layers with a correlation filter which allows an effective object specific description and fast computation in the Fourier domain. This significantly improves computational performance and shows that the branches are representable in various ways by combining layers of heterogeneous features. ### Connection of Branches All proposed methods except SINT connect the branches, SiamFC and CFNet with a single cross-correlation layer, YCNN and GOTURN with three fc layers. SINT omits the concatenating layer by using normalisation layers at the end of both branches. Fig. \[fig:net\] illustrates these three variations of network architecture. This Siamese network architecture of SiamFC, CFNet and SINT in combination with parameter sharing limit the feature selection to the spatial image domain. Instead, YCNN and GOTURN enable additional learning of spatiotemporal features in the concatenating layers, as argued by [@held-ECCV2016] such as “relationships between an object’s appearance change and its motion” which seems very general for different categories of objects. Parameter sharing has the consequence that all methods require appearance constancy between template and search region, hence [@chen-CORR2016]’s argument that YCNN’s deep features show “superiority of recognising an object with varying appearance” is questionable. SiamFC, Theoretically, the network of GOTURN generalises over the network of SiamFC which allows capturing features beyond sole visual features of the exemplar image and which allows regression of the bounding box instead of convoluting a final score map capturing potential positions of the exemplar image within the search image. The author’s argue that GOTURN learns a generic relationship between arbitrary motion and visual features, however this is not clear yet. Due to the more general Y-shaped architecture it might learn features beyond pure visual such as motion and their relationship in the fully-connected layers, however the network might also be able to learn context features as well. CFNet and SINT assume a specific function of similarity and the idea is to solely learn visual features to best match the given similarity. SINT even expresses similarity by the training loss which might have advantages in generalisation as particular different functions of similarity and training loss might derive adversary optimisation problems. This is not a problem for YCNN and GOTURN, as similarity and features are jointly learned, however, the interference with the particular training loss is unclear. ### Network Training Training is a crucial for sufficient performance. All methods describe basically two training phases, (i) a pre-training phase to transfer-learn generic features of objects from labeled datasets and (ii) a fine-tuning phase to adapt features to given video sets. The cross-correlation layer has here advantages as cross-correlation preserves the convolutional property of the whole network which introduces invariance to object translation. Therefore training samples must not contain translated versions which reduces significantly the effort for training. Less augmentation of training data is needed. The training loss and its choice has significant influence on the training result. [@held-ECCV2016] argue that $L^1$ is superior to $L^2$ as it penalises more harshly small errors near zero which increases substantially accuracy of the predicted bounding box. This argument is an exception, as none of the other studies show some insights into this important problem. [@held-ECCV2016] chose also different inputs for training and studied their influence on the mean error derived from VOT accuracy and robustness measures. They show that feeding the network with whole frames instead of template and search region pairs, the frames’ contexts are exploited which reduces significantly the mean error, especially in cases of occlusion. SINT is the only method following this insight but without any hints of their awareness. The reason is that their motivation comes from image retrieval where processing of frames is common. GOTURN allows the use of still images. ### Tracker Inference All proposed networks return in a single feed forward pass information about the bounding box in the search region. YCNN, SiamFC and CFNet return position and in a post-processing step then scale. SINT needs prior sampling of candidates in the search region and returns similarities to the template all at once thanks to the region proposal layer. The best candidate with maximum similarity defines the bounding box in the new frame. GOTURN is different as it regresses directly the bounding box. The idea of inferring direct position seems elegant and superior to the idea of candidate sampling as it allows e.g. in case of SiamFC and CFNet dense cross-correlation at pixel level. GOTURN regresses directly the bounding box and is the simplest recurrent network as it can be seen as unrolled recurrent network with Markov property. Although [@bertinetto-VOT2016] recognises this relationship, none of the works [@chen-CORR2016; @tao-CVPR2016; @bertinetto-VOT2016; @valmadre-CVPR2017] follow this fully sequential approach, as the intention is to learn a matching function. A significant advantage of YCNN, SINT, SiamFC and CFNet over GOTURN is that the predictions allows solutions for tracking in clutter, SINT however limited by the candidate samples. SiamFC and SINT consider a single initial template, while YCNN maintains the k-best templates, GOTURN keeps the last and CFNet a moving average template. [@held-ECCV2016] show that updating the template improves accuracy and robustness of the tracker. ### Tracker Results CFNet seems a promising method as it has state-of-the-art performance with 75fps[^5] with less than 4% of the parameters of other five layer methods (in total 600kB) such as SiamFC. This makes CFNet applicable to embedded applications. CFNet, SiamFC and SINT show comparable performance by reaching IoU/prec. 60/80% on OTB-2013 and one pass evaluation (OPE)[^6]. [@chen-CORR2016] reports significant lower IoU/prec. of 60/70%. [@held-ECCV2016] did not report results on OTB-2013. Conclusion {#sec:conclude} ========== The various combinations of possible inputs, outputs and features and their implementation as layers in the network need definitely future research work. There are strong pros for a fixed similarity function, nevertheless learning similarity with fixed features or learning similarity and features jointly might conceal success as shown in the fields of re-identification [@liao-CVPR2015] and sensor networks [@nguyen-ICML2004]. All methods keep for a good reason the tracking framework simple, namely to be able to better study the network’s properties. There is much room for improvement concerning the tracking, for example by combining the network with filtering methods. Seeing the Siamese network as matching function or seeing the network as simplest recurrent network poses important questions about the integration of the network into the tracking framework which have not been answered yet. More training data is needed as well as new ideas for combining supervised and unsupervised training approaches as labelled data will always be limited. There is currently little knowledge about the influence of training loss on the overall performance. Insights into these topics by in-depth ablative analysis such as done by [@held-ECCV2016; @valmadre-CVPR2017] are further needed. There are currently three lines of research: There is tracking research that assumes an initial label of the unknown object e.g. a bounding box and investigates tracking in the subsequent frames. These methods are combined with detection which on the one hand allows integrated perception but on the other hand the use of detectors unnecessarily restricts the tracker to certain object categories. The third line of research studies tracking jointly with attentional mechanism that does not assume any knowledge of the object. Acknowledgments {#acknowledgments .unnumbered} =============== Acknowledgment {#acknowledgment .unnumbered} ============== I thank all reviewers for reading the article and for their valuable comments which improved substantially the work. This research has received funding from the EU ARTEMIS Joint Undertaking under grant agreements no. 621429 (EM²) and from the FFG (Austrian Research Promotion Agency) on behalf of BMVIT, The Federal Ministry of Transport, Innovation and Technology. This work was supported by the AIT strategic research programme 2017 [*Visual Surveillance and Insight*]{}. [Roman Pflugfelder]{} is Scientist at the AIT Austrian Institute of Technology and lecturer at TU Wien. He received in 2002 a MSc degree in informatics at TU Wien and in 2008 a PhD in telematics at the TU Graz, Austria. In 2001, he was academic visitor at the Queensland University of Technology, Australia. His research focuses on visual motion analysis, tracking and recognition applied to automated video surveillance. He aims to combine sciences and theories in novel ways to gain theoretical insights into learning and inference in complex dynamical systems and to develop practical algorithms and computing systems. Roman contributed with more than 55 papers and patents to research fields such as camera calibration, object detection, object tracking, event recognition where he received awareness of media as well as several awards and grants for his scientific achievements. Roman is senior project leader at AIT where he has been managing cooperations among universities, companies and governmental institutions. Roman co-organised the Visual Object Tracking Challenges VOT’13-14 and VOT’16-17 and was program co-chair of AVSS’15. Currently he is steering committee member of AVSS. He is regular reviewer for major computer vision conferences and journals. For more details see <https://www.caa.tuwien.ac.at/cvl/staff/roman-pflugfelder>. [^1]: DSST is not a connectionist approach and was VOT’14 winner [@kristan-VOT2014]. [^2]: Other names are (hidden) state variables or variables. [^3]: A function of the class of Lipschitz functions $f: [-1,1]^d \to [-1,1]$[@shalev-shwartz-BOOK2014]. [^4]: Similarity is understood as decision function and features are known as experimental design in statistical decision theory [@berger-1985]. [^5]: GOTURN runs best at 100fps [@held-ECCV2016]. [^6]: SINT performs best with IoU/prec. 62.5/84.8% on OTB-2013 and OPE [@tao-CVPR2016].

[**PAC-BAYESIAN INDUCTIVE AND TRANSDUCTIVE LEARNING**]{}\ [Olivier Catoni]{}\ [*[May 28, 2006]{}*]{}\ [Abstract:]{} We present here a PAC-Bayesian point of view on adaptive supervised classification. Using convex analysis on the set of posterior probability measures on the parameter space, we show how to get local measures of the complexity of the classification model involving the relative entropy of posterior distributions with respect to Gibbs posterior measures. We then discuss relative bounds, comparing the generalization error of two classification rules, showing how the margin assumption of Mammen and Tsybakov can be replaced with some empirical measure of the covariance structure of the classification model. We also show how to associate to any posterior distribution an [*effective temperature*]{} relating it to the Gibbs prior distribution with the same level of expected error rate, and how to estimate this effective temperature from data, resulting in an estimator whose expected error rate converges according to the best possible power of the sample size adaptively under any margin and parametric complexity assumptions. Then we introduce a PAC-Bayesian point of view on transductive learning and use it to improve on known Vapnik’s generalization bounds, extending them to the case when the sample is made of independent non identically distributed pairs of patterns and labels. Eventually we review briefly the construction of Support Vector Machines and show how to derive generalization bounds for them, measuring the complexity either through the number of support vectors or through transductive or inductive margin estimates.\ [2000 Mathematics Subject Classification:]{} 62H30, 68T05, 62B10.\ [Keywords:]{} Statistical learning theory, adaptive statistics, pattern recognition, PAC-Bayesian theorems, VC dimension, local complexity bounds, randomized estimators, Gibbs posterior distributions, effective temperature, Mammen and Tsybakov margin assumption, transductive inference, compression schemes, Support Vector Machines, margin bounds.

--- abstract: 'The notion of being totally umbilic is considered for non-degenerate and degenerate submanifolds of semi-Riemanian manifolds. After some remarks on the general case, timelike and lightlike totally umbilic submanifolds of Lorentzian manifolds are discussed, along with their physical interpretation in view of general relativity. In particular, the mathematical notion of totally umbilic submanifolds is linked to the notions of photon surfaces and of null strings which have been used in the physics literature.' author: - | Volker Perlick\ [Institut für Theoretische Physik, Sekr. PN 7-1]{}\ [TU Berlin, 10623 Berlin, Germany]{}\ [Email: [email protected]]{} title: | On totally umbilic submanifolds of\ semi-Riemannian manifolds --- Introduction {#sec-intro} ============ In a Riemannian manifold, a submanifold is said to have an *umbilic point* at $p$ if, at this point $p$, the second fundamental form is a multiple of the first fundamental form. A submanifold is called *totally umbilic* if all of its points are umbilic. In $n$-dimensional Euclidean space, a $k$-dimensional complete connected submanifold with $2 \le k \le n-1$ is totally umbilic if and only if it is a $k$-sphere or a $k$-plane. If we generalize from Riemannian to semi-Riemannian manifolds (i.e., if the metric of the ambient space need not be positive-definite), a submanifold may be degenerate (i.e., the metric may induce a degenerate tensor field on this submanifold). In this case the standard text-book definition of the second fundamental form (or *shape tensor*) does not make sense; so many authors restrict the definition of totally umbilic submanifolds to the non-degenerate case, see, e.g., O’Neill [@o]. However, there is a fairly obvious way in which the definition of the second fundamental form can be generalized to include degenerate submanifolds. This is indicated in Exercise 9 on p.125 in O’Neill’s book [@o] but not used in the main part. The geometry of degenerate submanifolds of semi-Riemannian manifolds is discussed in detail in the books by Kupeli [@k] and by Duggal and Bejancu [@db] In this article I use the generalized definition of the second fundamental form, as suggested in the above-mentioned exercise of O’Neill’s book, and discuss, thereupon, some general properties of totally umbilic submanifolds which may or may not be degenerate. In the second part of the article I specialize to the case that the ambient space has Lorentzian signature and give several characterizations of totally umbilic submanifolds that are timelike or lightlike. The interpretation of these results in view of general relativity is also discussed. It is one of the main purposes of this article to link up the mathematical literature on totally umbilic submanifolds with some notions used in the physical literature; these are, in particular the notions of *photon surfaces* (introduced by Claudel, Virbhadra and Ellis [@cve] and also discussed by Foertsch, Hasse and Perlick [@fhp]) and of *null strings* (introduced by Schild [@sch]). Definition and general properties of totally umbilic submanifolds in semi-Riemannian manifolds {#sec:general} ============================================================================================== Let $(M,g)$ be a semi-Riemannian manifold and $\nabla$ the Levi-Civita connection of $g$. On every (immersed) submanifold $\tilde{M}$ of $M$, $g$ induces a second-rank tensor field $\tilde{g}$. If $\tilde{g}$ has a non-trivial kernel, the submanifold is called degenerate; otherwise it is called non-degenerate. In the non-degenerate case $(\tilde{M},\tilde{g})$ is a semi-Riemannian manifold in its own right; in particular, it defines a Levi-Civita connection on $\tilde{M}$. In the degenerate case, the ambient space induces, in general, no distinguished connection on $\tilde{M}$ Moreover, in the non-degenerate case at each point $p$ in $\tilde{M}$ the tangent space splits orthogonally, $$\label{eq:ortho} T_p M = T_p \tilde{M} + T_p^{\perp} \tilde{M}$$ where $T_p^{\perp} \tilde{M}$ denotes the set of all vectors in $T_pM$ that are perpendicular to $T_p\tilde{M}$ with respect to $g$. In the degenerate case, (\[eq:ortho\]) does not hold because $T_p \tilde{M}$ and $T_p^{\perp} \tilde{M}$ have a non-trivial intersection and do not span the whole tangent space $T_pM$. In other words, a vector in $T_pM$ cannot be decomposed uniquely into a component tangent to $\tilde{M}$ and a component perpendicular to $\tilde{M}$. This is the reason why the standard text-book definition of the second fundamental form, which makes use of this decomposition, does not work. However, this problem can be easily circumvented by using the quotient space $T_pM / T_p \tilde{M}$ instead of $T_p ^{\perp}\tilde{M}$. We denote the elements of this quotient space by square brackets, i.e., we write $$\label{eq:quotient} [Z_p] = \{ Z_p + Y_p \, \vert \, Y_p \in T_p \tilde{M} \}$$ for $Z_p \in T_pM$. If $Z$ is a vector field along $\tilde{M}$, we denote by $[Z]$ the map that assigns to each point $p \in T_p \tilde{M}$ the equivalence class $[Z_p ]$, where $Z_p$ is the value of $Z$ at $p$. Using this notation, we define the second fundamental form $\Pi$ of $\tilde{M}$ by the equation $$\label{eq:defPi} \Pi (X,Y) = [ \nabla _X Y ]$$ where $X$ and $Y$ are vector fields tangent to $\tilde{M}$. As $\nabla$ is torsion free, $\nabla _X Y - \nabla _Y X = [X,Y]$, and as with $X$ and $Y$ also the Lie bracket $[X,Y]$ must be tangent to $\tilde{M}$, it is clear that $\Pi$ is symmetric, $\Pi (X,Y) = \Pi (Y,X)$. This symmetry property implies that $\Pi$ is tensorial with respect to both arguments, because it is obviously tensorial with respect to the first one. This reasoning is quite analogous as for the standard text-book definition of the second fundamental form; the only difference to this standard definition lies in the fact that at each point $p \in \tilde{M}$ now $\Pi(X,Y)$ takes values in $T_pM/T_p\tilde{M}$ rather than in $T_p^{\perp} \tilde{M}$. For non-degenerate submanifolds, these two spaces can of course be identified in a natural fashion. With $\Pi$ given by (\[eq:defPi\]), we can now define the notion of being totally umbilic for submanifolds that may be degenerate or non-degenerate. \[def:umbilic\] A submanifold $\tilde{M}$ of a semi-Riemannian manifold is called *totally umbilic* if there is a vector field $N$ along $\tilde{M}$ such that $$\label{eq:umbilic} \Pi(X,Y) = [g(X,Y) N]$$ for all vector fields $X$ and $Y$ tangent to $\tilde{M}$. A totally umbilic submanifold with $[N]=[0]$ is called *totally geodesic*. Note that the property of being totally umbilic is invariant under conformal changes of $g$ whereas the property of being totally geodesic is not. Also note that for a totally umbilic submanifold the equivalence class $[N]$ is unique but the vector field $N$ is not. In the non-degenerate case we can make $N$ unique by requiring that it be perpendicular to $\tilde{M}$. In the totally geodesic case, we can of course choose $N=0$. For degenerate totally umbilic submanifolds that are not totally geodesic, however, there is no distinguished choice for the vector field $N$. Once $N$ has been chosen, the equation $$\label{eq:connection} \tilde{\nabla} _X Y = \nabla _X Y - g(X,Y) N$$ defines a torsion-free connection $\tilde{\nabla}$ on $\tilde{M}$. For non-degenerate totally geodesic submanifolds, $\tilde{\nabla}$ with the choice $N=0$ coincides with the Levi-Civita connection of $(\tilde{M}, \tilde{g})$. We now prove a proposition which is a simple consequence of the existence of the connection (\[eq:connection\]). In this proposition we use the following terminology. (This terminology comes from general relativity and is, actually, motivated only in the case that the metric of the ambient space has Lorentzian signature.) We call a vector field $X$ on $M$ lightlike if $g(X,X)=0$; a curve is called lightlike if it is the integral curve of a lightlike vector field. \[prop:geodesic\] - Let $\tilde{M}$ be a totally umbilic submanifold of $M$. Then a lightlike $\nabla$–geodesic that starts tangential to $\tilde{M}$ remains within $\tilde{M}$ $($for some parameter interval around the starting point$)$. - $\tilde{M}$ is totally geodesic if and only if every $\nabla$–geodesic that starts tangential to $\tilde{M}$ remains within $\tilde{M}$ $($for some parameter interval around the starting point$)$. From (\[eq:connection\]) we read that for lightlike vector fields on $\tilde{M}$ the equation $\tilde{\nabla} _X X =0$ is equivalent to $\nabla _X X =0$. In other words, the $\nabla$–geodesics with lightlike initial vectors tangent to $\tilde{M}$ are $\tilde{\nabla}$ geodesics and thus remain within $\tilde{M}$. This proves (a). In the totally geodesic case we may choose $N=0$, so the same argument works for non-lightlike initial vectors as well. This proves (b). Part (b) is, of course, the true justification for the name ’totally geodesic’. The following characterization is often useful. \[prop:tangent\] - $\tilde{M}$ is totally umbilic if and only if all vector fields $X$ and $Y$ which are tangent to $\tilde{M}$ with $g(X,Y)=0$ satisfy $\Pi (X,Y) = [0]$. - $\tilde{M}$ is totally geodesic if and only if all vector fields $X$ and $Y$ which are tangent to $\tilde{M}$ satisfy $\Pi (X,Y) = [0]$. Claim (b) and the ’only if’ part of claim (a) are obvious from Definition \[def:umbilic\]. To prove the ’if’ part of claim (a), we choose basis vector fields $E_1, \dots , E_l,L_1, \dots , L_m$ on $\tilde{M}$ such that the $E_i$ are pseudo-orthonormal and $g(L_{\mu},L_{\nu})=g(L_{\mu},E_i)=0$ for all $\mu,\nu = 1, \dots ,m$ and all $i=1, \dots ,l$. For each $i$, we define a vector field $N_i$ along $\tilde{M}$ by $\nabla _{E_i} E_i = g(E_i,E_i)N_i$. Considering for $X$ and $Y$ all linear combinations of $E_i$ and $E_j$ that are orthogonal to each other, our hypothesis implies that $[N_i]=[N_j]=:[N]$ for all $i,j=1, \dots , l$. With this information at hand, we consider for $X$ and $Y$ arbitray vector fields tangent to $\tilde{M}$, i.e., linear combinations of all $E_i$ and $L_{\mu}$; then our hypopthesis implies that $\Pi (X,Y)$ is, indeed, of the form (\[eq:umbilic\]). It follows directly from Definition \[def:umbilic\] that a non-degenerate one-dimensional submanifold is always totally umbilic and that a degenerate one-dimensional submanifold is totally umbilic if and only if it is the image of a geodesic. For this reason the notion of totally umbilic submanifolds is non-trivial only for $1 < \mathrm{dim} ( \tilde{M} ) < \mathrm{dim} (M)$. In an arbitrary semi-Riemannian manifold, the existence of non-trivial totally umbilic submanifolds is not guaranteed. For the case that the ambient space is Riemannian, existence criteria for totally umbilic foliations in terms of curvature conditions have been given by Walschap [@wa]. Generalizations to the semi-Riemannian case have, apparently, not been worked out so far. Here are two simple examples of semi-Riemannian manifolds that do admit non-trivial totally umbilic submanifolds. \[ex:flat\] O’Neill *[@o]*, p.$117$, considers the case where $(M,g)$ is pseudo-Euclidean, i.e. $\mathbb{R} ^n$ with a constant metric of arbitrary signature. He shows that for $n \ge 3$ a complete connected non-degenerate hypersurface is totally umbilic if and only if it is either a hyperplane or a hyperquadric. Every connected non-degenerate totally umbilic submanifold $\tilde{M}$ with $2 \le \mathrm{dim} ( \tilde{M} ) \le \mathrm{dim} (M) -2$ is a hypersurface in some pseudo-Euclidean subspace of dimension $\mathrm{dim} ( \tilde{M})+1$; this was proven by Ahn, Kim, and Kim *[@akk]* $($cf. Hong *[@ho]* for the Lorentzian case$)$. Thus, with the O’Neill result we know *all* non-degenerate totally umbilic submanifolds of pseudo-Euclidean space. \[ex:twisted\] Consider the case that $(M,g)$ is $($locally$)$ a *twisted product*. By definition, this means that $M$ admits coordinates $(u,v) = (u^1,\dots, u^m, v^1, \dots , v^{n-m})$ $($locally around any point$)$ such that the metric $g$ takes the form $$\label{eq:twist} g \, = \, h_{ij}(u) \, du^i \, du^j \, + \, \psi(u,v) \, k_{\mu \nu} (v) \, dv^{\mu} \, dv^{\nu}$$ with summation over $i,j$ from 1 to $m$ and over $\mu, \nu$ from 1 to $n-m$. The condition of $g$ being non-degenerate requires $\psi$ to be non-zero and $h_{ij}$ and $k_{\mu \nu}$ to be non-degenerate everywhere; otherwise, they are arbitrary. $($In the more special case that the “twisting function” $\psi$ is independent of $v$ one speaks of a [*warped product*]{}, cf. O’Neill *[@o]*.$)$ It is an elementary exercise to verify that for a metric of the form *(\[eq:twist\])* the submanifolds $u = {\mathrm{constant}}$ are totally umbilic and the submanifolds $v = {\mathrm{constant}}$ are totally geodesic. More generally, the following result is true. A semi-Riemannian manifold is $($locally$)$ a twisted product if and only if it $($locally$)$ admits two foliations ${\mathcal{F}}$ and ${\mathcal{G}}$ which are transverse and orthogonal to each other $($and thus non-degenerate$)$ with all leaves of ${\mathcal{F}}$ totally geodesic and all leaves of ${\mathcal{G}}$ totally umbilic, see Ponge and Reckziegel *[@pr]*, Theorem $1$. Recall that the notion of being totally umbilic is conformally invariant. So every metric that is $($locally$)$ conformal to a twisted product admits $($locally$)$ two foliations into non-degenerate totally umbilic submanifolds that are orthogonal to each other. This observation implies that a semi-Riemannian manifold can be $($locally$)$ foliated into non-degenerate totally umbilic hypersurfaces if and only if it admits $($locally$)$ coordinates $(u,v)=(u,v^1, \dots , v^ {n-1})$ such that $$\label{eq:hypersurface} g \, = \, \Phi(u,v) \big( \pm du^2 + \psi (u,v) \, k_{\mu \nu } (v) \, dv^{\mu} \, dv^{\nu} \, \big) \: .$$ Here we made use of the fact that, for a foliation into non-degenerate hypersurfaces, the orthocomplements of the leaves are one-dimensional and, thus. integrable. If we want to know if a given metric can be foliated into non-degenerate totally umbilic hypersurfaces, we may thus do this by checking whether it is isometric to *(\[eq:hypersurface\])*. Totally umbilic submanifolds of Lorentzian manifolds {#sec:Lorentzian} ==================================================== From now on we assume that the ambient space $(M,g)$ has Lorentzian signature $(+, \dots , + , -)$. We may then interpret $(M,g)$ as a spacetime in the sense of general relativity. (However, there is no need to restrict our discussion to the physically interesting case $\mathrm{dim} (M) =4$.) As usual in the Lorentzian case, we call the degenerate submanifolds *lightlike*. On a non-degenerate submanifold, the metric is either positive definite or again Lorentzian; in the first case, the submanifold is called *spacelike*, in the second case it is called *timelike*. It is our goal to discuss totally umbilic submanifolds that are timelike or lightlike. (The spacelike case is not very much different from the situation that the ambient space is Riemannian.) We begin with the timelike case. \[prop:timelike\] Let $\tilde{M}$ be a timelike submanifold with $2 \le \mathrm{dim} (\tilde{M}) \le \mathrm{dim} (M)$. Then $\tilde{M}$ is totally umbilic if and only if every lightlike geodesic that starts tangent to $\tilde{M}$ remains within $\tilde{M}$ $($for some parameter interval around the starting point$)$. The ’only if’ part is a special case of Proposition \[prop:geodesic\] (a). To prove the ’if’ part, choose a point $p \in \tilde{M}$ and two vectors $X_p$ and $Y_p$ in $T_p \tilde{M}$ with $g_p(X_p,Y_p)=0$. It is our goal to prove that $\Pi _p (X_p , Y_p) = [0]$ because then, by the tensorial property of $\Pi$, Proposition \[prop:tangent\] (a) proves that $\tilde{M}$ is totally umbilic. Owing to the tensorial property of $\Pi$, it suffices to consider the case that $g_p(X_p,X_p)=1$ and $g_p(Y_p,Y_p)=-1$. Then the vectors $L_p = \frac{1}{2} (Y_p + X_p)$ and $K_p = \frac{1}{2} ( Y_p - X_p)$ are lightlike. By hypothesis, we can find lightlike vector fields $L$ and $K$ on $\tilde{M}$ with $\nabla _L L = \nabla _K K = 0$ which take the values $L_p$ and $K_p$ at $p$. Writing $X=L-K$ and $Y=L+K$ we find $\Pi (X,Y) = [ \nabla _{(L-K)} (L+K)]= [ \nabla _L K - \nabla _K L]$ which is, indeed, equal to $[0]$ because with $L$ and $K$ also the Lie bracket $[L,K]$ is tangent to $\tilde{M}$. In general relativity, lightlike geodesics are interpreted as the wordlines of photons. Therefore, a timelike or lightlike submanifold $\tilde{M}$ is called a *photon surface* if each lightlike geodesic that starts tangent to $\tilde{M}$ remains within $\tilde{M}$ (for some parameter interval). In this terminology, Proposition \[prop:timelike\] says that a timelike submanifold of dimension $k \ge 2$ is totally umbilic if and only if it is a photon surface. The notion of a photon surface was discussed by Claudel, Virbhadra, and Ellis [@cve] for the case $k= \mathrm{dim} (M) -1$ and by Foertsch, Hasse and Perlick [@fhp] for the case $k =2$. A $k$-dimensional timelike submanifold can be interpreted as the history of a $(k-1)$-dimensional spatial manifold. Proposition \[prop:timelike\] says that this spatial manifold appears like a $(k-1)$-plane to the eye of every observer in $\tilde{M}$ if and only if its history is a totally umbilic submanifold. In particular, a 2-dimensional timelike submanifold can be interpreted as the history of a string; the condition of being totally umbilic means that the string looks like a straight line to an observer on the string. The best known non-trivial example is the surface $r=3m, \theta = \pi /2$ in Schwarzschild spacetime; it is the history of a circle which appears like a straight line to the eye of an observer who is situated on this circle. Other examples are worked out in [@fhp]. There the reader can also find a characterization of 2-dimensional timelike photon surfaces in terms of inertial forces and in terms of gyroscope transport. For axisymmetric and static situations, this connection was discussed already earlier in various articles by Abramowicz, see, e.g., [@a]. If a Lorentzian manifold admits a timelike conformal Killing field $K$ that is hypersurface-orthogonal, applying the flow of $K$ to a lightlike geodesic always gives a 2-dimensional timelike photon surface. The proof is worked out in [@fhp]. This demonstrates the existence of 2-dimensional timelike totally umbilic submanifolds in any conformally static spacetime. Actually, for the construction to work it is not necessary that $K$ be timelike; it suffices if it is nowhere orthogonal to the lightlike geodesic to which we want to apply the flow of $K$. Most known examples of 2-dimensional timelike photon surfaces are constructed in this way. However, this construction is not universal; with the help of Example \[ex:twisted\] one can construct 3-dimensional Lorentzian manifolds which do not admit any non-zero comformal Killing vector field but are foliated into 2-dimensional timelike photon surfaces. We emphasize again that, in an $n$-dimensional Lorentzian manifold the existence of a $k$-dimensional totally umbilic timelike submanifold is not guaranteed unless in the trivial cases $k=1$ and $k=n$. This implies that, in particular, the existence of totally geodesic timelike submanifolds of dimension $2 \le k \le n-1$ is not guaranteed. This has physical relevance for the case $k=2$ because 2-dimensional timelike submanifolds that are totally geodesic describe the history of self-gravitating strings, see Vickers [@v]. We now turn to the case of a lightlike submanifold $\tilde{M}$. We first observe that every such $\tilde{M}$ is ruled by a unique congruence of lightlike curves which are called the *generators* of $\tilde{M}$. This leads to the following characterization of totally umbilic lightlike submanifolds. \[prop:light\] A necessary condition for a lightlike submanifold to be totally umbilic is that the generators are geodesics. In a three-dimensional Lorentzian manifold, every two-dimensional lightlike submanifold $\tilde{M}$ is totally umbilic. The first statement is a special case of Proposition \[prop:geodesic\] (a). To prove the second statement, let $L$ be a vector field on $\tilde{M}$ that is tangent to the generators and $E$ any other vector field on $\tilde{M}$ that is linearly independent of $L$. Set $\nabla _E E =: g(E,E) N$. As $g(L,L)=0$, the vector field $\nabla _E L$ is perpendicular to $L$ and, thus, tangent to $\tilde{M}$. As $\nabla _L L$ is a multiple of $L$ and $g(E,L)=0$, the vector field $\nabla _L E$ is also perpendicular to $L$ and, thus, tangent to $\tilde{M}$. As a consequence, every linear combination $X=fE+hL$ satisfies $\Pi (X,X) = [f^2 g(E,E) N] = [g(X,X) N]$. As $\Pi$ is symmetric, this proves that $\tilde{M}$ is totally umbilic. For lightlike hypersurfaces, the generators are automatically geodesics. Schild [@sch] has defined a *null string* as a 2-dimensional lightlike submanifold whose generators are geodesics. Whereas every 2-dimensional lightlike submanifold can be interpreted as the history of a string whose individual points move at the speed of light, null strings are characterized by the additional condition that its individual points move on geodesics, i.e., freely like photons. In other words, we may visualize a null string as a one-parameter family of photons that are arranged like perls on a string. In a three-dimensional Lorentzian manifold, every two-dimensional lightlike submanifold is totally umbilic and it is a null string in the sense of Schild. In Schild’s article it is shown that null strings can be characterized by a variational principle. From this variational principle it is clear that null strings exist in every Lorentzian manifold. This can also be verified with the help of the following construction. Choose a one-dimensional spacelike submanifold $S$; at each point of $S$, choose a lightlike direction perpendicular to $S$ that depends smoothly on the foot-point; let $\tilde{M}$ be the union of the lightlike geodesics determined by these intial directions; in a neighborhood of $S$, this is indeed a submanifold and, by construction, it is a null string. (Farther away from $S$ the set $\tilde{M}$ constructed this way may form ’caustics’, i.e., it may fail to be a submanifold.) We can try to construct $k$-dimensional totally umbilic lightlike submanifolds in Lorentzian manifolds of arbitrary dimension by the same procedure, now starting with a $(k-1)$-dimensional spacelike submanifold $S$. However, even if we are lucky enough to find a totally umbilic initial submanifold $S$, it will not be guaranteed that the resulting lightlike submanifold will be totally umbilic everywhere. We end with an example where higher-dimensional totally umbilic lightlike submanifolds can be constructed owing to the existence of symmetries. \[ex:spheric\] Consider a $4$-dimensional Lorentzian manifold that is spherically symmetric and static, i.e., $$\label{eq:spheric} g \, = \, -A(r)^2 dt^2 + B(r)^2 dr^2 + r^2 ( \mathrm{sin}^2 \vartheta d \varphi ^2 + d \vartheta ^2 ) \, ,$$ e.g. the Schwarzschild metric with $A(r)^2=B(r)^{-2}=1- \frac{2m}{r}$. Let $S$ be the intersection of a hypersurface $r = \mathrm{constant}$ with a hypersurface $t= \mathrm{constant}$. This is a $2$-dimensional spacelike submanifold. Now choose at each point of $S$ a lightlike direction perpendicular to $S$, smoothly depending on the foot-point. $($You have to choose between two possibilities: the ingoing radial directions and the outgoing radial directions.$)$ Let $\tilde{M}$ be the union of all geodesics with the chosen initial direction. This is a submanifold near $S$. $($In the case at hand, the symmetry of the situation guarantees that $\tilde{M}$ is a submanifold everywhere, except at points where it meets the center of symmetry.$)$ It is easy to verify that $S$ is, indeed, a $3$-dimensional totally umbilic lightlike submanifold. The simplest examples of totally umbilic lightlike submanifolds constructed in this way are the light cones in Minkowski space. [99]{} B. O’Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983. D. N. Kupeli, Singular semi-Riemannian geometry, Kluwer, Dordrecht, 1996. K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Kluwer, Dordrecht, 1996. C. M. Claudel, K. S. Virbhadra, and G. F. R. Ellis, The geometry of photon surfaces, J. Math. Phys. 42 (2001) 818–838. T. Foertsch, W. Hasse, V. Perlick, Inertial forces and photon surfaces in arbitrary spacetimes, Class. Quantum Grav. 20 (2003) 4635–4652. J. A. G. Vickers, Generalized cosmic strings, Class. Quantum Grav. 4 (1987) 1–9. M. A. Abramowicz, Centrifugal force – a few surprises, Mon. Not. R. Astr. Soc. 245 (1990) 733–746. R. Ponge and H. Reckziegel, Twisted products in pseudo-Riemannian geometry, Geometriae Dedicata, 48 (1993) 15–25. G. Walschap, Umbilic foliations and curvature, Ill. J. Math. 41 (1997) 122–128. S. S. Ahn, D. S. Kim D-S, and Y. H. Kim, Totally umbilic Lorentzian sumbanifolds, J. Korean Math. Soc. 33 (1996) 507–512. S. K. Hong, Totally umbilic Lorentzian surfaces embedded in $L^n$, Bull. Korean Math. Soc. 34 (1997) 9–17. A. Schild, Classical null strings, Phys. Rev. D 16 (1977) 1722–1726.

--- abstract: | The Raychaudhuri equation has seen extensive use in general relativity, most notably in the development of various singularity theorems. In this rather technical article we shall generalize the Raychaudhuri equation in several ways. First an improved version of the standard timelike Raychaudhuri equation is developed, where several key terms are lumped together as a divergence. This already has a number of interesting applications, both within the ADM formalism and elsewhere. Second, a spacelike version of the Raychaudhuri equation is briefly discussed. Third, a version of the Raychaudhuri equation is developed that does not depend on the use of normalized congruences. This leads to useful formulae for the “diagonal” part of the Ricci tensor. Fourth, a “two vector” version of the Raychaudhuri equation is developed that uses two congruences to effectively extract “off diagonal” information concerning the Ricci tensor. Keywords: Raychaudhuri equation, timelike congruence, spacelike congruence, non-normalized congruence. author: - Gabriel Abreu - Matt Visser date: '22 December 2010; LaTeX-ed ' title: Some generalizations of the Raychaudhuri equation --- Introduction ============ The Raychaudhuri equation has become one of the standard workhorses of general relativity, particularly as applied to the singularity theorems. For textbook presentations see for instance [@Hawking-Ellis; @Wald; @Carroll; @Poisson]. Nevertheless, we feel that there are still some interesting ways in which the general formalism can be extended. There are four extended versions of the Raychaudhuri equation we wish to explore in this article: - *Single timelike unit vector field.*\ By collecting several terms in the usual formulation into a divergence, we obtain a particularly useful version that finds many applications in the ADM formalism and other situations. - *Single spacelike unit vector field.*\ This situation is most typically ignored. We will make a few hopefully clarifying comments. - *Single non-normalized vector field.*\ This somewhat simplifies the Raychaudhuri equation, at the cost of no longer having nice positivity properties. - *Two non-normalized vector fields.*\ This allows us to probe off-diagonal components of the Ricci tensor. These four extensions of the Raychaudhuri equation will soon be seen to each be useful in their own way, and to yield different information. We shall provide numerous examples below. Single unit timelike vector field ================================= This is the standard case. Let $u^a$ be a field of unit timelike vectors (a congruence). This does not have to be the 4-velocity of a physical fluid (though it might be), it applies just as well to the 4-velocities of an imaginary collection of “fiducial observers” \[FIDOs\]. Then it is a purely geometrical result (see for example Hawking and Ellis [@Hawking-Ellis], pp 82–84, or Wald [@Wald], or Carroll [@Carroll], or Poisson [@Poisson], or even [Wikipedia]{}, (note that there are sometimes minor disagreements of notation — typically just a factor of 2 in odd places) that: $$\label{E:R0} {\d\theta\over\d s} = - R_{ab} u^a u^b + \omega^2 - \sigma^2 - {1\over3}\theta^2 + \nabla_a \left( {\d u^a\over\d s} \right).$$ *This is the standard form of the Raychaudhuri equation.* The spatial projection tensor is $$h_{ab} = g_{ab} + u_a u_b.$$ This projection tensor has signature $\{0,+1,+1,+1\}$. Various shear and expansion related quantities are $$\theta_{ab} = h_{ac} \nabla^{(c} u^{d)} h_{db};$$ $$\theta= g^{ab}\theta_{ab} = h^{ab} \theta_{ab} = \nabla_a u^a;$$ $$\sigma_{ab} = \theta_{ab} - {1\over3} h_{ab} \theta;$$ $$\sigma^2 = \sigma_{ab} \sigma^{ab} \geq 0.$$ Vorticity related quantities are $$\omega_{ab} = h_{ac} \nabla^{[c} u^{d]} h_{db};$$ $$\omega^2 = \omega_{ab} \omega^{ab} \geq 0.$$ With these definitions we have the usual decomposition $$u_{a;b} = \omega_{ab} + \sigma_{ab} +{1\over3} \theta h_{ab} - {\d u_a\over\d s} \, u_b.$$ See (for example) pages 82–84 of Hawking and Ellis [@Hawking-Ellis]. Equation (\[E:R0\]) is Wald’s equation (9.2.11) [@Wald], supplemented with the $\nabla_a \left( {\d u^a\over\d s} \right)$ term due to allowing a non-geodesic congruence; you can deduce the presence of this term from the second line in his (9.2.10) by not assuming geodesic motion. Now consider the identity $${\d \theta\over \d s} = u \cdot \nabla \theta = \nabla\cdot (\theta u) - \theta \nabla\cdot u = \nabla\cdot (\theta u) - \theta^2.$$ Using this identity we can also write the Raychaudhuri equation in the slightly unusual forms $$\label{E:R1} \nabla_a \left( \theta u^a - {\d u^a\over\d s} \right)= - R_{ab} u^a u^b + \omega^2 - \sigma^2 + {2\over3}\theta^2 ,$$ or $$\label{E:R2} R_{ab} u^a u^b= \omega^2 - \sigma^2 + {2\over3}\theta^2 + \nabla_a \left(- \theta \; u^a + {\d u^a\over\d s} \right).$$ This extension of the usual Raychaudhuri equation is “close” to, but significantly more general than, a key technical result used by Padmanabhan and Patel in references [@paddy1; @paddy2; @paddy3]. Applications ============ We now consider several applications of the above formalism — these applications basically amount to strategically choosing an appropriate congruence. Vorticity-free congruence {#S:vorticity-free} ------------------------- Let $\Psi(x)$ be an arbitrary scalar field and define a set of fiducial observers \[FIDOs\] by $$u_a \propto \nabla_a \Psi.$$ Then normalizing we have $$u_a = - {\nabla_a \Psi\over ||\nabla \Psi||},$$ and furthermore $$\omega_{ab}=0.$$ The minus sign here is purely conventional, it guarantees that the $u^a$ is “future-pointing” in the direction of increasing $\Psi$. Conversely $$\omega_{ab}=0 \qquad \implies \qquad u_a \propto \nabla_a \Psi.$$ This is guaranteed by the Frobenius theorem. Then in this vorticity-free situation the extended Raychaudhuri equation reduces to $${\d\theta\over\d s} = - R_{ab} u^a u^b - \sigma^2 - {1\over3}\theta^2 + \nabla_a \left( {\d u^a\over\d s} \right),$$ or equivalently $$\label{E:18} R_{ab} u^a u^b= - \sigma^2 + {2\over3}\theta^2 + \nabla_a \left(- \theta u^a + {\d u^a\over\d s} \right),$$ or even $$R_{ab} u^a u^b= - \theta_{ab} \theta^{ab} + \theta^2 + \nabla_a \left(- \theta u^a + {\d u^a\over\d s} \right).$$ But since $u^a$ is now hyper-surface orthogonal we can use the slices of constant $\Psi$ to define a spacelike foliation — the scalar $\Psi$ serves (at least locally) as a “cosmic time” function. Then in terms of the extrinsic curvature $K_{ab}$ of the constant $\Psi$ hyper-surfaces we have, (using Misner, Thorne, and Wheeler [@MTW] sign conventions for the extrinsic curvature), the results: $$\theta_{ab} = -K_{ab}; \qquad \theta = - K;$$ $$\sigma_{ab} = - \left( K_{ab} - {1\over3} K h_{ab} \right);$$ $$\sigma^2 = {1\over2} \left[ K_{ab} K^{ab} -{1\over3} K^2 \right].$$ But then $$\label{E:paddy1} R_{ab} u^a u^b= - K_{ab} K^{ab} + K^2 + \nabla_a \left( K\; u^a + {\d u^a\over\d s} \right).$$ This is effectively one of the key technical results used by Padmanabhan and Patel in [@paddy1; @paddy2; @paddy3], but now we see that this result is actually a special case of a considerably more general result, and can be viewed as a relatively straightforward extension and then specialization of the Raychaudhuri equation. ADM formalism ------------- By definition, in any stably causal spacetime there is a globally defined “cosmic time” function $t(x)$ such that $\d t$ is always timelike. Then on the one hand the constant-$t$ slices are always spacelike and can be used to set up an ADM decomposition of the metric, while on the other hand $u = - (\d t)^\sharp /||\d t||$ is a vorticity-free unit timelike congruence, so that the results of the previous subsection apply. (As usual, $\d t^\sharp$ denotes the vector obtained form the one-form $\d t$ by “raising the index”, similarly $u^\flat$ will denote the one-form obtained from the vector $u$ by “lowering the index.) Consequently the extended Raychaudhuri equation can now be cast in the form $$\label{E:paddy2} R_{\hat t \hat t} = - K_{ab} K^{ab} + K^2 + \nabla_a \left( K\; u^a + {\d u^a\over\d s} \right).$$ This result complements and reinforces the information one obtains from the Gauss equations — see for example Misner, Thorne, and Wheeler [@MTW] pp 505–520, or Rendall [@Rendall] pp 23–24. The Gauss equations (for a spacelike hypersurface) are $$^{(4)} R_{abcd} = {}^{(3)} R_{abcd} + K_{ac} K_{bd} - K_{ad} K_{bc}.$$ Contracting once $$^{(4)} R_{ab} = {}^{(3)}R_{ab} - {}^{(4)}R_{acbd} u^c u^d + \tr(K) K_{ab} - (K^2)_{ab}.$$ Contracting a second time $$^{(4)} R = {}^{(3)}R - 2 \, {}^{(4)}R_{ab} u^a u^b + K^2 - \tr(K^2).$$ But now, since ${}^{(4)}R_{ab} u^a u^b$ has been given to us via the extended Raychaudhuri equation, we easily see that for a spacelike hypersurface $$^{(4)} R = {}^{(3)}R + \tr(K^2) - K^2 -2 \, \nabla_a \left( K\; u^a + {\d u^a\over\d s} \right).$$ Traditional derivations of this result are sometimes somewhat less transparent, and viewing it as an extension of the timelike Raychaudhuri equation is the cleanest derivation we have been able to develop. To see some of the deeper connections with the ADM formalism read (for example) §21.6 on pp 519–520 of Misner, Thorne, and Wheeler [@MTW]; note especially eq (21.88). See also exercise (21.10) on p 519. Also note the discussion by by Padmanabhan and Patel in references [@paddy1; @paddy2; @paddy3]. Also, we should warn the reader that Wald uses an opposite sign convention for the extrinsic curvature. See specifically Wald [@Wald] equation (10.2.13) on page 256. Static spacetimes ----------------- Let us now take the discussion in a rather different direction, and assume that the spacetime is [*static*]{}. That is, there exists a hypersurface-orthogonal Killing vector $k^a$ that is timelike at spatial infinity. Because it is hypersurface orthogonal then $k_a \propto \nabla_a \Psi$, and so $u^a = k^a/||k||$ is a set of FIDOs of the type considered in the previous section. But since $k^a$ is also a Killing vector we have $k_{(a;b)}=0$ and so obtain the quite standard result that $$\begin{aligned} \label{E:killing-shear} u_{(a;b)} &=& \nabla_{(a} \{k/||k||\}_{b)} = {k_{(a;b)}\over||k||} - {k_{(b} \nabla_{a)} ||k||\over ||k||^2} \nonumber\\ &=& - {k_{(b} \nabla_{a)} ||k||\over ||k||^2} = - {u_{(b} \nabla_{a)} ||k||\over ||k||} \nonumber\\ &=& - {u_{(b} ||k||_{,a)}\over ||k||} = {u_{(a} ||k||_{,b)}\over ||k||}.\end{aligned}$$ Hence $$\theta_{ab}=0 \quad\implies\quad K_{ab}=0 \quad\implies\quad K=0.$$ That is, in static spacetimes the extrinsic curvature of the time-slices is zero (in addition to the congruence being vorticity free). The Raychaudhuri equation then specializes to the particularly simple result $$R_{ab} \; u^a u^b= \nabla_a \left( {\d u^a\over\d s} \right).$$ This is essentially the technical result we used in our derivation of an entropy bound for static spacetimes [@Abreu1; @Abreu2], though in those articles we had derived it from an old result due to Landau and Lifshitz [@Landau]. (The original Landau–Lifshitz result is obtained via a straightforward but tedious series of index manipulations, with little geometrical insight.) Stationary spacetime — Killing congruence {#S:Killing} ----------------------------------------- What can we now do for *stationary*, as opposed to *static* spacetimes? (This distinction is relevant to “rotating spacetimes”, for example Kerr spacetimes versus Schwarzschild spacetimes. See for instance [@Kerr1; @Kerr2; @Kerr-book; @Kerr-survey].) The (asymptotically) timelike Killing vector $k= \partial_t$ \[that is, $k^a = (1; 0,0,0)^a$\] is no longer hypersurface orthogonal. Nevertheless we can still define the timelike Killing congruence $$u^a = {k^a\over ||k||}.$$ This timelike congruence corresponds to a class of FIDOs \[not ZAMOs, not zero angular momentum observers\] that sit at fixed spatial coordinate position [@membrane; @Abreu3]. This timelike congruence, even though it is *not* hypersurface orthogonal, still satisfies equation (\[E:killing-shear\]). So even though there is no longer any interpretation of the shear in terms of an extrinsic curvature, we still have $$\theta_{ab}=0,$$ whence both $$\sigma_{ab}=0; \qquad\hbox{and} \qquad \theta=0.$$ Therefore $$R_{ab} u^a u^b= \omega^2 + \nabla_a \left( {\d u^a\over\d s} \right).$$ However, unless further assumptions are made, we cannot do much with the $\omega^2$ term. Generically we have $$\begin{aligned} u_{[a;b]} &=& \nabla_{[a} \{k/||k||\}_{b]} = {k_{[a;b]}\over||k||} - {k_{[b} \nabla_{a]} ||k||\over ||k||^2} \nonumber\\ &=& {k_{[a;b]}\over||k||} - {k_{[b} \nabla_{a]} ||k||\over ||k||^2} = {k_{[a;b]}\over||k||} - {u_{[b} \nabla_{a]} ||k||\over ||k||} \nonumber\\ &=& {k_{[a;b]}\over||k||} - {u_{[b} ||k||_{,a]}\over ||k||}.\end{aligned}$$ This implies $$\omega^{ab} = h^{ac} h^{bd} \; {k_{[c;d]}\over||k||},$$ whence $$R_{ab} u^a u^b= +{h^{ac} h^{bd} \; k_{[a;b]} \; k_{[c;d]}\over||k||^2 } + \nabla_a \left( {\d u^a\over\d s} \right).$$ Unfortunately this does not simplify any further, and without further assumptions for the timelike Killing congruence on a stationary spacetime we should just be satisfied by the *inequality*: $$\label{E:inequality} R_{ab} u^a u^b \geq \nabla_a \left( {\d u^a\over\d s} \right).$$ Stationary axisymmetric spacetimes ---------------------------------- In a stationary axisymmetric spacetime consider the vorticity-free congruence of section \[S:vorticity-free\] (not the Killing congruence of section \[S:Killing\]). Because of the axisymmetry the congruence $u = - (\d t)^\sharp/||\d t||$ must then be a linear combination of the two Killing vectors, $k_t=\partial_t$ and $k_\phi=\partial_\phi$, in which case $\theta=\nabla\cdot u = 0$. In this case equation (\[E:18\]) reduces to $$\label{E:18b} R_{ab} u^a u^b= - \sigma^2 + \nabla_a \left( {\d u^a\over\d s} \right),$$ which implies, for the natural vorticity-free congruence on an stationary axisymmetric spacetime $$\label{E:18c} R_{ab} u^a u^b \leq \nabla_a \left( {\d u^a\over\d s} \right).$$ It is this particular inequality that we used in reference [@Abreu3] to place an entropy bound on rotating fluid blobs. (Note that the direction of the inequality has changed between equations (\[E:inequality\]) and (\[E:18c\]), but that is merely due to the fact that we are using different timelike congruences.) Single unit spacelike vector field ================================== In counterpoint, we now let $u^a$ be a field of unit spacelike vectors. The projection tensor becomes $$h_{ab} = g_{ab} - u_a u_b.$$ In contrast to the timelike situation the projection tensor is now of indefinite signature $\{-1,+1,+1,0\}$. One can still formally define the quantities $\theta_{ab}$, $\theta$, $\sigma_{ab}$, and $\omega_{ab}$, but they no longer have the same physical interpretation in terms of shear and vorticity. Furthermore since the projection tensor has indefinite signature we now *cannot* guarantee either $\sigma^2\geq0$ or $\omega^2 \geq 0$. On the other hand, the Raychaudhuri equation is formally unaffected. That is, the fundamental equations (\[E:R0\]), (\[E:R1\]), and (\[E:R2\]), continue to hold as they stand. If we now consider a vorticity-free spacelike congruence, it will be hypersurface orthogonal to a timelike hypersurface. (That is, the normal to the hypersurface is spacelike, while the tangent space to the hypersurface can be chosen to have a basis of one timelike and two spacelike tangent vectors.) In this situation we can without loss of generality set $u = (\d\Psi)^\sharp/||\d\Psi||$. Then $\omega_{ab}\to 0$, while in terms of the extrinsic curvature $\sigma_{ab} \to -K_{ab}$ as for vorticity free timelike congruencies. Thus equation (\[E:paddy1\]) is formally unaffected and can now be cast in the form $$\label{E:paddy3} R_{\hat n \hat n} = - K_{ab} K^{ab} + K^2 + \nabla_a \left( K\; u^a + {\d u^a\over\d s} \right).$$ On the other hand, because $u$ is now a spacelike normal to a timelike hypersurface there is a key sign flip in the Gauss equations, which now read $$^{(4)} R_{abcd} = {}^{(3)} R_{abcd} - K_{ac} K_{bd} +K_{ad} K_{bc} .$$ Contracting twice $$^{(4)} R = {}^{(3)}R + 2 \, {}^{(4)}R_{ab} u^a u^b + \tr(K^2) - K^2.$$ Therefore for a timelike hypersurface we have $$^{(4)} R = {}^{(3)}R - \tr(K^2) + K^2 +2 \, \nabla_a \left( K\; u^a + {\d u^a\over\d s} \right).$$ In summary, for spacelike congruences the Raychaudhuri equation itself is formally unaffected (though the projection tensor is slightly different and we can no longer rely on the non-negativity of $\sigma^2$ and $\omega^2$). However applications of the Raychaudhuri equation, specifically anything involving the Gauss equations for embedded hypersurfaces, typically exhibit a limited number of sign flips. Single non-normalized vector field ================================== Now consider an *non-normalized* vector field $u^a$, either spacelike, timelike, or null. What if anything can we say about the quantity $$R_{ab} u^a u^b = \quad ???$$ Following and modifying the discussion of Wald [@Wald], see (E.2.28) on page 464: $$\begin{aligned} R_{ab} u^a u^b &=& R^c{}_{acb} u^a u^b \nonumber\\ &=& - u^a \left[ \nabla_a \nabla_b - \nabla_b \nabla_a \right] u^b \nonumber\\ &=& - \nabla_a (u^a \nabla_b u^b) + (\nabla_a u^a)(\nabla_b u^b) \nonumber\\ && + \nabla_b (u^a \nabla_a u^b) - (\nabla_b u^a)(\nabla_a u^b) \qquad \nonumber\\ &=& \nabla_a (- u^a \nabla_b u^b + u^b \nabla_b u^a) \nonumber\\ &&+ (\nabla\cdot u)^2 - (\nabla_b u_a)(\nabla^a u^b) \nonumber\\ &=& \nabla\cdot\{ (u\cdot \nabla) u - (\nabla\cdot u) u \} \nonumber\\ &&+ (\nabla\cdot u)^2 - (\nabla_b u_a)(\nabla^a u^b) .\end{aligned}$$ In obvious notation, using $\theta=\nabla\cdot u$, this can be cast as $$\begin{aligned} R_{ab} u^a u^b &=& \nabla\cdot\{ \nabla_u u - \theta u \} + \theta^2 \nonumber\\ && - \nabla_{(a} u_{b)} \nabla^{(a} u^{b)} + \nabla_{[a} u_{b]} \nabla^{[a} u^{b]}.\end{aligned}$$ This result can be viewed as another generalization of the Raychaudhuri equation. The advantage of this particular formula is that we have not carried out any projections, and have not even committed ourselves to the nature of the congruence, be it spacelike, timelike, or null. One disadvantage is that because of the Lorentzian signature of spacetime we *cannot* (at least not without further assumption) guarantee $$\begin{aligned} \nabla_{(a} u_{b)} \; \nabla^{(a} u^{b)} \geq 0 \quad???\end{aligned}$$ $$\begin{aligned} \nabla_{[a} u_{b]} \; \nabla^{[a} u^{b]} \geq 0 \quad ???\end{aligned}$$ Two specific applications come readily to mind: - For any Killing vector $u^a = k^a$ we have $\nabla_{(a} u_{b)}=0$, and consequently $\theta=0$. Therefore for any Killing vector whatsoever $$\begin{aligned} R_{ab} k^a k^b &=& \nabla\cdot\{ \nabla_k k \} + \nabla_{[a} k_{b]} \nabla^{[a} k^{b]} . \end{aligned}$$ - For any one arbitrary exact one-form $u = d \Psi$, even a locally exact one-form, we have $\nabla_{[a} u_{b]}=0$, while $\theta=\nabla^2\Psi$ and $\nabla_{(a} u_{b)} \nabla^{(a} u^{b)} = \Psi_{;a;b} \Psi^{;a;b}$. Therefore for any locally exact one-form whatsoever $$\begin{aligned} R^{ab} (\d \Psi)_a (\d \Psi)_b &=& \nabla\cdot\{ \nabla_{\d\Psi}\d\Psi - (\nabla^2\;\Psi) \d\Psi \} \nonumber\\ &&+ (\nabla^2\Psi)^2 - \Psi_{;a;b} \,\Psi^{;a;b}.\end{aligned}$$ In fact, $\Psi$ could simply be one of the spacetime coordinates (defined on some suitable local coordinate patch) in which case this version of the Raychaudhuri equation turns into a statement about the diagonal components of the Ricci tensor in a coordinate basis $$\begin{aligned} R^{\Psi\Psi} &=& \nabla\cdot\{ \nabla_{\d\Psi}\d\Psi - (\nabla^2\;\Psi) \d\Psi \} \nonumber\\ &&+ (\nabla^2\Psi)^2 - \Psi_{;a;b} \,\Psi^{;a;b}.\end{aligned}$$ More boldly, if one chooses $\Psi$ to be a harmonic coordinate, ($\nabla^2 \Psi=0$), and this can always be done locally, then $$\begin{aligned} R^{\Psi\Psi} &=& \nabla\cdot\{ \nabla_{\d\Psi}\d\Psi \} - \Psi_{;a;b} \,\Psi^{;a;b}.\end{aligned}$$ In summary, this extension of the Raychaudhuri equation has given us some useful computational formulae. Two non-normalized vector fields ================================ We shall now ask if it is possible to extract any useful information by considering two different congruences simultaneously. Motivation ---------- To motivate this particular extension of the Raychaudhuri equation, recall that many decades ago Landau and Lifshitz had shown that in any stationary spacetime [@Landau] (§105 equation (105.22), for a recent application of this result see [@Abreu1; @Abreu2]): $$R_{0}{}^{0} = {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{0a} \, \Gamma^i{}_{a0} \right).$$ (Here $a\in\{0,1,2,3\}$; $i\in\{1,2,3\}$.) But because the metric is stationary ($t$ independent) we can also write this as $$R_{0}{}^{0} ={1\over\sqrt{-g_4}} \, \partial_b \left( \sqrt{-g_4} \, g^{0a} \, \Gamma^b{}_{a0} \right).$$ To begin converting this into a coordinate-free statement, note that $$R_{0}{}^{0} = R^a{}_b (\d t)_a (\partial_t)^b = R^{a}{}_{b} \; (\d t)_a \; k^b.$$ Here we have had to use *both* the timelike Killing vector $k$, for which $k^a = (\partial_t)^a =(1,0,0,0)^a$, *and* the one-form $\d t$, for which $(\d t)_a= (1,0,0,0)_a$. By direct computation $$\begin{aligned} g^{0a} \, \Gamma^b{}_{a0} &=& g^{ca} \, \Gamma^b{}_{ad} \; (\d t)_c k^d = \Gamma^b{}_{cd} \; (\d t)^c k^d \nonumber\\ &=& \Gamma^b{}_{cd} \; k^c (\d t)^d = \{\partial_d k^b +\Gamma^b{}_{cd} k^c \} \, (\d t)^d \nonumber\\ &=& (\nabla_d k^b) (\d t)^d = (\d t)^d (\nabla_d k^b).\end{aligned}$$ But then $$\begin{aligned} R_0{}^0 &=& {1\over\sqrt{-g_4}} \, \partial_b \left( \sqrt{-g_4} \, g^{0a} \, \Gamma^b{}_{a0} \right) \nonumber\\ &=& {1\over\sqrt{-g_4}} \, \partial_b \left( \sqrt{-g_4} (\d t)^d (\nabla_d k^b) \right) \nonumber\\ &=& \nabla_b \{ (\d t)^d (\nabla_d k^b) \}.\end{aligned}$$ So the Landau–Lifshitz result is equivalent to the statement that in any stationary spacetime $$R^{a}{}_{b} \; (\d t)_a k^b = \nabla_b \{ (\d t)^d (\nabla_d k^b) \} = \nabla \cdot ( \nabla_{\d t^\sharp} k ).$$ So some linear combination of Ricci tensor components is given by a pure divergence. Note that two *different* vector fields are involved. This observation naturally leads to the question: Is it possible to come up with a variant of the Raychaudhuri equation that depends on *two* congruences $u^a$ and $v^a$? Something of the form $$R_{ab} \; u^a v^b = \quad ???$$ We shall see how this is done below. For now, let us mention that $$\begin{aligned} (\nabla_d k^b) (\d t)^d &=& (\nabla^d k^b) (\d t)_d = - (\nabla^b k^d) (\d t)_d \nonumber\\ &=& - \nabla^b \{k^d (\d t)_d\} + k^d \nabla^b (\d t)_d \nonumber\\ &=& - \nabla^b \{ 1\} + k^d \nabla^b \nabla_d t \nonumber\\ &=& k^d \nabla^b \nabla_d t = k^d \nabla_d \nabla^b t.\end{aligned}$$ So the Landau–Lifshitz result can also be written in the alternative form $$R^{a}{}_{b} \; (\d t)_a k^b = \nabla_b \{ k^d \nabla_d \nabla^b t \} = \nabla\cdot(\nabla_k \d t^\sharp).$$ Finally note that $$(\nabla_d k^b) (\d t)^d (\d t)_b= (\nabla_d k_b) (\d t)^d (\d t)^b = 0,$$ so the vector $\nabla_{\d t^\sharp} k= \nabla_k \d t^\sharp$ is perpendicular to $\d t^\sharp$. Construction ------------ Following and modifying the discussion of Wald [@Wald], see equation (E.2.28) on page 464: $$\begin{aligned} R_{ab} u^a v^b &=& R^c{}_{acb} u^a v^b \nonumber\\ &=& - u^a \left[ \nabla_a \nabla_b - \nabla_b \nabla_a \right] v^b \nonumber\\ &=& - \nabla_a (u^a \nabla_b v^b) + (\nabla_a u^a)(\nabla_b v^b) \nonumber\\ && + \nabla_b (u^a \nabla_a v^b) - (\nabla_b u^a)(\nabla_a v^b)\qquad \nonumber\\ &=& \nabla_a (- u^a \nabla_b v^b + u^b \nabla_b v^a) \nonumber\\ && + (\nabla\cdot u) (\nabla\cdot v) - (\nabla_b u_a)(\nabla^a v^b).\end{aligned}$$ With minor notational changes and given the symmetry of the Ricci tensor this can also be written as $$\begin{aligned} R_{ab} u^a v^b &=& \nabla\cdot\{ (u\cdot \nabla) v - (\nabla\cdot v) u \} \nonumber\\ &&+ (\nabla\cdot u) (\nabla\cdot v) - (\nabla_b u_a)(\nabla^a v^b),\end{aligned}$$ and $$\begin{aligned} R_{ab} u^a v^b &=& \nabla\cdot\{ (v\cdot \nabla) u - (\nabla\cdot u) v \} \nonumber\\ &&+ (\nabla\cdot u) (\nabla\cdot v) - (\nabla_b u_a)(\nabla^a v^b) .\end{aligned}$$ Furthermore (in obvious notation) this can again be rewritten as $$\begin{aligned} R_{ab} u^a v^b &=& \nabla\cdot\{ \nabla_u v - \theta_v u \} + \theta_u \theta_v \nonumber\\ &&- \nabla_{(a} u_{b)} \nabla^{(a} v^{b)} + \nabla_{[a} u_{b]} \nabla^{[a} v^{b]}. \label{E:crossed1}\end{aligned}$$ and $$\begin{aligned} R_{ab} u^a v^b &=& \nabla\cdot\{ \nabla_v u - \theta_u v \} + \theta_u \theta_v \nonumber\\ &&- \nabla_{(a} u_{b)} \nabla^{(a} v^{b)} + \nabla_{[a} u_{b]} \nabla^{[a} v^{b]}. \label{E:crossed2}\end{aligned}$$ Note the similarities to the single-congruence case, and note particularly the presence of a divergence term generalizing the standard Raychaudhuri equation. To check the equivalence of these two formulae note $$\begin{aligned} (\nabla_u v - \theta_v u) &-& (\nabla_ v u- \theta_u v) \nonumber \\ &=& [\nabla_u v +\theta_u v] - [\nabla_v u +\theta_v u] \nonumber\\ &=& \nabla\cdot[u\otimes v - v\otimes u] \nonumber\\ &=& \nabla\cdot[u\wedge v].\end{aligned}$$ That is, the difference of these two currents is the divergence of a 2-form, which makes it automatically closed. Generalizing the Landau–Lifshitz result --------------------------------------- Let $u=k$ be any Killing vector, and let $v^\flat$ be an arbitrary (locally) exact one-form, so $v= (\d\Psi)^\sharp$ where $\Psi(x)$ is an arbitrary scalar. Then $ \nabla_{(a} u_{b)} = \nabla_{(a} k_{b)} = 0$, and so $\theta_u=0$. Furthermore $\nabla_{[a} v_{b]} = \nabla_{[a} \nabla_{b]} \Psi = 0$, so from equation (\[E:crossed1\]) we have $$R_{ab} \; k^a \nabla^b \Psi = \nabla\cdot \{ \nabla_k \d\Psi - (\nabla^2\Psi) k\},$$ while from equation (\[E:crossed2\]) we have $$R_{ab} \; k^a \nabla^b \Psi = \nabla\cdot \{ \nabla_{\d\Psi^\sharp} k\} .$$ This nicely generalizes the Landau–Lifshitz result to any arbitrary Killing vector and any arbitrary (locally) exact one form $\d\Psi$, not just $\d t$. (That these two formulae are equivalent follows from the discussion in the previous section above.) Note the (standard) Landau–Lifshitz result corresponds to $k^a\to(\partial_t)^a$ and $\Psi \to t$. Now choose a coordinate system adapted to the Killing vector $k$. Let $k = \partial_{\KK}$ define a Killing coordinate $\K$, so that all geometrical objects are independent of the coordinate $\K$. Let $\Psi$ also be viewed as a coordinate, relabel it as $x^a$, possibly distinct from $\K$, and with no claim that $x^a$ necessarily corresponds to a Killing vector. Then $$R_{\KK}{}^a = \nabla\cdot \{ \nabla_{(\d x^a)^\sharp} \partial_{\KK}\} .$$ Unwrapping the covariant derivatives we see $$R_{\KK}{}^{a} ={1\over\sqrt{-g_4}} \, \partial_b \left( \sqrt{-g_4} \, g^{a c} \, \Gamma^b{}_{c\KK} \right).$$ If we now let the index $i$ range over every coordinate except the Killing coordinate $\K$ then, because all geometrical objects are independent of the coordinate $\K$, we have $$R_{\KK}{}^{a} ={1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{a c} \, \Gamma^i{}_{c\KK} \right).$$ This equation, ultimately based on our two-congurence extension of the Raychaudhuri equation (\[E:crossed1\]), is now very much in Landau–Lifshitz form, but us definitely considerably more powerful than the original Landau–Lifshitz result. Landau–Lifshitz in axial symmetry --------------------------------- Since in a stationary spacetime with axial symmetry we have a second azimuthal Killing vector $k^a\to (\partial_\phi)^a$, and could also consider $\Psi \to \phi$, then there are three additional Landau–Lifshitz like results: $$R_\phi{}^t = R_{ab} (\partial_\phi) ^a \nabla^b t = \nabla\cdot \{ \nabla_{\d t^\sharp} \partial_\phi \};$$ $$R_t{}^\phi = R_{ab} (\partial_t)^a \nabla^b \phi= \nabla\cdot \{ \nabla_{\d\phi^\sharp} \partial_t\};$$ $$R_\phi{}^\phi = R_{ab} (\partial_\phi)^a \nabla^b \phi = \nabla\cdot \{ \nabla_{\d\phi^\sharp} \partial_\phi\}.$$ Let the indices $A,B\in\{t,\phi\}$ then we can collect these results (four of them altogether) as $$R_A{}^B = \nabla\cdot \{ \nabla_{(\d x^B)^\sharp} \partial_{A} \}.$$ Unwrapping the covariant derivatives $$R_{A}{}^{B} = {1\over\sqrt{-g_4}} \, \partial_b \left( \sqrt{-g_4} \, g^{Ba} \, \Gamma^b{}_{aA} \right).$$ If we now let the index $i$ range over every coordinate except the two Killing coordinates $t$ and $\phi$, then $$R_{A}{}^{B} = {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{Ba} \, \Gamma^i{}_{aA} \right).$$ Making this all very explicit, there are now four Landau–Lifshitz like results in total. They are: $$R_{t}{}^{t} = {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{ta} \, \Gamma^i{}_{at} \right);$$ $$R_{t}{}^{\phi} = {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{\phi a} \, \Gamma^i{}_{at} \right);$$ $$R_{\phi}{}^{t} = {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{ta} \, \Gamma^i{}_{a\phi} \right);$$ $$R_{\phi}{}^{\phi} = {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{\phi a} \, \Gamma^i{}_{a \phi} \right).$$ Furthermore, recall that in stationary axisymmetric spacetimes we can always choose coordinates to block diagonalize the metric: $g_{ab} = g_{AB} \oplus g_{ij}$. But then $$\begin{aligned} g^{Ba} \, \Gamma^i{}_{aA} &=& g^{BC} \, \Gamma^i{}_{CA} = g^{BC} \, g^{ij} \Gamma{}_{jCA} \nonumber\\ &=& - {1\over2} \, g^{BC} \, g^{ij} \; \partial_j g_{CA}.\end{aligned}$$ So finally we have the relatively compact result $$R_{A}{}^{B} = -{1\over2}\, {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{BC} \, g^{ij} \; \partial_j g_{CA} \right).$$ This can be rearranged in a number of different ways. As an illustration we point out $$\begin{aligned} R_{AB}&=& -{1\over2}\, {1\over\sqrt{-g_4}} \, \partial_i \left( \sqrt{-g_4} \, g^{ij} \; \partial_j g_{AB} \right) \nonumber\\ && + {1\over2} \, g^{ij} \; \partial_i g_{AC} \; g^{CD} \;\partial_j g_{DB}.\end{aligned}$$ We again see that our two-congruence extension of the Raychaudhuri equation has given us additional useful information regarding the Ricci tensor which might be difficult to extract by other means. Discussion and conclusions ========================== In this somewhat technical article we have developed several useful extensions of the usual Raychaudhuri equation. The main theme has been to relate various linear combinations of components of the Ricci tensor to divergences of suitably defined fluxes. We have worked with timelike congruences, spacelike congruences, and non-normalized congruences, in all cases being able to say just a little bit more (and sometimes much more) than standard the Raychaudhuri equation would imply. One potentially far-reaching result is the “two congruence” extension of the Raychauduri equation presented in equations (\[E:crossed1\]) and (\[E:crossed2\]). 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C. W. Misner, K. S. Thorne and J. A. Wheeler, *Gravitation*, (Freeman, San Francisco, 1973) A. D. Rendall, *Partial differential equations in general relativity*, (Oxford University Press, England, 2008) G. Abreu and M. Visser, “Tolman mass, generalized surface gravity, and entropy bounds”, Phys. Rev. Lett.  [**105**]{} (2010) 041302 \[arXiv:1005.1132 \[gr-qc\]\]. G. Abreu and M. Visser, “Entropy bounds for uncollapsed matter”, ERE2010 conference, arXiv:1011.4538 \[gr-qc\]. L. D. Landau and E. M. Lifschitz, *The Classical Theory of Fields* (Pergamon Press, Oxford, 2000), Chaps. 11, 12. See esp eq (105.22) on p 358 §105 \[4th revised english edition.\] R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics”, Phys. Rev. Lett.  [**11**]{}, 237 (1963). R. P. Kerr, “Gravitational collapse and rotation”, published in: *Quasi-stellar sources and gravitational collapse: Including the proceedings of the First Texas Symposium on Relativistic Astrophysics*, edited by Ivor Robinson, Alfred Schild, and E.L. Schücking (University of Chicago Press, Chicago, 1965), pages 99–102.\ The conference was held in Austin, Texas, on 16–18 December 1963. D. L. Wiltshire, M. Visser and S. M. Scott, *The Kerr spacetime: Rotating black holes in general relativity*, (Cambridge University Press, England, 2009) M. Visser, “The Kerr spacetime: A brief introduction”, arXiv:0706.0622 \[gr-qc\]. K. S. Thorne, R. H. Price and D. A. Macdonald, *Black holes: The membrane paradigm*, (Yale University Press, New Haven, 1986) G. Abreu and M. Visser, “Entropy bounds for uncollapsed rotating bodies”, arXiv:1012.2867 \[gr-qc\].

**$\delta$-derivations of semisimple finite-dimensional structurable algebras.** [^1] **Ivan Kaygorodov$^{a,b}$, Elizaveta Okhapkina$^{b,c}$** $^a$ Instituto de Matemática e Estatística, Universidade de São Paulo, Brasil,\ $^b$ Sobolev Institute of Mathematics, Novosibirsk, Russia,\ $^c$ Novosibirsk State University, Russia. [Abstract. ]{} In this paper we show the absence of nontrivial $\delta$-derivations of semisimple finite-dimensional structurable algebras over an algebraically closed field of characteristic not equal to 2,3,5. [**Keywords:**]{} $\delta$-derivation, structurable algebra. Introduction ============ The concept of $\delta$-derivation first appeared in papers by V. Filippov [@Fil]-[@Filll], as a generalization of ordinary derivations. Recall that for a fixed $\delta$ of the main field $F,$ under $\delta$-derivation of algebra $A$ we understand a linear mapping $\phi$, which satisfies the condition $$\begin{aligned} \label{delta} \phi(xy)=\delta(\phi(x)y+x\phi(y))\end{aligned}$$ for arbitrary elements $x,y \in A.$ In article [@Fill] V. Filippov proved that every prime Lie algebra has no nonzero $\delta$-derivations, if $\delta \neq -1,0,\frac{1}{2},1$. Subsequently in papers [@kay]-[@kay_de5] was given the description of $\delta$-derivations of semisimple finite-dimensional Jordan algebras and $\delta$-(super-)derivations of semisimple finite-dimensional Jordan superalgebras. Later, A. Shestakov generalized some results of [@kay] and described ternary derivations of simple finite-dimensional Jordan algebras [@shest]. $\delta$-derivations of Lie superalgebras were considered by I. Kaygorodov [@kay_lie; @kay_lie2] and P. Zusmanovich [@Zus]. The analogue of $\delta$-derivation for $n$-ary algebras was investigated in [@kay_nary; @kgind]. We note that some generalizations of $\delta$-derivations were proposed in articles [@kay_nary_ann]-[@kay_popov] and $\delta$-derivations were used for studying prime Lie algebras by V. Filippov [@Fil], for construction of post-Lie structures by D. Burde and K. Dekimpe [@BurdeDekimp], for studying speciality of Jordan superalgebas by I. Kaygorodov and V. Zhelyabin [@kay_zh], in analysis of the derivations of current Lie algebras by P. Zusmanovich [@Zusss], and for studying homotopes of Novikov algebras by V. Filippov and V. Sereda [@Filly]. Examples and definitions ======================== The class of structurable algebras was introduced in 1978 by B. Allison [@Allis]. It contains such objects as tensor multiplication of composition algebras, $56$-dimensional Freudenthal module over $E_{7}$ with natural binary operation, a $35$-dimensional algebra $T(C)$. Also, for these algebras a natural generalization of Kantor-Koecher-Tits construction holds. Structurable algebras are algebras with the unit, which we are going to designate as $e$, and involution  $\bar{},$ which satisfy the identity: $$\begin{aligned} \label{St.1}\left[T_{z},V_{x,y}\right]=V_{T_{z}x,y} - V_{x,T_{\overline{z}}y}, \text{ where } T_{z}, V_{x,y} \in End(A),\end{aligned}$$ $$V_{x,y}(z)=(x\overline{y})z+(z\overline{y})x-(z\overline{x})y, T_{z}=V_{z,1} \text{ for } x,y,z\in A.$$ Let ($A$, $\bar{}$ ) be an algebra with involution. We can consider vector space $A$ as a direct sum $A=H\oplus S$, where $H=\left\lbrace a\in A|\overline{a}=a\right\rbrace $ is the set of symmetric elements and $S=\left\lbrace a\in A|\overline{a}=-a\right\rbrace $ is the set of antisymmetric elements. Let us give some examples of structurable algebras (see [@Smi2; @Smi]): $S1)$ Let ($A$, $\bar{}$ ) be an unital associative algebra with the involution. Then ($A$, $\bar{}$ ) is a structurable algebra. $S2)$ Let $J$ be a Jordan algebra and   $\bar{ }=id$. For ($J$, $\bar{}$ ) the identity (\[St.1\]) turns into the following identity $$\left[L_{c},V_{a,b}\right]=V_{ca,b}-V_{a,cb}$$ which holds in an arbitrary Jordan algebra. Therefore ($J$, $\bar{}$ ) is a structurable algebra. $S3)$ Let ($E$, $\bar{}$ ) be an associative algebra with involution and $W$ a left $E$-module relative to the operation $\circ :E\times W\rightarrow W$. Let $W$ possesses a Hermitian form $h$, that is a bilinear mapping $h: W\times W \rightarrow E$ such that $$\overline{h(w_{1},w_{2})}=h(w_{2},w_{1}),$$ $$h(e_{1}\circ w_{1},w_{2})=e_{1}h(w_{1},w_{2})$$ for any $w_{1}, w_{2}\in W$ and $e_1\in E$. The operations of multiplication and involution on the space $A=E\oplus W$ are defined as follows: $$(e_{1},w_{1}) \bullet (e_{2},w_{2})=(e_{1}e_{2}+h(w_{2},w_{1}),e_{2}\circ w_{1}+\overline{e_{1}}\circ w_{2}),$$ $$\overline{(e,w)}=(\overline{e},w).$$ The algebra $(A, \bar{} )$ is a structurable algebra. It is a generalization of Jordan algebra of bilinear form and is called the algebra of Hermitian form. The algebra of Hermitian form is simple if the Hermitian form $h$ on the left $E$-module is nondegenerate and $(E, \bar{} )$ is a central simple algebra with involution. $S4)$ Let ($G_{1}$, $\bar{}$ ) and ($G_{2}$, $\bar{}$ ) be composition algebras with standard involution (for more about the composition algebras see, for example, [@ZSSS]). Let us define on algebra $A=G_{1}\otimes G_{2}$ the involution $\overline{x_{1}\otimes x_{2}}=\overline{x_{1}}\otimes \overline{x_{2}}$. Therefore $(A, \ \bar{} \ )$ is a structurable algebra. $S5)$ Let $J, J'$ be vector spaces over $F$ with trilinear forms $N$ and $N'$ which are related by a nondegenerate bilinear form $T:J\times J'\rightarrow F$. For $j,k\in J, j',k'\in J'$, we define $j\times k\in J'$ and $j'\times k'\in J$ by the condition $$T(l,j\times k)=N(j,k,l), T(j'\times k',l')=N'(j',k',l')$$ for any $l\in J, l'\in J'$. On $J$ and $J'$ we define the operation $\#$ as: $$j^{\#}=\frac{1}{2}j\times j, j'^{\#}=\frac{1}{2}j'\times j', \mbox{ for } j\in J, j'\in J'.$$ The triple $(T,N,N')$ is called an admissible triple defined on the pair of spaces $(J,J')$ if $N$ and $N'$ are nontrivial and satisfy the conditions: $$(j^{\#})^{\#}=\frac{1}{6}N(j,j,j)j,$$ $$(j'^{\#})^{\#}=\frac{1}{6}N'(j',j',j')j'$$ for $j\in J, j'\in J'$. Let $(T,N,N')$ be an admissible triple on a pair of spaces $(J,J')$. On a space of matrices $$A=\left\{\begin{bmatrix} \alpha& j\\ j'& \beta\\ \end{bmatrix}:\alpha,\beta\in F,j\in J,j'\in J'\right\}$$ the multiplication $$\begin{bmatrix}\alpha&j\\j'&\beta\\ \end{bmatrix} \begin{bmatrix}\gamma&k\\k'&\delta\\ \end{bmatrix}=\begin{bmatrix}\alpha \gamma+T(j,k')&\alpha k+\delta j+j'\times k'\\ \gamma j'+\beta k'+j\times k &T(k,j')+\beta\delta\\ \end{bmatrix}$$ and involution $$\overline{\begin{bmatrix} \alpha& j\\ j'& \beta\\ \end{bmatrix}}=\begin{bmatrix} \beta& j\\ j'& \alpha\\ \end{bmatrix}$$ are defined. The obtained structurable algebra ($A$, $\bar{}$ ) is called an algebra constructed from an admissible triple $(T,N,N')$ on a pair of spaces $(J,J')$. $S6)$ Let $C$ be a Cayley-Dickson algebra with an involution (about a Cayley-Dikson algebra see [@ZSSS]). The set $S$ of antisymmetric elements is a $7$-dimensional simple non-Lie Malcev algebra relative to operation commutation $[\cdot ,\cdot ]$ in $C$ (see [@kuz]); it is $S^{(-)}$. On $S^{(-)}$ the symmetrical nondegenerate invariant bilinear form $(\cdot ,\cdot )$ is such that for $x,y \in S$ the condition $$[[x,y],y]=(y,y)x-(x,y)y$$ is satisfied. Let M be a subspace of tensor product $S\otimes S$ which is generated by the set $\left\lbrace s\otimes r-r\otimes s:s,r\in S\right\rbrace $. We let $H=S\otimes S/M$ and on the direct sum of spaces $H\oplus S$ the commutative $\odot$ and anticommutative $[\cdot,\cdot ]$ operations are defined by the following conditions: $$[s_{1},s_{2}]=[s_{1},s_{2}],$$ $$[s,s_{1}\otimes s_{2}]=[s,s_{1}]\otimes s_{2}+s_{1}\otimes [s_{1},s_{2}],$$ $[s_{1}\otimes s_{2},s_{3}\otimes s_{4}]=$ $$=(s_{1},s_{3})[s_{2},s_{4}]+(s_{1},s_{4})[s_{2},s_{3}]+(s_{2},s_{3})[s_{1}.s_{4}]+(s_{2},s_{4})[s_{1},s_{3}],$$ $$s_{1}\odot s_{2}=s_{1}\otimes s_{2},$$ $$s\odot(s_{1}\otimes s_{2})=\frac{1}{2}(s_{1},s_{2})s+\frac{1}{4}(s,s_{1})s_{2}+\frac{1}{4}(s,s_{2})s_{1},$$ $(s_{1}\otimes s_{2})\odot(s_{3}\otimes s_{4})=$\ $$=\frac{1}{4}[s_{1},s_{3}]\otimes [s_{2},s_{4}]+\frac{1}{4}[s_{1},s_{4}]\otimes[s_{2},s_{3}]+\frac{1}{2}(s_{1},s_{2})s_{3}\otimes s_{4}+\frac{1}{2}(s_{3},s_{4})s_{1}\otimes s_{2}$$ for $s, s_{1}, s_{2}, s_{3}, s_{4}\in S$, where on the right side of equality under $[ \cdot , \cdot ]$ we understand the commutator in algebra $C$. The operations of multiplication and involution in $H\oplus S$ are defined in the following way: $$xy=x\odot y+\frac{1}{2}[x,y],$$ $$\overline{h+s}=h-s, \mbox{ where }x,y\in H\oplus S, h\in H, s\in S.$$ So obtained algebra with involution which is built by means of Cayley-Dikson algebra $C$ is designated as $T(C)$. Let us note that according to the results of [@Smi2; @Smi], the algebras of types $S1)$-$S6)$ exhaust all simple finite-dimensional structurable algebras over algebraically closed fields of characteristic not equal to $2,3,5.$ As was noted above, for a fixed element $\delta$ from the main field under $\delta$-derivation of algebra $A$ we understand a linear mapping $\phi:A\rightarrow A$ which for arbitrary $x,y \in A$ satisfies the condition $$\phi(xy)=\delta(\phi(x)y+x\phi(y)).$$ Centroid $\Gamma(A)$ of algebra $A$ is a set of linear mappings $\chi: A \rightarrow A$ with condition $\chi(ab)=\chi(a)b=a\chi(b).$ It is clear that any element of the centroid of an algebra is a $\frac{1}{2}$-derivation. Any endomorphism $\phi$ of algebra $A$ such that $\phi(A^{2})=0$ is a $0$-derivation. Any derivation of algebra $A$ is a $1$-derivation. Nonzero $\delta$-derivation $\phi$ is a nontrivial $\delta$-derivation if $\delta \neq 0,1$ and $\phi \notin \Gamma(A).$ Main lemmas =========== Here and below, all algebras are considered over an algebraically closed field $F$ with the characteristic not equal to $2, 3, 5$, although some of the results of the following lemmas will be correct with some relaxing of the required conditions on the field. Make a trivial direct calcucation and using [@kay Theorem 2.1], we can obtain that if $\phi$ is a nontrivial $\delta$-derivation of structurable algebra $A$, then $\delta=\frac{1}{2}$ and for any element $x \in A$ it is true that $\phi(x)=ax=xa,$ when $a \in A$ is a certain fixed element. **Lemma 1.** Let $A$ be an algebra of nondegenerate Hermitian form on a left $E$-module, where ($E$, $\bar{}$ ) is a simple central associative algebra with the involution. Then every $\frac{1}{2}$-derivation of $A$ is trivial. **Proof.** We will denote the unit of the algebra $A$ and of the algebra $E$ as $e$. Then we suppose $$\phi(e)=(\alpha e+e^{*},w^{*}),\mbox{ where } \alpha \in F, e^{*} \in E, w^{*}\in W$$ and the projection of $e^*$ on the vector space generated by $e$ is $0$. It is essential to prove $e^{*}=w^{*}=0$. Let us specify the mapping $\psi=\phi-\phi_{0}$, where $\phi_{0}(x)=\alpha x$. The mapping $\psi$ is a $\frac{1}{2}$-derivation as the sum of two $\frac{1}{2}$-derivations. It is easy to see, $$\psi(a,0)=\psi(e)\bullet(a,0)=(e^{*}a,a\circ w^{*}),$$ but from the other side $$\psi(a,0)=(a,0)\bullet \psi(e)=(ae^{*},\overline{a}\circ w^{*}).$$ We see that $e^{*}$ is in the commutative center of the central algebra ($E$, $\bar{}$ ) and hence, $e^{*} = \beta e$, where $\beta \in F$ and $\beta=0.$ Then we can consider $\psi(e)=(0,w^{*})$. Let us notice the following: $$\begin{aligned} \label{metka1} \psi((a,0)\bullet(0,w^{*}))=((a,0)\bullet(0,w^{*})) \bullet \psi(e)=(h(w^{*},\overline{a}\circ w^{*}),0) =(h(w^{*},w^{*})a,0),\end{aligned}$$ and also $$\psi((a,0)\bullet(0,w^{*}))=\psi(e)\bullet ((a,0)\bullet(0,w^{*}))=(h(\overline{a}\circ w^{*},w^{*}),0)=(\overline{a}h(w^{*},w^{*}),0).$$ Comparing the components from the algebra $E$ we get: $$\begin{aligned} \label{mark2}h(w^{*},w^{*})a=\overline{a}h(w^{*},w^{*}).\end{aligned}$$ Let us show that $h(w^{*},w^{*})$ is in the commutative center of the algebra $E$. It is easy to see that $$\psi((a,0)\bullet(0,w^{*}))=\frac{1}{2}\psi(a,0)\bullet(0,w^{*})+\frac{1}{2}(a,0)\bullet\psi(0,w^{*})= (\frac{1}{2}h(w^{*},\circ w^{*})a+\frac{1}{2}ah(w^{*},w^{*}),0).$$ Now, using (\[metka1\]) we can obtain that $h(w^{*},w^{*})$ is in the commutative center $Z(E)$. Let us show that $Eh(w^{*},w^{*})$ is an ideal in $E$. It is easy to see that $aEh(w^{*},w^{*})\subseteq Eh(w^{*},w^{*})$. Considering $Eh(w^{*},w^{*})a$ and using the associative property of the algebra $E$ we have $$(Eh(w^{*},w^{*}))a=E(h(w^{*},w^{*})a)=E(ah(w^{*},w^{*}))\subseteq Eh(w^{*},w^{*}).$$ As $E$ is a simple associative unital algebra, and for the ideal $Eh(w^{*},w^{*})$ either $Eh(w^{*},w^{*})=0$ or $Eh(w^{*},w^{*})=E$. The last case is impossible, because then the element $h(w^{*},w^{*})$ not equal to zero, and consequently, is invertible. Then from (\[mark2\]) we can conclude that from $(a-\overline{a})h(w^{*},w^{*})=0$ it follows that $a-\overline{a}=0$. If the involution is identity then the algebra reduces to the case of a Jordan algebra, which is described in [@kay]. If we have non-identity involution, then $h(w^{*},w^{*})=0$, which means that $\psi(w^{*})=0$. Let $u\in W$ be an arbitrary element. Then $$\psi((0,u) \bullet (0,w^{*}))=\frac{1}{2}\psi((0,u)) \bullet (0,w^{*})$$ and $$((0,u) \bullet (0,w^{*}))\bullet (0,w^{*})=0.$$ Let us notice that $h(W,Ew^{*})$ is an ideal in $E$. It is easy to see that: $$ah(W,E\circ w^{*})=h(a\circ W,E \circ w^{*})\subseteq h(W,E \circ w^{*})$$ and $$h(W,E\circ w^{*})a=\overline{\overline{a}h(E \circ w^{*},W)}\subseteq h(W,E\circ w^{*}).$$ As the Hermitian form $h:W\times W\rightarrow E$ is nondegenerate, the ideal is nonzero and as the algebra $E$ is simple, $h(W,E\circ w^{*})=E$. Then $\sum\limits_i (0,u_{i}) \bullet (0,a_{i} \circ w^{*})=e $, and we may conclude that $$e=\sum\limits_i (0,u_{i}) \bullet (0,a_{i}\circ w^{*})=\sum\limits_i h(a_{i}\circ w^{*},u_{i})=\sum\limits_i a_{i}h(w^{*},u_{i}).$$ We can say that $(0,w^*)$ is in the commutative center of algebra $A$. It is easy to see that $$(0,w^{*})= (0,(\sum\limits_i a_{i}h(w^{*},u_{i})) \circ w^{*})= \sum\limits_i (0, h(w^{*},u_{i})\circ w^*) \bullet ( a_{i},0)= \sum\limits_i (( (0,u_i) \bullet (0,w^*)) \bullet (0, w^*)) \bullet ( a_{i},0)=0.$$ And so we get $\phi(e)=\alpha e$. The lemma is proved. **Lemma 2.** The algebra of the tensor multiplication of composition algebras has no nontrivial $\frac{1}{2}$-derivations. **Proof**. Let $A,B$ be the composition algebras (about composition algebras see, for example, [@ZSSS]) with the units $1_A,1_B$ and $e$ be the unit of the algebra $A\otimes B$. If $\phi$ is a $\frac{1}{2}$-derivation of the algebra $A\otimes B$, then $\phi(e)=\sum\limits_i a_i \otimes b_i, a_i\in A, b_i\in B.$ It is easy to see that $a_i$ and $b_i$ are in the commutative centers of the algebras $A$ and $B$, respectively. It is easy to see that $$\phi(ac\otimes 1_B)=\dfrac{1}{2}\phi(a\otimes 1_B)(c\otimes 1_B)+\dfrac{1}{2}(a\otimes 1_B)\phi(c\otimes 1_B).$$ From the previous relation we get $$(ac\otimes1_B)\phi(1_A\otimes 1_B)=\phi(ac\otimes 1_B)=\phi((a\otimes 1_B)(c\otimes 1_B))=$$ $$\frac{1}{2}((a\otimes 1_B)\phi(1_A\otimes 1_B))(c\otimes 1_B)+\frac{1}{2}(a\otimes 1_B)((c\otimes 1_B)\phi(1_A\otimes 1_B)),$$ that is, $$\sum\limits_{i}(ac)a_{i}\otimes b_{i}=\dfrac{1}{2}\sum\limits_{i}((aa_{i})c\otimes b_{i}+a(ca_{i})\otimes b_{i}).$$ The last relation implies $\sum\limits_{i}(2(ac)a_{i}-(aa_{i})c-a(ca_{i}))\otimes b_{i}=0$ $2(ac)a_{i}-(aa_{i})c-a(ca_{i})=0$. Therefore, we can conclude that is, $(a,a_{i},c)=0.$ As composition algebras are alternative, $(a, c, a_i)=0=(a_i, a, c)$. Thus, $a_{i}$ is in commutative-associative center $Z(A)$. As the algebra $A\otimes B$ is isomorphic to algebra $B\otimes A$, analogously we get that $b_{i}$ is in commutative-associative center $Z(B)$. Now we are going to show that the mapping $\phi: A\otimes B \rightarrow A\otimes B $ which is defined by the rule $$\phi(x \otimes y)=(x \otimes y)(\sum\limits_i a_i \otimes b_i),\mbox{ where }a_i \in Z(A), b_i \in Z(B),$$ is the element of the centroid of the algebra $A\otimes B$, that is a trivial $\frac{1}{2}$-derivation. It is easy to see that $$\phi((a\otimes b)(c\otimes d))=\phi(ac\otimes bd)=(ac\otimes bd)(\sum\limits_i a_{i}\otimes b_{i})=$$ $$\sum\limits_{i}aca_{i}\otimes bdb_{i}=\sum\limits_{i}(aa_{i}\otimes bb_{i})(c \otimes d)=\phi(a\otimes b)(c\otimes d).$$ Analogously, we show that $\phi((a\otimes b)(c\otimes d))=(a\otimes b)\phi(c\otimes d).$ The lemma is proved. **Lemma 3.** The algebra of an admissible triple $(T,N,N')$ has no nontrivial $\frac{1}{2}$-derivations. **Proof**. Let $\phi(e)= \begin{bmatrix} \alpha & j \\ j'& \beta \\ \end{bmatrix} $ and $x=\begin{bmatrix} \gamma & k \\ k'& \delta \\ \end{bmatrix}$, where $\alpha,\beta,\gamma,\delta\in F$, $k,j\in J;k', j'\in J'; k,j, k', j' \neq 0$. We have $x\phi(e)=\phi(e)x$, so $\phi(e)x=\begin{bmatrix} \alpha & j \\ j'& \beta \\ \end{bmatrix}\cdot \begin{bmatrix} \gamma & k \\ k'& \delta \\ \end{bmatrix}=\begin{bmatrix} \alpha\gamma+T(j,k') & \alpha k+\delta j+j'\times k' \\ \gamma j'+\beta k'+j\times k & T(k,j')+\beta\delta \\ \end{bmatrix}$ and $x\phi(e)=\begin{bmatrix} \gamma & k \\ k'& \delta \\ \end{bmatrix}\cdot \begin{bmatrix} \alpha & j \\ j'& \beta \\ \end{bmatrix}=\begin{bmatrix} \gamma\alpha +T(k,j')& \gamma j+\beta k+k'\times j' \\ \alpha k'+\delta j'+k\times j & T(j,k')+\delta\beta \\ \end{bmatrix}.$ Thus, it is necessary that $\gamma j'+\beta k' +j\times k=\alpha k'+\delta j'+k\times j$. Let us assume $k=j$, $\gamma \neq \delta$. Then we get $\alpha=\beta$, $j'=0$. Similarly, from the condition $$\alpha k+\delta j+j'\times k'=\gamma j+\beta k+k'\times j',$$ assuming $k'=j'$ and $\delta \neq \gamma$, we get $j=0$. Hence, $\phi(e)=\alpha \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$. The lemma is proved. **Lemma 4.** The algebra $T(C)$ has no nontrivial $\frac{1}{2}$-derivations. **Proof**. Let $e_1, \ldots, e_8$ be the standard basis of algebra $C$ (see, for example, [@ZSSS]). Let us recall that $S^{(-)}$ is a simple $7$-dimensional non-Lie Malcev algebra relative to commutation operation $[\cdot , \cdot]$ in $C$ (about Malcev algebras see, for example, [@kuz]), $S$ is the set of elements of algebra $C$ that are antisymmetric relative to involution. It is easy to see that $e_{i}\in S$ if $i=2,3, \ldots, 8$. Let $\phi(e)=h^{*}+s^{*}$, where $h^{*}\in H,s^{*}\in S$, and $e$ is the unit of the algebra $T(C)$. Then: $[\phi(e),s]=0$ and $[h^{*},s]+[s^{*},s]=0$. Let us notice that $[s^{*},s]=0$. As the algebra $S^{(-)}$ is simple, its annihilator is trivial. It is easy to see that $s^{*}$ is in the annihilator of the algebra $S^{(-)}$, that is $s^{*}=0$. Consequently, $[h^{*},s]=0$. It is easy to see (see, for example, [@Poj]) that in Cayley-Dikson algebra $C,$ a basis can be chosen in such way that the multiplication of the basic elements of the algebra will be represented by the following table: $e_{2}=e_{1}e_{2}=e_{5}e_{6}=e_{7}e_{8}=e_{3}e_{4}$,\ $e_{3}=e_{1}e_{3}=e_{7}e_{6}=e_{4}e_{2}=e_{8}e_{5}$,\ $e_{4}=e_{1}e_{4}=e_{2}e_{3}=e_{6}e_{8}=e_{7}e_{5}$,\ $e_{5}=e_{1}e_{5}=e_{6}e_{2}=e_{4}e_{7}=e_{3}e_{8}$,\ $e_{6}=e_{1}e_{6}=e_{2}e_{5}=e_{8}e_{4}=e_{3}e_{7}$,\ $e_{7}=e_{1}e_{7}=e_{5}e_{4}=e_{8}e_{2}=e_{6}e_{3}$,\ $e_{8}=e_{1}e_{8}=e_{2}e_{7}=e_{4}e_{6}=e_{5}e_{3}$.\ Let the element $h^{*}$ be expressed by the basic elements in the following way: $h^{*}=\sum\limits_{i=2}\limits^{8}\alpha_{ij}e_{i}\otimes e_{j}$. We can say that $[h^{*},x]=0=[x,h^{*}]$. Let us analyze the product $[e_{j},h^{*}]$. From the condition $$[e_{2},\sum\limits_{i=2}\limits^{8}\alpha_{ij}e_{i}\otimes e_{j}]=0$$ we will get the conditions on $\alpha_{ij}$, and then the element of interest can be represented as follows: $h^{*}=\alpha_{22}e_{2}\otimes e_{2}+\alpha_{33}e_{3}\otimes e_{3}+\alpha_{33}e_{4}\otimes e_{4}+\alpha_{35}e_{3}\otimes e_{5}+\alpha_{45}e_{4}\otimes e_{5}+\alpha_{55}e_{5}\otimes e_{5}-\alpha_{45}e_{3}\otimes e_{6}+\alpha_{35}e_{4}\otimes e_{6}+\alpha_{55}e_{6}\otimes e_{6}+\alpha_{37}e_{3}\otimes e_{7}+\alpha_{47}e_{4}\otimes e_{7}+\alpha_{57}e_{5}\otimes e_{7}+\alpha_{67}e_{6}\otimes e_{7}+\alpha_{77}e_{7}\otimes e_{7}+\alpha_{28}e_{2}\otimes e_{8}-\alpha_{47}e_{3}\otimes e_{8}+\alpha_{37}e_{4}\otimes e_{8}-\alpha_{67}e_{5}\otimes e_{8}+\alpha_{57}e_{6}\otimes e_{8}+\alpha_{77}e_{8}\otimes e_{8}$. Performing the multiplication by $e_{j}$ $(j=3,4,5,6,7)$, we will get that all $\alpha_{ij}=0$ when $i\neq j$. Combining these results with those obtained from the condition $[e_{8},h^{*}]=0$, it is easy to notice that $\alpha_{ii}=\alpha_{jj}$ for any $i,j$. Therefore, $h^{*}= \sum\limits_{i=2}\limits^{8}\alpha e_{i}\otimes e_{i}$. Let us show that $(e_{i},e_{i})=-4$ and $(e_{i},e_{j})=0$ when $i\neq j$: $[[e_{j},e_{i}],e_{i}]=(e_{i},e_{i})e_{j}-(e_{j},e_{i})e_{i}$ ;\ $[e_{j},e_{i}]e_{i}-e_{i}[e_{j},e_{i}]=2[e_{j},e_{i}]e_{i}=2(e_{j}e_{i}-e_{i}e_{j})e_{i}=4(e_{j}e_{i})e_{i}=-4e_{j}.$ Using this fact and the multiplication in the algebra $T(C)$, making direct substitution and verification we can easily see that the element $e=-\frac{1}{16}\sum\limits_{i=2}\limits^{8}e_{i}\otimes e_{i}$ is the unit of the algebra. Let $\beta = -\frac{\alpha}{16}$. Then $h^{*}=\beta e$. The lemma is proved. Main results ============ **Theorem 5.** A simple finite-dimensional structurable algebra over an algebraically closed field with the characteristic $p \neq 2, 3, 5$ has no nontrivial $\delta$-derivations. [**Proof.**]{} Following [@Smi], we can say that a simple finite-dimensional structurable algebra over an algebraically closed field with the characteristic $p \neq 2, 3, 5$ is isomorphic to one of the algebras of the types $S1)$-$S6)$ listed in section $2$. Let us notice that nontrivial $\delta$-derivations are possible only when $\delta=\frac{1}{2}.$ We have that 1\) algebras of types $S1, S2$ do not admit nontrivial $\frac{1}{2}$-derivations (see [@Filll; @kay]); 2\) algebras of types $S3$–$S6$ do not admit nontrivial $\frac{1}{2}$-derivations (see lemmas $1$-$4$). Thus, the theorem is proved. **Theorem 6.** A semisimple finite-dimensional structurable algebra over an algebraically closed field of characteristic $p \neq 2, 3, 5$ has no nontrivial $\delta$-derivations. [**Proof.**]{} According to [@Smi], if $A$ is a semisimple finite-dimensional structurable algebra over an algebraically closed field with the characteristic not equal to 2,3,5, then $A=\oplus A_{i}$, where $A_{i}$ is a simple structurable algebra. Let $e_{k}$ be the unit of the algebra $A_{k}$. If $x_{i}\in A$, then $\phi(x_{i})={x_{i}}^{+}+{x_{i}}^{-}$, where ${x_{i}}^{+}\in A_{i}$, ${x_{i}}^{-}\notin A_{i}$. Let $e^{i}=\sum e_{k}-e^{i}$ and $\phi(e^{i})=e^{i+}+e^{i-}$, where $e^{i+}\in A_{i}$, $e^{i-}\notin A_{i}$. Then $$0=\phi(x_{i}e^{i})= \delta( \phi(x_{i})e^{i}+x_{i}\phi(e^{i}))= \delta(({x_{i}}^{+}+{x_{i}}^{-})e^{i}+x_{i}({e}^{i+}+{e}^{i-}))= \delta({x_{i}}^{-}+x_{i}{e}^{i+}),$$ whence ${x_{i}}^{-}=0$. Thus $\phi$ is invariant on $A_{i}$. According to theorem 6, $A_{i}$ has no nontrivial $\delta$-derivations. Then the semisimple algebra $A$ has no nontrivial $\delta$-derivations. The theorem is proved. Let us note that in [@kay_gendelta] the definition of the generalized $\delta$-derivation is introduced. The linear mapping $\chi$ is called a generalized $\delta$-derivation if it is related with $\delta$-derivation $\phi$ by the following correlations $$\chi(xy)=\delta(\chi(x)y+x\phi(y))=\delta(\phi(x)y+x\chi(y)).$$ According to the results of the work [@kay_gendelta], unital algebras have no generalized $\delta$-derivations, that are not either generalized derivations or $\delta$-derivations. It was also was shown that if the generalized $\delta$-derivation is the generalized derivation then it is the sum of the derivation and the element of the centroid. Therefore, from theorem 6 follows: **Theorem 7.** Let $\chi$ be the generalized $\delta$-derivation of the semisimple finite-dimensional structurable algebra $A$ over an algebraically closed field with the characteristic $p \neq 2, 3, 5$, then $\chi \in Der(A)+\Gamma(A),$ where $Der(A)$ is the space of derivations and $\Gamma(A)$ is the centroid. [**Acknowledgements.**]{} The authors are grateful to Prof. Ivan Shestakov (IME-USP, Brasil) and Prof. Alexandre Pozhidaev (Sobolev Inst. of Math., Russia) for interest and constructive comments; Mark Gannon (IME-USP, Brazil) for the translation and the referee for the correction of the first version of this paper. [10]{} On $\delta$-derivations of Lie algebras, Siberian Math. J., [**39**]{} (1998), 6, 1218–1230. $\delta$-derivations of prime Lie algebras, Siberian Math. J., [**40**]{} (1999), 1, 174–184. On $\delta$-derivations of prime alternative and Malcev algebras, Algebra and Logic, [**39**]{} (2000), 5, 354–358. 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[^1]: The authors were supported by RFBR 12-01-31016, 12-01-33031, by RF President Grant council for support of young scientists and leading scientific schools (project MK-330.2013.1), and FAPESP (Grant 2011/51132-9).

--- abstract: 'The new era of multimessenger astrophysics requires the capability of studying different aspects of the evolution of compact objects. In particular, the merger of neutron star binaries is a strong source of gravitational waves and electromagnetic radiation, from radio to $\gamma$-rays, as demonstrated by the detection of GW170817 and its electromagnetic counterparts. In order to understand the physical mechanisms involved in such systems, it is necessary to employ fully general relativistic magnetohydrodynamic (GRMHD) simulations able to include the effects of a composition and temperature dependent equation of state describing neutron star matter as well as neutrino emission and reabsorption. Here, we present our new code named `Spritz` that solves the GRMHD equations in 3D Cartesian coordinates and on a dynamical spacetime. The code can support tabulated equations of state, taking into account finite temperature effects and allowing for the inclusion of neutrino radiation. In this first paper, we present the general features of the code and a series of tests performed in special and general relativity to assess the robustness of the basic GRMHD algorithms implemented. Among these tests, we also present the first comparison between a non-staggered and a staggered formulation of the vector potential evolution, which is used to guarantee the divergence-less character of the magnetic field.' address: | $^{1}$INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Via Sommarive 14, I-38123 Trento, Italy\ $^{2}$Dipartimento di Fisica, Università di Trento, Via Sommarive 14, I-38123 Trento, Italy\ $^{3}$Center for Computational Relativity and Gravitation, School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, New York 14623, USA\ $^{4}$Università di Padova, Dipartimento di Fisica e Astronomia, Via Francesco Marzolo 8, I-35131 Padova, Italy\ $^{5}$INFN, Sezione di Padova, Via Francesco Marzolo 8, I-35131 Padova, Italy\ $^{6}$Dipartimento di Fisica G. Occhialini, Università di Milano - Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy\ $^{7}$INFN, Sezione di Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy\ $^{8}$INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy\ $^{9}$INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy author: - | F. Cipolletta$^{1, \, 2, \, 3}$, J. V. Kalinani$^{4, \, 5}$, B. Giacomazzo$^{6, \, 7, \, 8 \, \ast}$,\ R. Ciolfi$^{9, \, 5}$ bibliography: - 'references.bib' nocite: '[@*]' title: 'Spritz: a new fully general-relativistic magnetohydrodynamic code' --- December 2019 Introduction {#Intro} ============ Magnetic fields play a crucial role in several high-energy astrophysical scenarios at different scales, from Active Galactic Nuclei (AGN) to Gamma-Ray Bursts (GRBs). These phenomena involve compact objects such as neutron stars (NSs) and black holes (BHs) and therefore any attempt to model them requires a general relativistic treatment. As a consequence, studying this kind of systems demands to solve the full set of general relativistic magnetohydrodynamic (GRMHD) equations [@anton2006numerical]. In most situations, the GRMHD equations are to be solved numerically, often on dynamical spacetimes, and a number of GRMHD codes have been developed over the years for this purpose (e.g.,  [@giacomazzo2007whiskymhd; @mosta2013grhydro; @etienne2015illinoisgrmhd]). Some of them have been used, in particular, to study compact binary mergers (e.g., [@Liu2008; @Anderson2008; @giacomazzo2011accurate; @Kiuchi2018; @Ciolfi2019]) and accretion onto supermassive BHs (e.g., [@Palenzuela2010; @Giacomazzo2012; @Farris2012; @Gold2014; @Kelly2017; @dascoli2018; @EHTCodeComparison]). In the case of compact binary mergers, GRMHD codes have been used to simulate NS-NS and NS-BH mergers in order to study the effects of magnetic fields on the gravitational wave (GW) and electromagnetic (EM) emission (e.g., [@Kawamura2016; @Ciolfi2017]). For instance, GRMHD simulations have recently provided indications that, under certain conditions, the BH remnant of a NS-NS or NS-BH merger may be able to give rise to a relativistic jet and hence power a short GRB [@Paschalidis2015; @Ruiz2016]. This is a likely scenario to explain the connection between compact binary mergers and short GRBs, recently confirmed by the first simultaneous observation of GWs emitted by a NS-NS merger and a short GRB [@PhysRevLett.119.161101; @LVC-GRB]. Concerning the accretion onto supermassive BH mergers, current simulations aim at predicting the light curves of possible EM counterparts of future GW sources detected by LISA [@Schnittman2013; @Amaro-Seoane2017]. In this paper, we present our new fully GRMHD numerical code, named `Spritz`, that solves the GRMHD equations in 3D and on a dynamical spacetime. The code inherits a number of basic features from the `WhiskyMHD` code [@giacomazzo2007whiskymhd], but it also takes advantage of methods implemented and tested in the publicly available `GRHydro` [@mosta2013grhydro] and `IllinoisGRMHD` [@etienne2015illinoisgrmhd] codes. The `WhiskyMHD` code has been used successfully to simulate NS-NS mergers [@giacomazzo2011accurate; @Ciolfi2019; @Kawamura2016; @Ciolfi2017; @Giacomazzo2009; @Rezzolla2011; @GiacomazzoPerna2013; @Giacomazzo2015; @Endrizzi2016] and accretion onto supermassive black hole binaries [@Giacomazzo2012BBH], but it is limited to the use of simple piecewise polytropic equations of state [@Read2009] and it is not able to take into account neutrino emission. Moreover, this code can evolve the vector potential instead of the magnetic field, but employing a non-staggered formalism that may have undesired effects on the evolution (see discussion in the following sections). The new `Spritz` code can instead handle any equation of state for which the pressure is a function of rest-mass density, temperature, and electron fraction and therefore can also use modern tabulated equations of state. This has been possible by following a similar approach used in the `GRHydro` code, which can use finite temperature tabulated equations of state, but that still lacks of a magnetic field implementation able to handle correctly the use of mesh refinement techniques. `Spritz` also implements a staggered version of the vector potential formulation in a formalism that, as discussed later in the paper, recovers the original conservative flux-CT approach implemented in the original version of `WhiskyMHD`. This has been possible by using algorithms similar to those implemented in `IllinoisGRMHD`, which at the moment can only handle simple ideal fluid equations of state. Therefore, the `Spritz` code aims at merging together the main capabilities of the three codes mentioned above. Here, we present a series of extensive tests in 1D, 2D, and 3D, including, for the first time, a comparison between staggered and non-staggered schemes for the vector potential evolution and a rather demanding spherical explosion test. The `Spritz` code passes all the tests successfully and it will be soon used to carry out NS-NS and NS-BH merger simulations. We also plan to make the code publicly available in the near future within the Einstein Toolkit collaboration. The paper is organized as follows: in we present the GRMHD equations and the formulation used in the code; in the main numerical methods are discussed; in we present the results of our tests; and in we summarize the main results and discuss future developments. We use a system of units such that $G=c=1$ unless otherwise specified. Greek indices run from 0 to 3 and Latin indices run from 1 to 3. Equations {#sec2} ========= In this section we summarize the theoretical background and the equations implemented in `Spritz`, giving also the main references for the reader who wants to go deeper in the related details. In addition to these references, it is worth to mention the book [@baumgarte2010numerical] which presents an extensive theoretical introduction to numerical relativity approaches to solving Einstein’s Equations in several physical scenarios. 3+1 spacetime formulation {#3+1} ------------------------- Our numerical methods and implementation are largely based on the ones employed in `WhiskyMHD` [@giacomazzo2007whiskymhd], where a 3+1 formulation of the Einstein’s equations is adopted. In such a framework, the form of the line element is: $$\label{LinEl} ds^{2} = g_{\mu \nu} dx^{\mu} dx^{\nu} = -\left( \alpha^{2} - \beta^{i}\beta_{i} \right) dt^2 + 2 \beta_{i} dx^{i} dt + \gamma_{ij} dx^{i}dx^{j},$$ where the usual Einstein notation is adopted. Here $g_{\mu \nu}$ is the metric tensor, $\gamma_{ij}$ its purely spatial part, and $\alpha$ and $\beta^{i}$ are respectively the *lapse* and the *shift* vector. Our coordinate setting considers $x^{0} \equiv t$. `Spritz` makes use of the conservative formulation presented in [@anton2006numerical], which is the GRMHD version of the original general relativistic hydrodynamics Valencia formulation [@banyuls1997numerical; @marti1991numerical]. Here, the natural observer is called the *Eulerian observer* and its four–velocity $\bi{n}$ is normal to the 3–dimensional hypersurface of constant $t$ with the following components: $$\label{natvel} \eqalign{ n^{\mu} &= \frac{1}{\alpha} \left( 1, -\beta^{i} \right), \\ n_{\mu} &= \left( -\alpha,0,0,0 \right). }$$ When considering matter, the spatial components of the fluid velocity measured by the Eulerian observer read $$\label{spatfluidvel} v^{i} = \frac{h^{i}_{\mu} u^{\mu}}{-\bi{u} \cdot \bi{n}} = \frac{u^i}{\alpha u^t} + \frac{\beta^i}{\alpha} = \frac{u^i}{W} + \frac{\beta^i}{\alpha},$$ where $\bi{u}$ is the fluid four–velocity, $h_{\mu \nu} = g_{\mu \nu} + n_{\mu} n_{\nu}$ is the projector onto the aforementioned hypersurface at constant $t$, $W = 1/\sqrt{1-v^2}$ is the Lorentz factor, and $v^2\equiv\gamma_{ij} v^i v^j$ is the square norm of $\bi{v}$. Electromagnetic field {#sec:Maxwell} --------------------- The general relativistic formulation of [@anton2006numerical] describes the electromagnetic field via the Faraday tensor and its dual, given respectively by $$\label{Faraday} F^{\mu \nu} = U^{\mu} E^{\nu} - U^{\nu} E^{\mu} - \eta^{\mu \nu \lambda \delta} U_{\lambda} B_{\delta},$$ $$\label{FaradayDual} ^{*}F^{\mu \nu} = \frac{1}{2} \eta^{\mu \nu \lambda \delta} F_{\lambda \delta} = U^{\mu} B^{\nu} - U^{\nu} B^{\mu} - \eta^{\mu \nu \lambda \delta} U_{\lambda} E_{\delta},$$ being $E^{\mu}$ the electric field, $B^{\mu}$ the magnetic field, $U^{\mu}$ a generic observer’s four–velocity, and $\eta^{\mu \nu \lambda \delta} = \frac{1}{\sqrt{-g}} \left[ \mu \nu \lambda \delta \right]$ the volume element. The equations governing the electromagnetic field and its evolution are the well known Maxwell’s equations $$\label{eq:Maxwell} \eqalign{ {\nabla}_{\nu} ^{*}F^{\mu \nu} &= 0 \, , \\ {\nabla}_{\nu} F^{\mu \nu} &= 4 \pi \mathcal{J}^{\mu} \, , }$$ where $\bi{\mathcal{J}}$ is the four–vector current density, which can be expressed through the Ohm’s law as $$\label{4curr} \mathcal{J}^{\mu} = q u^{\mu} + \sigma F^{\mu \nu} u_{\nu} \, ,$$ with $q$ the proper charge density and $\sigma$ the electric conductivity. In the ideal MHD regime (i.e., when $\sigma \to \infty$ and $F^{\mu \nu} u_{\nu} \to 0$) and can be expressed as $$\label{FaradayIdeal} F^{\mu \nu} = \eta^{\alpha \beta \mu \nu} b_{\alpha} u_{\beta}, \qquad ^{*}F^{\mu \nu} = b^{\mu} u^{\nu} - b^{\nu} u^{\mu} = \frac{u^{\mu} B^{\nu} - u^{\nu} B^{\mu}}{W} \, ,$$ where $\bi{b}$ is the magnetic field according to the comoving observer, which can be written component–wise as follows [@giacomazzo2007whiskymhd]: $$\label{bsmall} b^0 = \frac{W B^i v_i}{\alpha}, \qquad b^i = \frac{B^i + \alpha b^0 u^i}{W}, \qquad b^2 \equiv b^\mu b_\mu = \frac{B^2 + \alpha^2 \left( b^0 \right)^2}{W^2} \, .$$ Here, $B^2 \equiv B^i B_i$, where $\bi{B}$ is now the magnetic field measured by the Eulerian observer (i.e., from now on $U^\mu=n^\mu$). With , the Maxwell’s equations considering the dual of Faraday tensor can be written as $$\label{MaxwellIdeal} {\nabla}_{\nu} ^{*}F^{\mu \nu} = \frac{1}{\sqrt{-g}} \partial_{\nu} \left( \sqrt{-g} \left(b^{\mu} u^{\nu} - b^{\nu} u^{\mu} \right) \right) = 0.$$ Splitted in its different components, provides the equations governing the magnetic field constraints and evolution, namely the *divergence–free condition* $$\label{divfree} \partial_i \tilde{B}^i = 0 \, ,$$ where $\tilde{B}^i \equiv \sqrt{\gamma} B^i$, and the magnetic field *induction equations* $$\label{magnind} \partial_t \tilde{B}^i = \partial_j \left( \tilde{v}^i \tilde{B}^j - \tilde{v}^j \tilde{B}^i \right) \, ,$$ where $\tilde{v}^i \equiv \alpha v^i - \beta ^i$. Conservative approach {#Cons} --------------------- The stress–energy tensor, considering a perfect fluid and the contribution of the magnetic field, can be written as $$\label{Tmunu} T^{\mu \nu} = \left( \rho h + b^2 \right) u^{\mu} u^{\nu} + \left( p_\mathrm{gas} + p_\mathrm{mag} \right) g^{\mu \nu} - b^{\mu} b^{\nu},$$ being $\rho$ the rest–mass density, $p_\mathrm{gas}$ the gas pressure, $p_\mathrm{mag} \equiv \frac{b^2}{2}$ the magnetic pressure, $h = 1 + \varepsilon + \frac{p_\mathrm{gas}}{\rho}$ the relativistic specific enthalpy, and $\varepsilon$ the specific internal energy. The energy-momentum conservation $$\label{consTmunu} \nabla_{\nu} T^{\mu \nu} = 0 \, ,$$ the conservation of baryon number $$\label{consbaryon} \nabla_{\nu} \left( \rho u^{\nu} \right) = 0 \, ,$$ Maxwell’s equations for the magnetic field (\[magnind\]), and an equation of state (EOS, see \[EOS\]) give together the complete set of equations describing the evolution of the primitive variables, i.e., $\bi{U} = \left[ \rho, v_j, \varepsilon, B^k \right]$. As usual, these equations can be written in the following conservative form: $$\label{consequat} \frac{1}{\sqrt{-g}} \left[ \partial_t \left( \sqrt{\gamma} \bi{F}^0 \right) + \partial_i \left( \sqrt{-g} \bi{F}^i \right) \right] = \bi{S},$$ being $\bi{F}^0 \equiv \left[ D, S_j, \tau, B^k \right]$ the vector of conserved variables, defined in terms of the primitive ones as $$\label{P2Csystem} \eqalign{ D &\equiv \rho W, \\ S_{j} &\equiv \left( \rho h + b^2 \right) W^2 v_{j} - \alpha b^0 b_{j}, \\ \tau &\equiv \left( \rho h + b^2 \right) W^2 - \left( p_\mathrm{gas} + p_\mathrm{mag} \right) - \alpha^2 \left( b^0 \right)^2 - D, \\ B^k &\equiv B^k, }$$ $\bi{F}^i$ the vector of fluxes defined as $$\label{conservedvector} \bi{F}^i \equiv \left[ \eqalign{ &\qquad D\tilde{v}^i / \alpha \\ S_j \tilde{v}^i / \alpha + &\left( p_\mathrm{gas} + p_\mathrm{mag} \right) \delta^i_j - b_j B^i / W \\ \tau \tilde{v}^i / \alpha + &\left( p_\mathrm{gas} + p_\mathrm{mag} \right) v^i - \alpha b^0 B^i / W \\ &B^k \tilde{v}^i / \alpha - B^i \tilde{v}^k / \alpha } \right]\,,$$ and $\bi{S}$ the vector of sources that reads $$\label{sources} \bi{S} \equiv \left[ \eqalign{ & \qquad 0 \\ T^{\mu \nu} &\left( \partial_{\mu} g_{\nu j} - \Gamma^{\delta}_{\nu \mu} g_{\delta j} \right) \\ \alpha &\left( T^{\mu 0} \partial_{\mu} \ln{\alpha} - T^{\mu \nu} \Gamma^{0}_{\nu \mu} \right) \\ & \qquad 0^k } \right].$$ In order to avoid time derivatives of the metric in the source terms, these are rewritten as done in the case of the `Whisky` code [@baiotti2005three] (see section 4.3.2 of  [@BaiottiPhDThesis] for details). Electromagnetic gauge conditions {#GaugeCond} -------------------------------- In order to accurately describe the magnetic field and its evolution, it can be convenient to formulate the problem in terms of the vector potential (see, e.g., [@feynman1979feynman]). Considering $\nabla$ as a purely spatial operator, one may write $$\label{rotB} \bi{B} = \nabla \times \bi{A} \, ,$$ so that $$\label{inductsatisfy} \nabla \cdot \bi{B} = \nabla \cdot \left( \nabla \times \bi{A} \right) = 0 \, ,$$ and thus evolving the vector potential $\bi{A}$ will automatically satisfy . As already done in [@baumgarte2003collapse; @baumgarte2003general; @etienne2012relativistic], we then introduce the four–vector potential $$\label{4vector} \mathcal{A}_{\nu} = n_{\nu} \Phi + A_{\nu} \, ,$$ being $A_{\nu}$ the purely spatial vector potential and $\Phi$ the scalar potential. With this, and become respectively $$\label{Adivfree} B^i = \epsilon^{ijk} \partial_j A_k \, ,$$ and $$\label{Ainduct} \partial_t A_i = \epsilon_{ijk} v^j B^k - \partial_i \left( \alpha \Phi - \beta^j A_j \right) = -E_i - \partial_i \left( \alpha \Phi - \beta^j A_j \right),$$ where $\epsilon^{ijk} = n_{\nu} \epsilon^{\nu ijk}$ is the three–dimensional spatial Levi–Civita tensor. However, the choice of the four-vector potential $\mathcal{A}^{\nu}$ is not unique and one has to choose a specific gauge. The first GRMHD simulations that employed the vector potential as an evolution variable were performed using the *algebraic* gauge [@etienne2012relativistic; @etienne2010relativistic] where the scalar potential satisfy the following equation: $$\label{algebraic} \Phi = \frac{1}{\alpha} \left( \beta^j A_j \right).$$ In this way  is strongly simplified, being reduced to $$\label{Ainduct-algebraic} \partial_t A_i = \epsilon_{ijk} v^j B^k = - E_i \, ,$$ and therefore it does not require to evolve the scalar potential $\Phi$. More recently, GRMHD simulations started to use the *Lorenz* gauge [@etienne2012relativistic], which consists of imposing the constraint $\nabla_{\nu} \mathcal{A}^{\nu} = 0$. This gauge requires now to solve also the evolution equation for the scalar potential: $$\label{lorenz} \partial_t \left( \sqrt{\gamma} \Phi \right) + \partial_i \left( \alpha \sqrt{\gamma} A^i - \sqrt{\gamma} \beta^i \Phi \right) = 0.$$ The *Lorenz* gauge has been shown to perform better in those simulations that implement adaptive mesh refinement, such as, for example, binary neutron star and neutron star–black hole mergers [@etienne2012relativistic]. The *algebraic* gauge may indeed cause interpolation errors at the boundaries between refinement levels and thus produce spurious magnetic fields (see [@etienne2012relativistic] for more details). An even more robust gauge choice has been introduced in [@Farris2012] with the name of *generalized Lorenz* gauge: $$\nabla_{\nu} \mathcal{A}^{\nu} = \xi n_\nu \mathcal{A}^{\nu} \, ,$$ where $\xi$ is a parameter that is typically set to be equal to $1.5/\Delta t_{\rm max}$, being $\Delta t_{\rm max}$ the timestep of the coarsest refinement level [@etienne2015illinoisgrmhd]. When employing this gauge choice the evolution equation for the scalar potential becomes $$\label{lorenz-generalized} \partial_t \left( \sqrt{\gamma} \Phi \right) + \partial_i \left( \alpha \sqrt{\gamma} A^i - \sqrt{\gamma} \beta^i \Phi \right) = -\xi \alpha \sqrt{\gamma} \Phi \, .$$ In `Spritz` we adopt the *generalized Lorenz* gauge which is also the gauge used in the latest `WhiskyMHD` simulations [@Ciolfi2019; @Kawamura2016; @Ciolfi2017; @GiacomazzoPerna2013; @Giacomazzo2015; @Endrizzi2016]. Numerical Implementation {#sec3} ======================== In the present section we summarize the main numerical methods implemented within the `Spritz` code. The code is based on the Einstein Toolkit [@ETKpaper; @EinsteinToolkit:2019_10; @ET] which provides a framework to automatically parallelize the code for the use on supercomputers as well as a number of open-source codes providing a number of useful routines, such as those for the evolution of the spacetime, adaptive mesh refinement, input and output of data, checkpointing, and many others. Riemann Solvers {#RieHRSC} --------------- The `Spritz` code adopts High Resolution Shock Capturing (HRSC) methods to solve . These methods are based on the choice of reconstruction algorithms, to compute the values of primitive variables at the interface between numerical cells, and of approximate Riemann solvers, to finally compute the fluxes. Our default Riemann solver is the Harten–Lax–van–Leer–Einfeldt (HLLE) [@harten1983upstream], where the numerical fluxes at cell interfaces are computed as follows: $$\label{HLLEfluxes} \bi{F}^i = \frac{c_\mathrm{min} \bi{F}^i_\mathrm{r} + c_\mathrm{max} \bi{F}^i_\mathrm{l} - c_\mathrm{max} c_\mathrm{min} \left( \bi{F}^0_\mathrm{r} - \bi{F}^0_\mathrm{l} \right)}{c_\mathrm{max} + c_\mathrm{min}} \, ,$$ where a subscript r (l) means that the function is computed at the right (left) side of the cell interface and $c_\mathrm{max} \equiv \max \left( 0, c_{+,\tiny{\mbox{l}}}, c_{+,\tiny{\mbox{r}}} \right)$, $c_\mathrm{min} \equiv -\min \left( 0, c_{-,\tiny{\mbox{l}}}, c_{-,\tiny{\mbox{r}}} \right)$, where $c_{\pm,\tiny{\mbox{r}}}$ ($c_{\pm,\tiny{\mbox{l}}}$) are the right-going ($+$) and left-going ($-$) maximum wave speeds computed from the primitive variables $\bi{U}$ at the right (left) side. We decided also to implement the Lax–Friedrichs (LxF) scheme [@toro2013riemann], that is $$\label{LxFfluxes} \bi{F}^i = \frac{\bi{F}^i_\mathrm{r} + \bi{F}^i_\mathrm{l} - c_\mathrm{wave} \left( \bi{F}^0_\mathrm{r} - \bi{F}^0_\mathrm{l} \right)}{2},$$ where $c_\mathrm{wave} = \max(c_\mathrm{max}, c_\mathrm{min})$ [@del2003efficient]. This scheme is a very dissipative one and it can be useful in cases where strong jumps in pressure must be considered. In order to compute the values of $\bi{F}^0$ at right and left sides of cell’s interfaces from the primitive variables, we adopt the third–order Piece-wise Parabolic Method (*PPM*) [@colella1984piecewise]. In addition, for those cases that require more dissipative methods, for example in presence of strong shocks, we also implemented the second–order total variation diminishing (TVD) *minmod* method [@toro2013riemann]. Electromagnetic Field Evolution {#Stagg} ------------------------------- As already mentioned in , the `Spritz` code is meant to deal with different electromagnetic gauge conditions for the vector potential. In order to accurately evolve the magnetic field, particular care must be taken in solving numerically , in the case of the algebraic gauge, or and , in case of the generalized Lorenz gauge. From now on, we will also use the following definition for simplicity: $$\label{psimhd} \Psi_\mathrm{mhd} \equiv \sqrt{\gamma} \Phi \,.$$ As in every numerical code, the spatial domain is divided in grid–cells of user specified dimensions. The fluid’s state variables (e.g., $\rho$, $p_{gas}$, $\bi{v}$) are stored in the grid–cell’s centers. Usually, the electric and magnetic fields ($\bi{E}$ and $\bi{B}$) are instead stored respectively on cell’s edges and faces. `Spritz` evolves the magnetic field as the curl of a given vector potential $\bi{A}$, whose components are staggered just like the electric field $\bi{E}$ (see ) and are usually evolved using the generalized Lorenz gauge. The precise storage location on the grid–cells of various quantities is reported in . [@ccc]{} Symbol&Definition&Location\ $\alpha$&lapse&$(i,j,k)$\ $\beta^m$&$m$–component of the shift vector&$(i,j,k)$\ $\gamma^{mn}$&$mn$–component of the spatial metric&$(i,j,k)$\ $\gamma$&determinant of the spatial metric&$(i,j,k)$\ $\rho$&rest-mass density&$(i,j,k)$\ $p_{gas}$&pressure&$(i,j,k)$\ $\varepsilon$&energy density&$(i,j,k)$\ $v_m$&$m$–component of fluid velocity&$(i,j,k)$\ $B^1$&$x$–component of magnetic field&$(i+\frac{1}{2},j,k)$\ $B^2$&$y$–component of magnetic field&$(i,j+\frac{1}{2},k)$\ $B^3$&$z$–component of magnetic field&$(i,j,k+\frac{1}{2})$\ $A_1$&$x$–component of vector potential&$(i,j+\frac{1}{2},k+\frac{1}{2})$\ $A_2$&$y$–component of vector potential&$(i+\frac{1}{2},j,k+\frac{1}{2})$\ $A_3$&$z$–component of vector potential&$(i+\frac{1}{2},j+\frac{1}{2},k)$\ $\Psi_\mathrm{mhd}$&scalar potential&$(i+\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2})$\ \ {width="0.5\linewidth"} Since $\bi{B}$ is computed from the curl of $\bi{A}$, the divergence–free character of the magnetic field is automatically satisfied. The `Spritz` code evolves the vector potential $A$ and, when employing the generalized Lorenz gauge, the scalar potential $\Psi_\mathrm{mhd}$ is also computed. Following [@etienne2015illinoisgrmhd], we can write the update terms for the vector potential’s components and for the scalar potential as follows: $$\label{Axrhs} \eqalign{ \partial_t A_m &= -E_m - \partial_m \left( G_{A} \right) = \cr &= -E_m - \partial_m \left( \alpha \frac{\Psi_\mathrm{mhd}}{\sqrt{\gamma}} - \beta^j A_j \right),}$$ for $m = 1$, $2$ and $3$, and $$\label{Psirhs} \eqalign{ \partial_t \Psi_\mathrm{mhd} &= -\partial_j \left( {F_{\Psi}}^j \right) - \xi \alpha \Psi_\mathrm{mhd} = \cr &= -\partial_j \left( \alpha \sqrt{\gamma} A^j - \beta^j \Psi_\mathrm{mhd} \right) - \xi \alpha \Psi_\mathrm{mhd},}$$ being $\xi$ the so–called damping factor, used for the generalized Lorenz gauge. As the reader may deduce from , , and , the terms on the right–hand sides in general have different storage locations and therefore we decided to follow this scheme: 1. At first we consider functions ${F_{\Psi}}^j$ and $G_{A}$, defined via and respectively, to be computed at cell centers, by interpolating $\Psi_\mathrm{mhd}$ and $A_j$ respectively from cell vertices and edges to the center. 2. Then we interpolate the values obtained at point (i) for ${F_{\Psi}}^j$ back to the cell edges and for $G_{A}$ back to cell vertices. 3. Finally we numerically differentiate the values at point (ii) via a centered difference scheme. For example, the derivative along $x$ ($m=1$) of $G_{A}$ in  on the edge $(i,j+1/2,k+1/2)$ is computed as $[G_{A}(i+1/2,j+1/2,k+1/2)-G_{A}(i-1/2,j+1/2,k+1/2)]/(\Delta x)$. A similar expression is used for the derivatives computed at the cell vertex $(i+1/2,j+1/2,k+1/2)$ in   where the two nearby edges are used. In details, if a variable $f$ is given at cell vertices, then we interpolate it at the center of the cell using a simple linear interpolation: $$\label{InterpVertices2Center} \eqalign{ f(i,j,k) =& \frac{1}{8} \left[ f\left(i-\frac{1}{2},j-\frac{1}{2},k-\frac{1}{2}\right) + f\left(i-\frac{1}{2},j+\frac{1}{2},k-\frac{1}{2}\right) \right. \cr &+ \left. f\left(i+\frac{1}{2},j+\frac{1}{2},k-\frac{1}{2}\right) + f\left(i+\frac{1}{2},j-\frac{1}{2},k-\frac{1}{2}\right) \right. \cr &+ \left. f\left(i-\frac{1}{2},j-\frac{1}{2},k+\frac{1}{2}\right) + f\left(i-\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2}\right) \right. \cr &+ \left. f\left(i+\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2}\right) + f\left(i+\frac{1}{2},j-\frac{1}{2},k+\frac{1}{2}\right) \right]}$$ is used to interpolate $\Psi_\mathrm{mhd}$ at step (i) of the aforementioned scheme. For quantities defined instead on edges, for example along the $x$–direction, the following interpolation is instead used: $$\label{InterpEdges2Center} \eqalign{ f(i,j,k) =& \frac{1}{4} \left[ f\left(i,j-\frac{1}{2},k-\frac{1}{2}\right) + f\left(i,j+\frac{1}{2},k-\frac{1}{2}\right) \right. \cr &+ \left. f\left(i,j-\frac{1}{2},k+\frac{1}{2}\right) + f\left(i,j+\frac{1}{2},k+\frac{1}{2}\right) \right]}$$ is used to interpolate $A_x$ at step (i) of the aforementioned scheme. Along other directions, the straightforward permutation of indices leads to the correct interpolating functions. The following expression is instead used to interpolate from the cell center to a cell edge: $$\label{InterpCenters2Edge} \eqalign{ f\left(i,j+\frac{1}{2},k+\frac{1}{2} \right) =& \frac{1}{4} \left[ f(i,j,k) + f(i,j+1,k) \right. \cr &+ \left. f(i,j,k+1) + f(i,j+1,k+1) \right]}$$ We use to obtain the values of ${F_{\Psi}}^j$ at point (ii). With the following interpolator we instead compute the values of $G_{A}$ at point (ii): $$\label{InterpCenters2Vertex} \eqalign{ f\left(i+\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2}\right) =& \frac{1}{8} \left[ f(i,j,k) + f(i,j+1,k) \right. \cr &+ \left. f(i,j,k+1) + f(i,j+1,k+1) \right. \cr &+ \left. f(i+1,j,k) + f(i+1,j+1,k) \right. \cr &+ \left. f(i+1,j,k+1) + f(i+1,j+1,k+1) \right]}$$ In order to finally be able to compute the right-hand-side of , one needs also to compute the electric field components $E_m$ that are stored at cell edges. Here we follow the same approach implemented in the `WhiskyMHD` code [@giacomazzo2007whiskymhd] and use the flux-CT method [@balsara1999staggered], in which the electric field is computed from the magnetic field HLLE fluxes. Our staggered formulation therefore benefits of the same properties of the constrained transport scheme [@evans88], but without the need of implementing special prolongation and restriction operators to ensure the divergence-free character of the magnetic field [@Balsara2001]. An alternative scheme could use a non–staggered formulation where $\bi{A}$ and $\bi{B}$ are both stored at the cell centers (e.g., as done in the `WhiskyMHD` code [@giacomazzo2011accurate]). An example of the different results for a shock–tube $1D$ test obtained via a staggered and a non–staggered scheme is shown in . {width="0.5\linewidth"} Boundary Conditions {#BC} ------------------- When developing new codes to work within the `EinsteinToolkit` framework, the treatment of boundary conditions (BC) is usually left to the generic thorn `Boundary` [@ET]. Through this approach, the `Spritz` code may consider “flat” or “none” BC, as already implemented in the `WhiskyMHD` [@giacomazzo2007whiskymhd] and `GRHydro` [@mosta2013grhydro] codes. The “flat” BC simply copies to the ghost zones the value that each variable has in the outermost grid point. The “none” BC instead does not update the ghost zones and keeps the value of the variables in the ghost zones equal to the ones set by the initial data routine. Although the “flat” and “none” BC have been successfully used in simulations with the aforementioned codes, we decided to modify the BC at the external boundary of the computational domain in order to provide a more accurate calculation of B. We followed in particular the work presented in [@etienne2015illinoisgrmhd] and we implemented the numerical extrapolation of $\bi{A}$ and $\Psi_\mathrm{mhd}$ at the outer boundary as described in the `IllinoisGRMHD` code. Basically, for each grid–point in the outer boundary we apply the following formula: $$\label{numextrap} F_{i}^{j} = \cases{2F_{i-1}^{j} - F_{i-2}^{j} \qquad \mbox{for } i = N^{j}-2,N^{j}-1,N^{j} \\ 2F_{i+1}^{j} - F_{i+2}^{j} \qquad \mbox{for } i = 3,2,1 }$$ where $F \in \left\lbrace \bi{A}, \Psi_\mathrm{mhd} \right\rbrace$, $j \in \left\lbrace 1,2,3 \right\rbrace$, $N$ is the number of grid–points in the $j$–direction, and we use 3 points in the ghost zones for each direction. In addition, the user may choose whether BC for $\bi{A}$ and $\Psi_\mathrm{mhd}$ should be given by or simply be obtained by the other two conditions provided by the `Boundary` thorn. Finally, we also successfully tested the implementation of periodic BC provided by the thorn `Periodic` [@ET] through the Loop Advection test (see ), in both uniform and mesh–refined grids. Primitive variables recovering {#C2P} ------------------------------ As mentioned in , the computation of fluxes at each time during the evolution depends on values of the primitive variables $\bi{U}$, although we evolve the conserved ones $\bi{F}^0$. As recurrent in many conservative approaches, one of the most delicate point is the inversion of , a problem that presents no analytic solution. Thus one has to apply a numerical method (usually a Newton-Raphson scheme). In the literature many methods have been presented in order to perform this step [@noble2006primitive; @Siegel2018]. In the `Spritz` code we implemented both the 2D method used in `WhiskyMHD` [@giacomazzo2007whiskymhd] and the one presented in [@noble2006primitive] and used in `GRHydro`. Atmosphere {#Atmo} ---------- As any GRMHD grid-based code, `Spritz` cannot handle zero values for the rest-mass density and a minimum value $\rho_\mathrm{atm}$ needs to be set. If at time $t$ the rest-mass density $\rho$ computed in our conservative-to-primitive routine is such that $\rho < \rho_\mathrm{atm}$, then its value is set to $\rho_\mathrm{atm}$, the pressure and specific internal energy are recomputed using a polytropic EOS, and the fluid’s three–velocity is set to zero. In the tests presented here we typically set $\rho_\mathrm{atm} = 10^{-7} \rho_{0,\mathrm{max}}$, being $\rho_{0,\mathrm{max}}$ the initial maximum value of the rest-mass density. Equation of State {#EOS} ----------------- To close the GRMHD system of equations, an equation of state that provides a relation between $\rho$, $\varepsilon$, and $p_\mathrm{gas}$ must be supplied. Many EOS exist, from analytical ones, such as that of an “ideal fluid” or of a “polytropic” gas [@horedt2004polytropes], to more complex ones that can only be expressed in a tabulated form [@COMPOSE]. One of the most challenging research fields in astrophysics is focussed on trying to understand how matter behaves in the core of NSs, where the rest-mass density may reach values as high as $\sim\!10^{15}$ g cm$^{-3}$, not reproducible in Earth laboratories. Different EOS result in different bulk properties of the star, e.g., different maximum mass or equatorial radius for both spherical (i.e., non–rotating) and rapidly–rotating equilibrium configurations of NS models (see [@cipolletta2015fast] for examples taking into account EOS with various stiffness). It is therefore crucial for any GRMHD code to be able to handle different EOS with different composition as well as different treatments of nucleon interactions, in order to improve the capabilities of comparison between theoretical models and observations. The `Spritz` code can implement both analytic and tabulated EOS. This is done via the `EOSOmni` thorn provided by the `EinsteinToolkit` which supports analytic EOS, such as “ideal fluid” and “piecewise polytropic” ones [@Read2009], and “tabulated” EOS. For the sake of clarity, we report the explicit equations for the “ideal fluid” EOS, that can be written as $$\label{IFEOS} p_\mathrm{gas} = \left( \Gamma - 1 \right) \rho \varepsilon,$$ where $\Gamma$ is the adiabatic index, and for the “polytropic” EOS, that reads $$\begin{aligned} \label{polyEOS} p_\mathrm{gas} &=& K \rho^\Gamma\,,\\ \varepsilon &=& K \rho^{\Gamma-1}/(\Gamma - 1)\,,\end{aligned}$$ being $K$ the polytropic constant. The tests presented in this paper will use only the “ideal fluid” EOS. A follow-up paper will present instead tests with cold and finite temperature equations of state, including also the evolution of the electron fraction and neutrino emission. Adaptive Mesh Refinement {#AMR} ------------------------ Adaptive Mesh Refinement (AMR) is very important in full 3D simulations of binary mergers because it allows for the optimization of the number of grid points by refining only interesting regions of the domain while maintaining a sufficiently large computational domain to reduce the effects of external boundaries and to allow for the extraction of gravitational wave signals far away from the source. In the `EinsteinToolkit` framework [@ETKpaper; @ET], this task is performed via the `Carpet` driver [@Carpet; @schnetter2004evolutions]. Particular care must be taken in case of staggered variables, like $\bi{A}$ and $\Psi_\mathrm{mhd}$ in the `Spritz` code, as already mentioned in . In particular, one needs to use separate restriction and prolongation operators with respect to variables located at the cell centers. Such operators are already provided by the `Carpet` driver and they are the same used by the `IllinoisGRMHD` code. In we show also some tests of our AMR implementation. Spacetime Evolution {#NumSTevo} ------------------- The spacetime evolution is performed using the `McLachlan` thorn [@Brown:2008sb; @Kranc:web; @McLachlan:web], which is part of the `EinsteinToolkit`. It adopts the BSSNOK formulation presented in [@baumgarte1998numerical; @nakamura1987general; @shibata1995evolution] and for which the numerical implementation has been presented in [@baiotti2005three; @alcubierre2003gauge; @alcubierre2000towards]. More details on the code can be found in [@ETKpaper]. Results {#sec4} ======= As already stressed in the Introduction, the central goal of the `Spritz` code is to perform simulations of BNS and NS-BH binary mergers. In order to address such a complex task with the necessary confidence, we need to assess the reliability of the code in a variety of physical conditions. In this Section, we report on the results of our extensive testing, including a number of 1–, 2– and 3–dimensional simulations. These simulations include critical tests that have been already considered in the literature in several previous papers (see, e.g., [@mosta2013grhydro; @etienne2015illinoisgrmhd; @balsara2001total; @beckwith2011second] and references therein), allowing for a direct comparison with other codes. 1D tests {#1D} -------- {width="\linewidth"} [@cD[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}D[.]{}[.]{}[4.1]{}]{} Test: & & & & &\ & L & R & L & R & L & R & L & R & L & R\ $\rho$ & 1.0 & 0.125 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.08 & 1.0\ $p_{gas}$ & 1.0 & 0.1 & 30.0 & 1.0 & 1000.0 & 0.1 & 0.1 & 0.1 & 0.95 & 1.0\ $v_x$ & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.999 & -0.999 & 0.4 & -0.45\ $v_y$ & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.3 & -0.2\ $v_z$ & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.2 & 0.2\ $B^x$ & 0.5 & 0.5 & 5.0 & 5.0 & 10.0 & 10.0 & 10.0 & 10.0 & 2.0 & 2.0\ $B^y$ & 1.0 & -1.0 & 6.0 & 0.7 & 7.0 & 0.7 & 7.0 & -7.0 & 0.3 & -0.7\ $B^z$ & 0.0 & 0.0 & 6.0 & 0.7 & 7.0 & 0.7 & 7.0 & -7.0 & 0.3 & 0.5\ \ {width="\linewidth"} In , we present the results for 1–dimensional (1D) relativistic shock–tube problems corresponding to the suite of tests of [@balsara2001total]. Here, our numerical solution of such problems can be directly compared with the exact solutions computed via the code presented in [@giacomazzo2006exact]. Initial data for such tests are described in . In all tests we employ an ideal fluid EOS, with $\Gamma = 2.0$ for test `Balsara 1` and $\Gamma = 5/3$ for the others. The final evolution time is $t = 0.55$ for test `Balsara 5` and $t = 0.4$ for the others. All tests show an excellent agreement between the numerical results and the exact solutions. We also compared the results of these 1D tests obtained with the `Spritz` code with those already published for the numerical code `GRHydro` [@mosta2013grhydro], finding a perfect match. In , we show an example of such comparison referring to the `Balsara 1` shock–tube test. Finally, shows our results on the most demanding `Balsara 3` test with different resolutions (200, 800, and 1600 grid points). Higher resolution leads to a significant increase in accuracy, which is particularly evident at the shock front (compare also with the exact solution in ). {width="\linewidth"} 2D tests {#2D} -------- We now move on to discuss 2D tests performed with the Spritz code. In this work, we considered three types of 2D tests, namely the cylindrical explosion, the magnetic rotor and the magnetic loop advection, all performed in Cartesian coordinates. We discuss them in some detail in the following subsections. ### Cylindrical Explosion {#CylExp} \ The cylindrical explosion (also known as the cylindrical blast wave) is a demanding multidimensional shock test, first introduced by [@komissarov1999godunov], and later modified and implemented in [@mosta2013grhydro; @etienne2010relativistic; @beckwith2011second; @del2007echo]. This test considers a uniformly magnetized domain consisting of a dense, over–pressured cylinder in the central region expanding in a surrounding ambient medium. Here, we adopt the parameters from the setup described in [@mosta2013grhydro]. For the cylinder, we set $$\label{CylBWintpar} r_\mathrm{in}= 0.8, \; r_\mathrm{out}= 1.0, \; \rho_\mathrm{in} = 10^{-2}, \; p_\mathrm{gas,in} = 1.0 ,$$ while for the surrounding ambient medium, we set $$\label{CylBWextpar} \rho_\mathrm{out} = 10^{-4}, \; p_\mathrm{gas,out} = 3 \times 10^{-5}.$$ Here, $r_\mathrm{in}$ and $r_\mathrm{out}$ are the radial parameters used for the density profile smoothening prescription (and similarly for the pressure profile smoothening prescription) considered in [@mosta2013grhydro], such that $$\label{CylBWdens} \rho(r) = \cases{ \rho_\mathrm{in} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ ; \ r \leq r_\mathrm{in} \\ \exp \Bigg[ \frac{(r_\mathrm{out} - r) \ln \rho_\mathrm{in} + (r-r_\mathrm{in}) \ln \rho_\mathrm{out}}{r_\mathrm{out} - r_\mathrm{in}} \Bigg] \, ; \ r_\mathrm{in} < r < r_\mathrm{out} \\ \rho_\mathrm{out} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad ; \ r \geq r_\mathrm{out} }$$ The fluid velocity is initially set to zero and the magnetic field is initially uniform with $B^x = 0.1$ and $B^y = B^z = 0$. The test is performed on a $200 \times 200$ grid with x- and y-coordinates spanning over the range $[-6,6]$. We adopt a Courant factor of 0.25 and an ideal fluid EOS with adiabatic index $\Gamma = 4/3$. We use the second order MINMOD reconstruction method along with the HLLE flux solver and the RK4 method for time-step evolution. The resulting structure of the blast wave is shown in for the final time $t=4.0$. In particular, we show the two-dimensional distribution of gas pressure $p_\mathrm{gas}$, Lorentz factor $W$ (together with magnetic field lines), and the $x$– and $y$–components of the magnetic field, $B^x$ and $B^y$. This Figure shows a very similar behavior as compared to the results already presented in the literature [@mosta2013grhydro; @etienne2010relativistic; @del2007echo]. {width="0.9\linewidth"} {width="0.95\linewidth"} {width="0.9\linewidth"} provides instead a quantitative indication of the accuracy of our code. In this case, we show a one-dimensional slice along $y=0$ of the final blast wave configuration at time $t=4.0$, in terms of rest-mass density and $B^x$. Two cases are considered: on the left, we compare the results obtained with low ($200$ grid–points) and high resolution ($400$ grid–points); on the right, we compare the high resolution test results obtained with `Spritz` with those obtained with `GRHydro` [@mosta2013grhydro]. For the first comparison, we notice that the peaks differ slightly in $\rho$ due to the fact that the ability to capture the peak sharpness depends significantly on resolution. The values of $B^x$ show a much smaller deviation, due to a smoother variation of this quantity. For the second comparison, the agreement between `Spritz` and `GRHydro` appears excellent, further verifying the robustness of our code. To validate the implementation of adaptive mesh refinement, we carried out another simulation including an inner refined grid covering the x- and y-coordinates in the range \[-3,3\] with grid-spacing $\Delta x = \Delta y = 0.03$ (while the rest of the domain has double grid spacing). shows the comparison with the uniform grid test in terms of final pressure distribution. No significant differences are found, nor specific effects at the inner grid separation boundary, demonstrating a correct implementation of the AMR infrastructure. ### Magnetic Rotor {#MagRot} \ The second two-dimensional test we consider is the magnetic cylindrical rotor, originally introduced for classic MHD in [@balsara1999staggered; @toth2000b] and later employed also for relativistic MHD in [@etienne2010relativistic; @del2003efficient]. The initial setup of this test consists of a dense, rapidly spinning fluid at the center, surrounded by a static ambient medium, where the entire domain is set with a uniform magnetic field and pressure. For setting the initial parameters, we take the radius of the inner rotating fluid as $r=0.1$, with inner rest-mass density $\rho_\mathrm{in} = 10.0$, uniform angular velocity $\Omega=9.95$, and therefore the maximum value of the fluid three–velocity is $v_\mathrm{max} = 0.995$. For the outer static ambient medium, we set the rest-mass density as $\rho_\mathrm{out} = 1.0$. The initial magnitudes of the magnetic field and gas pressure are $B^i=(1.0, 0, 0)$ and $p_{gas,\mathrm{in}} = p_{gas,\mathrm{out}} = 1.0$. The problem is set up on a $400 \times 400$ grid with $x$– and $y$–coordinates lying in range $[0,1]$. Here, we fix the Courant factor to 0.25 and consider an ideal fluid EOS with adiabatic index $\Gamma = 5/3$. For the system evolution, we use the second order MINMOD reconstruction method, the HLLE flux solver, and the RK4 method for time–stepping. shows the two-dimensional profiles of density $\rho$, gas pressure $p_\mathrm{gas}$, magnetic pressure $p_\mathrm{mag}= b^2/2$, and Lorentz factor $W$ along with magnetic field lines, all at the final time $t=0.4$. The rotation of the cylinder causes magnetic winding. As one can see in the bottom–right panel of , the field lines are twisted roughly by $\sim 90^\circ$ in the central region. This twisting of field lines eventually slows down the rotation of the cylinder. There is also a decrease in $\rho$, $p_\mathrm{gas}$, and $p_\mathrm{mag}$ in the central region, observed along with the formation of an oblate shell of higher density. Also for this test, the results are in good agreement with the ones in the literature [@mosta2013grhydro; @etienne2010relativistic; @del2003efficient]. {width="0.9\linewidth"} Similarly to the test discussed in , we perform a quantitative check by taking a one–dimensional slice along $y=0$ of the final rotor configuration at $t=0.4$. Again two cases are considered: (i) results comparison for the low and high resolution runs having $250$ and $400$ grid–points, respectively; (ii) results comparison for our high resolution test with the corresponding one preformed with `GRHydro` [@mosta2013grhydro]. shows this comparison made for the two quantities $\rho$ and $B^x$. For (i), as the resolution is increased, the peaks in $\rho$ as well as $B^x$ are better captured, showing signs of convergence towards the expected solution. For (ii), except for a minor difference in the peak values, the curves are comparable. {width="0.95\linewidth"} ### Loop Advection {#LoopAdv} \ The third and last two–dimensional test we performed is the advection of a magnetic field loop, which was first described in [@devore1991flux] and appeared later in a slightly modified version (the one we consider) in [@mosta2013grhydro; @beckwith2011second; @gardiner2005unsplit; @stone2008athena]. In this test, a magnetized circular field loop is propagated within the surrounding non–magnetized ambient medium with a constant velocity in a two–dimensional periodic grid. In particular, the analytical prescription for the initial imposed magnetic field (taken from [@mosta2013grhydro]) is given by $$\label{LoopAdvMag} B^x, \ B^y = \cases{ -A_\mathrm{loop}y/r, \ A_\mathrm{loop}x/r \ ; \quad r<R_\mathrm{loop} \\ \qquad \qquad \quad \qquad \ \; \; 0 \ ; \quad r\geq R_\mathrm{loop} }$$ where $A_\mathrm{loop}$ is the amplitude of the magnetic field, $r = \sqrt{x^2 + y^2}$ is the radial coordinate, $R_\mathrm{loop}$ is the loop radius, and $B^z$ is set to zero. The corresponding vector potential prescription from which can be obtained is given by $\bi{A}(r) = (0,0,\mathrm{max}[0,A_\mathrm{loop}(R_\mathrm{loop}-r)])$ [@gardiner2005unsplit]. For the initial parameters, we set the density as $\rho=1.0$ and pressure as $p_\mathrm{gas}=3.0$ throughout the computational domain. For the loop, we assume $A_\mathrm{loop}=0.001$ and $R_\mathrm{loop}=0.3$. The fluid 3-velocity is set to $v^i=(1/12, 1/24, 0)$ for a case where $v^z=0$ and $v^i=(1/12, 1/24, 1/24)$ for a more generic case in which the vertical component of the velocity is non-zero, i.e., $v^z\neq 0$. We run the test in both low resolution with a $128\times 128$ grid and high resolution with a $256\times 256$ grid, where the $x$– and $y$–components span the range \[-0.5,0.5\]. The Courant factor is 0.4 and the adiabatic index for the ideal EOS is $\Gamma=5/3$. Like the previous 2D tests, we utilize the MINMOD reconstruction method along with the HLLE flux solver and the RK4 method for time-step evolution. The outcome of the $v^z\neq 0$ test case is shown in . Here, the top row illustrates the initial configuration of the magnetic loop for the quantities $B^x$ and $p_\mathrm{mag}=b^2/2$ at $t=0$. After one entire cycle of the loop across the domain at $t=24$, the same quantities are depicted in the middle row for low resolution run and the bottom row for high resolution run. We notice a significant loss of magnetic pressure due to numerical dissipation for the low resolution test after one evolution cycle as also reported in [@mosta2013grhydro], which is however smaller for higher resolution. Our results are comparable with the ones presented in [@mosta2013grhydro]. It is worth noting that the expression for magnetic pressure used for is $p_\mathrm{mag}=b^2/2$ and differs from the expression used for figure 10 of [@mosta2013grhydro] by a factor of $1/2$ (in [@mosta2013grhydro] the authors actually plotted $b^2$). To consider a less dissipative numerical scheme, we also perform another run in low resolution employing the PPM reconstruction and compare the results with those obtained with MINMOD reconstruction. This is shown in , where the top and bottom panels represent the outcome of the runs with MINMOD reconstruction and PPM reconstruction, respectively. The first column depicts the initial data at $t=0$, the second column shows the loop at final time $t=24$, while the third column shows the logarithmic values of the absolute differences between the initial and final times. As expected, we find significantly lower dissipation in the PPM case. {width="0.85\linewidth"} {width="\linewidth"} 3D tests {#3D} -------- We now present the results of our 3D tests, mostly including a fully dynamical spacetime. ### Spherical Explosion {#SE} \ We present here the results of a very demanding GRMHD test which is not usually performed by other GRMHD codes and that is successfully passed by the `Spritz` code: the so–called Spherical Explosion. Usually, GRMHD codes based on Cartesian coordinates are tested with the Cylindrical Explosion test (refer to ), because the cylindrical symmetry can be well exploited in such a geometrical setting. Spherical Explosion tests, instead, have commonly been performed with GRMHD codes working in spherical coordinates [@cerda2008new; @cerda2007general], which are not well-suited for dealing with cylindrical symmetry. What make the Spherical Explosion test challenging in Cartesian coordinates are indeed the potential limitations in regions where the shock front is not parallel to the orientation of grid–cells’ faces. The test settings are an extension in spherical symmetry of the Cylindrical Explosion test of . We consider an inner dense sphere of radius $R_\mathrm{in} = 0.8$ centered in the domain’s origin with $\rho_\mathrm{in} = 10^{-2}$ and $p_{gas,\mathrm{in}} = 1.0$, surrounded by a spherical shell covering the radial range $R_\mathrm{in} < r < R_\mathrm{out} = 1.0$ where pressure and density are characterized by an exponential decay analogous to the prescription given in , except that here a spherical radius is considered instead of a cylindrical one. At $r > R_\mathrm{out}$, we have then a low-pressure uniform fluid with $\rho_\mathrm{out} = 10^{-4}$ and $p_{gas,\mathrm{out}} = 3.0 \times 10^{-5}$. In addition, following [@cerda2008new; @cerda2007general], a uniform magnetic field parallel to the z axis is added all over the domain. The domain extension is $ \left[ -6.0, 6.0 \right] $ and is covered by $160$ grid–cells, in all directions. Although a direct comparison with spherical coordinates settings of [@cerda2008new; @cerda2007general] can not be done in a straightforward way, it is worth noting that this choice for the resolution corresponds to considering $80$ cells in the radial direction along the polar axis, i.e., the low–resolution version of the results presented in the aforementioned papers. We decided to perform the evolutions for a total time of $t_\mathrm{final} = 6.0$, with a CFL factor of $0.25$. Our runs did not crash even at this late time, although the shock–front always reaches the boundary of the domain (that is treated with “none” BCs). We also note that in this case the imposed lower limit for the rest-mass density (defining the atmospheric floor, see ) is $\rho_\mathrm{atm} = 10^{-12}$. {width="0.8\linewidth"} In we report on separated rows the results on the $y = 0$ plane of the tests performed respectively with magnetic field strength $B^z = 0.0, 0.1$ and $1.0$. In particular, we show the gas pressure and Lorentz factor $W$ (respectively on the left and right columns) at time $t=4$. Looking a the top–right panel (Lorentz factor in the non–magnetized case), we can observe small deviations from spherical symmetry exactly aligned with the Cartesian axes, giving a hint of the geometrical issues brought by such a demanding test. In fact, as already noted by [@del2003efficient] for the Cylindrical Explosion, the biggest problems are due to the fluid velocity components along the diagonals. However, despite the accumulation of errors along the diagonals due to the non–perpendicularity of the fluxes, the spherical shape of the shock front seems to be very well preserved in this case, even at the relatively low resolution considered here. In presence of a dynamically important magnetic field oriented along the $z$ axis, the shock front deviates naturally from spherical symmetry (see middle row of ). Finally, when the magnetic field strength is very high (see bottom row), the central region gets completely evacuated. Even in such an extreme case, the evolution is still performed without any problem. A final important note is that all the tests for the Spherical Explosion here presented where performed with the minmod reconstruction and the LxF flux method, but without adopting any additional dissipation or ad-hoc fixes. ### TOV star {#TOV} \ Static, spherically symmetric stars in general relativity are best described by the Tolman–Oppenheimer–Volkoff (TOV) equations [@oppenheimer1939massive; @tolman1939static]. To further assess the stability and accuracy of our code, the next test we considered is the evolution of a non–rotating stable TOV configuration for both non–magnetised and magnetised cases. For the test setup, we adopt the model described in [@baiotti2005three] that we build using the TOVSolver thorn [@ET]. In particular, the initial TOV star configuration is generated using a polytropic EOS with adiabatic index $\Gamma=2.0$, polytropic constant $K=100$, and initial central rest-mass density $\rho=1.28\times10^{-3}$. We perform the evolution of this initial configuration adopting an ideal fluid EOS with the same value for $\Gamma$. For the magnetised version, we add the magnetic field to the computed TOV configuration using the analytical prescription of the vector potential $A_\phi$ given by $$\label{VecPot} A_\phi \equiv A_\mathrm{b} \varpi^2 {\rm max} \left( p - p_\mathrm{cut}, 0 \right)^{n_s} \ ,$$ where $\varpi$ is the cylindrical radius, $A_\mathrm{b}$ is a constant, $p_\mathrm{cut}=0.04p_\mathrm{max}$ determines the cutoff when the magnetic field goes to zero inside the NS, with $p_\mathrm{max}$ corresponding to the initial maximum gas pressure, and $n_s=2$ sets the degree of differentiability of the magnetic field strength [@giacomazzo2011accurate]. The value of $A_b$ is chosen such that the maximum value of the initial magnetic field strength is set to $\approx1\times 10^{16} \ \mathrm{G}$. This generates a dipole-like magnetic field confined inside the NS and zero magnetic field outside. The non-magnetised tests are run on a uniform grid with $x$–, $y$– and $z$–coordinates spanning over the range \[0, 20\] with low, medium and high resolution having $(32)^3$, $(64)^3$ and $(128)^3$ grid–cells respectively, and considering reflection symmetry with respect to every direction, i.e., the so–called octant symmetry. Furthermore, we perform two more tests for non-magnetised TOV NS in high resolution (i) employing the Cowling approximation (i. e. considering a fixed space–time) [@cowling1941non; @lindblom1990accuracy; @1969ApJ...158..997T] to check the accuracy of our code by evolving just the hydrodynamical equations on a static spacetime background, and (ii) implementing a mesh refinement composed by two nested boxes centered at the origin and extending up to $x,y,z=$20 and 40, respectively, both having $(128)^3$ grid–cells in each direction (therefore the inner box corresponds to the domain evolved in the unigrid run at high resolution while the outer box allows for a further out external boundary). As the `EinsteinToolkit` does not provide a way to handle reflection symmetry for staggered variables, we perform the magnetised TOV tests in low, medium and high resolution covering the entire domain with $x$–, $y$– and $z$–coordinates lying in the range \[-20, 20\] (considering no reflection symmetries) having the same respective grid-spacing as that of the non-magnetised simulations. All the test cases are simulated for $10$ ms using the PPM reconstruction method, the HLLE flux solver, and the RK4 method for time stepping with a CFL factor of $0.25$. ![Results of the non–magnetised TOV simulations. Top: Time evolution of the normalised central rest–mass density $\rho_\mathrm{c}/\rho_\mathrm{c,0}$ for the different resolution simulations inclusive of cases with Cowling approximation and AMR. Bottom: Comparison of results on the $\rho_\mathrm{c}/\rho_\mathrm{c,0}$ evolution with those obtained with `GRHydro` for low, medium, and high resolution, showing an exact match.[]{data-label="Fig13"}](TOV_NoB_rho1D.eps){width="1\linewidth"} The top panel of shows the central rest–mass density $\rho_\mathrm{c}$ evolution for all three resolutions, the high–resolution in Cowling approximation and the high–resolution with AMR, all for the non–magnetised TOV case. It is worth noting that the AMR case (orange curve) can reproduce perfectly the result in high–resolution (green curve), this proving once again the correctness of AMR implementation within the `Spritz` code. Periodic oscillations are initiated as a result of the truncation errors generated in the initial data, while the cause of dissipation is primarily due to the numerical viscosity of the finite differencing (FD) scheme [@baiotti2005three; @font2000non]. The results converge well after increasing the resolution, and the additional tests for the cases with Cowling approximation and AMR are also fully satisfactory. In order to further investigate the accuracy of our code, we compare the low, medium, and high resolution tests’ results on the $\rho_\mathrm{c}$ evolution with those obtained with `GRHydro`. As shown in the bottom panel of , we observe an exact match. {width="0.5\linewidth"} [\[Fig14\] ]{} The initial magnetic field configuration for the magnetised TOV test is illustrated in . Here, the magnetic field strength is shown along with representative magnetic field lines. ![Results of the magnetised TOV simulation. Top: Time evolution of the normalised central rest–mass density $\rho_\mathrm{c}/\rho_\mathrm{c,0}$ for the different resolution simulations; this gives a nearly exact match with that of the non-magnetised TOV case results (c.f., ). Bottom: Time evolution of the maximum value of the magnetic field strength for all three resolutions.[]{data-label="Fig15"}](MagTOV_Rho_Bmax.eps){width="1\linewidth"} The top panel of shows the evolution of the maximum of the rest-mass density $\rho_\mathrm{c}$ for the magnetised TOV case, which matches almost exactly the one for the non–magnetised case (see the top panel of ). This should be expected, since the imposed magnetic field represents only a small perturbation compared to the gravitational binding energy of the system. In addition, the time evolution of the maximum value of the magnetic field strength $B_\mathrm{max}$ is depicted in the bottom panel of . While $B_\mathrm{max}$ is highly damped for the lowest resolution test with a decrease by a factor of roughly $14.75$ in $10$ ms, its value stabilizes with increasing resolution, as observed for $\rho_\mathrm{c}$. We note again that here the damping is a numerical viscosity effect of the FD scheme. ![Power spectrum of the central rest–mass density evolution, normalized to maximum amplitude of the peaks of oscillations’ frequencies. Left–panel shows the results from the runs without magnetic field, while right–panel shows the results where also magnetic field is considered.[]{data-label="Fig17"}](TOV_NoB_OscFreqRhoMax_PostCactus_IF.eps){width="1\linewidth"} ![Power spectrum of the central rest–mass density evolution, normalized to maximum amplitude of the peaks of oscillations’ frequencies. Left–panel shows the results from the runs without magnetic field, while right–panel shows the results where also magnetic field is considered.[]{data-label="Fig17"}](TOV_B_OscFreqRhoMax_PostCactus_IF.eps){width="1\linewidth"} To conclude this section, we report in the oscillations’ peak frequencies for the evolution of the TOV star models that were simulated with our code, in order to validate our models with the literature results. In particular, we show the results of the high–resolution simulations in pure hydrodynamics with dynamical space–time both adopting uniform grid and AMR, with the Cowling approximation (see left-panel), as well as of the high–resolution run with magnetic field (see right-panel). The power spectrum of each simulation is computed via fast Fourier transform (FFT) in order to extract the amplitudes and frequencies of the oscillations of the central rest mass density, and then the amplitudes are normalized to the maximum one relatively to each simulation. also shows the peak frequencies of the oscillations from the literature taken from [@font2002three], that were obtained with an independent 2D code for fixed space–time and with a perturbative code in the case of hydrodynamics coupled to space–time evolution. An interesting point to note is that although the results of [@font2002three] were obtained with a polytropic EOS, our Ideal Fluid simulations seem to match perfectly the peak frequencies. The ideal fluid EOS produces indeed different results from a polytropic one only in presence of shocks, which in this case appears only on the low-density surface and therefore do not affect the oscillations of the core. Finally, it is worth noting that the peak frequencies of our non–magnetised and magnetised models are in perfect agreement as shown by the left panel of , proving the correctness of the magnetic field implementation. Conclusion and future developments {#sec5} ================================== We have presented a new fully general relativistic code, named `Spritz`, able to evolve the GRMHD equations in three spatial dimensions, on cartesian coordinates, and on dynamical backgrounds. The code is based and considerably improves over our previous `WhiskyMHD` code [@giacomazzo2007whiskymhd; @giacomazzo2011accurate; @GiacomazzoPerna2013]. The `Spritz` code benefits also from the publicly available `GRHydro` [@mosta2013grhydro] and `IllinoisGRMHD` [@etienne2015illinoisgrmhd] codes, in particular in the handling of different EOSs and in the use of a staggered formulation of the vector potential equations. In this paper, we presented in detail the equations and the numerical methods implemented in the code. We have adopted a conservative formulation of GRMHD equations, high-resolution shock-capturing schemes, and we guarantee the divergence-less character of the magnetic field by evolving the vector potential. We also presented a series of tests in special and general relativity. We started by showing the code capability of accurately solving 1D Riemann problems by comparing the numerical results with exact solutions [@giacomazzo2006exact]. We also showed, for the first time, a comparison between a non-staggered and a staggered formulation of the vector potential, demonstrating that the latter prevents the formation of spurious post-shock oscillations (see ) and therefore does not require to apply dissipation to the vector potential [@giacomazzo2011accurate]. We then performed a series of special relativistic MHD tests in 2D, including the cylindrical explosion, the magnetic rotor, and the loop advection tests. All tests showed very good agreement with the exact solution (loop advection) or with other GRMHD codes (cylindrical explosion and magnetic rotor). In the cylindrical explosion case we also tested the code capability of dealing with mesh refinement boundary and demonstrated that they have no effect in the correct evolution of MHD quantities. We also performed, for the first time for a fully GRMHD code, a demanding 3D spherical explosion test with different levels of magnetization. The code produced results in very good agreement with those produced by other codes. We concluded our series of tests with a standard 3D evolution of a stable TOV configuration (both with and without magnetic field) in order to show the code ability to handle fully general relativistic regimes. In particular we checked the frequency of TOV oscillations and compared them with results available in the literature. While the `Spritz` code can handle any equation of state, in this paper we focused on tests using simple gamma-law EOSs in order to check the robustness of our basic GRMHD routines. In a second paper we will present also tests involving the evolution of isolated NSs with finite temperature EOSs and neutrino emission with and without magnetic fields (Sala et al., in preparation). Once this second family of tests will be performed successfully, the `Spritz` code will be one of the very few codes worldwide able to evolve magnetised neutron stars with finite temperature EOSs and neutrino emission [@most2019beyond; @palenzuela2015effects]. In the multimessenger era it is indeed crucial to take into account different aspects of the microphysics in order to be able not only to compute a more accurate merger and post-merge GW signal, but also to provide reliable estimates of the EM emission, including both kilonova and short GRBs. The former requires indeed an accurate estimate of electron fraction and temperature in the post-merger remnant as well as in the ejected material, while the latter needs a precise description of the magnetic field evolution. Once the `Spritz` code will have been used for a first set of binary NS merger simulations, we plan to release it to the public and to ask for its inclusion in future releases of the Einstein Toolkit [@ETKpaper; @EinsteinToolkit:2019_10; @ET]. \ Acknowledgments {#acknowledgments .unnumbered} =============== Numerical calculations have been made possible through a CINECA-INFN agreement, providing access to resources on MARCONI at CINECA. F. C. acknowledges financial support from the INFN HPC\_HTC project. F. C. acknowledges the CCRG at the RIT for the computational resources provided there on the *Green Prairies* local Cluster. F.C. received also access to the NCSA *Blue Waters* Cluster via the NSF AST–1516150 grant and to the TACC `Frontera` Cluster via the NSF PHI–1707946 grant. F.C. has been partially supported by the NASA TCAN 80NSSC18K1488 grant for a three–months visiting period at RIT. F. C. acknowledges also Dr. V. Mewes, Prof. Y. Zlochower, Prof. M. Campanelli and Prof. C. Lousto for interesting scientific discussions. J.V.K. kindly acknowledges the CARIPARO Foundation (https://www.fondazionecariparo.it) for funding his PhD fellowship within the PhD School in Physics at the University of Padova. References {#references .unnumbered} ==========

--- abstract: | Производящие функции Харера-Цагира для эйлеровых характеристик пространств модулей кривых содержат $n$-ожерельные многочлены. Разложения Тейлора для этих многочленов зависят от количеств решений полулинейных сравнений Коэна. Harer-Zagier generating functions for Euler characteristics of moduli spaces of curves contain $n$-necklace polynomials. Taylor expansions for these polynomials depend on numbers of solutions of Cohen semilinear congruences. author: - 'Г. Г. Ильюта' title: 'Производящие функции Харера-Цагира, цикловой индекс Редфилда-Пойа и полулинейные сравнения Коэна' --- [^1] В производящих функциях для эйлеровых характеристик пространств модулей кривых [@6], Th. 4’, Th. 5’; [@3], Th. 4.5, Remark 4.6, появляются многочлены $\beta_{k,d}(t)$. Мы докажем в п. 1, что они отличаются простым преобразованием от обобщённых цикловых индексов Редфилда-Пойа, отвечающих регулярному представлению циклической группы $C_k$ порядка $k$ и её неприводимым характерам. Обобщённые цикловые индексы известны также как симметрические функции Шура конечной группы перестановок (характеры общей линейной группы) [@10] или как образы при отображении Фробениуса индуцированных характеров симметрической группы [@11], p. 395. Многочлены $\beta_{k,d}(t)$ были введены для алгебраического упрощения производящих функций для эйлеровых характеристик. Начальные варианты этих производящих функций содержали количества решений некоторых линейных сравнений (они появляются при гомоморфизме в циклическую группу как образы соотношений в исходной группе), которые затем были “упакованы” в многочлены $(\beta_{k,d}(t))^s$, $s\in\mathbb Z_{>0}$. Одна из частей равенства, приводящего к упрощению производящих функций [@6], р. 482, определяется количествами решений линейных сравнений – в п. 3 мы свяжем другую часть с количествами решений полулинейных сравнений Коэна. Возможно, поиск прямой связи между полулинейными сравнениями Коэна и комбинаторной топологией из [@6] и [@3] мог бы привести к сжатию информации не только на алгебраическом уровне производящих функций, но и на уровне комбинаторной топологии, что позволило бы упростить доказательства в [@6] и [@3]. Полулинейные сравнения Коэна можно рассматривать как семейства линейных сравнений в следующем смысле. Если $(x_1,\dots,x_m;y_1,\dots,y_m)$ – решение сравнения Коэна (7) для $r=1$, то $(x_1,\dots,x_m)$ – решение линейного сравнения, коэффициенты которого зависят от $(y_1,\dots,y_m)$ (формула для количества решений линейного сравнения с произвольными коэффициентами имеется в [@2], p. 138). Такая интерпретация может быть полезной для комбинаторных (биективных) доказательств формул из п. 3. Для $t\in\mathbb Z_{>0}$ специализация обобщённого циклового индекса в правой части формулы (4) совпадает с размерностью отвечающего неприводимому характеру класса симметрии тензоров, причём, это верно для обобщённого циклового индекса любого неприводимого характера любой конечной группы перестановок [@10], p. 226. Поэтому для регулярного представления циклической группы формулы из п. 3 связывают размерности классов симметрии тензоров, отвечающие неприводимым характерам циклической группы, с количествами решений полулинейных сравнений Коэна. Интересно было бы найти категорификацию этих формул. В п. 1 в качестве промежуточных объектов между обобщёнными цикловыми индексами Редфилда-Пойа и многочленами Харера-Цагира $\beta_{k,d}(t)$ вводятся $n$-ожерельные многочлены $M(t;n,k)$ (они определяются классическими суммами Рамануджана $c(n,k)$). Выделение этих многочленов мотивируется связями ожерельных многочленов $M(t;1,k)$ с различными разделами математики (обзор имеется в [@7]) – возникает вопрос о связях этих разделов с пространствами модулей кривых. Другие обобщения многочленов $M(t;1,k)$ [@7] определяют эйлеровы характеристики пространств модулей неприводимых многочленов от нескольких переменных над $\mathbb R$ и над $\mathbb C$ – вопрос о связи этих пространств с пространствами модулей кривых также остаётся открытым. Для $t\in\mathbb Z_{>0}$ производящий многочлен для чисел $M(t;n,k)$ $$\sum_{n=1}^kM(t;n,k)q^n$$ является специализацией симметрической функции Краскиевича-Веймана [@9], p. 9, [@1], p. 9 (полагаем $t$ переменных этой функции равными $1$, а остальные – равными $0$). В п. 2 мы покажем как зависит от многочленов $M(t;n,k)$ производящая функция для эйлеровых характеристик $e(\Gamma_g^1)$ групп классов отображений $\Gamma_g^1$ кривых рода $g$ с одной отмеченной точкой [@6], Th. 4’. Используется равенство (3), связывающее многочлены $M(t;n,k)$ и $\beta_{k,d}(t)$. Также из этого равенства вытекает, что появляющиеся в [@5], p. 447, многочлены являются обобщениями для любой конечной группы многочленов Харера-Цагира $\beta_{k,d}(t)$ (с заменой $t^a\to t^{k-a}$ для всех $a$). Роль сумм Рамануджана играют суммы значений неприводимого характера группы на всех элементах этой группы, имеющих фиксированный порядок. В п. 3 доказаны формулы, связывающие количества решений $Q_r(n,k,m)$ полулинейных сравнений Коэна с многочленами $M_r(t;n,k)$ (они определяются $r$-суммами Рамануджана $c_r(n,k)$ из [@4]). С многочленами Харера-Цагира $\beta_{k,d}(t)$ связан частный случай $Q_1(n,k,s)$ и $M_1(t;n,k)=M(t;n,k)$, но многочлены $M_r(t;n,k)$ позволяют использовать числа $Q_r(n,k,m)$ для всех $r$. Доказанные в п. 3 формулы вытекают из равенства $$\left.\left(t\frac{d}{dt}\right)^mM_r(t;n,k)\right|_{t=1}=\frac{Q_r(n,k,m)}{k^{rm}}. \eqno (1)$$ Соотношения между числами $Q_r(n,k,m)$ и многочленами $M_r(t;n,k)$ можно разными способами представить как соотношения между производящими функциями для них, мы рассмотрим один такой пример. В [@4], p. 548, получена формула для производящего ряда Дирихле чисел $Q_r(n,k,m)$, мы докажем аналогичную формулу для многочленов $M_r(t;n,k)$. Для соотношений между этими рядами Дирихле формула (1) сводится к равенству $$\left.\left(t\frac{d}{dt}\right)^mLi_s(t)\right|_{t=1}=\zeta(s-m),$$ где $\zeta(s)=Li_s(1)$ – дзета-функция Римана и $Li_s(t)$ – полилогарифм, $$Li_s(t):=\sum_{k\ge 1}\frac{t^k}{k^s}.$$ 1\. Многочлены Харера-Цагира и $n$-ожерельные многочлены. Для $n,k\in\mathbb Z_{>0}$ определим $n$-ожерельные многочлены $M(t;n,k)$ формулой $$M(t;n,k):=\frac{1}{k}\sum_{d|k}c(n,k/d)t^d,$$ где $c(n,k)$ – суммы Рамануджана, $$c(n,k):=\sum_{\substack{1\le m\le k\\ (m,k)=1}}\epsilon_k^{mn},$$ $\epsilon_k:=e^\frac{2\pi i}{k}$, $(n,k)$ – наибольший общий делитель чисел $n$ и $k$. В частности, $c(0,k)=\phi(k)$ – функция Эйлера, $c(1,k)=\mu(k)$ – функция Мёбиуса. Многочлены $\beta_{k,d}(t)$ определяются формулой [@6], p. 482, $$\beta_{k,d}(t):=\sum_{r=1}^{k-1}\epsilon_d^rt^{k-(k,r)}=\sum_{l|k,l< k}c(k,l,d)t^{k-l}, \qquad d|k,$$ где для $l|k$ и $d|k$ $$c(k,l,d):=\sum_{\substack{1\le m\le k\\ (m,k)=l}}\epsilon_d^m.$$ Для характера $\chi$ группы перестановок $H$ на множестве из $v$ элементов обобщённый цикловой индекс $Z_H^\chi(t_1,\dots,t_v)$ определяется формулой $$Z_H^\chi(t_1,\dots,t_v):=\frac{1}{|H|}\sum_{h\in H}\chi(h)\prod_{j=1}^vt_j^{c_j(h)},$$ где $c_j(h)$ – количество циклов длины $j$ в перестановке $h$. Для конечной группы $G$ обозначим через $R(G)$ её образ при регулярном представлении. Тогда $R(G)$ – группа перестановок на множестве $G$. Через $\chi_n$ обозначим неприводимый характер циклической группы $C_k$, значение которого на образующей группы равно $\epsilon_k^n$. \[prop1\] $$c(k,l,d)=c(k/d,k/l), \eqno (2)$$ $$\beta_{k,d}(t)=kt^kM(1/t;k/d,k)-1, \eqno (3)$$ $$M(t;n,k)=Z_{R(C_k)}^{\chi_n}(t,\dots,t). \eqno (4)$$ Доказательство. Полагая $a:=m/l$, получим $$c(k,l,d)=\sum_{\substack{1\le m\le k\\ (m,k)=l}}\epsilon_d^m=\sum_{\substack{1\le m\le k\\ (m/l,k/l)=1}}\epsilon_{k/l}^{mk/(dl)}.$$ $$=\sum_{\substack{1\le a\le k/l\\ (a,k/l)=1}}\epsilon_{k/l}^{ak/d}=c(k/d,k/l).$$ Из этого равенства следует формула (3). При регулярном представлении группы $G$ элемент $g\in G$ (обозначим через $o(g)$ его порядок) действует как циклическая перестановка на каждом классе смежности порождённой им подгруппы, а значит является произведением $|G|/o(g)$ циклов длины $o(g)$. Поэтому $$Z_{R(G)}^{\chi}(t_1,\dots,t_{|G|})=\frac{1}{|G|}\sum_{d||G|}t_{|G|/d}^d\sum_{o(g)=|G|/d}\chi(g).$$ В частности, для $G=C_k$ из определения сумм Рамануджана следует равенство $$Z_{R(C_k)}^{\chi_n}(t_1,\dots,t_k)=\frac{1}{k}\sum_{d|k}c(n,k/d)t_{k/d}^d. \blacksquare$$ 2\. Производящие функции Харера-Цагира. Согласно [@6], Th. 4’, производящая функция для эйлеровых характеристик $e(\Gamma_g^1)$ групп классов отображений $\Gamma_g^1$ кривых рода $g$ с одной отмеченной точкой имеет вид $$\sum_{g\ge 1}e(\Gamma_g^1)t^{2g-1}=\sum_{k\ge 1}\frac{\phi(k)}{k}\sum_{d|k}\mu(d)\Phi^1(\beta_{k,d}(t),kt^k), \eqno (5)$$ где $$\Phi^1(X,Y):=\sum_{s\ge 2}\frac{(-1)^sX^s}{s(s-1)Y} -\sum_{\substack{h\ge 1\\s\ge 0}}\binom{s+2h-2}{s}\frac{(-1)^sB_{2h}X^sY^{2h-1}}{2h}$$ $$=\frac{1}{Y}((1+X)\log(1+X)-X)+\mathfrak B\left(\frac{Y}{1+X}\right),$$ $$\mathfrak B(T)=-\sum_{h\ge 1}\frac{B_{2h}}{2h}T^{2h-1}\in\mathbb Q[[T]],$$ $B_{2h}$ – числа Бернулли. С помощью Предложения 1 формула (5) преобразуется к следующему виду. \[prop2\] $$\sum_{g\ge 1}e(\Gamma_g^1)t^{2g-1}=\sum_{k\ge 1}\frac{\phi(k)}{k}\sum_{d|k}\mu(d)\bar\Phi^1(M(1/t;k/d,k),kt^k),$$ где $$\bar\Phi^1(Z,Y)=\Phi^1(YZ-1,Y)=Z\log(YZ)-Z+1/Y+\mathfrak B(1/Z). \blacksquare$$ Аналогично зависит от многочленов $M(t;n,k)$ производящая функция для эйлеровых характеристик $e(\Gamma_g)$ групп классов отображений $\Gamma_g$ кривых рода $g$ без отмеченных точек [@6], Th. 5’, $$\sum_{g\ge 1}e(\Gamma_g)t^{2g-2}=\sum_{k\ge 1}\sum_{m,d|k}\frac{\phi(d)\mu(m)}{m^2}\Phi\left(\beta_{k/m,d/(d,m)}(t^m),\frac{kt^k}{m}\right),$$ где $$\Phi(X,Y):=\sum_{s\ge 3}\frac{(-1)^{s-1}X^s}{s(s-1)(s-2)}+\sum_{s\ge 1}\frac{(-1)^sX^sY^2}{12s}$$ $$+\sum_{h\ge 2}\frac{B_{2h}}{2h(2h-2)}\left(\frac{Y}{1+X}\right)^{2h-2}.$$ 3\. Полулинейные сравнения Коэна. Для делителей $l_1,\dots,l_s$ числа $k$ пусть $N_k(b;l_1,\dots,l_s)$ – количествo решений сравнения $$x_1+\dots+x_s=b\mod k,\quad (x_i,k)=l_i,\quad i=1,\dots,s.$$ Согласно [@2], p. 137, $$N_k(b;l_1,\dots,l_s)=\frac{1}{k}\sum_{d|k}c(b,d)\prod_{i=1}^sc(k/d,k/l_i).$$ В [@6], p. 479, появляются частные случаи чисел $N_k(b;l_1,\dots,l_s)$ и они удаляются из производящих функций [@6], p. 481, с помощью соответствующих частных случаев равенства (мы используем формулы (2) и (3)) $$\sum_{\substack{l_1,\dots,l_s|k\\l_i\ne k,i=1,\dots,s}}N_k(b;l_1,\dots,l_s)t^{ks-\sum l_i}$$ $$=\frac{1}{k}\sum_{d|k}c(b,d)\sum_{\substack{l_1,\dots,l_s|k\\l_i\ne k,i=1,\dots,s}}c(k/d,k/l_1)t^{k-l_1}\dots c(k/d,k/l_s)t^{k-l_s}$$ $$=\frac{1}{k}\sum_{d|k}c(b,d)(\beta_{k,d}(t))^s.$$ Аналогично для многочленов $M(t;n,k)$ имеем $$\sum_{l_1,\dots,l_s|k}N_k(b;l_1,\dots,l_s)t^{\sum l_i}$$ $$=\frac{1}{k}\sum_{d|k}c(b,d)\sum_{l_1,\dots,l_s|k}c(k/d,k/l_1)t^{l_1}\dots c(k/d,k/l_s)t^{l_s}$$ $$=\frac{1}{k}\sum_{d|k}c(b,d)(kM(t;k/d,k))^s. \eqno (6)$$ Пусть $Q_r(n,k,m)$ – количество решений $$x_i\mod k,\quad y_i\mod k^r,\quad i=1,\dots,m,$$ полулинейного сравнения Коэна $$a_1x_1^ry_1+\dots+a_sx_m^ry_m=n\mod k^r, \eqno (7)$$ где $(a_i,k)=1$ для всех $i$. Согласно [@4], p. 547, $$\frac{Q_r(n,k,m)}{k^{rm}}=\frac{1}{k^r}\sum_{d|k}c_r(n,k/d)d^m. \eqno (8)$$ В Предложении 3 мы свяжем числа $Q_r(n,k,m)$ с более общими многочленами $$M_r(t;n,k):=\frac{1}{k^r}\sum_{d|k}c_r(n,k/d)t^d,$$ где $c_r(n,k)$ – $r$-суммы Рамануджана [@4], $$c_r(n,k):=\sum_{\substack{1\le m\le k^r\\(m,k^r)_r=1}}\epsilon_{k^r}^{mn},$$ $(a,b)_r$ – наибольший общий делитель чисел $a$ и $b$, являющийся $r$-й степенью. В частности, $c_1(n,k)=c(n,k)$ и $M_1(t;n,k)=M(t;n,k)$. Числа Стирлинга первого рода $s(n,m)$ определяются равенством $$\prod_{i=0}^{n-1}(t-i)=\sum_{m=0}^ns(n,m)t^m.$$ Пусть $$\delta_{k^r|n}:=M_r(1;n,k)=\frac{1}{k^r}\sum_{d|k}c_r(n,k/d).$$ Согласно [@4], p. 546, сумма в правой части равна $1$, если $k^r|n$, и равна $0$ в других случаях. \[prop3\] $$\left.\frac{d^l}{dt^l}M_r(t;n,k)\right|_{t=1}=\sum_{m=1}^ls(l,m)\frac{Q_r(n,k,m)}{k^{rm}},\quad l\ge 1, \eqno (9)$$ $$M_r(t;n,k)=\delta_{k^r|n}+\sum_{l=1}^k\frac{(t-1)^l}{l!}\sum_{m=1}^ls(l,m)\frac{Q_r(n,k,m)}{k^{rm}}, \eqno (10)$$ $$M_r(e^\lambda;n,k)=\delta_{k^r|n}+\sum_{l=1}^{\infty}\frac{\lambda^l}{l!}\frac{Q_r(n,k,l)}{k^{rl}}, \eqno (11)$$ в частности, для $r=1$ $$\left.\frac{d^l}{dt^l}M(t;n,k)\right|_{t=1}=\sum_{m=1}^ls(l,m)\frac{Q_1(n,k,m)}{k^{m}}, \quad l\ge 1,$$ $$M(t;n,k)=\delta_{k|n}+\sum_{l=1}^k\frac{(t-1)^l}{l!}\sum_{m=1}^ls(l,m)\frac{Q_1(n,k,m)}{k^m},$$ $$M(e^\lambda;n,k)=\delta_{k|n}+\sum_{l=0}^{\infty}\frac{\lambda^l}{l!}\frac{Q_1(n,k,l)}{k^l}.$$ Доказательство. $$\left.\left(t\frac{d}{dt}\right)^mM_r(t;n,k)\right|_{t=1}=\frac{1}{k^r}\left.\left(t\frac{d}{dt}\right)^m\sum_{d|k}c_r(n,k/d)t^d\right|_{t=1}$$ $$=\left.\frac{1}{k^r}\sum_{d|k}c_r(n,k/d)d^mt^d\right|_{t=1}=\frac{Q_r(n,k,m)}{k^{rm}}.$$ Используя равенство [@8], p. 4, $$t^l\left(\frac{d}{dt}\right)^l=\sum_{m=1}^ls(l,m)\left(t\frac{d}{dt}\right)^m,$$ получим формулу (9). Формулы (10) и (11) представляют собой разложения Тейлора с учётом равенства $$\left.\left(\frac{d}{d\lambda}\right)^lM_r(e^\lambda;n,k)\right|_{\lambda=0}=\left.\left(t\frac{d}{dt}\right)^lM_r(t;n,k)\right|_{t=1}=\frac{Q_r(n,k,l)}{k^{rl}}.\blacksquare$$ Отметим ещё один способ связать числа $N_k(b;l_1,\dots,l_s)$ и $Q_1(n,k,s)$. Заменяя в правой части формулы (6) $t^d$ на $d^t$ для всех $d$, получим следующий её аналог $$\sum_{l_1,\dots,l_s|k}N_k(b;l_1,\dots,l_s)(\prod l_i)^t =\frac{1}{k}\sum_{d|k}c(b,d)(\sum_{\delta|k}c(k/d,k/\delta)\delta^t)^s.$$ Значения левой части этой формулы в точках $t\in\mathbb Z_{\ge 0}$, а значит и коэффициенты интерполяционного ряда Ньютона этой функции, определяются числами $Q_1(n,k,s)$ (это следует из формулы (8)). Напомним, что ряд Ньютона для функции $h(t)$ определяется по любой последовательности $a_0,a_1,a_2,\dots$ различных чисел (в нашем случае из $\mathbb Z_{\ge 0}$) $$h(t)=h(a_0)+\sum_{i=1}^\infty\Delta_h[a_0,\dots,a_i]\prod_{j=0}^{i-1}(t-a_j),$$ где $$\Delta_h[a_0,\dots,a_i]:=\sum_{j=0}^i\frac{h(a_j)}{\prod_{p=0,p\ne j}^{i}(a_j-a_p)}.$$ Согласно [@4], p. 548, для $p\in\mathbb C$ $$Q_r(n,m):=\sum_{k\ge 1}\frac{Q_r(n,k,m)}{k^{r(p+m)}}=\frac{\zeta(rp+r-m)}{\zeta(rp+r)}\sigma_{-p}(n,r),\eqno (9)$$ где $$\sigma_p(n,r):=\sum_{d^r|n}d^{rp}.$$ Докажем аналогичную формулу для многочленов $M_r(t;n,k)$. \[prop4\] $$M_r(t;n):=\sum_{k\ge 1}\frac{M_r(t;n,k)}{k^{rp}}=\frac{Li_{rp+r}(t)}{\zeta(rp+r)}\sigma_{-p}(n,k).\eqno (9)$$ Доказательство. Используя правило умножения рядов Дирихле (коэффициенты произведения являются свёртками Дирихле коэффициентов сомножителей) и равенство [@4], p. 548, $$\sum_{k\ge 1}\frac{c_r(n,k)}{k^{rp}}=\frac{\sigma_{1-p}(n,r)}{\zeta(rp)}$$ получим $$\sum_{k\ge 1}\frac{M_r(t;n,k)}{k^{rp}}=\sum_{k\ge 1}\frac{1}{k^{rp+r}}\sum_{d|k}c_r(n,k/d)t^d$$ $$=\sum_{k\ge 1}\frac{c_r(n,k)}{k^{rp+r}}\sum_{k\ge 1}\frac{t^k}{k^{rp+r}}=\frac{Li_{rp+r}(t)}{\zeta(rp+r)}\sigma_{-p}(n,k).\blacksquare$$ Для производящих рядов Дирихле $Q_r(n,m)$ и $M_r(t;n)$ Предложение 3 примет следующий вид (полагаем $M(t;n):=M_1(t;n)$ и $Q(n,m):=Q_1(n,m)$). \[prop3\] $$\left.\left(\frac{d}{dt}\right)^lM_r(t;n)\right|_{t=1}=\sum_{m=1}^ls(l,m)Q_r(n,m),\quad l\ge 1,$$ $$M_r(t;n)=\sum_{k:k^r|n}\frac{1}{k^{rp}}+\sum_{l\ge 1}\frac{(t-1)^l}{l!}\sum_{m=1}^ls(l,m)Q_r(n,m),$$ $$M_r(e^\lambda;n)=\sum_{k:k^r|n}\frac{1}{k^{rp}}+\sum_{l=1}^{\infty}\frac{\lambda^l}{l!}Q_r(n,l),$$ в частности, для $r=1$ $$\left.\left(\frac{d}{dt}\right)^lM(t;n)\right|_{t=1}=\sum_{m=1}^ls(l,m)Q(n,m),\quad l\ge 1,$$ $$M(t;n)=\sum_{k:k|n}\frac{1}{k^p}+\sum_{l\ge 1}\frac{(t-1)^l}{l!}\sum_{m=1}^ls(l,m)Q(n,m),$$ $$M(e^\lambda;n)=\sum_{k:k|n}\frac{1}{k^p}+\sum_{l=1}^{\infty}\frac{\lambda^l}{l!}Q(n,l).\blacksquare$$ [14]{} C. Ahlbach, J. P. Swanson, Cyclic sieving, necklaces, and branching rules related to Thrall’s problem, arXiv:1808.06043. K. Bibak, B. M. Kapron, V. Srinivasan, R. Tauraso, L. Toth, Restricted linear congruences, J. Number Theory 171 (2017), 128-144. G. Bini, J. Harer, Euler characteristics of moduli spaces of curves, J. Eur. Math. Soc. 13 (2011), 487–512. E. Cohen, An extension of Ramanujan’s sum. II. Additive properties, Duke Math. J. 22 (1955), 543-550. I. P. Goulden, J. H. Kwak, J. Lee, Distributions of regular branched surface coverings, European J. Combin. 25 (2004), 437-455. J. Harer, D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), 457–485. T. Hyde, Cyclotomic factors of necklace polynomials, arXiv:1811.08601. W. Lang, On generalizations of the Stirling number triangles, J. Integer Seq. 3 (2000), Article 00.2.4. A. Lascoux, B. Leclerc, J.-Y. Thibon, Ribbon tableaux, Hall-Littlewood functions and unipotent varieties, Sem. Lothar. Combin. 34 (1995), Art. B34g. R. Merris, Manifestations of Polya’s counting theorem, Linear Algebra Appl. 32 (1980), 299-234. R. P. Stanley, Enumerative Combinatorics, vol. II, Cambridge Univ. Press 1999. [^1]: Работа поддержана грантом РФФИ-16-01-00409

--- abstract: 'Within the relativistic Hartree-Fock-Bogoliubov (RHFB) theory, the structure properties of Carbon isotopes are systematically studied. To provide better overall description, the finite-range Gogny force D1S with an adjusted strength factor is adopted as the effective paring interaction in particle-particle channel. The self-consistent RHFB calculations with density-dependent meson-nucleon couplings indicate the single-neutron halo structures in both $^{17}$C and $^{19}$C, whereas the two-neutron halo in $^{22}$C is not well supported. It is also found that close to the neutron drip line there exists distinct odd-even staggering on neutron radii, which is tightly related with the blocking effects and correspondingly the blocking effect plays a significant role in the single-neutron halo formation.' author: - Xiao Li Lu - Bao Yuan Sun - Wen Hui Long title: 'Descriptions of Carbon isotopes within relativistic Hartree-Fock-Bogoliubov theory' --- Introduction ============ During the past decades, the radioactive ion beams (RIBs) have greatly extended our knowledge of nuclear physics, from which are obtained the critical data for nuclear physics, astrophysics, as well as for testing the standard model. With worldwide and rapid development of RIB facilities, the investigations of the nuclear systems under extreme conditions generate new frontiers in nuclear physics. For example, the exotic nuclei [@Chulkov1996; @Mueller2001; @Tanihata1995; @John1999] have fascinated more and more interests due to the unexpected exotic modes therein. One of the representatives is the nuclear halo structure characterized by a dilute matter distribution contributed by several (in general two) loosely bound valence neutrons (or protons) surrounding a condensed core, which was first found in $^{11}$Li [@Tanihata1985]. As the typical light nuclei, the Carbon isotopes have been devoted many efforts to probing the possible halo structure [@Bazin1998; @Fang2004; @Horiuchi2006] and specifically recent measured reaction cross section of $^{22}$C [@Tanaka2010] seems to assert a new two-neutron halo structure, which has also attracted fairly large interests from the community [@Sharma2011; @LuLu2012; @Fortune2012]. In fact, the exotic modes keeping found in the weakly bound nuclear systems also bring serious challenges on the reliability of the nuclear theoretical models. When extending to the limit of stability of isotopes or isotones, the single neutron or proton separation energies become comparable to the pairing gap energy, such that the continuum effects can be easily involved by pairing correlations and play a significant role in determining the structure properties of exotic nuclei [@Meng1996; @Meng1998; @Meng1998NPA]. In terms of Bogoliubov quasi-particle, the relativistic Hartree-Bogoliubov (RHB) theory [@Meng1998NPA; @Vretenar2005; @Meng2006] has unified the descriptions of relativistic Hartree (RH) mean field and pairing correlations, and consequently the continuum effects are involved automatically. Since the first self-consistent description of nuclear halo structure in $^{11}$Li [@Meng1996], the RHB theory has been successfully applied in predicting the giant halos in Ca [@Meng2002; @Long2010] and Zr [@Meng1998; @Grasso2006; @Long2010] isotopes, as well as the restoration of relativistic symmetry [@Meng1999PRC] and superheavy magic structures [@Zhang2005]. With the inclusion of Fock terms in the mean field, the relativistic Hartree-Fock-Bogoliubov (RHFB) theory with density-dependent meson-nucleon couplings [@Long2010RHFB] provides a new self-consistent platform for the exploration of exotic nuclei, e.g., predicting the giant halos in Cerium isotopes [@Long2010]. In addition, the inclusion of Fock terms has brought substantial improvements in the self-consistent description of nuclear shell structures [@Long2007] and the evolutions [@Long2008; @Long2009], the relativistic symmetry restorations [@Long2006; @Long2007; @Liang2010], and the low-energy excitation modes [@Liang2008]. In this work, the structure properties of Carbon isotopes, particularly the possible halo structures therein, will be studied systematically within the RHFB and RHB theories. The contents are organized as follows. In the Sec. \[Theory\], we introduce the general formalism of the RHFB equations with finite range (Gogny) pairing force. In Sec. \[result\] the discussions are concentrated on the halo structures and odd-even staggering (OES) on the neutron radii for Carbon isotopes. Finally, a brief summary and perspective are given in Sec. \[summary\]. Theoretical framework and numerical details {#Theory} =========================================== In relativistic nuclear models the effective force between the nucleons is mediated by the exchange of mesons and photons. Based on that, the model Lagrangian contains the system degrees of freedom associated with the nucleon $\psi$, isoscalar scalar $\sigma$-meson, isoscalar vector $\omega$-meson, isovector vector $\rho$-meson, isovector pseudo-scalar $\pi$-meson and the photon ($A$) fields [@Bouyssy1987; @Long2007]. Following the standard variational procedure, one can get the equations of motion for nucleons, mesons, and photons, namely the Dirac, Klein-Gordon, and Proca equations, as well as the continuity equation for energy-momentum tensor, from which is derived the system Hamiltonian. In the terms of the creation and annihilation operators $(c_\alpha^\dag,c_\alpha)$ defined by the stationary solutions of the Dirac equation, the Hamiltonian operator can be expressed as $$\begin{aligned} H=&\sum_{\alpha\beta}c_\alpha^\dag c_\beta T_{\alpha\beta}+\frac{1}{2}\sum_{\alpha\alpha'\beta\beta'}c_\alpha^\dag c_\beta^\dag c_{\beta'}c_{\alpha'}\sum_{\phi}V_{\alpha\beta\alpha'\beta'}^\phi,\label{H}\end{aligned}$$ where $T_{\alpha\beta}$ is the kinetic energy and the two-body terms $V_{\alpha\beta\alpha'\beta'}^\phi$correspond with the meson- (or photon-) nucleon couplings denoted by $\phi$, $$\begin{aligned} T_{\alpha\beta}&=\int d\bm{r}\bar{\psi}_\alpha(\bm{r})(-i{{\mbox{\boldmath${ \gamma}$}}}\cdot{{\mbox{\boldmath${ \nabla}$}}}+M)\psi_\beta(\bm{r}),\\ V_{\alpha\beta\alpha'\beta'}^\phi&=\int d\bm{r}d\bm{r}'\bar{\psi}_\alpha(\bm{r})\bar{\psi}_\beta(\bm{r}')\Gamma_\phi({{\mbox{\boldmath${ r}$}}},{{\mbox{\boldmath${ r}$}}}')\nonumber\\ &\hspace{8em}\times D_\phi(\bm{r},\bm{r}')\psi_{\beta'}(\bm{r}')\psi_{\alpha'}(\bm{r}).\end{aligned}$$ In above equations, $\Gamma_\phi(x,x')$ represent the interaction matrices associated with the $\sigma$-scalar, $\omega$-vector, $\rho$-vector, $\rho$-tensor, $\rho$-vector-tensor, $\pi$-pseudo-vector and photon-vector couplings, and $D_\phi(\bm{r}, \bm{r}')$ denotes relevant meson (photon) propagator, and $M$ is the nucleon mass (for details see Refs. [@Bouyssy1987; @Long2007; @Long2010RHFB]). Standing on the level of relativistic Hartree-Fock (RHF) approach, the contributions from the negative energy states in the Hamiltonian (1) are neglected as usual, i.e., the so-called no-sea approximation [@Bouyssy1987]. The Hartree-Fock ground state ${\left|\Phi_0\right>} $ is then determined and consequently is derived the energy functional $E$, i.e., the expectation of Hamiltonian with respect to ${\left|\Phi_0\right>} $, $$\begin{aligned} {\left|\Phi_0\right>} =& \prod_{i=1}^A c_i^\dag |0\rangle,& E=&{\left<\Phi_0\right|} H{\left|\Phi_0\right>} ,\end{aligned}$$ where the index $i$ denotes the positive energy states and $|0\rangle$ is the vacuum state. In the energy functional $E$, the two-body interactions $V^\phi$ lead to two types of contributions, i.e., the direct (Hartree) and exchange (Fock) terms. Within RHFB [@Long2010RHFB], the mean field part contains both types of the contributions, i.e., the RHF approach [@Long2006DDRHF], whereas within RHB the Fock terms are neglected just for simplicity. For the open-shell nuclei, the pairing correlations, which lead to valence particles spreading over the orbits around the Fermi level, have to be taken into account. Different from simple BCS method [@Bardeen1957], the Bogoliubov theory can unify the descriptions of mean field and pairing correlations in terms of Bogoliubov quasi-particle. It is of special significance in exploring the nuclei far from the $\beta$-stability line where the continuum effects become essential and the simple BCS method may break down. In the prior studies with both RHB and RHFB theories, it is already demonstrated that the scattering of the Cooper pairs into the continuum plays an essential role in the formation of the halo structures [@Long2010; @Meng1996; @Meng1998]. Following the standard procedure of the Bogoliubov transformation [@Gor1958; @Kucharek1991], the RHFB equation can be derived as, $$\begin{split} \int d{{\mbox{\boldmath${ r}$}}}'& {\left(\begin{matrix}h({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}') & \Delta({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}')\\[0.5em] -\Delta({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}')& h({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}')\end{matrix}\right)} {\left(\begin{matrix}\psi_U({{\mbox{\boldmath${ r}$}}}')\\[0.5em] \psi_V({{\mbox{\boldmath${ r}$}}}')\end{matrix}\right)}\\&\hspace{6em} = {\left(\begin{matrix} \lambda + E_q &0\\[0.5em] 0&\lambda-E_q \end{matrix}\right)}{\left(\begin{matrix}\psi_U({{\mbox{\boldmath${ r}$}}})\\[0.5em] \psi_V({{\mbox{\boldmath${ r}$}}})\end{matrix}\right)},\label{RHFB} \end{split}$$ where $\psi_U$ and $\psi_V$ are the quasi-particle spinors, $E_q$ denotes the single quasi-particle energy, and the chemical potential $\lambda$ is introduced to keep the particle number on the average. For the single-particle Hamiltonian $h(\bm{r}, \bm{r}')$, it consists of three parts, i.e., the kinetic energy $h^{kin}$, local potential $h^D$ and non-local one $h^E$, $$\begin{aligned} h^{kin}(\bm{r}, \bm{r}') &= \gamma^0{\left({{\mbox{\boldmath${ \gamma}$}}}\cdot{{\mbox{\boldmath${ p}$}}}+ M\right)}\delta(\bm{r}-\bm{r}'),\\ h^{D}(\bm{r}, \bm{r}') &= {\left[\Sigma_T(\bm{r})\gamma_5+\Sigma_0(\bm{r})+\gamma^0\Sigma_S(\bm{r})\right]}\delta(\bm{r}-\bm{r}'),\\ h^{E}(\bm{r}, \bm{r}') &= {\left(\begin{matrix} Y_G({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}') & Y_F({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}') \\[0.5em] X_G({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}') & X_F({{\mbox{\boldmath${ r}$}}}, {{\mbox{\boldmath${ r}$}}}') \end{matrix}\right)}.\end{aligned}$$ Detials are referred to Refs. [@Long2010RHFB; @Bouyssy1987]. The pairing potential in the RHFB equation (\[RHFB\]) reads as $$\begin{aligned} \Delta_\alpha(\bm{r},\bm{r}')=-\frac{1}{2}\sum_\beta V_{\alpha\beta}^{pp}(\bm{r},\bm{r}')\kappa_\beta(\bm{r},\bm{r}'),\end{aligned}$$ with the pairing tensor $\kappa$ $$\begin{aligned} \kappa_\alpha(\bm{r},\bm{r}')=\psi_{V_\alpha}(\bm{r})^*\psi_{U_\alpha}(\bm{r}').\end{aligned}$$ For the pairing interaction $V^{pp}$, it is generally taken as a phenomenological form with great success in RHB theory [@Vretenar2005; @Gonzalez1996] and conventional HFB theory [@Decharg1980; @Dobaczewski1984]. In this work, we utilize the finite-range Gogny force D1S [@Berger1984] with additional strength factor $f$ as the effective pairing interaction, $$\begin{aligned} V(\bm{r},\bm{r}')=&f\sum_{i=1,2}e^{((r-r')/\mu_i)^2}(W_i+B_iP^\sigma-H_iP^\tau-M_iP\sigma P^\tau),\end{aligned}$$ where $\mu_i,W_i,B_i,H_i,M_i(i=1,2)$ are the Gogny parameters and the factor $f$ will be adjusted to provide better overall description for the selected Carbon isotopes. Due to the numerical difficulties originating from both RHF mean field and finite-range pairing interactions, the integro-differential RHFB equation (\[RHFB\]) is solved by expanding the quasi-particle spinors on the Dirac Woods-Saxon (DWS) basis [@Shanggui2003], which can provide appropriate asymptotic behaviors for the continuum states in the weakly bound nuclei. For the calculations of Carbon isotopes, the DWS basis parameters are taken as follows: the spherical box-size is fixed to 30 fm and consistently the numbers of basis states with positive and negative energies are chosen as 48 and 12, respectively. =1em DD NL Fock term $\pi$ $\rho$-tensor -------- ----- ----- ----------- ------- --------------- PKA1 yes no yes yes yes PKO2 yes no yes no no PKO3 yes no yes yes no PKDD yes no no no no DD-ME2 yes no no no no PK1 no yes no no no NL2 no yes no no no : Details for the effective interactions PKA1, PKO2 PKO3, PKDD, DDME2, PK1 and NL2. The abbreviations DD and NL denote the density-depdent meson-nucleon couplings and the non-linear self-couplings, respectively.[]{data-label="tab:coupling"} =0.5em PKA1 PKO2 PKO3 PKDD DD-ME2 PK1 NL2 ---------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- $^{15}$C $\nu s_{1/2}$ $\nu d_{5/2}$ $\nu d_{5/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $^{17}$C $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $^{19}$C $\nu d_{5/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $^{21}$C $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ $\nu s_{1/2}$ : Blocked quasi-neutron ($\nu$) orbits of the ground states of the odd Carbon isotopes $^{15, 17, 19, 21}$C determined by the calculations of PKA1, PKO2, PKO3, PKDD, DD-ME2, PK1 and NL2.[]{data-label="tab:bl"} In this work, we performed systematical calculations for the Carbon isotopes from $^{10}$C to $^{22}$C by the RHFB and RHB theories, utilizing the effective interactions with density-dependent meson couplings, namely PKA1 [@Long2007], PKO2 [@Long2008] and PKO3 [@Long2008], PKDD [@Long2004] and DD-ME2 [@Lalazissis2005], and the ones with non-linear self-couplings, i.e., PK1 [@Long2004] and NL2 [@Lee1986]. The details of the selected effective Lagrangians are referred to Table \[tab:coupling\]. For the odd Carbon isotopes, the blocking effects are taken into account. In general, e.g., under the BCS scheme, several orbits around the Fermi surface are blocked separately and the blocking with the strongest binding corresponds to the ground state [@Junqing2002]. In present study, the self-consistent calculations are carried out within the Bogoliubov scheme and naturally the blocking effects are considered under the same scheme to keep the consistence of the theory itself. According to the mapping relation between the HF single-particle and Bogoliubov quasi-particle states (see Fig. 11 in Ref. [@Meng2006]), the blocked quasi-particle orbit can be determined as the lowest ones, e.g., the orbits $1s_{1/2}$ or $1d_{5/2}$ for $^{15,17,19,21}$C. In table \[tab:bl\] are shown the blocking configurations for the ground states of the odd Carbon isotopes close to the neutron drip line. Results and discussion {#result} ====================== ------ ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- --------- ------- ---------- $E_b$ $S_n$ $S_{2n}$ $E_b$ $S_n$ $S_{2n}$ $E_b$ $S_n$ $S_{2n}$ $E_b$ $S_n$ $S_{2n}$ $E_b$ $S_n$ $S_{2n}$ $E_b$ $S_n$ $S_{2n}$ $E_b$   $S_n$ $S_{2n}$ 1.00 1.34 0.88 0.81 2.14 0.43 1.02 2.10 0.65 1.66 2.83 0.78 1.72 3.00 0.97 1.92 1.38 0.71 1.42 12.29 1.77 3.87 1.10 1.58 0.78 0.75 1.78 0.41 0.84 1.35 0.59 1.72 2.46 0.65 1.59 2.49 0.78 1.71 0.90 0.66 1.28 11.18 1.64 3.59 1.15 1.71 0.95 1.10 1.43 0.46 0.80 0.90 0.62 1.79 2.17 0.63 1.52 2.29 0.63 1.68 0.60 0.78 1.27 10.54 1.57 3.43 1.20 2.08 1.03 1.22 1.02 0.55 0.81 0.51 0.69 1.80 1.81 0.66 1.48 1.90 0.64 1.73 0.46 0.80 1.27 9.83 1.51 3.25 1.25 2.60 1.14 1.34 0.65 0.67 0.99 0.67 0.79 1.79 1.38 0.72 1.57 1.45 0.68 1.80 0.81 0.92 1.44 9.05 1.46 3.05 ------ ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- ------- ------- ---------- --------- ------- ---------- {width="45.00000%"}   {width="45.00000%"} {width="45.00000%"}   {width="45.00000%"} To get appropriate pairing effects, firstly the systematical calculations with different pairing strength factors are performed for the Carbon isotopes. In Table \[tab:deviation\] are shown the root mean square deviations from the data [@Audi2011] for the binding energies $E_b$, single- ($S_n$) and two-neutron ($S_{2n}$) separation energies extracted from the calculations with the selected effective Lagrangians in Table \[tab:coupling\]. It is found that all the effective interactions present appropriate agreement with the data, except NL2 which fails to provide enough binding for Carbon isotopes with about 10% deviations. It can be also seen that the systematics on single- and two-neutron separation energies are improved quantitatively with the modified pairing interactions. Referred to the single-neutron separation energy $S_n$, the optimized strength factors are determined as $f=1.1$ for PKA1, PKO2 and PKO3, and $f=1.15$ for PKDD, DD-ME2 and PK1, and $f=1.25$ for NL2. Among the selected effective Lagrangians, one can find in Table \[tab:deviation\] that PKO2 provides the best agreement with the data on both $S_n$ and $S_{2n}$, which may imply the most reliable systematics. Figure \[fig:S\] presents the single-neutron separation energies $S_n$ (left panels) and two-neutron ones $S_{2n}$ (right panels) of Carbon isotopes calculated with PKA1, PKO2, PK1 and PKDD, as compared to the data (in filled squares) [@Audi2011]. The comparison is performed between the calculations with the original effective pairing interaction Gogny-D1S (open up-triangles) and the ones with the optimized strength factor $f$ (filled circles). From Fig. \[fig:S\] one can find that the modification on the pairing force brings some systematical improvements on both $S_n$ and $S_{2n}$, especially for the calculations with PKA1 and PKO2. The results calculated by PKO3 and DD-ME2 are omitted because of the similar systematics as PKO2 and PKDD, respectively. Specifically with the original Gogny-D1S, PKO2 can not reproduce the drip line $^{22}$C which becomes bound with enhanced pairing force (see Fig. \[fig:S\]). Combined with the results in Table \[tab:deviation\], we utilize PKO2 with optimized pairing force as the representative to analyze the detailed structure properties of Carbon isotopes in the following discussions. ![(Color online) Neutron and proton density distributions for even \[plot (a)\] and odd \[plot (b)\] Carbon isotopes. The results are calculated by RHFB with PKO2 and the optimized pairing strength factor is adopted as $f=1.10$. See the text for details.[]{data-label="fig:DEN"}](DEN-PKO2.pdf "fig:"){width="45.00000%"} ![(Color online) Neutron and proton density distributions for even \[plot (a)\] and odd \[plot (b)\] Carbon isotopes. The results are calculated by RHFB with PKO2 and the optimized pairing strength factor is adopted as $f=1.10$. See the text for details.[]{data-label="fig:DEN"}](DEN-PKO2.eps "fig:"){width="45.00000%"} ![(Color online) (a) Contributions to neutron density ($\rho_n$) from canonical neutron orbits ($\rho^{nlj}$) and the continuum in $^{19}$C, and (b) neutron numbers ($N_{R>r}$) beyond the sphere with radius $r$ for $^{17}$C, $^{18}$C, $^{19}$C and $^{20}$C. The results are extracted from the calculations of RHFB with PKO2. $r_{n, ^{17}\text{C}}$, $r_{n, ^{18}\text{C}}, r_{n, ^{19}\text{C}}$ and $r_{n, ^{20}\text{C}}$ denote the root mean square neutron radii of $^{17}$C, $^{18}$C,$^{19}$C and $^{20}$C, respectively.[]{data-label="fig:VV&halo"}](ration-halo.pdf "fig:"){width="45.00000%"} ![(Color online) (a) Contributions to neutron density ($\rho_n$) from canonical neutron orbits ($\rho^{nlj}$) and the continuum in $^{19}$C, and (b) neutron numbers ($N_{R>r}$) beyond the sphere with radius $r$ for $^{17}$C, $^{18}$C, $^{19}$C and $^{20}$C. The results are extracted from the calculations of RHFB with PKO2. $r_{n, ^{17}\text{C}}$, $r_{n, ^{18}\text{C}}, r_{n, ^{19}\text{C}}$ and $r_{n, ^{20}\text{C}}$ denote the root mean square neutron radii of $^{17}$C, $^{18}$C,$^{19}$C and $^{20}$C, respectively.[]{data-label="fig:VV&halo"}](ration-halo.eps "fig:"){width="45.00000%"} Aiming at the possible halo structure in Carbon isotopes, Fig. \[fig:DEN\] shows the neutron and proton density distributions provided by PKO2 calculations for even \[plot (a)\] and odd \[plot (b)\] Carbon isotopes. As shown in Fig. \[fig:DEN\](a), it seems that the neutron densities of the even isotopes tend to be more and more diffuse, while not distinct enough to support the occurrence of halo structure, when close to the drip line. From the recent data [@Audi2011] the two-neutron separation energy of $^{22}$C is 1.56 MeV, which implies that the last two valence neutrons are still bound too deep to spread over a fairly wide range. Hence $^{22}$C may not be a good candidate of well-developed two-neutron halo structure. Whereas in Fig. \[fig:DEN\] (b) distinct evidence is presented to demonstrate the halo occurrences in $^{17}$C and $^{19}$C, i.e., more diffused neutron distributions with less neutrons than $^{22}$C. In fact, the strong evidence of the halo occurrence in $^{19}$C can be found from the parallel momentum distribution of $^{18}$C after the breakup of $^{19}$C [@Bazin1998]. As shown in Fig. \[fig:S\] nearly zero single-neutron separation energies of $^{17}$C and $^{19}$C can be also treated as another evidence for the existence of single-neutron halo structure. For $^{21}$C the negative value of $S_n$ leads to a diverged matter distribution, which might not be a bound nucleus. ![(Color online) Canonical neutron single-particle energies for Carbon isotopes. The results are extracted from the RHFB calculations with PKO2. The length of thick bars corresponds with the occupation probabilities of neutron orbits in half and $E_F$ represent the Fermi levels.[]{data-label="fig:LEV"}](level.pdf "fig:"){width="45.00000%"} ![(Color online) Canonical neutron single-particle energies for Carbon isotopes. The results are extracted from the RHFB calculations with PKO2. The length of thick bars corresponds with the occupation probabilities of neutron orbits in half and $E_F$ represent the Fermi levels.[]{data-label="fig:LEV"}](level.eps "fig:"){width="45.00000%"} To further illustrate the halo occurrence, Fig. \[fig:VV&halo\](a) presents the contributions to the neutron density from different canonical single-particle orbits. It is clearly shown that the dilute matter distribution at large radial distance is dominated by low-$j$ state $2s_{1/2}$ and the continuum, in accordance with the evidence of halo occurrences in $^{11}$Li [@Meng1996] and Ca isotopes [@Meng2002]. Consistently Fig. \[fig:VV&halo\](b) presents another direct evidence, i.e., the number of neutrons $N_{R>r}$ located beyond the sphere with radius $r$. From Fig. \[fig:VV&halo\](b) it can be deemed that there exist evident single-neutron halo structures in $^{17,19}$C due to fairly large amount of neutrons spreading far beyond the neutron radii $r_n$. In contrast the values of $N_{R>r}$ in neutron-richer isotopes $^{18,20}$C drop sharply with the increase of radius $r$, consistent with the neutron distributions shown in Fig. \[fig:DEN\](a). Combining with the results in Fig. \[fig:VV&halo\](a), one can find that both canonical state $2s_{1/2}$ and the continuum present substantial contributions in the formation of halo, while dominated by the formal one due to its zero centrifugal barrier. As the complemented demonstration, Figure \[fig:LEV\] shows the neutron canonical single-particle energies for the Carbon isotopes from $^{15}$C to $^{22}$C, where the lengths of the ultra thick bar denote the occupation probabilities in half. From Fig. \[fig:LEV\] one can find that the valence orbits $2s_{1/2}$ and $1d_{5/2}$ are close to each another and such high level density in general leads to strong pairing effects. Although both valence orbits are fairly close to the continuum limit, the self-consistent RHFB and RHB calculations only support $^{17, 19}$C as the candidates of halo nuclei instead of even drip-line isotope $^{22}$C, which can be well understood from the blocking effects discussed later. It should be mentioned that the canonical single-particle states in Fig. \[fig:LEV\] are determined from the diagonalization of the density matrix constructed in the Bogoliubov quasi-particle space. In the calculations of $^{17,19}$C with PKO2, the neutron quasi-particle states near the Fermi surface are blocked, namely the lowest one $1s_{1/2}$. According to the mapping relation between the Bogoliubov quasi-particle and canonical single-particle states [@Meng2006; @Meng1998NPA], the corresponding contributions of the blocked quasi-neutron orbits will be mainly mapped into the canonical ones near the Fermi surface, i.e., the canonical $2s_{1/2}$ and $3s_{1/2}$ states as shown in Fig. \[fig:LEV\]. Compared to the even isotopes, the neutron staying on the canonical orbit $2s_{1/2}$ then becomes much less bound in the odd Carbons due to lacking the extra binding from pairing correlations, which is also illustrated by nearly zero values of $S_n$ in Fig. \[fig:S\]. As a result, the probability density of the valence state $2s_{1/2}$ tends to be much diffuser than those in even isotopes to develop the halo structure in $^{17,19}$C. Due to the blocking of $s$ orbit, the continuum effects are also enhanced relatively in the odd isotopes because the neutron Cooper pairs in $d_{5/2}$ orbit can be only scattered into the continuum by pairing correlations. In addition, the odd-even staggering on the position of the canonical state $3s_{1/2}$ (see Fig. \[fig:LEV\]) can be also interpreted by the blocking effects. In the odd Carbon isotopes, the odd quasi-neutron can be mapped partially into the canonical $3s_{1/2}$ orbits, little while still visible, e.g., $v^2 = 0.034$ for $^{19}$C. As a result relatively enhanced couplings with the core will remarkably lower the $3s_{1/2}$ orbit. Whereas in even isotopes the pairing correlations constrain the valence neutrons spreading mostly over the valence orbits $2s_{1/2}$ and $1d_{5/2}$, and much less neutron can be scattered into the $3s_{1/2}$ states in the continuum, e.g., $v^2 = 0.004$ for $^{20}$C, which therefore become high-lying ones. As we know, the pairing correlations play significant roles for the halo occurrences in the even nuclear systems, not only in stabilizing nucleus itself but also in developing the halos by scattering the Cooper pairs into the low-lying $s$ or $p$ orbits. The typical examples are $^{11}$Li, the drip line isotopes of Ca, Zr and Ce. While for the even Carbons particularly $^{22}$C it seems that the extra binding from the pairing correlations makes $s$ orbit too bound to get dilute matter distribution, which also leads to a fairly large two-neutron separation energy $S_{2n}$ (see Fig. \[fig:S\]). On the contrary for $^{17, 19}$C, due to the absence of extra pairing binding the odd neutron in $s$ orbit presents substantial contribution in the formation of halo structure, which also results in the odd-even staggering (OES) on the neutron radii of Carbon isotopes. $^{14}$C $^{15}$C $^{16}$C $^{17}$C $^{18}$C $^{19}$C $^{20}$C --------------------- ----------------- ----------------- ----------------- ------------------ ------------------ ------------------ ----------------- PKA1 2.53 2.74 2.73 2.89 2.91 3.02 3.08 PKO2 2.43 2.55 2.63 2.92 2.81 3.11 2.97 PKO3 2.47 2.59 2.68 2.91 2.86 3.21 3.02 PKDD 2.43 2.75 2.66 2.91 2.86 3.26 3.03 DD-ME2 2.55 2.68 2.76 2.93 2.94 3.24 3.10 PK1 2.42 2.72 2.65 2.86 2.84 3.06 3.01 Ref. [@Liatard1990] 2.62[[(6)]{}]{} 2.78[[(9)]{}]{} 2.76[[(6)]{}]{} 3.04[[(11)]{}]{} 2.90[[(19)]{}]{} 2.74[[(96)]{}]{} $-$ 2.30[[(7)]{}]{} 2.48[[(3)]{}]{} 2.70[[(3)]{}]{} 2.72[[(3)]{}]{} 2.82[[(4)]{}]{} 3.13[[(7)]{}]{} 2.98[[(5)]{}]{} $-$ 2.50[[(8)]{}]{} $-$ 2.73[[(4)]{}]{} $-$ 3.23[[(8)]{}]{} $-$ : Matter radii (fm) for Carbon isotopes extracted from the calculations of PKA1, PKO2 PKO3, PKDD, DD-ME2 and PK1, as compared to the experimental data [@Liatard1990; @Ozawa2001].[]{data-label="tab:radius"} Before discussing the OES of neutron radii, it is worthwhile to check the quantitative precision for the theoretical description of the radius. In Table \[tab:radius\] are shown the matter radii of neutron-rich Carbon isotopes obtained from the calculations of PKA1, PKO2, PKO3, PKDD, DD-ME2 and PK1, as compared to the experimental data [@Liatard1990; @Ozawa2001]. It is found that both RHFB and RHB calculations with the selected effective Lagrangians provide appropriate agreement with the data, which to some extent demonstrates the theoretical reliability. ![(Color online) (a) Neutron root mean square radii calculated by RHFB with PKA1, PKO2 and PKO3, and by RHB with PKDD and PK1, as compared with the data \[for $^{19}$C it reads as 2.86(1.4) fm\] [@Liatard1990], and (b) corresponding contributions from the neutron core orbits ($1s_{1/2}$, $1p_{3/2}$ and $1p_{1/2}$), valence orbits ($2s_{1/2}$ and $1d_{5/2}$) and the continuum. The results are provided by the calculations of RHFB with PKO2. See the text for details.[]{data-label="fig:Rn"}](blocking.pdf "fig:"){width="45.00000%"} ![(Color online) (a) Neutron root mean square radii calculated by RHFB with PKA1, PKO2 and PKO3, and by RHB with PKDD and PK1, as compared with the data \[for $^{19}$C it reads as 2.86(1.4) fm\] [@Liatard1990], and (b) corresponding contributions from the neutron core orbits ($1s_{1/2}$, $1p_{3/2}$ and $1p_{1/2}$), valence orbits ($2s_{1/2}$ and $1d_{5/2}$) and the continuum. The results are provided by the calculations of RHFB with PKO2. See the text for details.[]{data-label="fig:Rn"}](blocking.eps "fig:"){width="45.00000%"} In fact not only for the total ones, the selected effective Lagrangians with optimized pairing forces also present proper quantitative descriptions for the neutron radii. As shown in Fig. \[fig:Rn\](a) and referred to the data [@Ozawa2001], the neutron root mean square radii from $^{14}$C to $^{22}$C are well reproduced by both RHFB and RHB calculations to certain quantitative precision. Evidently as shown in Fig. \[fig:Rn\](a) that all the theoretical calculations present distinct OES on the neutron radii, in accordance with the experimental systematics. Specifically, as seen from Fig. \[fig:Rn\] (b), such OES is determined by the valence neutrons lying in the canonical orbit $2s_{1/2}$ and also depends on the fact whether the corresponding Bogoliubov quasi-particle $s$ orbit is blocked or not. Referring to Table \[tab:bl\], one can find that the blocking configurations are consistent with the OES in Fig. \[fig:Rn\](a). In $^{15}$C which has larger neutron radius than $^{16}$C in the calculations with PKA1, PKDD and PK1, the quasi-particle $s$ orbit is blocked. Due to similar reason and consistently with the halo occurrence, the neutron radii of halo nuclei $^{17,19}$C are distinctly larger than the even neighbors. The exceptions are the calculations with PKO2 and PKO3 at $^{15}$C and the one with PKA1 at $^{19}$C, where the neutron radii change smoothly. As seen from Table \[tab:bl\] such exceptional cases correspond with the blocking of $d_{5/2}$ orbit, in which the odd neutron is localized mostly inside the nucleus by the centrifugal barrier. As a result, the ground state of $^{19}$C determined by PKA1 does not correspond with a halo structure since the odd neutron blocks the $d_{5/2}$ orbit and due to the pairing effects the paired neutrons in the low-$j$ $s$ state are bound too strongly to distribute extensively. =0.25em\[tab:bindings&radii\] PKA1 PKO2 PKO3 PKDD DD-ME2 PK1 -- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- [**$-$107.48**]{} [ $-$104.98]{} [ $-$105.95]{} [**$-$105.30**]{} [**$-$105.79**]{} [**$-$105.91**]{} [ $-$106.63]{} [**$-$105.69**]{} [**$-$106.11**]{} [ $-$105.23]{} [ $-$105.35]{} [ $-$105.79]{} [**$-$112.99**]{} [**$-$110.32**]{} [**$-$110.81**]{} [**$-$109.83**]{} [**$-$110.11**]{} [**$-$110.76**]{} [ $-$112.44]{} [ $-$110.06]{} [ $-$110.30]{} [ $-$109.30]{} [ $-$109.50]{} [ $-$110.21]{} [ $-$117.03]{} [**$-$114.17**]{} [**$-$114.43**]{} [**$-$113.35**]{} [**$-$113.21**]{} [**$-$114.78**]{} [**$-$117.21**]{} [ $-$113.13]{} [ $-$113.65]{} [ $-$112.72]{} [ $-$112.71]{} [ $-$114.16]{} [**2.92**]{} [ 3.05]{} [ 2.99]{} [**2.98**]{} [**2.93**]{} [**2.94**]{} [ 2.66]{} [**2.67**]{} [**2.71**]{} [ 2.70]{} [ 2.72]{} [ 2.88]{} [**3.10**]{} [**3.15**]{} [**3.14**]{} [**3.15**]{} [**3.16**]{} [**3.09**]{} [ 3.04]{} [ 2.91]{} [ 2.98]{} [ 2.99]{} [ 3.04]{} [ 2.97]{} [ 3.42]{} [**3.37**]{} [**3.48**]{} [**3.57**]{} [**4.20**]{} [**3.31**]{} [**3.23**]{} [ 4.85]{} [ 3.43]{} [ 3.31]{} [ 3.60]{} [ 3.19]{} : Binding energies in MeV (upper panel) and neutron radii in fm (lower panel) for $^{15, 17, 19}$C calculated by the effective Lagrangians PKA1, PKO2, PKO3, PKDD, DD-ME2 and PK1 with different blocking configurations. For each odd isotopes, the first and second rows correspond with blocking neutron ($\nu$) orbits $s_{1/2}$ and $d_{5/2}$, respectively. The bold types denote the ground states. As the further illustration of the consistent relation between the OES and blocking configurations, Table \[tab:bindings&radii\] shows the binding energies and neutron radii of $^{15, 17, 19}$C extracted from the self-consistent calculations with the blockings of $s_{1/2}$ and $d_{5/2}$ orbits, respectively. As shown in the lower panel, the blockings of the low-$j$ $s$ orbit in general lead to more extensive neutron distributions, from which are also well demonstrated the blocking effects in the formation of single-neutron halo structure of $^{17, 19}$C. Specifically for the calculations of $^{19}$C with different blocking, the binding energies determined by PKA1 are close to each another and in fact when $s$ orbit is blocked PKA1 also supports the halo occurrence in $^{19}$C. In contrast the others present distinct differences on the binding energies, especially for PKO2 which confirms the halo emergence in ground state evidentally. It is well known that pairing correlation plays an important role in stabilizing the finite nuclei, especially the exotic ones. For $^{11}$Li, the neutron drip-line isotopes of Ca, Zr and Ce, it is already demonstrated that the pairing correlations show positive effects in both stabilizing and developing the halo structures. While in the RHFB and RHB calculations of $^{17, 19}$C, the quasi-particle $s$ orbit is blocked and the corresponding contributions are mainly mapped to the canonical orbit $2s_{1/2}$, which plays the dominate role in the single-neutron halo formation of $^{17, 19}$C. This implies that the unpaired odd neutron in low-$j$ orbit may also contribute to develop a halo structure when it is not so deeply bound. From previous analysis it is just due to the lack of extra binding from the pairing correlations that the odd-neutron in $s$ orbit can spread over far beyond the center of nucleus. Summary and Perspectives {#summary} ======================== In this work we have systematically calculated the Carbon isotopes using the relativistic Hartree-Fock-Bogliubov (RHFB) theory with PKA1, PKO2 and PKO3 as well as the relativistic Hartree-Bogliubov (RHB) theory with PKDD, DD-ME2, PK1 and NL2. It is found that with the optimized pairing force the selected effective Lagrangians except NL2 can properly describe the structure properties of the Carbon isotopes, e.g., reproducing the binding energies and matter radii by certain quantitative precision. Specifically the distinct evidences have demonstrated for the single-neutron halo occurrences in $^{17,19}$C, as well as the odd-even staggering (OES) of neutron radii in the vicinity of neutron drip line. While the self-consistent RHFB or RHB calculations do not support the emergence of two-neutron halo structure in $^{22}$C as indicated by the experimental reaction cross section measurement [@Tanaka2010]. Further detailed analysis shows that the halo emergences in $^{17, 19}$C, as well as the OES of neutron radii, are essentially concerned with the blocking effects in the odd Carbon isotopes. Different from even nuclear systems, in which the pairing correlations play significant roles in both developing and stabilizing the halo structures, the unpaired odd neutron in weakly bound low-$j$ $s$ orbit dominates the halo formation in $^{17, 19}$C, as well as reproducing the OES of neutron radii for the drip-line Carbon isotopes. It should be noticed that for the odd Carbons the blocking treatment in this work is just the first-order evaluation of the blocking effects and the current effects induced by the odd neutron are neglected as well. In addition, due to the limit of the present theoretical platform, we only performed the spherical calculations for the Carbon isotopes within the relativistic Hartree and Hartree-Fock theories, while some Carbon isotopes are potentially deformed. After considering the shape fluctuations in both $\beta$ and $\gamma$ deformations, the average neutron quadrupole deformations ${\left(\langle \beta\rangle_n, \langle \gamma\rangle_n\right)}$ of $^{16,18,20}$C are $(0.50, 21^\circ)$, $(0.49, 29^\circ)$ and $(0.50, 21^\circ)$, respectively [@Jiangming2011]. It is then expected that the shape fluctuations will bring some influence on the structure properties of the Carbon isotopes, especially in the vicinity of drip line. Therefore, the self-consistent treatment of the deformation as well as the odd-particle effects is perspected to be considered carefully for more reliable description of Carbon isotopes. This work is partly supported by the National Science Foundation of China under Grant Nos. 11075066 and 11205075, the Fundamental Research Funds for the Central Universities under Contracts No. lzujbky-2012-k07 and No. lzujbky-2012-7, and the Program for New Century Excellent Talents in University. 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--- abstract: 'The Leggett-Garg inequality holds for any macrorealistic system that is being measured noninvasively. A violation of the inequality can signal that a system does not conform to our primal intuition about the physical world. Alternatively, a violation can simply indicate that “clumsy” experimental technique led to invasive measurements. Here, we consider a recent Leggett-Garg test designed to try to rule out the mundane second possibility. We tailor this Leggett-Garg test to the IBM 5Q Quantum Experience system and find compelling evidence that qubit $Q_2$ of the system cannot be described by noninvasive macrorealism.' author: - Emilie Huffman - Ari Mizel bibliography: - 'ref.bib' title: 'Leggett-Garg test of superconducting qubit addressing the clumsiness loophole' --- Introduction ============ The field of quantum computation has stimulated interest in tests of quantum behavior. Such tests have provided standardized protocols to showcase control over qubit systems [@Ansmann:2009aa; @Vlastakis:2015aa]. They can provide metrics for qubit performance. Moreover, as a result of experimental advances associated with the quantum computation era, it has become possible to close loopholes in foundational tests of quantum mechanics [@Hensen:2015aa; @Giustina2015]. While Bell inequality violations [@Bell64] retain their canonical status among tests of quantum behavior, they are ill-suited for many experimental systems. To apply a Bell inequality test to a system under investigation, the system must possess two parts that can retain quantum coherence while being segregated until they have a spacelike separation. An alternative to the Bell inequality, one that doesn’t make this demand, is the Leggett-Garg inequality [@Leggett1985; @Emary2014]. The Leggett-Garg inequality holds for any macrorealist system that is being measured noninvasively. It has recently been applied to a number of systems [@Palacios10; @Xu:2011aa; @Athalye2011; @Dressel2011; @Goggin2011; @Waldherr2011; @Souza2011; @Suzuki2012; @Knee:2012aa; @George2013; @Groen2013; @Katiyar2013; @Zhou2015; @Knee16; @Formaggio2016]. Unfortunately, a clumsiness loophole [@Wilde2012] can thoroughly undermine the significance of any violation of the Leggett-Garg inequality. It is essential to address the clumsiness loophole if one wishes to draw meaningful conclusions from a Leggett-Garg test. In this paper, we study the IBM 5Q Quantum Experience system [@IBM5Q] by carefully implementing a Leggett-Garg program designed to address the clumsiness loophole. The IBM 5Q Quantum Experience is a publicly accessible system of five superconducting qubits that can be controlled via a website interface. Earlier papers have exhibited the capabilities of the IBM 5Q [@Devitt2016; @Alsina2016; @Berta2016]. Our aim is to execute a particularly careful and persuasive demonstration that at least one of the qubits of the IBM 5Q is genuinely quantum, or at least not a macrorealistic system being measured noninvasively. Our Leggett-Garg test, which is structured to address the clumsiness loophole in the deliberate manner formulated in [@Wilde2012], can also productively inform the design of future tests of other systems. The paper is organized as follows. Section \[Protocols\] frames the six experimental protocols that make up our Leggett-Garg test. These protocols must be tailored to accommodate constraints in the IBM 5Q system; section \[Experiment\] describes details. Results are supplied in section \[Results\], and we conclude in section \[Conclusion\]. Protocols {#Protocols} ========= We test for violations of macrorealism by adapting the proposal in [@Wilde2012]. Consider an experimental program comprised of the six protocols depicted in Fig. \[exp\]. Focus initially on the first and last protocols, (a) and (f), appearing in the figure. Protocol (a) shows a physical system measured by some operation $O_1$ that is arranged to yield a dichotomous result $1$ or $-1$. This system is then subjected to operation $O_3$ that yields another dichotomous result $1$ or $-1$. One can compute the correlator $\left\langle O_1 O_3\right\rangle_a$ by repeating protocol (a) many times and taking the average value of the product $O_1 O_3$. Protocol (f) shows an interleaved series of manipulations and measurements of the two kinds appearing in protocol (a). In particular, operation $O_2$ is a manipulation and measurement equivalent to $O_3$ but occurring earlier in the series. A single run of protocol (f) yields measurement results $\pm 1$ for operations $O_1$, $O_2$, and $O_3$. One can compute the correlators $\left\langle O_1 O_3\right\rangle_f$, $\left\langle O_1 O_2\right\rangle_f$, and $\left\langle O_2 O_3\right\rangle_f $ by repeatedly executing protocol (f), taking the products $O_1 O_3$, $O_1 O_2$, and $O_2 O_3$ each run, and averaging over runs. For any given run of protocol (f), all 8 possible values of the triplet $(O_1, O_2, O_3) = (\pm 1, \pm 1,\pm 1)$ satisfy the inequality $O_1 O_3 + O_1 O_2 + O_2 O_3 + 1 \geq 0$. Taking the average of this inequality over repeated runs yields an inequality on correlators $\left\langle O_1 O_3\right\rangle_f + \left\langle O_1 O_2\right\rangle_f + \left\langle O_2 O_3\right\rangle_f + 1 \geq 0$. Suppose that our physical system is macrorealistic and that all of the operations in protocol (f) measure it noninvasively. Then, $$\label{eq:correlatorequality} \left\langle O_1 O_3\right\rangle_f = \left\langle O_1 O_3\right\rangle_a$$ since $O_2$ and the other operations before $O_3$ in protocol (f) do not perturb the system. Substituting this into our correlator inequality, we obtain the Leggett-Garg inequality $$LG = \left\langle O_1 O_3\right\rangle_a + \left\langle O_1 O_2\right\rangle_f + \left\langle O_2 O_3\right\rangle_f + 1 \geq 0. \label{eq:LG}$$ If a system violates this inequality, it is not a macrorealistic system undergoing noninvasive measurement. One exciting possibility is that the system is impossible to describe correctly using any macrorealistic noninvasive theory. For instance, perhaps the system is quantum mechanical, exhibiting the strange properties described by quantum theory. But there is a mundane possibility as well. Perhaps the system is macrorealistic, and can be measured noninvasively, but our measurements are invasive simply because of our experimental clumsiness. This entirely plausible circumstance is termed the “clumsiness loophole" in [@Wilde2012]. To address the clumsiness loophole, our full experimental program includes verification protocols (b) - (e) in Fig. \[exp\] in addition to protocols (a) and (f). Each protocol (b) - (e) is designed to place a limit, called the $\epsilon$-*adroitness*, on the invasiveness of an operation. ![Proposed Leggett-Garg experimental program. The symbols $\sigma_z$ and $\sigma_{\theta}$ should be ignored while deriving the Leggett-Garg inequality, which does not presume any quantum mechanics. The symbol $\sigma_z$ below, say, $O_3$ indicates that, in quantum theory, $O_3$ consists of a measurement along the qubit initialization direction, $\hat{z}$. The symbol $\sigma_{\theta}$ below, say, $O_1$ indicates that $O_1$ consists of manipulations and a measurement that are equivalent to a measurement along a direction $\sin \theta \hat{y} + \cos \theta \hat{z}$ oriented at an angle $\theta$ with respect to $\hat{z}$. []{data-label="exp"}](expfig2.pdf){width="3in"} For the $O_2$ measurement in the middle of the experiment in protocol (b), for example, we say that it is $\epsilon_b$-*adroit* if $$\left|\left\langle O_1 O_3\right\rangle_b - \left\langle O_1 O_3\right\rangle_a\right| \leq \epsilon_b. \label{eq:epsilonb}$$ Similarly, the $O_2$ measurement in protocol (c) is said to be $\epsilon_c$-*adroit* if $$\left|\left\langle O_1 O_3\right\rangle_c - \left\langle O_1 O_3\right\rangle_a\right| \leq \epsilon_c.$$ We define $\epsilon_d$ and $\epsilon_e$ analogously based on protocols (d) and (e). Assuming that several of these measurements together cannot somehow collude nonlinearly to have an unexpectedly dramatic effect on the system, the maximum effect that the four intermediate measurements in part (f) could have on the correlation function $\left\langle O_1 O_3\right\rangle_f$ is $$\epsilon_{total} = \epsilon_b + \epsilon_c + \epsilon_d+ \epsilon_e. \label{eq:epsilontotal}$$ By separately testing every single operation that appears in Fig. \[exp\] protocol (f), we have direct experimental evidence that none of these operations is causing a mundane violation of the Leggett-Garg inequality by clumsy invasiveness. Designing a Leggett-Garg program with this feature is subtle. If an experiment yields a value for $LG$ satisfying both $$LG < 0 \text{ and } \left|LG\right| \geq \epsilon_{total}, \label{eq:bothLG}$$ we have evidence that the system can never be correctly characterized by any macrorealistic noninvasive theory. Suppose that we believe that our system is a qubit correctly described by quantum mechanics. Will it actually exhibit a violation of eq. (\[eq:LG\])? We can derive a quantum mechanical expression for $LG$ in this set of experiments by using the formulae below, where $\sigma_\theta = \sin \theta \sigma_y + \cos \theta \sigma_z$ and the superoperators $\bar{\Delta}$ and $\bar{\Delta}_\theta$, are defined as $\bar{\Delta}\left(\rho\right) = \frac{1}{2}\left(\rho + \sigma_z \rho \sigma_z\right)$ and $\bar{\Delta}_\theta\left(\rho\right) = \frac{1}{2} \left(\rho + \sigma_\theta \rho \sigma_\theta\right)$: $$\begin{aligned} \left\langle O_1 O_3\right\rangle_a &= \frac{1}{2} {\rm Tr}\left(\sigma_z, \left\{\sigma_\theta,\rho\right\}\right) \\ \left\langle O_1 O_2\right\rangle_f &= \frac{1}{2} {\rm Tr}\left(\sigma_z, \left\{\sigma_\theta,\rho\right\}\right) \\ \left\langle O_2 O_3\right\rangle_f &= \frac{1}{2} {\rm Tr}\left(\sigma_z \left(\bar{\Delta}_\theta\circ \bar{\Delta} \circ \bar{\Delta}_\theta\right)\left(\left\{\sigma_z, \bar{\Delta}_\theta\left(\rho\right)\right\}\right)\right). \end{aligned}$$ These formulae imply $$LG = 2 \cos\theta + \cos^4\theta + 1.$$ This value is negative if we choose $\theta$ between $.683\pi$ and $\pi$ or between $-.683\pi$ and $-\pi$. We therefore do expect to be able to see a violation of our Leggett-Garg inequality for a qubit. Note also that protocols (b) - (e) were designed with a qubit in mind such that the intermediate measurements should not change $\left\langle O_1 O_3\right\rangle$, and the $\epsilon$-*adroitness* parameters should be small. We now tailor this experimental protocol so that it can be implemented on the IBM 5Q. Experiment {#Experiment} ========== The IBM 5Q consists of five superconducting transmon qubits patterned on a silicon substrate. The qubits are labeled $Q_0,Q_1,Q_2,Q_3,$ and $Q_4$. There are several constraints on the current qubit setup that are relevant to our proposed experimental program. ![Protocol (a) of the experiment implemented in a circuit acting on qubit $Q_2$.[]{data-label="parta"}](parta.pdf){width="14.8cm"} First, the IBM 5Q permits only one measurement on a given qubit each experimental run. Fig. \[exp\] involves multiple measurements on a single qubit. Rather than performing an additional measurement directly on a single qubit, we therefore perform the measurement by transmitting the qubit’s state to an ancilla qubit using $CNOT$ gates and measuring the ancilla qubit. This is just an alternate realization of the measurement operations in Fig. \[exp\]; it does not invalidate our carefully constructed Leggett-Garg test. This modification does force us to consider a second constraint on the IBM 5Q system. For a 5 qubit system, one might imagine $5 \times 4 = 20$ different types of $CNOT$ gates, targeting any one of the 5 qubits and controlled by any of the remaining 4 qubits. For the IBM 5Q system, only 4 different types of $CNOT$ gates are available: every $CNOT$ must have $Q_2$ as the target qubit and $Q_0,Q_1,Q_3,$ or $Q_4$ as the control qubit. To reduce the number of $CNOT$ gates necessary for our experimental program, we choose $Q_2$ to play the role of the qubit that appears in Fig. \[exp\] and the other qubits as the internal degrees-of-freedom of the measurement devices in Fig. \[exp\]. The third and final constraint we consider arises from the fact that there are only five qubits in the IBM 5Q. Since each qubit can be measured at most once, any IBM 5Q circuit can only make five total measurements. Protocol (f) of Fig. \[exp\] involves six measurements. To deal with this issue, we treat $O_1$ in Fig. \[exp\] not as a measurement but instead as an initialization into $\rho_\theta = \left|1 \right\rangle_\theta \left\langle 1\right|_\theta$, a state that should give result $+1$ when subjected to the $O_2$ measurement in protocols (c) or (e) of Fig. \[exp\]. (If one wishes to make this as parallel as possible to our discussion of the experimental protocol above, it may be helpful to think of initialization as measurement followed by a postselection on result $1$ rather than $-1$.) The IBM 5Q device initializes qubits into state $\left| 0 \right\rangle$, so our $O_1$ will consist of a $NOT$ gate followed by a rotation gate. We note that the correlation function in part (a) for $\rho_\theta = \left|1\right\rangle_\theta \left\langle 1\right|_\theta$ may be rewritten as $$\begin{aligned} \left\langle O_1 O_3\right\rangle_a & = & p_{-1,-1} + p_{1,1} - p_{-1,1} - p_{1,-1} \nonumber \\ & = & p_{1,1} - p_{1,-1} = \left\langle O_3\right\rangle_a,\end{aligned}$$ with similar results for every correlator that involves $O_1$. The significance of the Leggett-Garg conditions (\[eq:bothLG\]) and the quantum mechanical predictions are unchanged by this alternation in our experimental program. It turns out to be convenient to choose $\theta=-3\pi/4$ for our $\theta$-measurements in Fig. \[exp\]. The IBM 5Q currently permits single qubit gates $X,Z,Y,H,S,S^\dagger,T,T^\dagger$ and measurement in the $z$-direction. To perform a $\theta=-3\pi /4$ measurement, we thus rotate the basis noting that the rotation matrix for $\theta=-3\pi / 4$ obeys the identity $$e^{-i3\pi \sigma_x / 8} = H e^{-i3\pi \sigma_z/8} H.$$ This product has the form $e^{-3i\pi/8} H T^3 H = e^{-3i\pi/8} H T S H$. It turns out that the IBM 5Q system exhibits better performance on our experimental program if we re-express the product as $e^{-i\pi/8} H T H S^\dagger H S^\dagger$ using the identity $e^{i\pi/4} (H S^\dagger)^2 = S H$. Up to an overall phase, we arrive at the rotation gate $$\label{eq:R} R = H T H S^\dagger H.$$ We were permitted to remove the $S^\dagger$ gate on the right end because this matrix $R$ still rotates the eigenstates of $\sigma_z$ into the eigenstates of $\sigma_\theta$ – the resulting eigenstates of $\sigma_\theta$ just have different overall phases when the $S^\dagger$ on the right end is removed. This is clear from the equations $\sigma_\theta = R S^\dagger \sigma_z S R^\dagger = R \sigma_z R^\dagger$. We note that a measurement in the $\theta$ direction when our qubit is in state $\left|\psi\right\rangle$ is given by $$\begin{aligned} \left\langle O_\theta\right\rangle &= \left|\prescript{}{\theta}{\left\langle 1 \right|}\left.\psi\right\rangle\right|^2 - \left|\prescript{}{\theta}{\left\langle 0 \right|}\left.\psi\right\rangle\right|^2\\ &= \left|\prescript{}{z}{\left\langle 1 \right|}R^\dagger\left|\psi\right\rangle\right|^2 - \left|\prescript{}{z}{\left\langle 0 \right|} R^\dagger\left|\psi\right\rangle\right|^2. \end{aligned}$$ Thus, to take a $\theta$ measurement, we simply apply $R^\dagger$ to our state, make a $z$ measurement, and then apply $R$ to the result. Now that we have tailored Fig. \[exp\] to the IBM 5Q, we can run the Leggett-Garg test. Fig. \[parta\] gives the circuit for protocol (a). The circuit begins with operation $O_1$ comprised of an $X$ gate that flips state $\left|0\right\rangle_z$ to state $\left|1\right\rangle_z$ and the set of gates $R$ that rotates the state to $\left|1\right\rangle_\theta$. At the end of protocol (a), the $z$-directional measurement $O_3$ is taken. Moving on to determine $\epsilon_b$ from protocol (b) and $\epsilon_d$ from protocol (d), we have the two circuits shown in Fig. \[partbd\]. Note the use of the $CNOT$ gate to record the intermediate state on the second qubit. Because the $CNOT$ gates can only have $Q_2$ as the target qubit, we must add $H$ gates directly before and after the application of the $CNOT$ gate to both of the qubit states in each experiment. This causes the target qubit and control qubit to exchange roles. While Fig. \[parta\] and \[partbd\] show exactly which gates are placed in the circuit and exactly where they are placed in the circuit, several additional gates are placed in the IBM 5Q interface to prevent the IBM 5Q complier from changing these circuits during execution. In protocol (b), for example, we have two Hademard gates in a row, $HH$. To keep the IBM 5Q from collapsing the two gates into an identity gate, we inserted the operator combination $T T^\dagger$ between the two Hademard gates [@JayGambetta]. This $T T^\dagger$ combination prevents the compiler from combining $HH$ into an identity gate but does not have any other effect on the circuit execution since $T$ and $T^\dagger$ gates physically correspond to timing delays rather than actual pulses. Whenever an instance of $HH$ is found in a protocol, we actually insert $H T T^\dagger H$ into the IBM 5Q interface. Additionally, the IBM 5Q allows use of the $Id$ gate, or identity gate. To ensure that the IBM 5Q compiler applies the second $H$ gate on $Q_1$ ($Q_4$) directly after the $CNOT$ gate, we fill the space between the second $H$ gate applied to $Q_1$ ($Q_4$) and the $z$-directional measurement at the end with identity gates. Otherwise, the IBM 5Q compiler would apply the $H$ gate immediately before the final measurements [@JayGambetta]. This technique is used whenever we wish to impose a fixed time interval between a gate operating on a qubit and the final $z$-measurement. Protocols (c) and (e), shown in Fig. \[partce\], contain intermediate measurements in the $\theta$-direction. By combining these with the results from the circuit in Fig. \[parta\], we determine $\epsilon_{c}$ and $\epsilon_{e}$. In these protocols the instances of $HH$ on either side of the $CNOT$ gates are collapsed into identities. (The same collapse occurs in protocol (f), so that we really are individually testing the operations in (f) as required by our Leggett-Garg program.) This allows for a reduced number of gates necessary in the circuits. Finally, Fig. \[partf\] gives us the circuit necessary to measure $\left\langle O_1 O_2\right\rangle_f$ and $\left\langle O_2 O_3\right\rangle_f$. All five qubits are used and $R^\dagger$ and $R$ are applied in an interleaved pattern to alternate measurements back and forth between the $\theta$- and $z$- directions. The qubit $Q_1$ is chosen for the measurement $O_2$ because it has the longest relaxation times of the five qubits of the IBM 5Q and in our experience gave the most reliable results. (This is one of many specific choices in Fig. \[parta\] - \[partf\] and in the definition (\[eq:R\]) that permitted us to achieve a Leggett-Garg violation. The fidelity of the gates in the IBM 5Q system is currently too low to achieve a violation for generic implementations of Fig. \[exp\].) Results {#Results} ======= Results are summarized by the tables in Fig. \[tab\]. We performed 10 repetitions of the complete experimental program, all six protocols given by Figs. \[parta\]-\[partf\]. The data from these 10 repetitions allowed us to compute the error bounds given in the tables. Every time we took a measurement, it was actually the output of $r$ repeated executions of the IBM 5Q hardware, where we set $r=8192$ in the IBM 5Q interface. The Leggett-Garg quantity table gives the average measurements obtained from experiments (a) and (f). The second table, labelled Adroitness Test Results, gives the correlation function measurements from protocols (b)-(e). We evaluate the adroitness of each measurement using equations like eq. (\[eq:epsilonb\]) and total them according to eq. (\[eq:epsilontotal\]). For reference, we include the quantum mechanical prediction for each value in the table. The data confirm that both of the conditions specified in eq. (\[eq:bothLG\]) are met: the calculated $LG$ is indeed negative and $\left|LG\right| \geq \epsilon_{total} $. ![Protocol (f) of the experiment program implemented in circuits.[]{data-label="partf"}](partf.pdf){width="14cm"} [|c|c|c|c|c|]{}\ &$\left\langle O_1 O_3\right\rangle_a$ & $\left\langle O_1 O_2\right\rangle_f$ & $\left\langle O_2 O_3\right\rangle_f$ & $LG$\ Measured & $-0.70 \pm .01$ & $-0.69 \pm .01$ & $0.18 \pm .02$ & $-0.21 \pm .03$\ $\begin{array}{c}\text{Quantum}\\ \text{Prediction}\end{array}$ & $-\frac{1}{\sqrt{2}} \approx -0.70$ & $-\frac{1}{\sqrt{2}} \approx -0.70$ & $\frac{1}{4} = 0.25$ & $-\sqrt{2} + \frac{1}{4} +1 \approx -0.16$\ \ [|c|c|c|c|c|c|]{}\ & $\left\langle O_1 O_3\right\rangle_{b}$&$\left\langle O_1 O_3\right\rangle_{c}$&$\left\langle O_1 O_3\right\rangle_{d}$&$\left\langle O_1 O_3\right\rangle_{e}$&$\epsilon_{total} $\ Measured & $-.69 \pm .02$ & $-0.71 \pm .02$ & $-0.68 \pm .01$ & $-0.67 \pm .02$ & $.08 \pm .04$\ $\begin{array}{c}\text{Quantum}\\ \text{Prediction}\end{array}$ & $-\frac{1}{\sqrt{2}} \approx -0.70$ & $-\frac{1}{\sqrt{2}}\approx -0.70$ & $-\frac{1}{\sqrt{2}} \approx -0.70$ & $-\frac{1}{\sqrt{2}} \approx -0.70$ & 0\ Conclusion {#Conclusion} ========== We have carefully framed a Leggett-Garg program that (a) addresses the clumsiness loophole and (b) is suited for execution on the IBM 5Q Quantum Experience. This program demonstrated that qubit $Q_2$ of IBM 5Q is not a macrorealistic system being measured noninvasively. It also supplies compelling evidence that noninvasiveness in the measurements does not exclusively derive from mundane experimental clumsiness. This suggests that it is impossible to formulate a noninvasive macrorealistic description of $Q_2$. Some recent papers have stressed the role of equalities rather than inequalities in testing macrorealism [@Kofler2013; @Clemente2016]. One might consider reframing our Leggett-Garg program in the future by directly checking the equality (\[eq:correlatorequality\]) rather than inserting it into the inequality (\[eq:LG\]). Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Ben Palmer for suggesting that we analyze the quantum character of the IBM 5Q. We acknowledge use of the IBM Quantum Experience for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum Experience team.

--- abstract: 'We propose a procedure to build a decision tree which approximates the performance of complex machine learning models. his single approximation tree can be used to interpret and simplify the predicting pattern of random forests . in medical questionnaires . We study the asymptotic behavior of splits and introduce an improved splitting method designed to stabilize tree structure. Empirical studies both simulation and real data sets illustrate that our method can simultaneously achieve high approximation power and stability.' author: - | [**Yichen Zhou**]{}\ [**Giles Hooker**]{}\ \ Department of Statistical Science\ Cornell University\ Ithaca, NY 14850, USA bibliography: - 'citation.bib' title: Interpreting Models via Single Tree Approximation --- =1 Introduction ============ Random forests [@breiman2001random] have been widely used across different disciplines and are acknowledged for their outstanding predictive power [@caruana2006empirical]. However, machine learning models suffer from trade-off between predictive accuracy and model interpretability. RFs, as @breiman2001statistical pointed out, are “A+ predictors” but “on interpretability they rate F”. This paper presents a method to resolve this issue by constructing a single tree approximation that mimics the behavior of a machine learning model; interpret . In this paper we particularly focus on the RF models, . Interpreting a --------------- his paper methods developed in @gibbons2013computerized where the authors have introduced the single tree approximation of a RF to produce an adaptive screening tool for depression diagnosis in psychiatry. A RF was initially built to classify the patients based on their responses to an 82-question survey, after which a decision tree was learned from pseudo samples generated by the RF to construct an adaptive screening tool with an average length of 7 questions. The aim is to reduce response burden on patients while at the same time mimicking the predictions made by the RF as much as possible. This idea is related to @domingos1997knowledge in which the author introduced Combined Multiple Models(CMM) to learn a base learner from a multiple model that is complex, incomprehensible but accurate. Approximation trees are a useful expedient when we need to trace or reproduce the computerized decision making procedure. The predictive pattern of a tree is easy to follow and guarantees a clear predicting path which can be viewed as a guideline for reasoning. In addition, Stablizing a Decision Tree -------------------------- Approximating an existing model by a tree is different from learning directly from data. The existing model serves as an oracle that the sample space from which the tree is intended to learn, thus over-fitting @gibbons2013computerized @gibbons2013computerized Gini Indices ------------ There are multiple choices of defining the information gain in the literature [@breiman1984classification; @loh1997split]. In this paper, we will focus on the Gini information associated with the Gini index as its empirical estimator. For the distribution of $(X,Y)$ where $Y \in \{1,\dots,k\}$ the category labels of $X$, one way to write the Gini gain $g$ is $$\begin{aligned} g = \sum_{i\not=j}P(Y=i)P(Y=j) = 1-\sum_{i=1}^k P(Y=i)^2. \label{eqn:gini}\end{aligned}$$ The empirical version of this formula defines the corresponding Gini index. It is worth noticing that this conventional definition more information with a smaller value. The formula further indicates its relation with the sample variance. In other words, smaller Gini indices implies less discrepancy among responses. When splitting a node in the decision tree, we divide the sample space into two subsets within each the responses are more uniform , increasing the total information gain. This value is estimated by the weighted sum of the two Gini indices after splitting, hence the split with the maximal Gini index implies the best information gain and is therefore employed. In the following sections we will show that, in our approximation setting, we can determine our sample size to get more precise estimate of the Gini indices, stabilizes the split at each node . A test of better split ====================== Consider a multi-category classification problem. The original sample consist of covariates and responses $\{(\tilde{X}_i, \tilde{Y}_i)\}_{i=1}^{n_0}$ where $\tilde{X}_i \in \mathbb{R}^m, \tilde{Y}_i \in \{1,2,\dots, k\}$, $m$ the dimension of covariate space, and $k$ the levels of responses. We obtain a RF $\mathcal{F}$ from the sample. $\mathcal{F}$ will later serve as the oracle we try to mimic, generating points (pseudo sample) $\{(X_i, Y_i)\}_{i=1}^{n}$ of arbitrary size $n$. Here $X_i=(X_i^1,\dots,X_i^m) \in \mathbb{R}^m$, and $Y_i=(Y_i^1,\dots,Y_i^k) \in \mathbb{R}^k$ are the $\mathcal{F}$-predicted class probabilities over responses. To approximate the RF performance, our tree classifiers will be constructed from $\{(X_i, Y_i)\}_{i=1}^{n}$. Asymptotic Distribution of Gini Indices --------------------------------------- A theoretical discussion of the evaluation of splits can be found in @banerjee2007confidence. In our specific , we compare the Gini indices of splits: examine their asymptotic behavior and obtain a central limit theorem (CLT) so normal based tests can be developed. (\[eqn:gini\]) implies an averaging over all samples when calculating the Gini index, suggesting the existence of the CLT. To examine two perspective splits $G_1$ and $G_2$ with the same samples, recall their Gini gains $$\begin{aligned} g_1 &= 1-\pi_{1,l} \left(\sum_{j=1}^k \theta_{1,l,j}^2\right) - \pi_{1,r} \left(\sum_{j=1}^k \theta_{1,r,j}^2\right),\\ g_2 &= 1-\pi_{2,l}\left(\sum_{j=1}^k \theta_{2,l,j}^2\right) - \pi_{2,r}\left(\sum_{j=1}^k\theta_{2,r,j}^2\right),\end{aligned}$$ where $\pi$ represents the covariate distribution of $\tilde{X}$ and $\theta$ the conditional probability of $\hat{\tilde{Y}}$ given $\tilde{X}$. Subscripts are arranged in the order of the split, the left or right child, and the class label. For instance, $$\begin{aligned} \pi_{1,l} &= P(G_1(X)=0), \\ \pi_{1,r} &= P(G_1(X)=1), \\ \theta_{1,l,j} &= P(Y=1|G_1(X)=j),\end{aligned}$$ and respectively for $G_2$. The empirical versions, Gini indices, are $$\begin{aligned} \hat{g}_{1,n} &= 1-\frac{n_{1,l}}{n} \sum_{j=1}^k \left(\hat{\theta}_{1,l,j}\right)^2 - \frac{n_{1,r}}{n} \sum_{j=1}^k \left(\hat{\theta}_{1,r,j}\right)^2,\\ \hat{g}_{2,n} &= 1-\frac{n_{2,l}}{n} \sum_{j=1}^k \left(\hat{\theta}_{2,l,j}\right)^2 - \frac{n_{2,r}}{n} \sum_{j=1}^k \left(\hat{\theta}_{2,r,j}\right)^2.\end{aligned}$$ Moving to the left and right children of both splits, we denote the numbers of samples and the ratios of class labels in each child by, for $p \in \{1,2\}, j \in \{1,\dots,k\},$ $$\begin{aligned} n_{p,l} &= \sum_{i=1}^n 1_{\{G_p(X_i)=0\}}, \\ \hat{\theta}_{p,l,j} &= \frac{1}{n_{p,l}} \sum_{i=1}^n Y_i^j\cdot 1_{\{G_p(X_i)=0\}}, \\ n_{p,r} &= \sum_{i=1}^n 1_{\{G_p(X_i)=1\}}, \\ \hat{\theta}_{p,r,j} &= \frac{1}{n_{p,r}} \sum_{i=1}^n Y_i^j\cdot 1_{\{G_p(X_i)=1\}}.\end{aligned}$$ For simplicity we write, for $p \in \{1,2\},q\in\{l,r\},$ $$N_{p,q} = \begin{bmatrix} n_{p,q}\hat{\theta}_{p,q,1}\\ \vdots \\ n_{p, q}\hat{\theta}_{p,q,k} \end{bmatrix}, \quad \Theta_{p,q} = \begin{bmatrix} \pi_{p,q} \theta_{p,q,1}\\ \vdots \\ \pi_{p,q}\theta_{p,q,k} \end{bmatrix}.$$ Employing a multivariate CLT we obtain $$\begin{aligned} \sqrt{n} \left(\frac{1}{n}\begin{bmatrix} N_{1,l} \\ N_{1,r} \\ N_{2,l} \\ N_{2,r} \end{bmatrix} - \begin{bmatrix} \Theta_{1,l} \\ \Theta_{1,r} \\ \Theta_{2,l} \\ \Theta_{2,r} \end{bmatrix} \right) \longrightarrow N(0, \Sigma).\end{aligned}$$ To relate this limiting distribution to the difference of Gini indices we shall employ the $\delta$-method. Consider the analytic function $f: \mathbb{R}^{4k} \to \mathbb{R}$ s.t. $$\begin{aligned} f(x_1, \dots, x_{4k}) = &-\frac{1}{\pi_{1,l}}\sum_{i=1}^k x_i^2 -\frac{1}{\pi_{1,r}} \sum_{i=k+1}^{2k} x_i^2 +\frac{1}{\pi_{2,l}} \sum_{i=2k+1}^{3k} x_i^2 +\frac{1}{\pi_{2,r}} \sum_{i=3k+1}^{4k} x_i^2.\end{aligned}$$ The $\delta$-method imples that $$\begin{aligned} \sqrt{n} \left(f\left(\frac{1}{n}\begin{bmatrix} N_{1,l} \\ N_{1,r} \\ N_{2,l} \\ N_{2,r} \end{bmatrix}\right) - f\left( \begin{bmatrix} \Theta_{1,l} \\ \Theta_{1,r} \\ \Theta_{2,l} \\ \Theta_{2,r} \end{bmatrix}\right)\right) \nonumber \longrightarrow N(0, \Theta^T\Sigma\Theta). \label{eqn:main}\end{aligned}$$ Here we write $$\begin{gathered} \Theta = f'\left(\begin{bmatrix} \Theta_{1,l} \\ \Theta_{1,r} \\ \Theta_{2,l} \\ \Theta_{2,r} \end{bmatrix} \right) = 2 \begin{bmatrix} -\Theta_{1,l} \\ -\Theta_{1,r} \\ \Theta_{2,l} \\ \Theta_{2,r} \end{bmatrix} \in \mathbb{R}^{4k},\\ \Sigma = cov \begin{bmatrix} \Theta_{1,l} \\ \Theta_{1,r} \\ \Theta_{2,l} \\ \Theta_{2,r} \end{bmatrix} = cov \begin{bmatrix} Y\cdot 1_{\{G_1(X)=0\}}\\ Y\cdot 1_{\{G_1(X)=1\}}\\ Y\cdot 1_{\{G_2(X)=0\}}\\ Y\cdot 1_{\{G_2(X)=1\}} \end{bmatrix} \in \mathbb{R}^{4k \times 4k}.\end{gathered}$$ We should point out that (\[eqn:main\]) provides us with the CLT we the difference between two Gini indices. After expanding (\[eqn:main\]), $$\sqrt{n} \left ((\hat{g}_{1,n}-\hat{g}_{2,n}) - (g_1 - g_2)\right) \longrightarrow N(0, \Theta^T \Sigma \Theta).$$ or asymptotically, $$(\hat{g}_{1,n} - \hat{g}_{2,n}) - (g_1 - g_2) \sim N \left(0, \frac{\Theta^T \Sigma \Theta}{n}\right).$$ Hence, by replacing $\Theta, \Sigma$ by the empirical versions from the pseudo samples, we write $$\begin{gathered} \hat{g}_{1,n}-\hat{g}_{2,n} \sim N\left(g_1 - g_2, \frac{\hat{\Theta}^T \hat{\Sigma}\hat{\Theta}}{n}\right). \label{eqn:bs2}\end{gathered}$$ Comparing Two Splits -------------------- where $Z$ is the inverse c.d.f. of a standard normal. With a sufficiently large $n$ it is possible to always determine the better split between $G_1$ and $G_2$ should they have any difference. In addition, by combining this test with pairwise comparisons , we are capable of finding the best split among multiple prospective splits. Sequential Testing ------------------ The power of better split test increases with $n$. Since we need to determine $n$ to reveal any detectable difference between two splits, when no prior knowledge is given regarding the magnitude of the difference, we need an adaptive approach to increasing $n$ accordingly. confidence level $\alpha$, suppose we have tested at sample size $n$ and get p-value $p_n > \alpha$. Referring to (\[eqn:bs\]), we have $$\sqrt{n}\cdot \frac{\hat{g}_{1,n}-\hat{g}_{2,n}}{\sqrt{2\hat{\Theta}^T \hat{\Sigma} \hat{\Theta}}} = Z_{p_n}.$$ Notice that $\dfrac{\hat{g}_{1,n}-\hat{g}_{2,n}}{\sqrt{2\hat{\Theta}^T \hat{\Sigma} \hat{\Theta}}}$ is the estimator of $\dfrac{g_1-g_2}{\sqrt{2\Theta^T \Sigma \Theta^T}}$ which is an intrinsic constant with respect to the pairwise comparison. Hence in order to reach a p-value less than $\alpha$ we may increase sample size to $n'$ such that $$\begin{aligned} \sqrt{n'}\cdot \frac{\hat{g}_{1,n}-\hat{g}_{2,n}}{\sqrt{2\hat{\Theta}^T \hat{\Sigma} \hat{\Theta}}} = Z_{\alpha},\end{aligned}$$ which yields that $$\begin{gathered} \sqrt{\frac{n}{n'}} = \frac{Z_{p_n}}{Z_{\alpha}}. \label{eqn:ad1}\end{gathered}$$ Due to pseudo sample randomness, a few successive increments are required before we land in the confidence level. We also need an upper bound for $n'$ in case the difference between two splits is too to identify. Multiple Testing ---------------- So far we have obtained a method to compare a pair of splits. When splitting a certain node, however, it is always the case that we need to choose the best split among multiple $G_1,\dots G_m$. In order to adapt our pairwise better split test to this situation, we consider modifying the problem slightly into deciding whether the split with the minimal Gini index is intrinsically superior than any other splits. This problem can be resolved by conducting pairwise comparisons of the split with minimal Gini index against the rest. If we still want to test at a certain significance $\alpha$ whether the with the lowest estimated Gini index, i.e, $\hat{g}_{n,(1)}$, is the optimal, we can still work within the scheme of the pairwise comparison with an additional procedure controlling the familywise error rate (FWER). Here we make an analogue of the Bonferroni correction [@dunnett1955multiple]. $\bullet$ Test the hypotheses $H_{i,0}: g_{(1)} = g_{(i)}, i=2,\dots, t$. Get the $p$-values $p_2,\dots,p_t$. $\bullet$ Analogous to a Bonferroni correction, use $\sum_{i=2}^t p_t$, the upper bound for making at most one Type I error, as the $p$-value of the multiple comparison. This test aggregates all significance levels into one. The Bonferroni correction will result in a conservative as we ignore much correlation structure of the splits. In this scenario, the updates of sample size made in sequential testing should also be adapted as we are now taking the aggregated significance level. A quick and feasible fix is to replace the $p_n$ in (\[eqn:ad1\]) by the aggregated significance level. Alternatively, we may just test between the best two splits. Because of the computational cost, when we have two splits that , the sequential and multiple testing may end up demanding an extremely large number of points to make the test significant. In practice, we halt the testing early at a cutoff of certain amount $N_{ps}$ of points, and choose the current best split. This compensation for computation time might lower the real power of the test, leading to a less stable result. Choice of Prospective Splits ---------------------------- e look at the original samples that have been carried along the path and take the possible combinations of the covariates and their middle points of adjacent values that have appeared in those samples. Although this method will generate a large number of prospective splits, because of the sequential testing scheme, most of those splits will be identified far worse than the best after a few tests and can be discarded, leaving a negligible effect on the overall performance. In practice, we implement a scheme [@benjamini1995controlling] to adaptively discard splits that perform far worse than the current best. All splits are ordered by their p-values against the current best split, and the splits fall below the threshold are discarded. Approximate a RF ================ To build an approximation tree, we replace the greedy splitting criterion by our stabilized version within the CART construction algorithm. At each node, we generate an initial of sample points from the RF, guaranteeing that those points belong to this node. Then we compare prospective splits simultaneously based on this set and decide whether we choose the one with the smallest Gini index with certain confidence or request more pseudo sample points. In the latter case, we keep generating until the pseudo sample size reaches what required by the sequential testing . This is repeated until we distinguish the best split. We perform this procedure on any node that needs to split during construction to get the final approximation tree. $\mathcal{T} \leftarrow$ There are several parameters to tune for this algorithm. We first need all the parameters for CART construction, i.e., the maximal depth of the tree, or maximal and minimal number samples in each leaf node. $\alpha$ the significance of the test of better split, and $N_{ps}$ which controls the maximal amount of psuedo samples we require at each node. Generating Points ------------------ To generate the pseudo sample, we first generate pseudo covariates then to get the responses. It is worth noticing that the first step here may encounter the obstacle that, in practice, we do not have the prior distribution of covariates. Conventional statistical techniques apply here. Some methods focus on estimating the underlying distribution by smoothers [@wand1994kernel], while the others, for instance, bootstrapping or residual permutation, attempt to directly manipulate and reorganize the samples to generate more samples. In the purpose of exploring more of the covariate space, we take the first approach and use a Gaussian kernel smoother upon the empirical distribution of the samples. This translates to generating pseudo covariates from observed covariates plus random noise. When we go further down the approximation tree, the covariate space may as well be narrowed down by the splits along the path. A feasible covariate generator can thus be produced by only smoothing the empirical distribution of those original samples that have been carried on by this path. We further check the boundary condition to ensure that the covariates we generated agree within the region divided by the splits along the path. Empirical Study =============== We have conducted empirical studies on both simulation and real data to show the performance approximation tree compares with both decision trees and the RF. The performance is assessed in two : consistency with the RF, and stability. In order to evaluate consistency, we generate new covariates and measure how much the predictions of approximation tree agree with of the RF. In this paper we are more interested in the mimicking ability rather than the predictive power of the approximation tree. However, we will still compare the of approximation trees with decision trees . To measure stability, which is defined in our case as the structural uniqueness, we construct multiple approximation trees out of a single RF and look into the variation in their structures. The better split test does not always guarantee a consistent pick through multiple trials due to the pseudo sample randomness, hence we hope to see small variation among all the trees built. We also examine the trees at different depths to capture the variation along the tree growth. Simulated Data -------------- We assume that the original samples have $\tilde{X} \in \mathbb{R}^5$ and $\tilde{Y} \in \{0, 1\}$. The covariate distribution is given by $\tilde{X} = (x_1,\dots,x_5) \sim \mbox{Unif}[0,1]^5,$ and $\mbox{logit}(P(\tilde{Y}=1|\tilde{X}))$ is given by $$\begin{gathered} \begin{cases} 2, &x_1>0.5, \quad x_2>0.7,\\ -3, &x_1>0.5, \quad 0.7 \geq x_2>0.2,\\ -4, &x_1>0.5, \quad x_2 \leq 0.2, \\ 3, &x_1 \leq 0.5, \quad x_5 \leq 0.5, \quad x_3+x_4^2 \geq 1.4, \\ 2, &x_1 \leq 0.5, \quad x_5 \leq 0.5, \quad 1.4 > x_3+x_4^2 \geq 0.5, \\ -2, &x_1 \leq 0.5, \quad x_5 \leq 0.5, \quad x_3+x_4^2 < 0.5, \\ 2, &x_1 \leq 0.5, \quad x_5 > 0.5. \end{cases}\end{gathered}$$ ### Predictive and Mimicking Accuracy We compare across three methods: classification trees (CART), random forests (RF) and our proposed single tree approximation (STA). We generate 1,000 sample points from this distribution and a standard RF consisting of 100 trees and a classification tree . Then we build our approximation tree via the algorithm above. The significant level $\alpha$ for the test of better split is set to be 0.1, and the maximal number of pseudo samples at each node $N_{ps}$ is set to be 10,000, 100,000, and 1,000,000 respectively. Both CART and STA set the maximal tree depth to be 5 including the root. ![Predictive Accuracy. Three methods here are classification tree (CART), random forest (RF) and single tree approximation (STA) respectively.[]{data-label="fig:PA1"}](PA1.png){width="4.5in"} Figure \[fig:PA1\] shows the predictive accuracy of the three methods on new test points. On average they share similar predictive accuracy the RF has the smallest variance, followed by the single tree approximation. This coincides with our expectation that STA is capable of inheriting stability from the random forest. Since the relation between the covariates and the responses is relatively simple, the increase of $N_ps$ has not introduced a significant improvement in performance. ![Mimicking Accuracy. PROB compares the output of RF and STA by the $L^1$ difference of their class probabilities. CLASS compares the output by the predicted class labels.[]{data-label="fig:MA1"}](MA1.png){width="4.5in"} Figure \[fig:MA1\] shows the comparison between RF and STA in terms of the $L^1$ difference of their predicted class probability, and the disagreement of their class labels. Again the increase of $N_{ps}$ has no significant improvement in performance. The approximation trees, which are solely built to their fifth layer, already have achieved [around 95%]{} agreement on average with the RFs. It is reasonable to believe that by expanding the trees to larger sizes the mimicking accuracy can still be marginally increased by “overfitting” the RFs. ### Stability ![STA stability. Four settings: RF+STA with $N_{ps}=100,000$, RF+STA with $N_{ps}=1,000,000$, RF+CART with $N=100,000$ sample points, RF+CART with $N=1,000,000$ sample points. In each plot, the columns represent the depths of the trees (from 1 to 4 excluding the root). In each column, a single black bar represents a unique structure of the tree, while the height of the bar represents the number of occurrence of that structure out of 100 replications.[]{data-label="fig:syntree1"}](final_unsym.png){width="4.5in"} ![Mimicking Accuracy. PROB compares the output of RF and STA by the $L^1$ difference of their class probabilities. CLASS compares the out by the predicted class label. From left to right represent $N_{ps}=100,000$ and $N_{ps}=1,000,000$.[]{data-label="fig:mddacc"}](mddacc.png){width="4.5in"} We compare the stability of STA in contrast to CART under similar setting. Following the same setting as the previous experiment, we first employ a RF to learn the sample points, then draw pseudo sample points from the RF to generate both STA and CART. In terms of CART, to obtain a fair comparison, we generate $N_{ps}$ pseudo samples to build a complete CART tree, the maximal number of pseudo sample points STA can utilize CART will slight advantages when generating the top layer splits. Figure \[fig:syntree1\] shows the result the two methods labeled as RF+STA and RF+CART. Defining stability as structural uniqueness, we examine each split by both the covariate and the value . CART has very poor stability when $N$ is small when the trees are . Our STA, in contrast, has much fewer structures. The variation of the approximation trees shrinks along with the increase of $N_{ps}$. Major Depression Disorder (MDD) Data ------------------------------------ We now demonstrate our method using the MDD data set which motivated @gibbons2013computerized. In this study, we focus on examining the stability of approximation trees and the consistency between the trees and the RF. The data set consists of 836 patients and 88 integer-valued covariates representing their responses to 88 survey questions. Among 88 questions there are 27 yes-no questions, 11 four-choose-one’s and 50 five-choose-one’s. Patients are classified as either with severe depression (257 out of 836) or without. e obtain a RF from these samples, and then mimic the RF with 100 approximation trees (STA). We set $N_{ps}$ to be 100,000 and 1,000,000 respectively. The significant level $\alpha$ for the test of better split is set to be 0.1. Figure \[fig:mddacc\] shows the mimicking power of the approximation trees. Notice that by constructing the trees to the fifth layer, the approximation trees already agree on about 90% of the classification results $L^1$-norm difference in class probabilities . Note that by increasing $N_{ps}$ the accuracy does not significantly increase, which demonstrates that the decision tree prediction performs well number of pseudo samples. Figure \[fig:mdd0\] the stability . The stability increases along with the increasing $N_{ps}$, the maximal amount of pseudo samples generated for deciding splits. In addition, the case $N_{ps}=1,000,000$ brought up several . Our $\alpha$, the significance level of the test of better split, works conservatively here as all trees have the same root and second-layer splits. This disagreement ratio among all the 100 trees we built is decently low. Following @gibbons2013computerized we improve the depression diagnosis from an 88-question survey plus a RF predictor, to a unique adaptive screening tool with at most 4 questions, while retaining 90% consistency. ![STA stability. STA with $N_{ps}=1,000,000$ reports 10 structures when the trees are build to the fifth layer, with the major structure occurring 68 out of 100 times.[]{data-label="fig:mdd0"}](final_mdd.png){width="4.5in"} we only have 836 original patients, we still need over 1,000,000 pseudo points to a split. One reason to this is that we have many variables and values to choose as a splitting rule, which is likely to cause the best splits indistinguishable a stopping rule . remedy is to obtain a prior set of fewer splits of interests and split by this set. However, in general we still require a large amount of points. Another observation is that the significance level $\alpha$ controls the stability at a split-wise level. It is possible extend this to further stabilize the tree by again introducing the FWER at the tree level. Notice this procedure may also increase the number of pseudo samples we need at each split. Conclusion ========== In this paper we have proposed a procedure to mimic a complex model by a single decision tree approximation. The asymptotic behavior of differences of Gini indices was studied and a normal based test of better split was developed and inspected under this oracle learning setting. We further incorporated this test in tree building to ensure the performance and the stability of the approximation trees. Empirical studies were done based on both simulation and real data and the results met our expectation. We in addition presented the interpretability of our procedure on the real data. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by grants NIH R03DA036683, NSF DMS-1053252 and NSF DEB-1353039. The authors would like to thank Robert Gibbons for providing the CAT-MDD data.

--- abstract: 'We present an extension of the recombination formalism to analyze the effects from the variation of the hadron wavefunctions. The hadron spectra are sensitive to the shape of the wavefunctions. However, when we fit the wavefunction parameters to the physical observables, such as the average charge radius, the final spectra are very similar each other. We discuss our numerical results in comparison with the published PHENIX and STAR data at RHIC. In the hadron spectra, the recombination of thermal partons dominates at intermediate transverse momentum ($P_{T}$ = 2 $\sim$ 5 GeV), and the fragmentation dominates at high $P_{T}$ ($>$ 5 GeV). The yield ratios and the nuclear modification factors for various hadron species are also estimated and compared to the experimental data. We present a new prediction on $\bar{p}/p$ and $K^{-}/K^{+}$ ratios, including the jet quenching effects to the fragmentation mechanism.' author: - Byungsik Hong - 'Chueng-Ryong Ji' - 'Dong-Pil Min' title: 'Light-front wavefunction dependence of the quark recombination' --- Introduction ============ Although no one doubt the existence of quarks and gluons, they have not yet been detected individually at the zero temperature. The current vigorous efforts at the Relativistic Heavy Ion Collider (RHIC) and the future plans of the Large Hadron Collider (LHC) may reveal the temperature dependence of the confinement mechanism and the chrial symmetry restoration [@lee1]. The high-energy nuclear collisions compress and heat the heavy nuclei so much that their individual protons and neutrons may overlap and, in addition, a lot of pions may arise to ultimately create the quark-gluon plasma (QGP). The QGP may have existed ten millionths of second after the big bang and created the primordial matter of the universe. The RHIC and the future LHC may yield the QGP in the laboratory. It has been reported that the four experiments at RHIC already obtained distinguished results from the lower energy heavy ion collisions at CERN SPS [@b-wp; @p-wp; @ph-wp; @s-wp]. The future LHC experiments (ALICE as well as CMS and ATLAS) would require theoretical predictions at the 30-fold energy increase from the RHIC. Among many others, the effects from the jet quenching and the bulk hadronization may be regarded as the important new results. Especially, the elliptic flow analysis revealed that the differential second-harmonic Fourier moment ($v_{2}$) of the azimuthal distribution with respect to the reaction plane had a remarkable saturation property in the intermediate transverse momentum ($P_{T}$) range between 2 and 6 GeV for all hadrons including multistrange baryons. This saturation effect and eventual decrease of $v_{2}$ at high $P_{T}$ have been qualitatively interpreted to be the results of partonic energy loss in an opaque parton system created by nuclear collisions [@wang-1; @wang-2]. Furthermore, the estimated $v_{2}$ parameter as a function of $P_{T}$ are scaled by the number of constituent quarks of particles. Together with an enhanced proton production in the intermediate $P_{T}$ region, this agreement has been taken seriously as one piece of evidence for the quark recombination process and the presence of partonic collectivity at the early stage of a collision [@fries; @mola-1; @grec-1; @qc-1; @qc-2; @hong-1]. In this work, we utilize the previous recombination formalism, and extend it to analyze the light-front (LF) wavefunction dependence in the theoretical predictions from this formalism [@fries]. Typical forms of the LF wavefunctions such as the Gaussian form and the power-law form are applied to this extended formulation. The numerical results are contrasted to each other, and compared with the available single invariant spectra by PHENIX and STAR for various mesons and baryons. We also discuss the production ratios of various hadrons, including ${\bar p}/p$ and $K^-/K^+$, in the fragmentation region. While we include the jet quenching effects as others do, we get a rather distinguished results compared to the previous ones. For the high $P_T$ regions, we get a dramatic suppression of the antiparticles $\bar p$ and $K^-$ compare to the corresponding particles $p$ and $K^+$, respectively. The paper is organized as follows. In Sec. \[formulation\], we present the recombination formalism which is extended from the previous one to explicitly include the intrinsic transverse momenta of the constituents inside the hadron. Rather than an extensive review of the previous formalism, we focus on what has been extended from the previous model. In Sec. \[result\], we present the numerical results of the $P_T$ spectra for various mesons and baryons to contrast the results between the Gaussian form and the power-law form. The results are compared with the available experimental data from PHENIX and STAR collaborations. We also discuss the production ratios of various hadrons and the nuclear modification factor $R_{CP}$ in this section. Conclusions and discussions follow in Sec. \[conc\]. Formulation =========== Recombination and Light-Front Wavefunction Dependence {#recombi} ----------------------------------------------------- The current data from the RHIC experiments seem to indicate two distinguished mechanisms of hadronization: (1) quark recombination for a rather low and intermediate $P_T$ region and (2) quark fragmentation for a high $P_T$ region. In this section, we present an extension of the recombination formalism to analyze the effects from the variation of the hadron wavefunctions. Introducing the density matrix $\hat\rho$ for the system of partons, the number of quark-antiquark states that one may interprete as mesons is given by $$N_M = \Sigma_{a b} \int \frac{d^3 P}{(2\pi)^3} <M;P|{\hat\rho}_{ab}|M;P>, \label{number}$$ where $|M;P>$ is a meson state with the momentum $P$ and the sum is over all combinations of quantum numbers such as flavor, helicity, and color of the valence quarks that contribute to the given meson $M$. Inserting complete sets of coordinates and using the Wigner function formalism, one can derive the formula for the invariant spectrum of the meson $M$ as follows [@fries]: $$\begin{aligned} E \frac{d^{3}N_M}{dP^{3}}&=&C_M \int_\Sigma \frac{d^3 R P\cdot u(R)}{(2\pi)^3} \int \frac{d^3 q}{(2\pi)^3} \nonumber \\ &&\times w_a(R; \frac{P}{2}-q) \Phi^W_M (q) w_b(R;\frac{P}{2}+q) \nonumber \\ &=& C_M \int_\Sigma \frac{d^3 R P\cdot u(R)}{(2\pi)^3} \int \frac{dx P^+ d^2 {\vec k}_\perp}{(2\pi)^3} \nonumber \\ &&\times w_a(R; xP^+, {\vec k}_\perp) \Phi_M (x, {\vec k}_\perp) w_b(R;(1-x)P^+, -{\vec k}_\perp), \label{spectrum-meson}\end{aligned}$$ where $\Phi^W_M(q) = \int d^{3}r \Phi^W_M(r,q)$ in the Wigner function formalism and $\Phi_M(x,{\vec k}_\perp) = |\psi_M (x, {\vec k}_\perp)|^2$ using the LF wavefunction of the meson $\psi_M (x, {\vec k}_\perp)$. Here, $x$ and ${\vec k}_{\perp}$ are the momentum fraction and the respective intrinsic transverse momentum of each quark. Similarly, the invariant spectrum of the baryon $B$ can be obtained as follows [@fries]: $$\begin{aligned} E \frac{d^{3}N_B}{dP^{3}}&=& C_B \int_\Sigma \frac{d^3 R P\cdot u(R)}{(2\pi)^3} \int \frac{dx_{1}P^{+} d^{2}{\vec k}_{1\perp}}{(2\pi)^{3}} \int \frac{dx_{2}P^{+} d^{2}{\vec k}_{2\perp}}{(2\pi)^{3}} \int \frac{dx_{3}P^{+} d^{2}{\vec k}_{3\perp}}{(2\pi)^{3}} \nonumber \\ && \times w_a(R; x_{1} P^{+}, {\vec k}_{1\perp}) w_b(R; x_{2} P^{+}, {\vec k}_{2\perp}) w_c(R; x_{3} P^{+}, {\vec k}_{3\perp}) \nonumber \\ && \times \Phi_B (x_{1}, x_{2}, x_{3}, { \vec k}_{1\perp}, {\vec k}_{2\perp}, {\vec k}_{3\perp}), \label{spectrum-baryon}\end{aligned}$$ where $x_{1} + x_{2} + x_{3} = 1$ and ${\vec k}_{1\perp}+{\vec k}_{2\perp}+{\vec k}_{3\perp} = 0$. The previous work by Fries and collaborators used a factorized ansatz for the LF wavefunction, for example [@fries], $$\psi_M (x,{\vec k}_\perp) = \phi_M(x) \Omega({\vec k}_\perp), \label{ansatz}$$ for mesons with a longitudinal distribution amplitude $\phi_M (x)$ and a transverse part $\Omega({\vec k}_\perp)$. However, we note that such factorization ansatz cannot be justified in free space since the LF wavefunction depends on the LF invariant mass of the particle, e.g., for the meson $(m_a^2 + {\vec k}_\perp^2)/x + (m_b^2 + {\vec k}_\perp^2)/(1-x)$ (here the meson is composed of quark $a$ and $b$), which cannot be factorized as Eq.(\[ansatz\]). In general, the assumption of wavefunction factorization such as Eq.(\[ansatz\]) is not acceptable in free space because LF wavefunctions should be solutions of LF bound-state equations and the LF energy-momentum dispersion relation is rational, i.e. $k^- = ({k_\perp}^2 + m^2)/{k^+}$ for the particle with mass $m$. Both the LF kinetic energy (i.e. the LF invariant mass of the bound-state) and the LF kernel (or the inverse of the LF energy difference between the initial and intermediate states) involved in the LF bound-state equations are not factorizable due to the rational energy-momentum dispersion relations. Thus, the solutions of the LF bound-state equations cannot be factorizable and we do not integrate over ${\vec k}_\perp$ in Eqs.(\[spectrum-meson\]) and (\[spectrum-baryon\]) but leave ${\vec k}_\perp$ explicitly in the formulation. On the other hand, since it is not yet known if the LF bound-state solution in free space is also applicable to the dynamical recombination process in quark matter without any modification, we note that the factorization ansatz used in Ref. [@fries] may be equally valuable as one of the model calculations in this work. The usual parton spectrum at a given temperature is given by [@fries] $$w_a (R; p) = \gamma_a e^{-p \cdot v(R)/T} \cdot e^{-\eta^2/{2 \Delta^2}} f(\rho,\phi), \label{parton-spectrum}$$ where $\rho$ and $\phi$ are the radial and the azimuthal angle coordinates, respectively. In addition, $v(R)$ and $\eta$ represent the velocity four vector and the rapidity of the quark $a$, respectively. Here, $\gamma_a = \exp(\mu_{a}/T)$ is the fugacity factor for each quark species $a$ for which we adopt the results from the statistical analysis for the hadron production at RHIC [@pbm-1]: the chemical potential $\mu_{a}$’s are 9, 6.7, and $-$3.9 MeV for $a$ = $u$(or $d$), $s$, and $c$, respectively. Note that Ref. [@pbm-1] gives resulting chemical potentials for isospin, strangeness and charmness as well as baryon chemical potential estimated by the statistical model at RHIC. Since statistical analysis of hadron production provides only chemical potentials of hadrons (not quarks), we obtained the quark fugacities by using the following formula for the fugacity of hadron $i$ [@torri]:[^1] $$\Upsilon_{i} = \gamma_{I_{3}^{i}} \prod_{a} \gamma_{a}^{N_{a}^{i}} \label{fugacity}$$ where $N_{a}^{i}$ is the number of quark species $a$ in hadron $i$. In Eq.(\[fugacity\]), the fugacity $\gamma_{I_{3}^{i}}$ is close to 1, as the isospin chemical potential $\mu_{I_{3}^{i}}$ and the assumed freeze-out temperature $T$ are -0.96 MeV and 175 MeV, respectively [@pbm-1]. We assume that the temperature $T$ for hadronization occurs at 175 MeV. The lattice QCD predicts that the phase transition occurs at $T =$ 175 MeV at vanishing baryon chemical potential [@lqcd]. It should be reasonable that the temperature of the partonic phase is assumed to be the same as that of the phase transition. The space-time structure of the parton source in Eq.(\[parton-spectrum\]) is given by a transverse distribution $f(\rho,\phi)$ and a wide Gaussian rapidity distribution with a width $\Delta$. Also, one may assume $f(\rho,\phi) \approx \Theta(\rho_0 -\rho)$ especially for the analysis of the central collisions. With these assumptions, we find $$\begin{aligned} \frac{d^{3}N_M}{dP_{T}^{2}dy}|_{y=0}&=& C_M M_T \frac{V}{(2\pi)^3} 2 \gamma_a \gamma_b I_0 \left[\frac{P_T \sinh \eta_T}{T}\right] \nonumber \\ &&\times \int_0^1 dx \int_0^{\infty} d^2 {\vec k}_\perp |\psi_M (x, {\vec k}_\perp)|^2 k_M (x, {\vec k}_\perp, P_T), \label{extend-spectrum}\end{aligned}$$ where $V = \tau A_{T}$ ($\tau$ is the hadronization time and $A_{T}$ is the transverse size) is the volume of the parton system and $$k_M (x, {\vec k}_\perp, P_T) = K_1 \left[\frac{\cosh{\eta_T}}{T} \{\sqrt{m_a^2 + (x P_T + {\vec k}_\perp)^2} + \sqrt{m_b^2 + \{(1-x) P_T - {\vec k}_\perp\}^2} \} \right]. \label{kM}$$ We note that the particular combination of $P_T$ and ${\vec k}_\perp$ for each constituent quark in Eq.(\[kM\]) is consistent with the boost invariance of $k_M$ in light-front dynamics. Extension to the baryon case is straightforward as $$\begin{aligned} \frac{d^{3}N_B}{dP_{T}^{2}dy}|_{y=0}&=& C_B M_T \frac{V}{(2\pi)^3} 2 \gamma_a \gamma_b \gamma_c I_0 \left[\frac{P_T \sinh \eta_T}{T}\right] \int_0^1 dx_{1} dx_{2} \int_0^{\infty} d^2 {\vec k}_{1\perp} d^2 {\vec k}_{2\perp} \nonumber \\ &&\times |\psi_B (x_{1},x_{2}, {\vec k}_{1\perp},{\vec k}_{2\perp})|^2 k_B (x_{1}, x_{2}, {\vec k}_{1\perp}, {\vec k}_{2\perp}, P_T), \label{Bextend-spectrum}\end{aligned}$$ and $$\begin{aligned} k_B (x_{1}, x_{2}, {\vec k}_{1\perp}, {\vec k}_{2\perp}, P_T) &=& K_1 [ \frac{\cosh{\eta_T}}{T} \{ \sqrt{m_a^2 + (x_{1} P_T + {\vec k}_{1\perp})^2} + \sqrt{m_b^2 + (x_{2} P_T + {\vec k}_{2\perp})^2} \nonumber \\ && +\sqrt{m_c^2 + \{(1-x_{1}-x_{2})P_T - ({\vec k}_{1\perp} + {\vec k}_{2\perp})\}^2}\} ]. \label{kB}\end{aligned}$$ In the following analysis, we take $V$ as a free parameter to fit all invariant spectra of hadrons simultaneously for a given collision centrality. With this extension, we can now explicitly include the effect from the intrinsic transverse momentum of each quark, and vary the form of the LF wavefunction such as the Gaussian form and the power-law form [@ji-1]. In this analysis, we use the following typical LF wavefunctions for mesons and contrast the results between the two: $$\psi_{Gaussian} (x, {\vec k}_\perp) = \exp \left[-(\frac{m_a^2 + {\vec k}_\perp^2}{x} + \frac{m_b^2 + {\vec k}_\perp^2}{1-x})/\beta^2 \right], \label{Gaussian}$$ and $$\psi_{Power-law} (x, {\vec k}_\perp) = 1/\left[\frac{m_a^2 + {\vec k}_\perp^2}{x} + \frac{m_b^2 + {\vec k}_\perp^2}{1-x} + \alpha^2 \right]^{n}, \label{power-law}$$ where $\beta$, $\alpha$, and $n$ are the parameters that can be fixed from the physical observables such as the size and the mass spectrum of meson, etc.. The extension of Eqs.(\[Gaussian\]) and (\[power-law\]) to baryons is rather straightforward: $$\psi_{Gaussian} (x_{1}, x_{2}, {\vec k}_{1\perp}, {\vec k}_{2\perp}) = \exp \left[-(\frac{m_a^2 + {\vec k}_{1\perp}^2}{x_{1}} + \frac{m_b^2 + {\vec k}_{2\perp}^2}{x_{2}} + \frac{m_c^2 + ({\vec k}_{1\perp} + {\vec k}_{2\perp})^2} {1-x_{1}-x_{2}})/\beta^2 \right], \label{B_Gaussian}$$ and $$\psi_{Power-law} (x_{1}, x_{2}, {\vec k}_{1\perp}, {\vec k}_{2\perp}) = 1/\left[\frac{m_a^2 + {\vec k}_{1\perp}^2}{x_{1}} + \frac{m_b^2 + {\vec k}_{2\perp}^2}{x_{2}} + \frac{m_c^2 + ({\vec k}_{1\perp} + {\vec k}_{2\perp})^2} {1-x_{1}-x_{2}} + \alpha^2 \right]^{n}. \label{B_power-law}$$ In this calculation, we used 260 MeV for the masses of $u$ and $d$ quarks and 460 MeV for the mass of $s$ quark. In a relativized quark model with chromodyamics, spectra of both mesons and baryons have been well analyzed. As shown in typical references ([@mass-m] for mesons and [@mass-b] for baryons), the potentials among constituents such as confinement, hyperfine, spin-orbit, etc. work together to reproduce the hadron spectra comparable to the experimental values. For instance, proton mass was predicted as 960 MeV and the mass difference between nucleon and delta was obtained around 300 MeV while the light quark mass was taken as 220 MeV. The same light quark mass was used to predict the meson spectra which were overall in good agreement with data. These support our light-front quark model calculations (see e.g. [@mass-3]). Although we took the light quark mass 260 MeV as used in Ref. [@fries] for the present analysis, the essential predictions from a relativized quark model (or light-front quark model) remain intact. Just to illustrate how typical LF wavefunctions look like, we plot $\psi_{Gaussian} (x, {\vec k}_\perp)$ for different $\beta^{2}$ values in Fig. \[Fig:wf\_evolve\]. As expected, the LF wavefunctions are symmetric around $x$ = 0.5, if the masses of constituent quark and antiquark are the same for pions. As $\beta^{2}$ increases, $\psi_{Gaussian} (x, {\vec k}_{\perp})$ becomes broader in $x$ as well as in ${\vec k}_\perp$. When the mass of constituent quark and antiquark are not the same like $K$ and $D$, $\psi_{Gaussian} (x, {\vec k}_{\perp})$’s are clearly skewed in $x$. In order to constrain $\beta$, $\alpha$, and $n$ in Eqs.(\[Gaussian\]) - (\[B\_power-law\]), the average values of ${\vec k}_{\perp}$ are fixed by the measured average charge radius square $<r^{2}>$ of each hadron. If the experimental data for $<r^{2}>$ are not available, we adopt the calculated ones by a relativistic quark model [@ji-2; @schl-1]. As an example, $<r^{2}>$ = 0.44 fm$^{2}$ for pions, and the corresponding $\beta^{2}$ is 0.825 GeV$^2$ for $\psi_{Gaussian}$. The average values of the charge radius square and the corresponding values of $\beta^{2}$ are summarized in Table \[Tab:beta2\]. In addition, the deduced $\alpha^{2}$ and $n$ of $\psi_{Power-law}$ are 0.5 (1.53) GeV$^2$ and 2 (6), respectively, for pions (protons). The left two panels of Fig. \[Fig:wf\_comp\] show the comparison of $\psi_{Gaussian}$ and $\psi_{Power-law}$ of pions by using the adjusted wavefunction parameters. They demonstrate that the LF wavefunctions are very similar in shape when the parameters are determined by some physical observables such as the charge radii of hadrons. However, if we use some arbitrary values for those parameters, the shape of LF wavefunctions can be quite different, which is demonstrated in the lower right panel of Fig. \[Fig:wf\_comp\]. The importance of the proper choice of the LF wavefunction parameters in the hadron spectra are explained in Fig. \[Fig:wf\_check\] for $\pi$’s and protons. The invariant yields of the recombined hadrons are quite different for different sets of parameters. However, the hadron yields from the recombination process are quite similar for $\psi_{Gaussian}$ and $\psi_{Power-law}$, once the wavefunction parameters are fixed by some physical observables (See solid vs. dashed lines in Fig. \[Fig:wf\_check\].). In the following analysis, we use only the Gaussian wavefunction with proper $\beta^2$ for each hadron. For the comparison, the wavefunctions used in Ref. [@fries] are also included in Fig. \[Fig:wf\_check\]. The wavefunctions used in the present analysis (solid lines) are lower than the factorized wavefunctions used in Ref. [@fries] shown by dotted lines especially for relatively low $P_{T}$ region. Even in the logarithmic scale, the differences are as visible as the ones with arbitrary wavefunction parameters (dash-dotted lines). Since the essence of this work is to study the effect of the proper treatment of the LF wavefunctions to the recombination yields, we also compare the invariant spectra estimated by using full LF wavefunctions with those estimated by factorized wavefunctions. For the approximation that uses factorized wavefunctions for mesons, we tested $\psi_{Gaussian} (x, k_{\perp})$ of Eq.(\[Gaussian\]) with $<{\vec k_{\perp}}^2> =$ 0.088 GeV$^2$, multiplied by $\Omega ({\vec k_{\perp}}) = \exp{(-{\vec k_{\perp}}^{2}/\beta^{2})}$ with $\beta^{2} =$ 0.176 GeV$^2$, which gave us the right $<{\vec k_{\perp}}^2>$ value. The resulting invariant spectrum for the recombined $\pi^{+}$’s is very similar to the PL1 option in Fig. \[Fig:wf\_check\]. Since it duplicates the result of PL1 option, we do not display it explicitly in Fig. \[Fig:wf\_check\]. However, this result indicates that the above factorized wavefunction may be as useful as the full LF wavefunctions developed in this paper for the phenomenological analysis of mesons, when the parameters are properly chosen. Similarly, for baryons, we tested $\psi_{Gaussian} (x_{1}, x_{2}, k_{1\perp}, k_{2\perp})$ of Eq.(\[B\_Gaussian\]) with $<{\vec k_{i\perp}}^2> =$ 0.0512 GeV$^2$, multiplied by $\Omega ({\vec k_{\perp}}) = \exp{[-({\vec k_{1\perp}}^{2}+{\vec k_{2\perp}}^{2})/2\beta^{2}]}$ with $\beta^{2} =$ 0.495 GeV$^2$. However, we could not find proper $\beta^2$ value for the factorized wavefunction to get the correct $<k_\perp^2>$ = 0.0512 GeV$^2$ since the results were not at all stable (too large standard deviations). Thus, we instead varied $\beta^2$ value in a wide range starting from 0.03 GeV$^2$ all the way even above 1 GeV$^2$, and compared the resulting proton spectra. Results for $\beta^2$ below 0.03 GeV$^2$ couldn’t be obtained due to a numerical instability. The results for $\beta^2$ above 1 GeV$^2$ were about the same as the result of $\beta^2 =$ 1 GeV$^2$ and very stable as expected from the form of the above factorized wavefunction. Also, what we obtained for $\beta^2$ values from 0.03 GeV$^2$ to 1 GeV$^2$ was that the proton spectrum results were fairly insensitive to the $\beta^2$ value. The resulting invariant spectrum of the recombined protons are shown by the coarsely-dotted line (typically for $\beta^2 =$ 0.495 GeV$^2$ but the result doesn’t change much for other $\beta^2$ values) in the bottom panel of Fig. \[Fig:wf\_check\]. The recombined proton yield by the factorized wavefunction is significantly lower than the one by nonfactorizable wavefunction. This comparison indicates that it may be more significant in the baryon case than in the meson case what form of the wavefunction is taken for the prediction of the recombination process. Thus, the proper treatment of the wavefunction seems particularly important for the baryon production in the recombination process. We note that there are also other approaches for the recombination process: Refs. [@grec-1] and [@hwa-yang] considered soft-hard recombination with a covariant coalescence model and the fragmentation as a part of recombination, respectively. In addition, a recent work by A. Majumder, E. Wang, and X.-N. Wang [@frag-re] discussed a derivation of the recombination model from field theory description of jet fragmentation. They noted that an ad hoc formulation of the recombination model is only valid under some strict conditions on the hadron wavefunction. Therefore, we note that the present development is not the first one to consider the dependence of hadron wavefunctions in the recombination process. Fragmentation and Jet Quenching {#frag} ------------------------------- Inclusive hadron production by fragmentation at large momentum transfer can be described well by perturbative quantum chromodynamics (pQCD). In the framework of pQCD, the invariant yield of hadron $h$ with momentum $P$ is given by [@fries] $$E{{d^{3}N_{h}^{frag}}\over{dP^{3}}} = \sum_{a}\int_{0}^{1} {{dz}\over{z^{2}}} D_{a \rightarrow h}(z) E_{a}{{d^{3} N_{a}^{pert}}\over{dp_{a}^{3}}}, \label{fragmentation}$$ where the sum runs over all constituent quark species $a$ in $h$. For the spectrum of parton $a$ with momentum $p_{a} = P/z$ at midrapidity, we use the parameterization by pQCD: $$E_{a}{{d^{3}N_{a}^{pert}}\over{dp_{a}^{3}}} = {{d^{2} N_{a}^{pert}}\over{2\pi p_{aT}dp_{aT}dy}}|_{y=0} = {{K}\over{\pi}} {{C}\over{(1+p_{aT}/B)^{\kappa}}}, \label{pqcd}$$ where the parameters $C$, $B$, and $\kappa$ are taken from a leading order pQCD calculations [@sriva-1]. $K$ = 1.5 is taken in order to consider higher order corrections approximately [@fries]. Note that the number of partons in different collision centralities are obtained by scaling the number of binary nucleon-nucleon collisions ($N_{coll}$) or, equivalently, by the nuclear thickness function ($T_{AA}$). The probability that parton $a$ fragments into hadron $h$ is taken into account by the fragmentation function $$D_{a \rightarrow h}(z) = N z^{\gamma} (1 - z)^{\delta}, \label{dah}$$ where the numerical values of $N$, $\gamma$, and $\delta$ are taken from the parameterization by Kniehl, Kramer, and Pötter (Tabel 2 of Ref. [@kkp] and the website “<http://www.desy.de/~poetter/kkp.f>”) for fragmentation of pions, kaons, protons and antiprotons. We call this parametrization as the KKP parametrization. $\Lambda$ fragmentation function is taken from the work by de Florian, Stratmann and Vogelsang (Tabel 1 of Ref. [@flori-1]). Finally, the energy loss of energetic partons (so called jet quenching), especially, in central collisions is considered by the following parameterization [@baier-1; @mueller-1] $$\Delta p_{T}(b, p_{T}) = \epsilon(b) \sqrt{p_{T}} {{\langle L \rangle(b)}\over{R_{A}}}, \label{quenching}$$ where $R_{A}$ is the radius of nucleus $A$, $\langle L \rangle(b)$ is the geometrical factor of the overlap zone of two nuclei, and $\epsilon(b)$ is the energy loss parameter of the hot medium with impact parameter $b$. The detailed functional forms of $\langle L \rangle(b)$ and $\epsilon(b)$ are the same as Ref. [@fries]: $$\epsilon(b) = \epsilon_{0}{{1-\exp[-(2R_{A}-b)/R_{A}]} \over{1-\exp(-2)}} \label{epsilon}$$ and $$\langle L \rangle(b)={{\sqrt{R_{A}^{2}-(b/2)^{2}}+(R_{A}-b/2)} \over {2}}, \label{loverb}$$ but, practically speaking, it is reasonable to assume that $\langle L \rangle(b) \simeq R_{A}$ and $\epsilon(b) \simeq \epsilon_{0}$ = 0.82 GeV$^{1/2}$ for the most central collisions as $b \rightarrow 0$. results {#result} ======= Invariant Spectra {#inva} ----------------- Fig. \[Fig:meson\] shows the numerical results on the invariant spectra of various mesons at midrapidity for central Au + Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. In Fig. \[Fig:meson\], we compare our calculations for the meson spectra with the published PHENIX and STAR data [@p1; @p2; @p3; @s1; @s2] up to $P_{T}$ = 10 GeV in order to show the overall shapes, especially, the transition regions near 5 GeV. The neutral pion spectrum was measured by PHENIX up to 10 GeV in $P_{T}$, but lacks data in a low $P_{T}$ region. In contrary, the charged pions were measured only at low $P_{T}$ up to 3 GeV with high precision. However, the high $P_{T}$ spectra of charged pions are expected to be very similar to those of neutral pions. For charged kaons, PHENIX measured up to 2 GeV, and STAR measured up to about 0.7 GeV in $P_{T}$. In general, the data by PHENIX and STAR agree quite well in the overlapped phase space except $\phi$: the STAR data is about a factor of three larger than the PHENIX data. In Fig. \[Fig:meson\], the dashed and dotted lines represent the model calculations from the recombination and the fragmentation, respectively, and the solid lines are the sum of two contributions. In the $\pi^{0}$ spectra, the two distinguished $P_{T}$ regions of hadron production are manifest. Although the transient $P_{T}$ depends on the particle species, the recombination process is dominant between $\sim$ 2 and 5 GeV, and the fragmentation is dominant above 5 GeV. Our calculation is not expected to reproduce the hadron spectra below about 2 GeV in $P_{T}$. In such a very low $P_{T}$ region, the calculation underestimates the experimental data significantly, implying that other processes like the transverse flow, the secondary decay of hadron resonances, and the binding energy effect become important. Our calculation reproduces the measured meson $P_{T}$ spectra larger than 2 GeV reasonable well, including the strange mesons. Note that we do not plot the fragmentation contribution for $\phi$ due to the lack of the fragmentation function. Fig. \[Fig:baryon\] shows the numerical results on the invariant spectra of baryons up to $P_T$ = 10 GeV at midrapidity for central Au + Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. Fig. \[Fig:baryon\] also compares the calculations with the published experimental data with the open circles by PHENIX [@p2; @p4] and the solid triangles by STAR [@s1; @s3; @s4]. The left most column of Fig. \[Fig:baryon\] is for protons and antiprotons; PHENIX and STAR measured up to about 5 and 1.2 GeV, respectively. Note that the published $P_{T}$ spectra for protons and antiprotons by STAR [@s1] are about 40 % higher than those by PHENIX [@p2]. This difference comes from the fact that the contributions from the $\Lambda$ and $\Sigma^{0}$ decays are removed only for PHENIX. For a fair comparison between the two data sets, the $p$ and $\bar{p}$ spectra by the STAR collaboration are scaled by 0.6 in Fig. \[Fig:baryon\]. After scaling down, the STAR spectra agree quite well with the PHENIX spectra in the overlapped phase space. The present model reproduces the measured proton and antiproton spectra reasonably well. The model also predicts that the transient $P_{T}$ for baryons from recombination to fragmentation is somewhat higher than that for mesons. In Fig. \[Fig:baryon\], the experimental invariant spectra of $\Lambda + \Sigma^{0}$, $\Xi^{-}$, $\Omega^{-}$, and their antiparticles are for $\sqrt{s_{NN}}$ = 130 GeV. But all model calculations are for $\sqrt{s_{NN}}$ = 200 GeV because all input parameters of the model calculations are available only for $\sqrt{s_{NN}}$ = 200 GeV. Furthermore, due to the lack of the fragmentation functions, we do not plot the fragmentation contribution for $\Xi$ and $\Omega$. Because of the difference in beam energy, the model overestimates the yields of $\Lambda + \Sigma^{0}$, $\Xi^{-}$, $\Omega^{-}$, and their antiparticles, and the discrepancy is larger for a larger number of strange quarks in a given baryon. Yield Ratios {#ratio} ------------ One of the most interesting data from RHIC is the yield ratio of protons(or antiprotons) to pions at the intermediate transverse momentum region (2 $< P_{T} <$ 5 GeV) in central heavy-ion collisions. The $p/\pi$ and $\bar{p}/\pi$ ratios rise steeply with $P_{T}$ up to about 2.5 GeV, but levels off at about 1 and 0.7, respectively, in 2.5 $< P_{T} <$ 5 GeV for the most central 10 % Au + Au collisions [@p2; @p5]. At $P_{T} >$ 2 GeV, $p/\pi$ and $\bar{p}/\pi$ for peripheral collisions are similar to those for elementary $p + p$ and $e^{+} e^{-}$ collisions, and the ratios increase from peripheral to central collisions. Since the hydrodynamic model, which had been rather successful in describing the low $P_{T}$ hadron spectra, could not explain the centrality dependence of limiting values, a recombination mechanism of hadronization at intermediate $P_{T}$ was proposed as a possible resolution [@fries; @grec-1; @hwa-yang; @vg; @hy]. The recombination process naturally explains that the $p/\pi$ and $\bar{p}/\pi$ ratios level off in 2 $< P_{T} <$ 5 GeV, and fall sharply near $P_{T} \simeq$ 5 GeV where the fragmentation takes over the recombination. Similar trends can also be found in the present calculation. The top row of Fig. \[Fig:ratio\] shows the results from our calculations for the $p/\pi^{0}$, $\bar{p}/\pi^{0}$, and $\bar{p}/p$ ratios in comparison with the published PHENIX data [@p2]. As $P_{T}$ increases, the $p/\pi^{0}$ and $\bar{p}/\pi^{0}$ ratios rises, reach the maximum values around 3 GeV, decrease sharply, and, finally, become constant at about 0.1 for $P_{T} >$ 6 GeV. In addition, the $\bar{p}/p$ ratio is almost constant at about 0.9 for $P_{T} <$ 5 GeV. However, it also decreases with $P_{T}$, and become almost constant at about 0.1 for $P_{T} >$ 7 GeV, which is very different from the previous calculation by Fries [*et al.*]{} (the dashed line in Fig. \[Fig:ratio\]) [@fries]. Although there is no dispute on KKP parametrizations [@kkp] for the gluon fragmentation, the quark fragmentation is a problem because the KKP fragmentation functions are not fully flavor seperated and one has to make additional assumptions to seperate contributions from different flavors. Incidentally, the recent STAR data on identified hadrons [@star2006] reveal a poor job of the KKP fragmentation functions for $p$ and ${\bar p}$ yields, possibly due to the lack of flavor seperation. It seems that the sea quark contributions in $K^\pm$ and ${\bar p}/p$ are particularly problematic. For instance, as stated in Ref. [@kkp], the $d$ quark in $K^\pm$ does not behave sea-like contrary to expectations. Thus, even a slight difference in handling sea quark contributions could make a large difference in the predictions of ratios for $K^-/K^+$ and ${\bar p}/p$. Our results are based on maintaining expected smallness of sea quark contributions consistently not only in the pion case but also in other hadron cases. Our low ratio for $P_{T} >$ 7 GeV in ${\bar p}/p$ of Fig. \[Fig:ratio\] is consequently due to the dominance of valence contributions. It is a fact that incident heavy ions possess valence quarks, but not antiquarks. In other words, the charge conjugation symmetry is already broken in RHIC environment due to the initial nuclei carry only nucleons (not antinucleons). Since the baryon number (or, equivalently, the quark number) must be conserved throughout the reactions, more protons than antiprotons are expected in the fragmentation region. For more comparisons on the hadron yield ratios, the bottom row of Fig. \[Fig:ratio\] shows $K^{+}/\pi^{+}$, $K^{-}/\pi^{-}$, and $K^{-}/K^{+}$. Although the measured $P_{T}$ range of $K^{\pm}$ is limited, the present estimates are in reasonable agreements with the data. The $K^{+}/\pi^{+}$ and $K^{-}/\pi^{-}$ ratios increase with $P_{T}$, and reach maximum around $P_{T}$ = 3 GeV. If $P_{T}$ further increases, $K^{+}/\pi^{+}$ and $K^{-}/\pi^{-}$ decrease, and level off at some constants. The $P_{T}$ dependence of the $K^{-}/K^{+}$ ratio is very similar to that of $\bar{p}/p$. Especially, we note that the present results on $K^{-}/\pi^{-}$ and $K^{-}/K^{+}$ at high $P_{T}$ region, where the fragmentation is dominant, are quite different from the previous model calculations by Fries [*et al.*]{} [@fries]. As discussed above, even a slight difference in handling sea quark contributions could make a large difference in the predictions of ratios for $K^-/K^+$ as well as $K^-/\pi^-$. The forthcoming RHIC data at high $P_{T}$, e.g., the PHENIX data with newly installed aerogel detector, may help to further clarify the flavor seperation issue in the KKP fragmentation functions [@kkp]. Nuclear Modification Factors {#rcp} ---------------------------- Another important feature of the RHIC data can be identified in the nuclear modification factor $R_{CP}$, which is defined by the $N_{coll}$ scaled central to peripheral yield ratios: $$R_{CP} = {{Yield^{central}/<N_{coll}^{central}>}\over {Yield^{peripheral}/<N_{coll}^{peripheral}>}}. \label{Eq:rcp}$$ The RHIC experiments observed that the $R_{CP}$ parameters of various mesons in $P_{T} >$ 2 GeV in central collisions were suppressed with respect to the $N_{coll}$ scaled $p + p$ and peripheral collision data. Moreover, the suppression in the intermediate transverse momentum region (2 $< P_{T} <$ 4 GeV) was only for mesons, but not for baryons. The experimental $R_{CP}$ parameter of protons in intermediate transverse momentum region is unity, which is completely consistent with $N_{coll}$ scaling. The $R_{CP}$ of $\Lambda$ and $\bar{\Lambda}$ are also close to unity in an intermediate $P_{T}$ region, but somewhat smaller than protons. One of possible explanations for the suppression of hadron yields at high $P_{T}$ and a distinguished behavior of mesons and baryons at the intermediate $P_{T}$ region is the combined effect of recombination and fragmentation. Fig. \[Fig:rcp\] shows the estimated $R_{CP}$ parameters of $\pi$, $p$, and $\Lambda + \bar{\Lambda}$ as a function of $P_{T}$. For pions, we plot charged and neutral pions together, as almost no difference is expected from the present model. In the present model calculation, we scaled the hadron yields due to the fragmentation by the number of binary collisions. For the recombination part, the ratio of $V \displaystyle{\prod_{a}}\gamma_{a}$ in Eq.(\[extend-spectrum\]) for peripheral collisions to that in central collisions was assumed as 40 % of the number of participant ($N_{part}$) ratio: $${{(V\displaystyle{\prod_{a}}\gamma_{a})^{peripheral}}\over {(V\displaystyle{\prod_{a}\gamma_{a}})^{central}}} = c_{1}~~ {{N_{part}^{peripheral}}\over{N_{part}^{central}}}, \label{volume}$$ where $c_1$ = 0.4 fitted by the $R_{CP}$ parameters of $\pi$, $p$, and $\Lambda + \bar{\Lambda}$, simultaneously, by fixing the temperature at 175 MeV as it is almost independent of the collision centrality [@fwang]. Since the quark fugacities are also almost constant except for the very peripheral collisions [@fwang], the factor $c_1$ mostly reflects the effect of volume. As a result, the fact that $c <$ 1 is understandable as the flow velocity is larger for more central collisions. The agreement between the present calculations and the experimental data are reasonable for all considered hadron species. conclusions {#conc} =========== We have presented an extended formalism of the recombination model to analyze the effects from the variation of the hadron’s light-front wavefunctions. Two different functional forms of the light-front wavefunction, which are the Gaussinan form and the power-law form, are tested in detail. The hadron spectra are indeed sensitive to the shape of the wavefunctions. However, when we fit the wavefunction parameters to the physical observables, such as the average charge radius, the final spectra are very similar each other. We discuss our numerical results in comparison with the published RHIC data, especially, from the PHENIX and STAR collaborations. In the hadron spectra, the recombination of thermal partons dominates at the intermediate transverse momentum region between 2 and 5 GeV, and the fragmentation dominates at high $P_{T}$ larger than 5 GeV. The yield ratios and the nuclear modification factors for various hadron species are also estimated. In general, the present model, which combines the recombination and fragmentation processes, are quite consistent with the experimental data. We have also discussed new predictions on $\bar{p}/p$ and $K^{-}/K^{+}$ ratios, including the jet quenching effects to the fragmentation mechanism. This work is supported in part by a grant from the U.S. Department of Energy(DE-FG02-96ER40947) and a brain pool program from the KOFST. CRJ thanks to the faculty and staff at the School of Physics at Seoul National University for the hospitality during the Sabbatical visit while this work was made. We would like to thank B. Mueller, R. Fries, S. Bass and X.-N. 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Particle $\beta^2$ (GeV$^2$) $<r^{2}>$ (fm$^2$) ----------- --------------------- -------------------- $\pi$ 0.825 0.44 $K$ 1.06 0.34 $\phi$ 1.02 0.34 $p$ 0.495 0.76 $\Lambda$ 0.45 0.76 $\Xi$ 0.47 0.76 $\Omega$ 0.48 0.76 : The deduced $\beta^2$ for the Gaussian LF wavefunctions used in this paper and the corresponding average charge radius square for various hadrons. \[Tab:beta2\] ![Shapes of the Gaussian wavefunctions as functions of $x$ and $k_{\perp}$ for different $\beta^{2}$ (normalization is not performed in this figure). The top, middle, and bottom rows represent the wavefunctions for $\pi$, $K$, and $D$, respectively.[]{data-label="Fig:wf_evolve"}](wavefunction_evolve.eps){width="16cm"} ![Comparison of the pion wavefunctions between the Gaussian form and the power-law form. The Gaussian wavefunction with $\beta^{2}$ = 0.825 GeV$^{2}$ (top) and the second order power-law wavefunction with $\alpha^{2}$ = 0.5 GeV$^{2}$ (bottom left) are adjusted to the average charge radius of pions. But the first order power-law wavefunction with $\alpha^{2}$ = 0.825 GeV$^{2}$ (bottom right) is not adjusted.[]{data-label="Fig:wf_comp"}](wavefunction_comp.eps){width="13cm"} ![Comparison of the invariant spectra of $\pi^{+}$ (top) and $p$ (bottom) by the recombination process for various assumptions on the wavefunction parameters. Left panels show the spectra in linear scale in order to emphasize the difference in relatively low $P_T$ region, while right panels show them in log scale for the comparison of the overall shapes. G represents the Gaussian wavefunction used in this paper. PL1 represents the power-law wavefunction whose parameters are adjusted by the known average charge radius, whereas PL2 represents the power-law wavefunction with arbitrary values for the parameters. The coarsely-dotted lines are calculated by a factorized Gaussian form of the LF wavefunctions with $\beta^2 =$ 0.495 GeV$^2$ (see text for details). For the comparison, the wavefunctions used in Ref. [@fries] are also plotted by the dotted lines. []{data-label="Fig:wf_check"}](fdraft_wf_all.eps){width="12cm"} ![Invariant spectra of mesons at midrapidity for central Au + Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. The dashed and dotted lines represent the model calculations from the recombination and the fragmentation, respectively. The solid lines are the sum of two contributions. The open circles are the published data by the PHENIX collaboration [@p1; @p2; @p3], and the solid triangles are those by the STAR collaboration [@s1; @s2].[]{data-label="Fig:meson"}](fdraft_meson.eps){width="13cm"} ![Invariant spectra of baryons at midrapidity for central Au + Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. The dashed and dotted lines represent the model calculations from the recombination and the fragmentation, respectively. The solid lines are the sum of two contributions. The open circles are the published data by the PHENIX collaboration [@p2; @p4] whereas the solid triangles are those by the STAR collaboration [@s1; @s3; @s4]. For a fair comparison between two sets of the data, the published $p$ and $\bar{p}$ spectra by the STAR collaboration [@s1] are scaled by 0.6, which removes the contribution by the weak decays of $\Lambda$, $\Sigma^{0}$, and their antiparticles. Note that the experimental invariant spectra of hyperons are for $\sqrt{s_{NN}}$ = 130 GeV, whereas all model calculations are for $\sqrt{s_{NN}}$ = 200 GeV. []{data-label="Fig:baryon"}](fdraft_baryon.eps){width="16cm"} ![Calculated hadron yield ratios (solid lines) as a function of $P_{T}$ in comparison with the PHENIX data [@p2]. For the comparison, we also show the model calculations by Fries [*et al.*]{}, in $\bar{p}/p$, $K^{-}/\pi^{-}$, and $K^{-}/K^{+}$ by dashed lines [@fries].[]{data-label="Fig:ratio"}](fdraft_ratio.eps){width="13cm"} ![Nuclear modification factors $R_{CP}$ as a function of $P_{T}$ for $\pi$, $p$, and $\Lambda + \bar{\Lambda}$. The solid lines are from the present calculation. []{data-label="Fig:rcp"}](fdraft_rcp.eps){width="8.5cm"} [^1]: The quark fugacity $\gamma_{a}$ in this paper means $\lambda_{a}$ in Eq.(3) of Ref. [@torri] with the saturation factor one.

--- abstract: 'The primary problem in property testing is to decide whether a given function satisfies a certain property, or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the [*uniform*]{} distribution on the domain. This restriction to uniformity is more a matter of convenience than of necessity, and it is important to investigate distances induced by more general distributions. In this paper, we make significant strides in this direction. We give simple and optimal testers for [*bounded derivative properties*]{} over [*arbitrary product distributions*]{}. Bounded derivative properties include fundamental properties such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing.We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show this sum-of-depths is also a lower bound. A fundamental technical contribution of this work is an [*optimal dimension reduction theorem*]{} for all bounded derivative properties, which relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity for the past 15 years, and our theorem is an exponential improvement to the previous best known result. **Don’t expect a pretty answer, but actually get one.** More precisely, we give an intimate connection between bounded derivative property testing and optimal binary search trees. Binary search trees (BST) of a total order are fundamental data structures which have been studied for more than 40 years. We exhibit a tester whose query complexity is precisely the sum of depths of optimal binary search trees of each marginal distribution. This sum is at most the [*entropy*]{} of the product distribution, and can be smaller. The [*same*]{} tester works for all bounded derivative properties! To complement the upper bound, we show that the sum-of-optimal-depths is indeed a lower bound on the query complexity as well. This cements the connection between the two areas. Technically, a fundamental contribution of this work is an [*optimal dimension reduction theorem*]{}. Given any product distribution, and any bounded derivative property, we show that the distance of any function over the whole hypergrid is upper-bounded by the sum of the expected distances of the function restricted to a random line along the various dimensions. Such a theorem has been elusive, even for the special case of monotonicity, for the past 15 years and our theorem is an [*exponential*]{} improvement over what was the best known result. This dimension reduction allows us to focus on testing on the lines where we provide a conceptual insight – we show how [*any*]{} BST can provide a tester for a bounded derivative property. Finally, we use specific BSTs to construct ‘hard functions’ to obtain our lower bounds for the line. We then ‘stitch’ these together to get lower bounds' author: - 'Deeparnab Chakrabarty[^1]' - 'Kashyap Dixit[^2]' - 'Madhav Jha[^3]' - 'C. Seshadhri[^4]' bibliography: - 'derivative-testing.bib' title: | Property Testing on Product Distributions:\ Optimal Testers for Bounded Derivative Properties --- Introduction ============ The field of property testing formalizes the following problem: how many queries are needed to decide if a given function satisfies a certain property? Formally, a *property* $\cP$ is a subset of functions. A tester solves the relaxed membership problem of distinguishing functions in $\cP$ from those ‘far’ from $\cP$. To formalize ‘far’, one requires a notion of *distance*, $\dist(f,g) \in [0,1]$, between functions. A function $f$ is *$\eps$-far from $\cP$* if $\dist(f,g)\geq \eps$ for all functions $g \in \cP$. The notion of distance is central to property testing. The most prevalent notion of distance in the literature is the Hamming distance over the [*uniform*]{} distribution, that is, $\dist(f,g) := \Pr_{x \sim \cU} [f(x) \neq g(x)]$, where $\cU$ is the uniform distribution over the input domain. But the restriction to uniformity is more a matter of convenience than of necessity, and it is important and challenging to investigate distances induced by more general distributions. This was already underscored in the seminal work of Goldreich et. al. [@GGR98] who “stressed that the generalization of property testing to arbitrary distributions" is important for applications. Nevertheless, a vast majority of results in property testing have focused solely on the uniform distribution. In this paper we investigate property testing of functions defined over the hypergrid $[n]^d$ with respect to distances induced by [*arbitrary product distributions*]{}. Product distributions over this domain form a natural subclass of general distributions where each individual coordinate is an arbitrary distribution independent of the other coordinates. They arise in many applications; the following are a couple of concrete ones. [*Differential privacy:*]{} Recent work on testing differential privacy [@DiJh+13] involve product distributions over the domain $[n]^d$. In this application, each domain point represents a database and each coordinate is a single individual’s data. A distribution on databases is given by independent priors on each individual. The goal in [@DiJh+13] is to distinguish private mechanisms from those that aren’t private on ‘typical’ databases. [*Random testing of hardware:*]{} Given an actual silicon implementation of a circuit, it is standard practice for engineers to test it on a set of random instances. Coordinates represent entities like memory addresses, data, control flow bits, etc. One chooses an independent but not identical distribution over each input to generate realistic set of test cases. There are specific commands in hardware languages like VHDL and Systemverilog [@VHDL; @verilog] that specify such coordinate-wise distributions. From a theoretical perspective, the study of property testing over non-uniform distributions has mostly led to work on specific problems. For uniform distributions, it is known that broad classes of algebraic and graphic properties are testable [@AS05; @AENS06; @KaSu08; @BSS]. But little is known in this direction even for product distributions. One reason for this may be aesthetics: a priori, one doesn’t expect a succinct, beautiful answer for testing over an arbitrary product distribution. In this paper we make significant strides in property testing under arbitrary product distributions. We give [*simple*]{}, [*optimal*]{} testers for the class of *bounded derivative* properties. This class contains (and is inspired by) the properties of monotonicity and Lipschitz continuity, which are of special interest in property testing. In fact, the [*same*]{} tester works for all such properties. Furthermore, our ‘answer’ is aesthetically pleasing: the optimal query complexity with respect to a product distribution is the sum of optimal binary search tree depths over the marginals. In particular, our results resolve a number of open problems in monotonicity testing, and subsume all previous upper and lower bounds over any product, including the uniform, distribution. #### Previous Work. We set some context for our work. The property of monotonicity is simple. There is a natural coordinate-wise partial order over $[n]^d$. For a monotone function, $x \prec y$ implies $f(x) \leq f(y)$. Monotonicity is one of the most well-studied properties in the area [@EKK+00; @GGLRS00; @DGLRRS99; @LR01; @FLNRRS02; @AC04; @E04; @HK04; @PRR04; @ACCL07; @BRW05; @BGJRW09; @BCG+10; @BBM11; @ChSe13; @ChSe13-2; @BlJh+13]. A function is $c$-Lipschitz continuous if for all $x,y$, $|f(x) - f(y)| \leq c\|x-y\|_1$. Lipschitz continuity is a fundamental mathematical property with applications to differential privacy and program robustness. The study of Lipschitz continuity in property testing is more recent [@JR11; @AJMS12; @ChSe13; @DiJh+13; @BlJh+13]. With the exception of [@HalevyK07; @HK04; @AC04; @DiJh+13], all the previous works are in the uniform distribution setting, for which the story is mostly clear: there is an $O({\eps}^{-1}d\log n)$-query tester for both properties [@ChSe13], and this is optimal for monotonicity [@ChSe13-2]. For Lipschitz continuity, an $\Omega(d\log n)$ non-adaptive lower bound has been proved [@BlJh+13] recently. For general product distributions, the story has been far less clear. Ailon and Chazelle [@AC04] design an $O(2^dH/\eps)$-query tester for monotonicity over product distributions, where $H$ is the Shannon entropy of the distribution. This work connects property testing with information theory, and the authors explicitly ask whether the entropy is the “correct answer”. There are no non-trivial lower bounds known for arbitrary product distributions. For Lipschitz continuity, no upper or lower bounds are known for general hypergrids, although an $O(d^2)$-query tester is known for the hypercube ($\{0,1\}^d$) domain [@DiJh+13]. Halevy and Kushilevitz [@HK04; @HalevyK07] study monotonicity testing in the *distribution-free setting*, where the tester does not know the input distribution but has access to random samples. Pertinent to us, they show a lower bound of $\Omega(2^d)$ for monotonicity testing over *arbitrary distributions*; for product distributions, Ailon and Chazelle [@AC04] give an $O(\eps^{-1}d2^d\log n)$-query distribution-free tester. Bounded Derivative Properties ----------------------------- To describe the class of bounded derivative properties, we first set some notation. For an integer $k$, we use $[k]$ to denote the set $\{1,2,\ldots, k\}$. Consider a function $f:[n]^d \mapsto \R$ and a dimension $r \in [d]$. Define $\partial_r f(x) := f(x+{{\bf e}}_r) - f(x)$, where ${{\bf e}}_r$ is the unit vector in the $r$th dimension ($\partial_r f$ is defined only on $x$ with $x_r < n$.). \[def:bound\] An ordered set ${\mathbf{B}}$ of $2d$ functions $l_1, u_1, l_2, u_2, \ldots, l_d, u_d: [n-1] \mapsto \R$ is called a *bounding family* if for all $r \in [d]$ and $y \in [n-1]$, $l_r(y) < u_r(y)$. Let ${\mathbf{B}}$ be a bounding family of functions. The property of being *${\mathbf{B}}$-derivative bounded*, denoted as $\cP({\mathbf{B}})$, is the set of functions $f:[n]^d \mapsto \R$ such that: for all $r \in [d]$ and $x \in [n]^d$, $$\label{eq:defnbnd} l_r(x_r) \leq \partial_r f(x) \leq u_r(x_r).$$ This means the $r$th-partial derivative of $f$ is bounded by quantities that only depend on the $r$th coordinate. Note that this dependence is completely arbitrary, and different dimensions can have completely different bounds. This forms a rich class of properties which includes monotonicity and $c$-Lipschitz continuity. To get monotonicity, simply set $l_r(y) = 0$ and $u_r(y) = \infty$ for all $r$. To get $c$-Lipschitz continuity, set $l_r(y) = -c$ and $u_r(y) = +c$ for all $r$. The class also includes the property demanding monotonicity for some (fixed) coordinates and the $c$-Lipschitz continuity for others; and the non-uniform Lipschitz property that demands different Lipschitz constants for different coordinates. \[def:tester\] Fix a bounding family ${\mathbf{B}}$ and product distribution $\cD = \prod_{r\leq d}\cD_r$. Define $\dist_\cD(f,g) = \Pr_{x \sim \cD}[f(x) \neq g(x)]$. A property tester for $\cP({\mathbf{B}})$ with respect to $\cD$ takes as input proximity parameter $\eps > 0$ and has query access to function $f$. If $f \in \cP({\mathbf{B}})$, the tester accepts with probability $> 2/3$. If $\dist_\cD(f,\cP({\mathbf{B}})) > \eps$, the tester rejects with probability $> 2/3$. Main Results ------------ Our primary result is a property tester for all bounded-derivative properties over any product distribution. The formal theorem requires some definitions of search trees. Consider any binary search tree (BST) $T$ over the universe $[n]$, and let the depth of a node denote the number of [*edges*]{} from it to the root. For a distribution $\cD_r$ over $[n]$, the [*optimal BST for $\cD_r$*]{} is the BST minimizing the expected depth of vertices drawn from $\cD_r$. Let $\Delta^*(\cD_r)$ be this optimal depth: a classic dynamic programming solution finds this optimal tree [@Knuthvol3; @YAO82] in polynomial time. Given a product distribution $\cD = \prod_{r \leq d}\cD_r$, we abuse notation and let $\Delta^*(\cD)$ denote the sum $\sum_{r=1}^d \Delta^*(\cD_r)$. [theorem]{}[distknown]{} \[thm:distknown-weak\]\[thm:distknown\] [**\[Main upper bound\]**]{} Consider functions $f:[n]^d \mapsto \R$. Let ${\mathbf{B}}$ be a bounding family and $\cD$ be a product distribution. There is a tester for $\cP({\mathbf{B}})$ w.r.t. $\cD$ making $100\eps^{-1}\Delta^*(\cD)$ queries. The tester is non-adaptive with one-sided error, that is, the queries don’t depend on the answers, and the tester always accepts functions satisfying the property. Furthermore, the [*same*]{} tester works for all bounding families, that is, the set of queries made by the tester doesn’t depend on ${\mathbf{B}}$. Interestingly, the “worst" distribution is the uniform distribution, where $\Delta^*(\cD)$ is maximized to $\Theta(d\log n)$. We remark that the class of bounded derivative properties was not known to be testable even under uniform distributions. Results were known [@ChSe13] (only under the uniform distribution) for the subclass where all $l_r$ (and $u_r$) are the same, constant function. To give perspective on the above result, it is instructive to focus on say just monotonicity (one can repeat this for Lipschitz). Let $H(\cD)$ denote the Shannon entropy of distribution $\cD$ over the hypergrid. It is well-known that $\Delta^*(\cD_r) \leq H(\cD_r)$ (see [@Melhorn75] for a proof), so $\Delta^*(\cD)\leq H(\cD)$ for product distribution $\cD$. [corollary]{}[disthyp]{}\[cor:ub-hg\]\[thm:ub-hg\] Consider functions $f:[n]^d \mapsto \R$. Monotonicity testing over a product distribution $\cD$ can be done with $100 H(\cD)/\eps$ queries. This is an *exponential* improvement over the previous best result of Ailon and Chazelle [@AC04], who give a monotonicity tester with query complexity $O(2^d H(\cD)/\eps)$. Observe that for uniform distributions, $H(\cD) = \Theta(d\log n)$, and therefore the above result subsumes the optimal testers of [@ChSe13]. Now consider the monotonicity testing over the boolean hypercube. [corollary]{}[distknowncube]{}\[cor:ub-hc\]\[cor:ub-cube\] Consider functions $f:\{0,1\}^d \mapsto \R$. Monotonicity testing over any product distribution $\cD = \prod_{r=1}^d \cD_r$, where each $\cD_r = (\mu_r,1-\mu_r)$, can be done with $100\eps^{-1}\sum_{r=1}^d \min(\mu_r,1-\mu_r)$ queries. Given that monotonicity testing over the hypercube has received much attention [@GGLRS00; @DGLRRS99; @LR01; @FLNRRS02; @BBM11; @ChSe13; @ChSe13-2], it is somewhat surprising that *nothing non-trivial* was known even over the $p$-biased distribution for $p\neq 1/2$; our result implies an $O({\eps}^{-1}pd)$-query tester. The above corollary also asserts that entropy of a distribution [*doesn’t capture the complexity of monotonicity testing*]{} since the entropy, $\sum_r \mu_r\log(1/\mu_r) + (1-\mu_r)\log(1/(1-\mu_r))$, can be larger than the query complexity described above by a logarithmic factor. For example, if each $\mu_r = 1/\sqrt{d}$, the tester of requires $O(\sqrt{d}/\eps)$ queries, while $H(\cD) = \Theta(\sqrt{d}\log d)$. We complement with a matching lower bound, cementing the connection between testing of bounded-derivative properties and optimal search tree depths. This requires a technical definition of stable distributions, which is necessary for the lower bound. To see this consider a distribution $\cD$ for which there exists a product distribution $\cD'$ such that $||\cD'-\cD||_{\TV} \leq {\eps}/2$ but $\Delta^*(\cD') \ll \Delta^*(\cD)$. One could simply apply with $\cD'$ to obtain a tester with a much better query complexity than $\Delta^*(\cD)$. $\cD$ is called $({\eps}',\rho)$-stable if $\|\cD-\cD\|\leq {\eps}'$ implies $\Delta^*(\cD')\geq\rho\Delta^*(\cD)$, for any product distribution $\cD'$. [theorem]{}[mainlowerbound]{} [**\[Main lower bound\]**]{} \[thm:the-lower-bound\] For any parameter ${\eps}$, there exists ${\eps}' = \Theta({\eps})$ such that for any bounding family ${\mathbf{B}}$ and $(\eps',\rho)$-stable, product distribution $\cD$, any (even adaptive, two-sided) tester for $\cP({\mathbf{B}})$ w.r.t. $\cD$ with proximity parameter $\eps$ requires $\Omega(\rho\Delta^*(\cD))$ queries. This lower bound is new even for monotonicity testing over one dimension. Ailon and Chazelle [@AC04] explicitly ask for lower bounds for monotonicity testing for domain $[n]$ over arbitrary distributions. Our upper and lower bounds completely resolve this problem. For Lipschitz testing, the state of the art was a *non-adaptive* lower bound of $\Omega(d\log n)$ for the uniform distribution [@BlJh+13]. Since the uniform distribution is stable, the previous theorem implies an optimal $\Omega(d\log n)$ lower bound even for adaptive, two-sided testers over the uniform distribution. The previous upper bounds are in the setting where the tester knows the distribution $\cD$. In the *distribution-free* setting, the tester only gets random samples from $\cD$ although it is free to query any point of the domain. As a byproduct of our approach, we also get results for this setting. The previous best bound was an $O(\eps^{-1}d2^{d}\log n)$ query tester [@AC04]. [theorem]{}[distfree]{} \[thm:main-free\] Consider functions $f:[n]^d \mapsto \R$. There is a distribution-free (non-adaptive, one-sided) tester for $\cP({\mathbf{B}})$ w.r.t. $\cD$ making $100\eps^{-1}d\log n$ queries. Technical highlights {#sec:ourtechs} -------------------- [**Optimal dimension reduction.**]{} The main engine running the upper bounds is an optimal dimension reduction theorem. Focus on just the uniform distribution. Given $f:[n]^d \mapsto \R$ that is $\eps$-far from $\cP({\mathbf{B}})$, what is the expected distance of the function restricted to a uniform random line in $[n]^d$? This natural combinatorial question has been at the heart of various monotonicity testing results [@GGLRS00; @DGLRRS99; @AC04; @HK04]. The best known bounds are that this expected distance is at least $\eps/(d2^d)$ [@AC04; @HK04]. Weaker results are known for the Lipschitz property [@JR11; @AJMS12]. We given an optimal resolution (up to constant factors) to this problem not only for the uniform distribution, but for any arbitrary product distribution, and for any bounded derivative property. In $[n]^d$, an $r$-line is a combinatorial line parallel to the $r$-axis. Fix some bounding family ${\mathbf{B}}$ and product distribution $\cD = \prod_r \cD_r$. Note that $\cD_{-r} = \prod_{i \neq r} \cD_i$ is a distribution on $r$-lines. If we restrict $f$ to an $r$-line $\ell$, we get a function $f|_\ell:[n] \mapsto \R$. It is meaningful to look at the distance of $f|_\ell$ to $\cP({\mathbf{B}})$ (though this only involves the bounds of $l_r, u_r \in {\mathbf{B}}$). Let $\dist^r_\cD(f,\cP({\mathbf{B}})) := \EX_{\ell \sim \cD_{-r}} [\dist_{\cD_r}(f|_\ell,\cP({\mathbf{B}}))]$. [**\[Optimal Dimension Reduction\]**]{}\[thm:dimred\] Fix bounding family ${\mathbf{B}}$ and product distribution $\cD$. For any function $f$, $$\sum_{r=1}^d \dist^r_\cD(f,\cP({\mathbf{B}})) \geq \dist_\cD(f,\cP({\mathbf{B}}))/4.$$ Let us give a short synopsis of previous methods used to tackle the case of monotonicity in the uniform distribution case. For brevity’s sake, let ${\eps}^r_f$ denote $\dist^r_\cU(f,\MON)$ and ${\eps}_f$ denote $\dist_\cU(f,\MON)$. That is, ${\eps}^r_fn^d$ modifications makes the function monotone along the $r$-dimension, and the theorem above states that $4\sum_r{\eps}^r_fn^d$ modifications suffice to make the whole function monotone. Either explicitly or implicitly, previous attempts have taken a constructive approach: they use the modifications along the $r$th dimensions to correct the whole function. Although in principle a good idea, a bottleneck to the above approach is that correcting the function along one dimension may potentially introduce significantly larger errors along other dimensions. Thus, one can’t just “add up” the corrections in a naive manner. The process is even more daunting when one tries this approach for the Lipschitz property. Our approach is completely different, and is ‘non-constructive’, and looks at all bounded-derivative properties in a uniform manner. We begin by proving for $\cP({\mathbf{B}})$ over the uniform distribution. The starting point is to consider a weighted violation graph $G$, where any two domains point forming a violation to $\cP({\mathbf{B}})$ are connected (the weight is a “magnitude" of violation). It is well-known that the size of a maximum matching $M$ in $G$ is at least $\eps_f n^d/2$. The main insight is to use different matchings to get handles on the distance ${\eps}^r_f$ rather than using modifications that correct the function. More precisely, we construct a sequence of special matchings $M = M_0,M_1,\ldots, M_d = \emptyset$, such that the drop in size $|M_{r-1}| - |M_r|$ is at most $2{\eps}^r_fn^d$, which proves the above theorem. This requires structural properties on the $M_r$’s proven using the *alternating path machinery* developed in [@ChSe13]. What about a general product distribution $\cD$? Suppose we ‘stretch’ every point in every direction proportional to its marginal. This leads to a ‘bloated’ hypergrid $[N]^d$ where each point in the original hypergrid corresponds to a high-dimensional cuboid. By the obvious association of function values, one obtains a $\fext:[N]^d \mapsto \R$. If $\cP({\mathbf{B}})$ is monotonicity, then it is not hard to show that $\dist_\cD(f) = \dist_\cU(\fext)$. So we can apply dimension reduction for $\fext$ over the uniform distribution and map it back to $f$ over $\cD$. However, such an argument breaks down for Lipschitz (let alone general ${\mathbf{B}}$), since $\dist_\cU(f')$ can be much smaller than $\dist_\cD(f)$. The optimal fix for $\fext$ could perform non-trivial changes within the cuboidal regions, and this cannot be mapped back to a fix for the original $f$. This is where the generality of the bounded-derivative properties saves the day. For any ${\mathbf{B}}$ and $\cD$, we can define a new bounding family ${\mathbf{B}}_{\tt ext}$ over $[N]^d$, such that $\dist_\cD(f,\cP({\mathbf{B}})) = \dist_\cU(\fext,\cP({\mathbf{B}}_{\tt ext}))$. Now, dimension reduction is applied to $\fext$ for $\cP({\mathbf{B}}_{\tt ext})$ over $\cU$ and translated back to the original setting. [**Search trees and monotonicity.**]{} An appealing aspect of our results is the tight connection between optimal search trees over product distributions to bounded-derivative properties. The dimension reduction lemma allows us (for the upper bounds) to focus on just the line domain $[n]$. For monotonicity testing on $[n]$ over an arbitrary distribution $\cD$, Halevy and Kushilevitz gave an $O(\eps^{-1}\log n)$-query distribution free tester [@HK04], and Ailon and Chazelle gave an $O(\eps^{-1}H(\cD))$-query tester [@AC04]. Pretty much every single result for monotonicity testing on $[n]$ involves some analogue of binary search [@EKK+00; @BRW05; @ACCL07; @PRR04; @HK04; @AC04; @BGJRW09]. But we make this connection extremely precise. We show that [*any*]{} binary search tree can be used to get a tester with respect to an arbitrary distribution, whose expected query complexity is the expected depth of the tree with respect to the distribution. This argument is extremely simple in hindsight, but it is a significant conceptual insight. Firstly, it greatly simplifies earlier results – using the completely balanced BST, we get an $O(\eps^{-1}\log n)$-distribution free tester; with the optimal BST, we get $O(\eps^{-1}H(\cD))$-queries. The BST tester along with the dimension reduction, provides a tester for $[n]^d$ whose running time can be better than $H(\cD)$ (especially for the hypercube). Most importantly, optimal BSTs are a crucial ingredient for our lower bound construction. [**Lower Bounds for Product Distributions.**]{} The first step to general lower bounds is a simple reduction from monotonicity testing to any bounded-derivative property. Again, the reduction may seem trivial in hindsight, but note that special sophisticated constructions were used for existing Lipschitz lower bounds [@JR11; @BlJh+13]. For monotonicity, we use the framework developed in [@E04; @ChSe13-2] that allows us to focus on comparison based testers. The lower bound for $[n]$ uses a convenient near-optimal BST. For each level of this tree we construct a ‘hard’ non-monotone function, leading to (roughly) $\Delta^*(\cD)$ such functions in case of stable distributions. These functions have violations to monotonicity lying in ‘different regions’ of the line, and any bonafide tester must make a different query to catch each function. In going to higher dimensions, we face a significant technical hurdle. The line lower bound easily generalizes to the hypergrid [*if each marginal distribution is individually stable.*]{} However, this may not be the case – there are stable product distributions whose marginals are unstable. As a result, each dimension may give ‘hard’ functions with very small distance. Our main technical contribution is to show how to [*aggregate*]{} functions from various dimensions together to obtain hard functions for the hypergrid in such a way that the distances add up. This is rather delicate, and is perhaps the most technical portion of this paper. In summary, we show that for stable distributions, the total search-tree depth is indeed the lower bound for testing monotonicity, and via the reduction mentioned above, for any bounded-derivative property. Other Related Work. {#sec:other_related_work} ------------------- Monotonicity testing has a long history, and we merely point the reader to the discussions in [@ChSe13; @ChSe13-2]. The work on testing over non-uniform distributions was performed in [@HK04; @HalevyK07; @AC04], the details of which have been provided in the previous section. Goldreich et al. [@GGR98] had already posed the question of testing properties of functions over non-uniform distributions, and obtain some results for dense graph properties. A serious study of the role of distributions was undertaken by Halevy and Kushilevitz [@HalevyK07; @HK04; @HalevyK05; @HalevyK08], who formalized the concept of distribution-free testing. (Refer to Halevy’s thesis [@Hal-thesis] for a comprehensive study.) Kopparty and Saraf extend the classic linearity test to classes of distributions, including product distributions [@KoSa08]. Glasner and Servedio [@GlasnerS09] and Dolev and Ron [@DolevR11] give various upper and lower bounds for distribution-free testers for various properties over ${\{0,1\}}^n$. Non-uniform distributions were also considered recently in the works of Balcan et al. [@BalcanBBY12] and [@GoldreichR13] which constrain the queries that can be made by the tester to samples drawn from the distribution. Recent work of Berman et al. [@BeRaYa14] introduces property testing over $\ell_p$-distances. We believe work along these lines studying richer notions of distance is critical to the growth of property testing. #### Note to the reader. The paper is rather long, although, we hope the extended introduction above will allow the reader to choose the order in which to peruse the paper. We give a brief outline of remainder. In , we define a particular [*quasi-metric*]{} corresponding to a bounding family ${\mathbf{B}}$ and give an equivalent definition of the bounded-derivative property with respect to it. This definition is convenient and will be the one used for the rest of the paper. This section must be read next. The dimension reduction theorem is presented in its full glory in . In , we describe the tester when the domain is just the line, and the easy generalization to the hypergrid via dimension reduction is presented in . For lower bounds, we prove the reduction to monotonicity in , and describe the approach to montonicity lower bounds in . The hard families for the line is given in , for the hypercube in , and the general hypergrid lowerbound is described in . Quasimetric induced by a Bounding Family {#sec:abstract} ======================================== It is convenient to abstract out $\cP({\mathbf{B}})$ in terms of a *metric-bounded property*. Such ideas was used in [@ChSe13] to give a unified proof for monotonicity and Lipschitz for the uniform distribution. The treatment here is much more general. We define a quasimetric depending on ${\mathbf{B}}$ denoted by $\pdi(x,y)$. \[def:dist\] Given bounding family ${\mathbf{B}}$, construct the weighted directed hypergrid $[n]^d$, where all adjacent pairs are connected by two edges in opposite directions. The weight of $(x+{{\bf e}}_r,x)$ is $u_r(x_r)$ and the weight of $(x,x+{{\bf e}}_r)$ is $-l_r(x_r)$. $\pdi(x,y)$ is the shortest path weight from $x$ to $y$. Note that $\pdi$ is asymmetric, can take negative values, and $\pdi(x,y) = 0$ does not necessarily imply $x=y$. For these reasons, it is really a possibly-negative-pseudo-quasi-metric, although we will refer to it simply as a metric in the remainder of the paper. Since ${\mathbf{B}}$ is a bounding family, any cycle in the $[n]^d$ digraph has positive weight, and $\pdi(x,y)$ is well-defined. Therefore, a shortest path from $x$ to $y$ is given by the rectilinear path obtained by decreasing the coordinates $r$ with $x_r > y_r$ and increasing the coordinates $r$ with $x_r< y_r$. A simple calculation yields $$\label{eq:supergeneralLip} \pdi(x,y) := \sum_{r:x_r > y_r} \sum_{t = y_r}^{x_r-1}\! u_r(t) - \sum_{r:x_r < y_r}\sum_{t = x_r}^{y_r-1}\!l_r(t)$$ If a function $f\in \cP({\mathbf{B}})$, then applying on every edge of the path described above (the upper bound when we decrement a coordinate and the lower bound when we increment a coordinate), we get $f(x) - f(y) \leq \pdi(x,y)$ for any pair $(x,y)$. Conversely, if $\forall x,y, f(x) - f(y) \leq \pdi(x,y)$, then considering neighboring pairs gives $f\in \cP({\mathbf{B}})$. This argument is encapsulated in the following lemma. \[lem:dist\] $f \in \cP({\mathbf{B}})$ iff $~\forall x, y \in [n]^d$, $f(x) - f(y) \leq \pdi(x,y)$. When $\cP({\mathbf{B}})$ is monotonicity, $\pdi(x,y) = 0$ if $x \prec y$ and $\infty$ otherwise. For the $c$-Lipschitz property, $\pdi(x,y) = c\|x-y\|_1$. The salient properties of $\pdi(x,y)$ are documented below and can be easily checked. \[lem:dist-prop\] $\pdi(x,y)$ satisfies the following properties. [*(Triangle Inequality.)*]{} For any $x,y,z$, $\pdi(x,z) \leq \pdi(x,y) + \pdi(y,z)$. [*(Linearity.)*]{} If $x,y,z$ are such that for every $1\leq r\leq d$, either $x_r \leq y_r \leq z_r$ or $x_r \geq y_r \geq z_r$, then $\pdi(x,z) = \pdi(x,y) + \pdi(y,z)$. [*(Projection.)*]{} Fix any dimension $r$. Let $x,y$ be two points with $x_r = y_r$. Let $x'$ and $y'$ be the projection of $x, y$ onto some other $r$-hyperplane. That is, $x'_r = y'_r$, and $x'_j = x_j$, $y'_j = y_j$ for $j\neq r$. Then, $\pdi(x,y) = \pdi(x',y')$ and $\pdi(x,x') = \pdi(y,y')$. $\pdi(x,x) = 0$ follows since the RHS of is empty. Triangle inequality holds because $\pdi(x,y)$ is a shortest path weight. Linearity follows by noting $\sum_{t = y_r}^{x_r - 1} u_r(t) = \sum_{t = y_r}^{z_r - 1} u_r(t) + \sum_{t = z_r}^{x_r - 1} u_r(t)$. For projection, note that if $x_r = y_r$, the RHS of has no term corresponding to $r$. Thus, $\pdi(x,y) = \pdi(x',y')$. Suppose $x'_r > x_r$. Then, $\pdi(x,x') = \sum_{t = x_r}^{x'_r} u_i(t)$ $= \pdi(y,y')$. A similar proof holds when $x'_r < x_r$. Henceforth, all we need is and . We will interchangably use the terms $\cP({\mathbf{B}})$ and $\cP(\pdi)$ where $\pdi$ is as defined in . In fact, since ${\mathbf{B}}$ and therefore $\pdi$ will be fixed in most of our discussion, we will simply use $\cP$ including the parametrization wherever necessary. The violation graph of a function $f$ with respect to property $\cPdd$, denoted as $\VG(f,\cPdd)$, has $[n]^d$ as vertices, and edge $(x,y)$ if it forms a violation to $\cPdd$, that is either $f(x) - f(y) > \pdi(x,y)$ or $f(y) - f(x) > \pdi(y,x)$. The triangle inequality of $\pdi$ suffices to prove the following version of a classic lemma [@FLNRRS02] relating the distance of a function to $\cP$ to the vertex cover of the violation graph. \[lem:characterization\]\[lem:matchmatch\] For any distribution $\cD$ on $[n]^d$, any bounded-derivative property $\cP$, and any function $f$, $\dist_\cD(f,\cP) = \min_X \mu_\cD(X)$ where the minimum is over all vertex covers of $\VG(f,\cP)$. Thus, if $M$ is [*any*]{} maximal matching in $\VG(f,\cP)$, then for the [*uniform distribution*]{}, $|M|\geq \dist_\cU(f,\cP)n^d/2$. The Dimension Reduction Theorem {#sec:dimred} =============================== For any combinatorial line $\ell$ in $[n]^d$, $f|_\ell:[n] \mapsto \R$ is $f$ restricted to $\ell$. It is natural to talk of $\cP$ for any restriction of $f$, so $\dist_{\cD_r}({{f}_{|\ell}}, \cP)$ is well-defined for any $r$-line $\ell$. For any $1\leq r\leq d$, define the $r$-distance of the function: $$\dist^r_\cD(f,\cP):=\Exp_{\ell \sim \cD_{-r}} [\dist_{\cD_r}({{f}_{|\ell}}, \cP)]$$ Call a function $f$ $r$-good if there are no violations along $r$-lines, that is, for any $x$ and $y$ on the same $r$-line, we have $f(x) - f(y) \leq \pdi(x,y)$. Observe that $\dist^r_\cD(f,\cP)$ is the minimum $\mu_\cD$-mass of points on which $f$ needs to be modified to make it $r$-good. The following is the optimal dimension reduction theorem which connects the $r$-distances to the real distance. \[thm:dimred\] For any function $f$, any bounded-derivative property $\cP$, and any product distribution $\cD = \prod_{1\leq r\leq d}\cD_i$, [$$\sum_{r=1}^d \dist^r_\cD(f,\cP) \geq \dist_\cD(f,\cP)/4.$$]{} (It can be easily shown that $\sum_{r=1}^d \dist^r_\cD(f,\cP) \leq \dist_\cD(f,\cP)$, by simply putting the same 1D function of all, say, $1$-lines.) We first prove the above theorem for the uniform distribution. Recall the violation graph $\VG(f,\cP)$ whose edges are violation to $\cP$. We define weights on the edges $(x,y)$. $$\label{eq:wtdefn} w(x,y) := \max(f(x) - f(y) - \pdi(x,y), f(y) -f(x) - \pdi(y,x))$$ Note that $w(x,y) > 0$ for all edges in the violation graph. Let $M$ be a maximum weight matching of minimum cardinality (MWmC). (Introduce an arbitrary tie-breaking rule to ensure this is unique.) A pair $(x,y)\in M$ is an [*$r$-cross pair*]{} if $x_r \neq y_r$. The following theorem [(proof defered to )]{} establishes the crucial structural result about these MWmC matchings in violated graphs of $r$-good functions. [theorem]{}[noviol]{} \[thm:noviol\] Let $f$ be an $r$-good function. Then there exists an MWmC matching $M$ in $\VG(f,\cP)$ with no $r$-cross pairs. We proceed with the proof of over the uniform distribution starting with some definitions. Given an $r$-dimensional vector $\a\in [n]^r$, the [*$a$-slice*]{} is ${S}_\a := \{x\in [n]^d: x_j = \a_j, ~1\leq j\leq r\}$. Each $\a$-slice is a $(d-r)$-dimensional hypergrid, and the various $\a$-slices for $\a\in[n]^r$ partition $[n]^d$. Let ${{f}_{|\a}}$ denote the restriction of $f$ to the slice ${S}_\a$. For two functions $f,g$ we use ${\Delta}(f,g) := |\{x: f(x) \neq g(x)\}| = \dist_\cU(f,g)\cdot n^d$. The following claim relates the sizes of MWmC matchings to ${\Delta}(f,g)$. \[clm:matchdiff\] Let $f,g: [n]^d \mapsto \R$. Let $M$ and $N$ be the MWmC matchings in the violation graphs for $f$ and $g$, respectively. Then, $||M|-|N|| \leq {\Delta}(f,g)$. The symmetric difference of $M$ and $N$ is a collection of alternating paths and cycles. $||M|-|N||$ is at most the number of alternating paths. Each alternating path must contain a point at which $f$ and $g$ differ, for otherwise we can improve either $M$ or $N$, either in weight or cardinality. Define a sequence of $d+1$ matchings $(M_0,M_1,\ldots,M_d)$ in $\VG(f,\cP)$ in non-increasing order of cardinality as follows. For $0 \leq r\leq d$, $M_r$ is the MWmC matching in $\VG(f,\cP)$ among matchings that [*do not contain any $i$-cross pairs for $1\leq i\leq r$*]{}. By , we have $|M_0| \geq \dist_\cU(f,\cP)n^d/2$. The last matching $M_d$ is empty and thus has cardinality $0$. \[lem:mi\] For all $1\leq r\leq d$, we have $|M_{r-1}| -|M_{r}| \leq 2\cdot \dist^r_\cU(f,\cP)\cdot n^d$. Adding the inequalities in the statement of for all $r$, we get $\dist_\cU(f,\cP)n^d/2 \leq|M_0| - |M_d| \leq 2\sum_{r=1}^d \dist^r_\cU(f,\cP)\cdot n^d.$ This completes the proof of for the uniform distribution. Now we prove . Since $M_{r-1}$ has no $j$-cross pairs for $1\leq j\leq r-1$, all pairs of $M_{r-1}$ have both endpoints in the same slice ${S}_\a$ for some $\a\in [n]^{r-1}$. Thus, $M_{r-1}$ partitions into sub-matchings in each ${S}_\a$. Let $M^\a_{r-1}$ be the pairs of $M_{r-1}$ with both endpoints in slice ${S}_\a$, so $|M_{r-1}| = \sum_{\a\in [n]^{r-1}} |M^\a_{r-1}|$. Similarly, $M^\a_r$ is defined. Since $M_r$ has no $r$-cross pairs either, $\forall \a\in [n]^{r-1}$, $|M^\a_{r}| = \sum_{i=1}^n |M^{(\a\circ i)}_{r}|$, where $(\a\circ i)$ is the $r$-dimensional vector obtained by concatenating $i$ to the end of $\a$. Observe that for any $\a\in [n]^{r-1}$, $M^\a_{r-1}$ is an MWmC matching in ${S}_\a$ w.r.t. ${{f}_{|\a}}$. Furthermore, for any $i\in [n]$, $M^{(\a\circ i)}_r$ is an MWmC matching in ${S}_{(\a\circ i)}$ w.r.t. ${{f}_{|(\a\circ i)}}$. Let $\fii$ be the closest function to $f$ with no violations along dimension $r$. By definition, ${\Delta}(f,\fii) = \dist^r(f,\cP)\cdot n^d$. Now comes the crucial part of the proof. Fix $\a\in [n]^{r-1}$ and focus on the $\a$-slice ${S}_\a$. Since $\fii$ has no violations along the $r$-lines, neither does ${{f^{(r)}}_{|\a}}$. By , there exists an MWmC matching $N^\a$ in ${S}_\a$ w.r.t. ${{f^{(r)}}_{|\a}}$ which has no $r$-cross pairs. Therefore, $N^\a$ partitions as $N^\a = \bigcup_{i=1}^n N^{(\a\circ i)}$. Furthermore, each matching $N^{(\a\circ i)}$ is an MWmC matching in ${S}_{(\a\circ i)}$ with respect to the weights corresponding to the function $f^{(r)}_{|(\a\circ i)}$. Since $M^\a_{r-1}$ is an MWmC matching w.r.t. ${{f}_{|\a}}$ and $N^\a$ is an MWmC matching w.r.t. $\fia$ in ${S}_\a$, gives $$\label{eq:3} |N^\a| \geq |M^\a_{r-1}| - {\Delta}({{f}_{|\a}},{{f^{(r)}}_{|\a}})$$ Since $M^{(\a\circ i)}_r$ is an MWmC matching w.r.t. $f_{|(\a\circ i)}$ and $N^{(\a\circ j)}$ is an MWmC matching w.r.t. $f^{(r)}_{|(\a\circ i)}$ in ${S}_{(\a\circ i)}$, gives us $|M^{(\a\circ i)}_r| \geq |N^{(\a\circ i)}| - {\Delta}({{f}_{|(\a\circ i)}}, {{f^{(r)}}_{|(\a\circ i)}})$. Summing over all $1\leq i\leq n$, $$\label{eq:4} |M^\a_r| \geq |N^\a| - {\Delta}({{f}_{|\a}},{{f^{(r)}}_{|\a}})$$ Adding , over all $\a\in [n]^{r-1}$, $|M_r| \geq |M_{r-1}| - 2\sum_{\a\in [n]^{r-1}}{\Delta}({{f}_{|\a}},{{f^{(r)}}_{|\a}}) $ $= |M_{r-1}| - 2\cdot\dist^r(f,\cP)\cdot n^d$. Reducing from arbitrary product distributions {#sec:uniftogen} --------------------------------------------- [We reduce arbitrary product distributions to uniform distributions on what we call the bloated hypergrid.]{} Assume without loss of generality that all $\mu_{\cD_r}(j) = q_r(j)/N$, for some integers $q_r(j)$ and $N$. Consider the $d$-dimensional $N$-hypergrid $[N]^d$. There is a natural many-to-one mapping from $\Phi: [N]^d\mapsto [n]^d$ defined as follows. First fix a dimension $r$. Given an integer $1\leq t \leq N$, let $\phi_r(t)$ denote the index $\ell \in [1,n]$ such that $\sum_{j <\ell} q_r(j) < t \leq \sum_{j\leq \ell} q_r(j)$. That is, partition $[N]$ into $n$ contiguous segments of lengths $q_r(1),\ldots,q_r(n)$. Then $\phi_r(t)$ is the index of the segment where $t$ lies. The mapping $\Phi: [N]^d\mapsto [n]^d$ is defined as $$\Phi(x_1,x_2\ldots,x_d) = \left(\phi_1(x_1), \phi_2(x_2), \ldots, \phi_\pdi(x_d) \right).$$ We use $\Phi^{-1}$ to define the set of preimages, so $\Phi^{-1}$ maps a point in $[n]^d$ to a ‘cuboid’ in $[N]^d$. Observe that for any $x\in [n]^d$, $$\label{eq:obs} |\Phi^{-1}(x)| = N^d\prod_{r=1}^d \mu_{\cD_r}(x) = N^d\mu_\cD(x).$$ \[clm:XtoZ\] For any set $X\subseteq [n]^d$, define $Z \subseteq [N]^d$ as $Z := \bigcup_{x\in X}\Phi^{-1}(x)$. Then $\mu_\cD(X) = \mu_\cU(Z)$. The set $Z = \bigcup_{x\in X}\Phi^{-1}(x)$ is the union of all the preimages of $\Phi$ over the elements of $X$. Since preimages are disjoint, we get $|Z| = \sum_{x\in X}|\Phi^{-1}(x)| = N^d\mu_\cD(X)$. Therefore, $\mu_\cU(Z) = \mu_\cD(X)$. Given $f:[n]^d \mapsto \R$, we define its extension $\fext:[N]^d \mapsto \R$: $$\label{eq:fext}\fext(x_1,\ldots,x_d) = f(\Phi(x_1,\ldots,x_d)).$$ Thus, $\fext$ is constant on the cuboids in the bloated hypergrid corresponding to a point in the original hypergrid. Define the following metric on $[N]^d$. $$\label{eq:dext} \textrm{For $x,y \in [N]^d$,} \quad \pdiext(x,y) =\pdi(\Phi(x),\Phi(y))$$ The following statements establish the utility of the bloated hypergrid, and the proof of the dimension reduction of $f$ over $[n]^d$ w.r.t. $\cD$ follows easily from these and the proof for the uniform distribution. \[lem:distconsistency\] If $\pdi$ satisfies the conditions of over $[n]^d$, then so does $\pdiext$ over $[N]^d$. Consider $x,y,z \in [N]^d$. Triangle inequality and well-definedness immediately follow from the validity of $\pdi$. Now for linearity. If $x_r\leq y_r\leq z_r$, then so is $\phi_r(x_r) \leq \phi_r(y_r)\leq \phi_r(z_r)$. Thus, $\Phi(x), \Phi(y),\Phi(z)$ satisfy linearity w.r.t. $\pdi$. So, $\pdiext(x,z) = \pdi(\Phi(x),\Phi(z)) = \pdi(\Phi(x),\Phi(y)) + \pdi(\Phi(y),\Phi(z)) = \pdiext(x,y) + \pdiext(y,z)$. Now for projection. Suppose $x_r = y_r$ and $x'_r = y'_r$. Note that $\Phi(x)$ and $\Phi(y)$ have same $r$th coordinate, and so do $\Phi(x')$ and $\Phi(y')$. Furthermore, $\Phi(x')$ (resp. $\Phi(y')$) is the projection of $\Phi(x)$ (resp. $\Phi(x)$). Thus we get $\pdiext(x,y) = \pdi(\Phi(x),\Phi(y)) = \pdi(\Phi(x'),\Phi(y')) = \pdiext(x',y')$, and similarly $\pdiext(x,x') = \pdiext(y,y')$. \[thm:bhg\] $\dist_\cD(f,{\cP({\pdi})}) = \dist_\cU(\fext,\cP(\pdiext))$. ($\geq$). Let $X\subseteq [n]^d$ be a vertex cover in $\VG(f,{\cP({\pdi})})$ minimizing $\mu_\cD(X)$. From , $\dist_\cD(f,{\cP({\pdi})}) = \mu_\cD(X)$. We claim $Z = \bigcup_{x\in X}\Phi^{-1}(x)$ is a vertex cover of $\VG(\fext,\cP(\pdiext))$. This implies $\dist_\cU(\fext,\cP(\pdiext)) \leq \mu_\cU(Z) = \mu_\cD(X) = \dist_\cD(f,{\cP({\pdi})})$, where the first equality follows from . Consider a violated pair $(u,v)$ in this graph and so wlog $\fext(u) - \fext(v) > \pdiext(u,v)$. Hence, $f(\Phi(u)) - f(\Phi(v)) > \pdi(\Phi(u),\Phi(v))$ implying $(\Phi(u),\Phi(v))$ is an edge in $\VG(f,{\cP({\pdi})})$. Thus, either $\Phi(u)$ or $\Phi(v)$ lies in $X$ implying either $u$ or $v$ lies in $Z$.($\leq$). Let $Z \subseteq [N]^d$ be a vertex cover in $\VG(\fext,\cP(\pdiext))$ minimizing $\mu_\cU(Z)$. Therefore, $\dist_\cU(\fext,\cP(\pdiext)) = \mu_\cU(Z)$. Define $X\subseteq [n]^d$ as $X = \{x\in [n]^d: \Phi^{-1}(x)\subseteq Z\}$. Therefore, $Z \supseteq \bigcup_{x\in X}\Phi^{-1}(x)$ and from we get $\mu_\cU(Z)\geq \mu_\cD(X)$. It suffices to show that $X$ is a vertex cover of $\VG(f,{\cP({\pdi})})$. Consider a violated edge $(x,y)$ in this graph such that $f(x) - f(y) > \pdi(x,y)$. Suppose neither $x$ nor $y$ are in $X$. Hence, there exists $u\in \Phi^{-1}(x) \setminus Z$ and $v\in \Phi^{-1}(y) \setminus Z$. So $\fext(u) - \fext(v) = f(\Phi(u)) - f(\Phi(v)) = f(x) - f(y) > \pdi(x,y) = \pdiext(\Phi(u),\Phi(v))$, implying $(u,v)$ is a violation in $\VG(\fext,\cP(\pdiext))$. This contradicts the fact that $Z$ is a vertex cover. Fix a dimension $r$ and $r$-line $\ell$. Abusing notation, let $\Phi^{-1}(\ell)$ denote the collection of $r$-lines in $[N]^d$ that are mapped to $\ell$ by $\Phi$. Note that $|\Phi^{-1}(\ell)| = N^{d-1} \mu_{\cD_{-r}}(\ell)$. A proof identical to one above yields the following theorem. For any $r$-line, $\dist_{\cD_r}({{f}_{|\ell}},{\cP({\pdi})}) = \dist_{\cU_r}({{\fext}_{|\ell'}},\cP(\pdiext))$ for all $\ell' \in \Phi^{-1}(\ell)$. Now we can complete the proof of . $$\begin{aligned} \dist^r_\cD(f,{\cP({\pdi})}) & = & \sum_{\textrm{$r$-line $\ell$}} \mu_{\cD_{-r}}(\ell) \cdot \dist_{\cD_r}({{f}_{|\ell}},{\cP({\pdi})}) \\ & = & \frac{1}{N^{d-1}} \sum_{\textrm{$r$-line $\ell$}} |\Phi^{-1}(\ell)| \cdot \dist_{\cD_r}({{f}_{|\ell}},{\cP({\pdi})}) \\ & = & \frac{1}{N^{d-1}}\sum_{\textrm{$r$-line $\ell$}} \sum_{\ell' \in \Phi^{-1}(\ell)} \dist_{\cU_r}({{\fext}_{|\ell'}},\cP(\pdiext)) \\ & = & \Exp_{\ell' \sim \cU_{-r}} [\dist_{\cU_r}({{\fext}_{|\ell'}},\cP(\pdiext))] = \dist^r_{\cU}(\fext,{\cP({\dext})}).\end{aligned}$$ We can apply the dimension reduction to $\fext$ for property $\cP(\pdiext)$ over the uniform distribution. The proof of for $f$ follows directly. No $r$-violations imply no $r$-cross pairs. {#sec:noviol} ------------------------------------------- In this subsection we prove . This closely follows the techniques and proofs from [@ChSe13]. This requires the alternating path setup of [@ChSe13]. Recall the weight function $w(x,y) = \max(f(x) - f(y) - \pdi(x,y), f(y) - f(x) - \pdi(y,x))$ defined on pairs of the domain. Note that $(x,y)$ is a violation iff $w(x,y) > 0$. Let $M$ be a maximum weight minimum cardinality (MWmC) matching of $\VG(f,{\cP({\pdi})})$ with the [*minimum number*]{} of $r$-cross pairs. Recall an $r$-cross pair $(x,y)$ has $x_r \neq y_r$. We will prove that this minimum value is $0$. Let $\cross(M)$ be the set of $r$-cross pairs in $M$. Let $\str(M) := M\setminus \cross(M)$. For contradiction’s sake, assume $\cross(M)$ is nonempty. Let $(x,y)\in \cross(M)$ be an arbitrary $r$-cross pair with $x_r = a$ and $y_r = b$ with $a \neq b$. Define matching $H := \{(u,v): u_r = a, v_r = b, u_j = v_j, j\neq i\}$. This is a matching by projection between points with $r$th coordinate $a$ and $b$. For convenience, we denote the points with $r$th coordinate $a$ (resp. $b$) as the *$a$-plane* (resp. $b$-plane). Consider the alternating paths and cycles in $H \Delta \;\str(M)$. The vertex $y$ is incident to only an $H$-pair, since $(x,y)\in \cross(M)$. Let $y = s_1,s_2,\ldots,s_t$ be the alternating path starting from $y$, collectively denoted by $S$. We let $s_0 := x$. The end of $S$, $s_t$, may be either $M$-unmatched or $\cross(M)$-matched. In the latter case, we define $s_{t+1}$ to be such that $(s_t,s_{t+1})\in \cross(M)$. For even $i$, $(s_{i-1},s_i)$ is an $H$-pair and $(s_i,s_{i+1})$ is an $M$-pair. We list out some basic claims about $S$. \[clm:alt\] If strictly positive $j \equiv 0,1 \mod 4$, then $s_j$ is in the $b$-plane. Otherwise, $s_j$ is in the $a$-plane. \[clm:w\] For strictly positive even $i$, $f(s_{i-1}) - f(s_{i}) - \pdi(s_{i-1},s_{i}) \leq 0$ and $f(s_{i}) - f(s_{i-1}) - \pdi(s_{i},s_{i-1}) \leq 0$. Since $f$ is $r$-good and $H$-pairs differ only in the $r$th coordinate, $w(s_{i-1},s_{i}) \leq 0$ for all even $i$. The definition of $w(s_{i-1},s_{i})$ completes the proof. \[clm:d\] For strictly positive $i\equiv 0\mod 4$, $\pdi(s_{i-1},s_i) = \pdi(s_2,s_1)$. For $i\equiv 2\mod 4$, $\pdi(s_{i},s_{i-1}) = \pdi(s_2,s_1)$. The point $s_0$ (which is $x$) lies in the $a$-plane. Hence, for any $i\equiv 2\mod 4$, $s_i$ lies in the $b$-plane. Similarly, for $i \equiv 0\mod 4$, $s_i$ lies in the $a$-plan. For strictly positive even $i$, $(s_{i-1},s_i)$ is an $H$-pair. An application of the projection property completes the proof. \[clm:d2\] For strictly positive even $i$, $\pdi(s_i,s_{i+1}) = \pdi(s_{i-1},s_{i+2})$ and $\pdi(s_{i+1},s_i) = \pdi(s_{i+2},s_{i-1})$. Consider $\str(M)$-pair $(s_i, s_{i+1})$. Both points are on the same ($a$ or $b$-)plane. Observe that $s_{i-1}$ is the projection of $s_i$ and $s_{i+2}$ is the projection of $s_{i+1}$ onto the other plane. Apply the projection property of $d$ to complete the proof. Now we have all the ingredients to prove the theorem. The strategy is to find another matching $M'$ such that either $w(M') > w(M)$ or $w(M')=w(M)$ and $M'$ has strictly fewer cross pairs. Let us identify certain subsets of pairs to this end. For even $k$, define $$E_-(k) := (s_0,s_1), (s_2,s_3),\ldots, (s_k,s_{k+1}) = \{(s_j,s_{j+1}): j \textrm{ even, } 0\leq j\leq k\}$$ These are precisely the $\str(M)$-pairs in $S$ in the first $k$-steps. Note that $|E_-(k)| = k/2 + 1$. Now we define $E_+(k)$. In English: first pick pair $(s_0,s_2)$; subsequently pick the first unpaired $s_i$ and pair it with the next unpaired $s_j$ of the [*opposite*]{} parity. More precisely, for even $k$, $$E_+(k) := (s_0,s_2), (s_1,s_4), (s_3,s_6),\ldots, (s_{k-3},s_k) = (s_0,s_2) \cup \{(s_{j-3},s_j) : j \textrm{ even, }4\leq j\leq k\}$$ Note that $|E_+(k)| = k/2$. Wlog, assume that $w(x,y) = f(x) - f(y) - \pdi(x,y)$. It turns out the weights of all other $M$-pairs in $S$ are determined. We will assert that the pattern is as follows. $$\label{eq:cond} \tag{$\clubsuit$} w(s_i,s_{i+1}) = \begin{cases} f(s_i) - f(s_{i+1}) - \pdi(s_i,s_{i+1}) & \text{if } i\equiv 0\mod 4 \\ f(s_{i+1}) - f(s_{i}) - \pdi(s_{i+1},s_{i}) & \text{if } i\equiv 2\mod 4 \end{cases}$$ The following lemma determines the weights of all other $M$-edges in the alternating path $S$. Recall $(s_i,s_{i+1}) \in \str(M)$ for even $i$. \[lem:match\] Suppose $s_i$ exists. If holds for all even indices $< i$, then $s_i$ is matched in $M$. Assume $i \equiv 2 \mod 4$. (The other case is analogous and omitted.) We prove by contradiction, so suppose $s_i$ is not matched in $M$. We set $M' := M - E_-(i-2) + E_+(i)$. Note that $M'$ is a valid matching, since $s_i$ is not matched. We compare $w(M')$ and $w(M)$. By , we can express $w(E_-(i-2))$ exactly. $$\begin{aligned} w(E_-(i-2)) & = & \sum_{j: \textrm{even, } 0\leq j\leq i-2} w(s_j,s_{j+1}) \nonumber \\ & = & [f(s_0) - f(s_1) - \pdi(s_0,s_1)] + [f(s_3) - f(s_2) -\pdi(s_3,s_2)] + \nonumber \\ & & [f(s_4) - f(s_5) - \pdi(s_4,s_5)] + [f(s_7) - f(s_6) - \pdi(s_7,s_6)] + \cdots \nonumber \\ & & [f(s_{i-2}) - f(s_{i-1}) - \pdi(s_{i-2},s_{i-1})] \label{eq:w-}\end{aligned}$$ We lower bound $w(E_+(i))$. Since each individual weight term is a maximum of two expressions, we can choose either. We set the expression up to match $w(E_-(i-2))$ as best as possible. $$\begin{aligned} w(E_+(i)) & \geq & [f(s_0) - f(s_2) - \pdi(s_0,s_2)] + [f(s_4) - f(s_1) - \pdi(s_4,s_1)] + \nonumber \\ & & [f(s_3) - f(s_6) - \pdi(s_3,s_6)] + [f(s_8) - f(s_5) - \pdi(s_8,s_5)] + \nonumber \\ & & [f(s_{i-3}) - f(s_{i}) - \pdi(s_{i-3},s_{i})] \label{eq:w+}\end{aligned}$$ Note that $w(M') - w(M) = w(E_+(i)) - w(E_-(i-2))$. Observe that any $f$ term that occurs in both and has the same coefficient. By , $\pdi(s_3,s_2) = \pdi(s_4,s_1)$, $\pdi(s_4,s_5) = \pdi(s_3,s_6)$, etc. $$w(E_+(i)) - w(E_-(i-2)) \geq f(s_{i-1}) - f(s_i) - \pdi(s_0,s_2) + \pdi(s_0,s_1)$$ The points $s_0$ and $s_1$ lie is different planes, and $(s_1,s_2) \in H$. We can apply the linearity property to get $\pdi(s_0,s_1) = \pdi(s_0,s_2) + \pdi(s_2,s_1)$. Plugging this in, applying and for $i$, $$w(E_+(i)) - w(E_-(i-2)) \geq f(s_{i-1}) - f(s_i) + \pdi(s_2,s_1) = -[f(s_i) - f(s_{i-1}) - \pdi(s_i,s_{i-1})] \geq 0$$ Hence $w(M') \geq w(M)$. Note that $|M'| - |M|$ $= |E_+(i)| - |E_-(i-2)| $ $= i/2 - ((i-2)/2 + 1) = 0$. Finally, observe that $E_+(i)$ has no $r$-cross pairs, but $E_-(i-2)$ has one (pair $(s_0,s_1)$). This contradicts the choice of $M$ as a MWmC matching with the least $r$-cross pairs. \[clm:distinct\] If holds for all even indices $< i$, then $s_0, s_1, \ldots, s_{i+1}$ are all distinct. (This is trivial if $i < t$. The non-trivial case if when $S$ ends as $s_i$.) The points $s_1, \ldots, s_{i}$ are all distinct. If $s_i \neq x$, the claim holds. So assume $s_i = x = s_0$. By , $i \equiv 2 \mod 4$. Replace pairs $A = \{(s_0,s_1), (s_{i-2},s_{i-1})\}$ by $(s_{i-2},s_1)$. Note that $\pdi(s_0,s_1) = \pdi(s_0,s_{i-1}) + \pdi(s_{i-1},s_1)$. By , $$\begin{aligned} w(A) & = & [f(s_0) - f(s_1) - \pdi(s_0,s_1)] + [f(s_{i-2}) - f(s_{i-1}) - \pdi(s_{i-2},s_{i-1})] \\ & = & [f(s_{i-2}) - f(s_1) - \pdi(s_{i-2},s_{i-1}) - \pdi(s_{i-1},s_1)] + [f(s_0) - f(s_{i-1}) - \pdi(s_0,s_{i-1})] \\ & \leq & [f(s_{i-2}) - f(s_1) - \pdi(s_{i-2},s_1)] \leq w(s_{i-2},s_1)\end{aligned}$$ The total number of pairs has decreased, so we complete the contradiction. \[lem:cond\] Suppose $s_i$ exists. If holds for all even indices $< i$, then holds for $i$. We prove by contradiction, so is false for $i$. (Again, assume $i \equiv 2 \mod 4$. The other case is omitted.) By , $E_+(i-2) \cup (s_{i-3},s_{i+1})$ is a valid set of matched pairs. Let $M' := M- E_-(i) + (E_+(i-2) \cup (s_{i-3},s_{i+1}))$. Observe that $|M'| = |M| - 1$ and the vertices $s_{i-1}$ and $s_i$ are left unmatched in $M'$. By for even indices $< i$ and the opposite of for $i$, $$\begin{aligned} w(E_-(i)) & = & [f(s_0) - f(s_1) - \pdi(s_0,s_1)] + [f(s_3) - f(s_2) -\pdi(s_3,s_2)] + \nonumber \\ & & [f(s_4) - f(s_5) - \pdi(s_4,s_5)] + [f(s_7) - f(s_6) - \pdi(s_7,s_6)] + \cdots \nonumber \\ & & [f(s_{i-2}) - f(s_{i-1}) - \pdi(s_{i-2},s_{i-1})] + [f(s_{i}) - f(s_{i+1}) - \pdi(s_{i},s_{i+1})] \label{eq:w-2}\end{aligned}$$ We stress that the last weight is “switched". We lower bound $w(E_+(i-2) \cup (s_{i-3},s_{i+1}))$ appropriately. $$\begin{aligned} w(E_+(i-2) \cup (s_{i-3},s_{i+1})) & \geq & [f(s_0) - f(s_2) - \pdi(s_0,s_2)] + [f(s_4) - f(s_1) - \pdi(s_4,s_1)] + \nonumber \\ & & [f(s_3) - f(s_6) - \pdi(s_3,s_6)] + [f(s_8) - f(s_5) - \pdi(s_8,s_5)] + \cdots \nonumber \\ & & [f(s_{i-7}) - f(s_{i-4}) - \pdi(s_{i-7},s_{i-4})] + [f(s_{i-2}) - f(s_{i-5}) - \pdi(s_{i-2},s_{i-5})] + \nonumber \\ & & [f(s_{i-3}) - f(s_{i+1}) - \pdi(s_{i-3},s_{i+1})] \label{eq:w+2}\end{aligned}$$ As before, we subtract from . All function terms from cancel out. By , all $\pdi$-terms except the first and last cancel out. $$w(M') - w(M) \geq f(s_{i-1}) - f(s_i) - \pdi(s_0,s_2) - \pdi(s_{i-3},s_{i+1}) + \pdi(s_0,s_1) + \pdi(s_{i-2},s_{i-1}) + \pdi(s_i,s_{i+1})$$ By linearity, $\pdi(s_0,s_1) = \pdi(s_0,s_2) + \pdi(s_2,s_1)$. Furthermore, by , $\pdi(s_2,s_1) = \pdi(s_{i},s_{i-1})$. By , $\pdi(s_{i-2},s_{i-1}) = \pdi(s_{i-3},s_i)$. By triangle inequality, $-\pdi(s_{i-3},s_{i+1}) + \pdi(s_{i-3},s_i) + \pdi(s_i,s_{i+1}) \geq 0$. Putting it all together and applying , $$w(M') - w(M) \geq -[f(s_i) - f(s_{i-1}) - \pdi(s_i,s_{i-1})] \geq 0$$ So $M'$ has at least the same weight but lower cardinality than $M$. Contradiction. \[lem:str\] Suppose $s_i$ exists. If holds for all even indices $< i$, then $s_i$ is matched in $\str(M)$. Suppose not. (Again, assume $i \equiv 2 \mod 4$.) By , $s_i$ is matched in $M$, so $(s_i,s_{i+1}) \in \cross(M)$. We set $M' = M - E_-(i) + (E_+(i) \cup (s_{i-1},s_{i+1}))$. By , $M'$ is a valid matching. We have $|M'| = |M|$. $M$ has two $r$-cross pairs $(s_0,s_1)$ and $(s_i,s_{i+1})$, but $M'$ has at most one $(s_{i-1},s_{i+1})$. It suffices to show that $w(M') \geq w(M)$ to complete the contradiction. By and , $$\begin{aligned} w(E_-(i)) & = & [f(s_0) - f(s_1) - \pdi(s_0,s_1)] + [f(s_3) - f(s_2) -\pdi(s_3,s_2)] + \nonumber \\ & & [f(s_4) - f(s_5) - \pdi(s_4,s_5)] + [f(s_7) - f(s_6) - \pdi(s_7,s_6)] + \cdots \nonumber \\ & & [f(s_{i-2}) - f(s_{i-1}) - \pdi(s_{i-2},s_{i-1})] + [f(s_{i+1}) - f(s_{i}) - \pdi(s_{i+1},s_{i})] \nonumber\end{aligned}$$ $$\begin{aligned} w(E_+(i) \cup (s_{i-1},s_{i+1})) & \geq & [f(s_0) - f(s_2) - \pdi(s_0,s_2)] + [f(s_4) - f(s_1) - \pdi(s_4,s_1)] + \nonumber \\ & & [f(s_3) - f(s_6) - \pdi(s_3,s_6)] + [f(s_8) - f(s_5) - \pdi(s_8,s_5)] + \cdots \nonumber \\ & & [f(s_{i-3}) - f(s_{i}) - \pdi(s_{i-3},s_{i})] + [f(s_{i+1}) - f(s_{i-1}) - \pdi(s_{i+1},s_{i-1})] \nonumber \end{aligned}$$ All function terms and all but the first and last $\pdi$-terms cancel out. The second inequality below holds by linearity and triangle inequality. The last equality is an application of . $$\begin{aligned} w(M') - w(M) & \geq & \pdi(s_0,s_1) - \pdi(s_0,s_2) + \pdi(s_{i+1},s_i) - \pdi(s_{i+1},s_{i-1}) \\ & \geq & \pdi(s_2,s_1) - \pdi(s_i,s_{i-1}) = 0\end{aligned}$$ Finally, we prove . We started with a MWmC matching $M$ with the minimum number of $r$-cross pairs. If there exists at least one such cross pair $(x,y)$, we can define the alternating path sequence $S$. Wlog, we assumed holds for $i=0$. Applications of and imply that $S$ can never terminate. Contradiction. Search Trees and Bounded Derivative Property Testing. {#sec:ub} ===================================================== As a result of dimension reduction, we can focus on designing testers for the line $[n]$. Our analysis is simple, but highlights the connection between bounded-derivative property testing and optimal search trees. Testers for the Line $[n]$. {#sec:line-ub} --------------------------- Let $T$ be [*any*]{} binary search tree (BST) with respect to the totally ordered domain $[n]$. Every node of $T$ is labeled with a unique entry in $[n]$, and the left (resp. right) child, if it exists, has a smaller (resp. larger) entry. The [*depth*]{} of a node $v$ in the tree $T$, denoted as $\depth_T(v)$, is the number of [*edges*]{} on its path to the root. So the root has depth $0$. Given a distribution $\cD$ on $[n]$, the expected depth of $T$ w.r.t. $\cD$ is denoted as $\Delta(T;\cD) = \Exp_{v\sim \cD}[\depth_T(v)]$. The depth of the optimal BST w.r.t. $\cD$ is denoted by $\Delta^*(\cD)$. It has long been observed that the transitivity of violations is the key property required for monotonicity testing on $[n]$ [@BRW05; @EKK+00; @ACCL07; @JR11]. We distill this argument down to a key insight: Given [*any*]{} BST $T$, there exists the following tester $\textrm{BST}(T)$ for $\cP$ on the line. [**BST Tester ($T$)**]{} 0em Sample $v \sim \cD$. If $v$ is the root of $T$, do nothing. Else, query $f(u)$ for all vertices lying on the path from $v$ to root (including the root and $v$). Reject if any pair of these vertices form a violation to $\cP$. [It is clear that the tester never rejects a function satisfying $\cP$. ]{} (To connect with previous work, observe that the list of ancestor-descendant pairs forms a 2-Transitive Closure spanner [@BGJRW09].) \[lem:tree-tester\] For any bounded derivative property $\cP$, $\Pr[\textrm{BST tester rejects}]\geq \dist_\cD(f,\cP)$. Let $X$ be the set of non-root nodes $v$ of $T$ with the following property: $(u,v)$ is a violation to $\cP$ for some node $u$ on the path from $v$ to the root of $T$. The probability of rejection of the BST tester is precisely $\mu_\cD(X)$. We claim that $X$ is a vertex cover of $\VG(f,\cP)$ which proves the lemma using . Pick any violation $(x,y)$ and assume without loss of generality $f(x) - f(y) > \pdi(x,y)$. Let $z$ be the lowest common ancestor of $x$ and $y$ in $T$. By the BST property, either $x < z < y$ of $x>z>y$. By the linearity property of $\pdi$, we get $\pdi(x,y) = \pdi(x,z) + \pdi(z,y)$. This implies either $f(x) - f(z) > \pdi(x,z)$ or $f(z) - f(y) > \pdi(z,y)$, that is, either $(x,z)$ or $(y,z)$ is a violation implying one of them is in $X$. [lemma]{}[optbst]{} \[thm:tree-optbst\]\[lem:optbst\] For any BST $T$, there is a $24\eps^{-1}\Delta(T;\cD)$-query line monotonicity-tester. The expected number of queries made by the BST tester is $\sum_{v: \textrm{ non-root}} \Pr[v]\cdot (\depth_T(v) + 1) = (1-\Pr[\textrm{root}]) + \Delta(T;\cD) \leq 2\cdot\Delta(T;\cD)$. [$$\sum_{v: \textrm{ non-root}} \Pr[v]\cdot (\depth_T(v) + 1) = (1-\Pr[\textrm{root}]) + \Delta(T;\cD) \leq 2\cdot\Delta(T;\cD)$$]{} The expected depth is at least $(1-\Pr[\textrm{root}])$ since non-roots have depth at least $1$. To get a bonafide tester with deterministic query bounds, run the BST tester $2/\eps$ times, aborting (and accepting) if the total number of queries exceeds $24\Delta(T;\cD)/\eps$. The expected total number of queries is at most $4\Delta(T;\cD)/\eps$. By Markov’s inequality, the probability that the tester aborts is $\leq 1/6$. By , if $\dist_\cD(f;\cP) > \eps$, the probability that this tester does not find a violation is at most $(1-\eps)^{2/\eps} \leq 1/6$. With probability $\geq (1-1/6-1/6) = 2/3$, the tester rejects an $\eps$-far function. Choose $T$ to be the optimal BST to get the following theorem. \[thm:ub-line-dist-known\] There exists a $24\eps^{-1}\Delta^*(\cD)$-query tester for any bounded derivative property over the line. Note that once the tree is fixed, the BST tester only needs random samples from the distribution. Pick $T$ to be the balanced binary tree of depth $O(\log n)$ to get a [*distribution-free*]{} tester. There exists a $24\eps^{-1}\log n$-query [*distribution free*]{} tester for any bounded-derivative property over the line. Testers for the Hypergrid. {#sec:hg-ub} -------------------------- Given a series of BSTs $T_1, T_2, \ldots, T_d$ corresponding to each dimension, we have the following hypergrid BST tester. [**Hypergrid BST Tester ($T_1, T_2, \ldots, T_d$)**]{} 0em Sample $x \sim \cD$. Choose dimension $r$ u.a.r. and let $\ell$ be the $r$-line through $x$. Run **BST Tester($T_r$)** on ${{f}_{|\ell}}$. \[lem:hyper-bst\] For any set of BSTs $T_1, T_2, \ldots, T_d$, the probability of rejection is at least $\dist_\cD(f,\cP)/4d$. Condition on an $r$-line being chosen. The probability distribution over $r$-lines for this tester is $\cD_{-r}$. By , the rejection probability is at least $\Exp_{\ell \sim \cD_{-r}} [\dist_{\cD_r}({{f}_{|\ell}},\cP)] = \dist^r_{\cD}(f,\cP)$. The overall rejection probability is at least $\sum_{r=1}^d \frac{\dist^i_{\cD}(f,\cP)}{d} \geq \frac{\dist_\cD(f,\cP)}{4d}$, by . The expected number of queries made by this procedure is at most $\frac{1}{d}\!\cdot\!\sum_{r=1}^d \!2\Delta(T_r;\cD_r)$. Repeating it $O(d/\eps)$ times to get the desired tester. The proof of the following is identical to that of and is omitted. [lemma]{}[optbst-hg]{} \[thm:tree-optbst\]\[lem:optbst-hg\] For any collections of BSTs $(T_1,\ldots,T_d)$, there is a $100\eps^{-1}\sum_{i=1}^d\Delta(T;\cD)$-query tester for any bounded derivative property. As in the case of the line we get the following as corollaries. The upper bound $\sum_{r=1}^d \Delta^*(\cD_r) $ is at most $H(\cD)$, but can be much smaller, and it is clearest in the case of the hypercube. In the hypercube, each $\cD_r$ is given by $(\mu_r,1-\mu_r)$. Set $\theta_r := \min(\mu_r,1-\mu_r)$. The optimal BST places the point of larger mass on the root and has expected depth $\theta_r$. It is instructive to open up this tester. It samples a point $x$ from the distribution $\cD$ and picks a dimension $r$ uniformly at random. With probability $\theta_r$, it queries both endpoints of $(x,x\oplus {{\bf e}}_r)$. With probability $(1-\theta_r)$, it does [*nothing*]{}. This process is repeated $O(d/\eps)$ times. When $\theta_r = \mu_r =1/2$, this is the standard edge tester. Lower Bounds {#sec:lb} ============ We prove that the upper bounds of are tight up to the dependence on the distance parameter $\eps$. As alluded to in , we can only prove lower bounds for [*stable*]{} product distributions. These are distributions where small perturbations to the mass function do not change $\Delta^*$ drastically. \[def:stable\] A product distribution $\cD$ is said to be $({\eps},\rho)$-stable if for all product distributions $\cD'$ with $||\cD - \cD'||_{\TV} \leq \eps$, $\Delta^*(\cD') \geq \rho\Delta^*(\cD)$. The uniform distribution on $[n]^d$ is $(\eps,1-o(1))$-stable, for any constant $\eps < 1$. The Gaussian distribution also shares the same stability. An example of an unstable distribution is the following. Consider $\cD$ on $[n]$, where the probability on the first $k = \log n$ elements is $(1-\eps)/k$, and is $\eps/(n-k)$ for all other elements. Let $\cD'$ have all its mass uniformly spread on the first $k$ elements. We have $||\cD - \cD'||_\TV = {\eps}$ but $\Delta^*(\cD) \approx \eps\log n$ and $\Delta^*(\cD') \approx \log k = \log\log n$. Reduction from monotonicity to bounded-derivative property {#sec:bdp-to-mono} ---------------------------------------------------------- Consider a function $f: [n]^d \mapsto [R]$ with where $R \in \NN$. Let $\pdi$ be the distance function obtained by bounding family ${\mathbf{B}}$. We let ${{\bf 0}}\in [n]^d$ be $(0,0,\ldots,0)$. We use $\prec$ to denote the natural partial order in $[n]^d$, and let ${{\tt hcd}}(x,y)$ be the highest common descendant of $x,y \in [n]^d$. We first prove an observation about triangle equality. \[obs:equal\] If $\pdi({{\bf 0}},x) + \pdi(x,y) = \pdi({{\bf 0}},y)$, then $x \prec y$. By linearity, $\pdi(x,y) = \pdi(x,{{\tt hcd}}(x,y)) + \pdi({{\tt hcd}}(x,y),y)$. Since ${{\tt hcd}}(x,y) \prec x$, by linearity again, $\pdi({{\bf 0}},x) = \pdi({{\bf 0}},{{\tt hcd}}(x,y)) + \pdi({{\tt hcd}}(x,y),x)$. (Similarly for $y$.) Putting it all into the ‘if’ condition, $$\begin{aligned} \pdi({{\bf 0}},{{\tt hcd}}(x,y)) + \pdi({{\tt hcd}}(x,y),x) + \pdi(x,{{\tt hcd}}(x,y)) + \pdi({{\tt hcd}}(x,y),y) = \pdi({{\bf 0}},{{\tt hcd}}(x,y)) + \pdi({{\tt hcd}}(x,y),y)\end{aligned}$$ This yields $\pdi({{\tt hcd}}(x,y),x) + \pdi(x,{{\tt hcd}}(x,y)) = 0$. Suppose ${{\tt hcd}}(x,y) \neq x$. The length (in terms of ${\mathbf{B}}$) of the path from ${{\tt hcd}}(x,y)$ to $x$ involves a sum of $u_i(t)$ terms, and the reverse path involves corresponding $-l_i(t)$ terms. Since $u_i(t) > l_i(t)$, the total path length from ${{\tt hcd}}(x,y)$ to $x$ and back is strictly positive. Therefore, ${{\tt hcd}}(x,y) = x$ and $x \prec y$. Let $U$ be the set of incomparable (ordered) pairs in $[n]^d$. Define $\delta := \min_{(x,y) \in U} \{\pdi({{\bf 0}},x)+\pdi(x,y) - \pdi({{\bf 0}},y)\}$. By , $\delta > 0$. Define $$g(x) := \frac{\delta}{2R}\cdot f(x) - \pdi({{\bf 0}},x)$$ \[lem:bijective\] $\dist_\cD(g,\cP) =\dist_\cD(f,\MON)$. We show that $(u,v)$ violates ${\cP({\pdi})}$ of $g$ iff it violates monotonicity of $f$. First, the ‘only if’ case. Assume $g(u)-g(v)>\pdi(u,v)$. Plugging in the expression for $g(\cdot)$ and rearranging, $$\begin{aligned} \frac{\delta}{2R}(f(u)-f(v))> \pdi({{\bf 0}},u) + \pdi(u,v) - \pdi({{\bf 0}},v)\end{aligned}$$ By triangle inequality on the RHS, $f(u) > f(v)$. Note that $f(u)-f(v)\le R$ so $\frac{\delta}{2R}(f(u)-f(v))\leq \delta/2$. So $\delta/2 > \pdi({{\bf 0}},u) + \pdi(u,v) - \pdi({{\bf 0}},v)$. By choice of $\delta$, the RHS must be zero. By , $u \prec v$, and $(u,v)$ is a violation to monotonicity of $f$. Now the ‘only if’ case, so $u \prec v$ and $f(u)>f(v)$. Note that $\pdi({{\bf 0}}, v)=\pdi({{\bf 0}},u)+\pdi(u,v)$. We deduce that $(u,v)$ is also a violation to ${\cP({\pdi})}$ for $g$. $$g(u)-g(v)=\frac{\delta}{2R}(f(u)-f(v))+\pdi({{\bf 0}},v)-\pdi({{\bf 0}},u)=\frac{\delta}{2R}(f(u)-f(v))+\pdi(u,v)>\pdi(u,v)$$ Our main reduction theorem is the following. \[thm:red\] Fix domain $[n]^d$ and a product distribution $\cD$. Suppose there exists a $Q$-query tester for testing a bounded-derivative property $\cP$ with distance parameter ${\eps}$. Then there exists a $Q + 10/{\eps}$-query tester for monotonicity for functions $f:[n]^d\mapsto\NN$ over $\cD$ with distance parameter $2{\eps}$. The monotonicity tester first queries $10/\eps$ points of $[n]^d$, each i.i.d. from $\cD$. Let the maximum $f$-value among these be this $M$. Consider the truncated function $f':[n]^d \mapsto [M]$, where $f'(x) = M$ if $f(x) \geq M$ and $f'(x) = f(x)$ otherwise. If $f$ is monotone, $f'$ is monotone. Note that $\dist_\cD(f,f') < \eps$. So if $f$ is $2\eps$-far from monotone, $f'$ is $\eps$-far from monotone. We can apply the $\cP({\mathbf{B}})$ tester on the function $g$ obtained from . Monotonicity Lower Bound Framework. {#sec:monotone-lb-framework} ----------------------------------- The lower bound for monotonicity testing goes by the proof strategy set up in [@ChSe13-2]. This is based on arguments in [@E04; @ChSe13-2] that reduce general testers to comparison-based testers. We encapsulate the main approach in the following theorem, proven implicitly in [@ChSe13-2]. (We use $\MON$ to denote the monotonicity property.) \[thm:lb-frame\] Fix domain $[n]^d$, distribution $\cD$, proximity parameter $\eps$, and positive integer $L$ possibly depending on $\cD$ and $\eps$. A pair $(x,y)$ distinguishes function $g$ from $h$ if $h(x) < h(y)$ and $g(x) > g(y)$. Suppose there is a collection of ‘hard’ functions $h,g_1,\ldots,g_L: [n]^d \mapsto \NN$ such that -0.1em The function $h$ is monotone. Every $\dist_\cD(g_i,\MON) \geq {\eps}$. Pairs in any set $Q \subset [n]^d$, can distinguish at most $|Q|$ of the $g_i$’s from $h$. Then any (even adaptive, two-sided) monotonicity tester w.r.t. $\cD$ for functions $f:[n]^d\mapsto\NN$ with distance parameter ${\eps}$ must make $\Omega(L)$ queries. In and , we first describe hard functions for the line and the hypercube domain, respectively. The general hypergrid is addressed in . The Line {#sec:lb-line} -------- \[thm:lb-line\] Fix a parameter $\eps$. If $\cD$ is $(2\eps,\rho)$-stable, then any ${\eps}$-monotonicity tester w.r.t. $\cD$ for functions $f:[n]^d\mapsto\NN$ requires $\Omega(\rho\Delta^*(\cD))$ queries. Not surprisingly, the lower bound construction is also based on BSTs. We specifically use the *median BST* [@Melhorn75]. When $n=1$, then the tree is the singleton. For a general $n$, let $t \in [n]$ be the smallest index such that $\mu({\{1,\cdots,t\}}) \geq 1/2$ (henceforth, in this section, we use $\mu$ to denote $\mu_\cD$). The root of $T$ is $t$. Recur the construction on the intervals $[1,t-1]$ and $[t+1,n]$. By construction, the probability mass of any subtree together with its parent is greater than the probability mass of the sibling subtree. This [*median property*]{} will be utilized later. We follow the framework of to construct a collection of hard functions. The monotone function $h$ can be anything; $h(i) = 3i$ works. We will construct a function $g_j$ ($j \geq 1$) for each non-root level of the median BST. Consider the nodes at depth $j-1$ (observe the use of $j-1$, and not $j$). Each of these corresponds to an interval, and we denote this sequence of intervals by ${\I^{1}_{j}}, {\I^{2}_{j}}, \ldots$. (Because internal nodes of the tree are also elements in $[n]$, there are gaps between these intervals.) Let $L_{\geq j} := \{x: \depth_T(x)\geq j\}$ be the nodes at depth $j$ and higher. We have the following simple claim. \[clm:line-interval\] ${\I^{k}_{j}}$ can be further partitioned into ${\I^{k,\lleft}_{j}}$ and ${\I^{k,\rright}_{j}}$ such that $\sum_k \min\left(\mu({\I^{k,\lleft}_{j}}), \mu({\I^{k,\rright}_{j}})\right)\!\geq\! \frac{\mu(L_{\geq j})}{2}.$ Consider the node $u_k$ corresponding ${\I^{k}_{j}}$, and let the nodes in the left and right subtrees be $S_\ell$ and $S_r$. If $\mu(S_\ell) \leq \mu(S_r)$, then ${\I^{k,\lleft}_{j}} = S_\ell\cup u_k$ and ${\I^{k,\rright}_{j}} = S_r$. Otherwise, ${\I^{k,\lleft}_{j}} = S_\ell$ and ${\I^{k,\rright}_{j}} = u_k \cup S_r$. By the median property of the BST, $\min(\mu({\I^{k,\lleft}_{j}}), \mu({\I^{k,\rright}_{j}})) = \max(\mu(S_\ell),\mu(S_r))$ $\geq (\mu(S_\ell) + \mu(S_r))/2$. We describe the non-monotone $g_j$’s and follow up with some claims. Let ${{\tt lca}}(x,y)$ denote the least common ancestor of $x$ and $y$ in $T$. $$\label{eq:try} g_{j}(x) = \left\{ \begin{array} {l l l} 2x \quad \textrm{if $x \notin \bigcup_k {\I^{k}_{j}}$}\\ 2x + 2(b-m) + 1 \quad \textrm{if $x \in {\I^{k,\lleft}_{j}} = [a,m]$, ~~where ${\I^{k}_{j}} = [a,b]$.}\\ 2x - 2(m-a) - 1 \quad \textrm{if $x \in {\I^{k,\rright}_{j}} = [m+1,b]$, where ${\I^{k}_{j}} = [a,b]$.} \end{array} \right.$$ [claim]{}[gj]{} \[clm:gj\] [(i)]{} ${\dist_{\cD}(g_j,\MON)}\!\geq\!\frac{\mu(L_{\geq j})}{2}$. [(ii)]{} If $(x,y)$ distinguishes $g_j$ from $h$, then ${{\tt lca}}(x,y)$ lies in level $(j-1)$. All elements in ${\I^{k,\lleft}_{j}}$ are in violation with all elements in ${\I^{k,\rright}_{j}}$ for all $k$. To see this, let $x\in {\I^{k,\lleft}_{j}}$ and $y\in {\I^{k,\rright}_{j}}$, and so $x \prec y$. Denote ${\I^{k}_{j}} = [a,b]$, $$g_j(x) - g_j(y) = 2x + 2(b-m) + 1 - 2y + 2(m-a) +1 = 2(x-a) +2(b-y) + 2 > 0$$ The vertex cover of the violation graph of $g_i$ has mass at least $\sum_k \min(\mu({\I^{k,\lleft}_{j}}),\mu({\I^{k,\rright}_{j}})) \geq \mu(L_{\geq j})/2$ (). This proves part (i). To prove part (ii), let $x \prec y$ distinguish $g_j$ from $h$, so $g_j(x) > g_j(y)$. We claim there exists a $k^*$ such that $x \in {\I^{k^*,\lleft}_{j}}$ and $y\in {\I^{k^*,\rright}_{j}}$. For any ${\I^{k}_{j}} = [a,b]$, the $g_j$ values lie in $[2a+1,2b+1]$. Hence, if $x \in {\I^{k}_{j}}$ and $y \notin {\I^{k}_{j}}$ (or vice versa), $(x,y)$ is not a violation. So $x$ and $y$ lie in the same ${\I^{k^*}_{j}}$, But the function restricted to ${\I^{k^*,\lleft}_{j}}$ or ${\I^{k^*,\rright}_{j}}$ is increasing, completing the proof. The following claim is a simple combinatorial statement about trees. [claim]{}[lcaa]{}\[clm:lca\] Given a subset $Q$ of $[n]$, let ${{\tt lca}}(Q) = {\{{{\tt lca}}(x,y) : x, y \in Q\}}$. Then $|{{\tt lca}}(Q)| \leq |Q| - 1$. The proof is by induction on $|Q|$. The base case of $|Q| = 2$ is trivial. Suppose $|Q| > 2$. Consider the subset $P \subseteq Q$ of all elements of $Q$, none of whose ancestors are in $Q$. Also observe that if $P = Q$, then ${{\tt lca}}(Q)$ are precisely the internal nodes of a binary tree whose leaves are $Q$, and therefore $|{{\tt lca}}(Q)| \leq |Q|-1$. If $P$ is a singleton, then ${{\tt lca}}(Q) = {{\tt lca}}(Q \setminus P) + 1 \leq |Q \setminus P| -1 + 1 = |Q|-1$ (inequality from induction hypothesis). So assume $P \subset Q$ and $|P| \neq 1$. For $p \in P$, let $S_p$ be the set of elements of $Q$ appearing in the tree rooted at $p$. For every $x \in S_p$ and $y \in S_{p'}$ ($p \neq p'$), ${{\tt lca}}(x,y) = {{\tt lca}}(p, p')$. Furthermore, the sets $S_p$ non-trivially partition $Q$. Therefore, ${{\tt lca}}(Q) = {{\tt lca}}(P) \cup \bigcup_{p \in P}{{\tt lca}}(S_p)$. Applying the induction hypothesis, $|{{\tt lca}}(Q)| \leq |P| - 1 + \sum_{p \in P} |S_p| - |P| = |Q| - 1$. Let $\ell_\eps$ be the largest $\ell$ such that $\mu(L_{\geq \ell}) \geq 2\eps$. By .(i), the collection of functions $\{g_1,\ldots,g_{\scriptscriptstyle \ell_\eps}\}$ are each $\eps$-far from monotone. By .(ii) and , a subset $Q \subseteq [n]$ can’t distinguish more than $|Q|$ of these functions from $h$. gives an $\Omega(\ell_\eps)$ lower bound and follows from . \[clm:lepsilon\_is\_large\] $\ell_{\eps}\geq \rho\Delta^*(\cD)$. Consider the distribution $\cD'$ that transfers all the mass from $L_{\geq \ell_\eps+1}$ to the remaining vertices proportionally. That is, if $\nu:=\mu(L_{\geq \ell_\eps+1})$, then $\mu_{\cD'}(i) = 0$ for $i\in L_{\geq \ell_\eps+1}$, and $\mu_{\cD'}(i) = \mu_{\cD}(i)/(1-\nu)$ for the rest. Observe that $||\cD - \cD'||_{\TV} = \mu_{\cD}(L_{\geq \ell_\eps+1}) < 2\eps$. Also observe that since $T$ is a binary tree of height $\ell_\eps$, $\ell_\eps \geq \Delta^*(\cD')$: the LHS is the max depth, the RHS is the (weighted) average depth. Now, we use stability of $\cD$. Since $\cD$ is $(2\eps,\rho)$-stable, $\Delta^*(\cD') \geq \rho\Delta^*(\cD)$. The Boolean hypercube {#sec:lb-cube} --------------------- For the boolean hypercube, the lower bound doesn’t require the stability assumption. Any product distribution over $\{0,1\}^d$ is determined by the $d$ fractions $(\mu_1,\ldots,\mu_d)$, where $\mu_r$ is the probability of $0$ on the $r$-th coordinate. Let $\theta_r := \min(\mu_r,1-\mu_r)$. \[thm:lb-cube\] Any monotonicity tester w.r.t. $\cD$ for functions $f:\{0,1\}^d\mapsto\NN$ with distance parameter ${\eps}\leq 1/10$ must make $\Omega\left(\sum_{r=1}^d \min(\mu_r,1-\mu_r)\right)$ queries. We begin with the basic setup. A tester for non-trivial $\eps$ makes at least $1$ query, so we can assume that $\sum_{r=1}^d \theta_r > 1$. For ease of exposition, assume $\theta_r = \mu_r$, for all $1\leq r \leq d$. (If not, we need to divide into two cases depending on $\theta_r$ and argue analogously for each case.) Assume wlog $\theta_1\leq \theta_2 \leq \cdots \leq \theta_d$. Partition $[d]$ into contiguous segments $I_1,\ldots,I_b,I_{b+1}$ such that for each $1\leq a\leq b$, $\sum_{r\in I_a}\theta_r \in [1/2,1)$. Observe that $b = \Theta\left(\sum_r\theta_r\right)$. For $1\leq a\leq b$, define the indicator functions $\chi_a:\{0,1\}^d\mapsto\{0,1\}$ as follows: $$\chi_a(x) = \left\{ \begin{array} {l l} 1& \quad \textrm{if $\forall i \in I_a, x_i=1$}\\ 0& \quad \textrm{otherwise \quad ($\exists i \in I_a, x_i = 0$)} \end{array} \right.$$ By , we need to define the set of functions with appropriate properties. The monotone function $h(\cdot)$ is defined as $h(x) = \sum_{a=1}^b \chi_a(x)2^a$. The functions $g_1,\ldots,g_b$ are defined as $$g_a(x) = \left\{ \begin{array} {l l} h(x) - 2^{r} - 1& \quad \textrm{if $\chi_a(x) = 1$}\\ h(x) & \quad \textrm{if $\chi_a(x) = 0$} \end{array} \right.$$ We prove all the desired properties. For all $a$, $\dist_\cD(g_a,\MON) \geq 1/10$. Let $I$ denote $I_a$, and $J = [n] \setminus I$. Think of $x = (x_{I}, x_{J})$. Fix $\v$ in $\{0,1\}^{|J|}$, and define sets $X_1(\v) := \{x | \chi_a(x) = 1, x_{J}=\v\}$ (a singleton) and $X_0(\v) = \{x | \chi_a(x) = 0, x_{J}=\v\}$. Note that $\bigcup_\v(X_1(\v)\cup X_0(\v))$ forms a partition of the cube. For $c \neq a$, $\chi_c(x)$ is the same for all $x \in (X_0(\v) \cup X_1(\v))$. Hence, for [*any*]{} $x \in X_0(\v)$ and $y \in X_1(\v)$, $x \prec y$ and $g_a(x) > g_a(y)$. Any vertex cover in the violation graph must contain either $X_1(\v)$ or $X_0(\v)$, for each $\v$. Let $\cD_I$ be the conditional distribution on the $I$-coordinates. In the following, we use the inequalities $\sum_{i \in I}\theta_i \in [1/2,1)$ and $1 - t \in [e^{-2t},e^{t}]$ for $t\leq 1/2$. $$\begin{aligned} \mu_{\cD_I}(X_1(\v)) & = & \prod_{i\in I}(1-\theta_i) \geq \exp(-2\sum_{i\in I}\theta_i) \geq e^{-2} > 1/10 \\ \mu_{\cD_I}(X_0(\v)) & = & 1 - \prod_{i\in I}(1-\theta_i) \geq 1 - \exp(-\sum_{i\in I}\theta_i) \geq 1 - e^{-1/2} > 1/10.\end{aligned}$$ For each $\v$, the conditional mass of the vertex cover is at least $1/10$, and therefore, the $\mu_\cD$ mass of the vertex cover is at least $1/10$. A pair $x,y$ in $[n]^d$ *captures* index $a$ if $a$ is the largest index such that $\chi_a(x) \neq \chi_a(y)$. Furthermore, a set $Q$ captures $a$ if it contains a pair capturing $a$. [claim]{}[hccap]{} \[clm:hccap\] If $Q$ distinguishes $g_a$ from $h$, then $Q$ must capture $a$. Consider $x, y \in Q$ where $h(x) < h(y)$ but $g_a(x) > g_a(y)$. It must be that $\chi_a(x) = 0$ and $\chi_a(y) = 1$. Suppose this pair does not capture $a$. There must exist index $c > a$ (let it be the largest) such that $\chi_c(x) \neq \chi_c(y)$. Because $h(x) < h(y)$, $\chi_c(x) = 0$ and $\chi_c(y) = 1$. By definition, $g_a(y) - g_a(x) = (h(y) -2^a - 1) - h(x)$. We have $h(y) - h(x) = \sum_{t=1}^c (\chi_t(y) - \chi_t(x))2^t$. Since $\chi_a(x) = \chi_c(x) = 0$ and $\chi_a(y) = \chi_c(y) = 1$, $h(y) - h(x) \geq 2^c + 2^a - \sum_{t < c:t\neq a} 2^t$. Combining, $$g_a(y) - g_a(x) \geq 2^c + 2^a - \sum_{t < c:t\neq a} 2^t - 2^a - 1 = 2^a > 0.$$ \[clm:cap\]\[Lifted from [@ChSe13-2].\] A set $Q$ captures at most $|Q|-1$ coordinates. We prove this by induction on $|Q|$. When $|Q| = 2$, this is trivially true. Otherwise, pick the largest coordinate $j$ captured by $Q$ and let $Q_0 =\{x: x_j = 0\}$ and $Q_1 = \{x:x_j = 1\}$. By induction, $Q_0$ captures at most $|Q_0|-1$ coordinates, and $Q_1$ captures at most $|Q_1| -1$ coordinates. Pairs $(x,y)\in Q_0\times Q_1$ only capture coordinate $j$. The total number of captured coordinates is at most $|Q_0| - 1 + |Q_1| - 1 + 1 = |Q| - 1$. We can now invoke to get an $\Omega(b)=\Omega(\sum_r\theta_r)$ lower bound thereby proving . The hypercube lower bound can be generalized to give a weak lower bound for hypergrids, which will be useful for proving the stronger bound. Fix a dimension $r$. For any $1\leq j\leq n$, define $\theta^j_r := \min(\sum_{k\leq j} \mu_{\cD_r}(k), 1 - \sum_{k\leq j} \mu_{\cD_r}(k))$. Define $\theta_r := \max_{1\leq j\leq n} \theta^j_r$. Note that $\theta_r$ generalizes the above definition for the hypercube. The following theorem follows by a reduction to the hypercube lower bound. [theorem]{}[hgpart]{} \[thm:hg-part1\]\[thm:lb-hg-part1\] Any monotonicity tester on the hypergrid with distance parameter $\eps\leq 1/10$, makes $\Omega\left(\sum_{r=1}^d \theta_r\right)$ queries. For $1\leq r \leq d$, let $1\leq j_r\leq n$ be the $j$ such that $\theta_r = \theta^j_r$. Project the hypergrid onto a Boolean hypercube using the following mapping $\psi:[n]^d\to\{0,1\}^d$: for $x\in [n]^d$, $\psi(x)_r = 0$ if $x_r \leq j_r$, and $1$ otherwise. The corresponding product distribution $\cD'$ on the hypercube puts $\mu_{\cD'_r}(0) = \sum_{k\leq j_r}\mu_{\cD_r}(k)$, for all $r$. Note that $\min(\mu_r,1-\mu_r) = \theta_r$. Given any function $f$ on $\{0,1\}^d$, extend it to $g$ over the hypergrid in the natural way: for $x\in [n]^d$, $g(x) = f(\psi(x))$. Note that $\dist_{\cD'}(f,\MON) = \dist_{\cD}(g,\MON)$. (This is akin to .) Any tester for $g$ over $[n]^d$ induces a tester for $f$ on $\{0,1\}^d$ with as good a query complexity: whenever the hypergrid tester queries $x\in [n]^d$, the hypercube tester queries $\psi(x)$. Therefore, the lower bound for the hypercube implies . The Hypergrid. {#sec:lb-hypergrid} -------------- \[sec:lb-hg\] Our main lower bound result is the following, which implies via . \[thm:lb-hg\] For any parameter ${\eps}< 1/10$, and for any $(\const\eps,\rho)$-stable, product distribution $\cD$, any (even adaptive, two-sided) montonicity tester w.r.t. $\cD$ for functions $f:[n]^d \mapsto \NN$ with proximity parameter $\eps$ requires $\Omega(\rho\Delta^*(\cD))$ queries. ### The intuition {#sec:lb-intuit} Since we already have a proof for $d=1$ in , an obvious approach to prove is via some form of induction on the dimension. Any of the $g_j$-functions on $[n]$ in can be extended the obvious way to a function on $[n]^d$. Given (say) $g_j:[n] \mapsto \NN$, we can define $f:[n]^d \mapsto \NN$ as $f(x) = g_j(x_1)$. Thus, we embed the hard functions for $\cD_1$ along dimension $1$. One can envisage a way do the same for dimension $2$, and so on and so forth, thereby leading to $\sum_i \Delta^*(\cD_i)$ hard functions in all. There is a caveat here. The construction of for (say) $\cD_1$ requires the stability of $\cD_1$. Otherwise, we don’t necessarily get $\Omega(\Delta^*(\cD_1))$ functions with distance at least ${\eps}$. For instance, if the root of the median BST has more than $(1-{\eps})$ fraction of the weight, we get at most one hard function of distance at least ${\eps}$. So, the above approach requires stability of all the *marginals* of $\cD$. Unfortunately, there exist stable product distributions with all marginals unstable. Consider $\cD = \prod_r \cD_r$, where each $\cD_r = (\frac{1}{(n-1)d},\ldots,\frac{1}{(n-1)d}, 1-\frac{1}{d})$. Note that $\Delta^*(\cD) \approx \log n$. Each $\cD_r$ is individually unstable (for $\eps > 1/d$), since there is a $\cD'_i$ with all the mass on the $n$th coordinate, such that $\|\cD_i - \cD'_i\|_{TV} = 1/d$ and $\Delta^*(\cD'_i) = 0$. On the other hand, it is not hard to see that $\cD$ is $(1/100,1/100)$-stable. A new idea is required to construct the lower bound. [To see this, suppose there is a product distribution $\cD'$ such that $\Delta^*(\cD') < \Delta^*(\cD)/100 = (\log n)/100$. Markov’s inequality implies that for $\Omega(d)$ dimensions, $\|\cD'_r - \cD_r\|_{\TV} = \Omega(1/d)$. A calculation shows that $\|\cD - \cD'\|_{\TV}$ must be at least $1/100$. In sum, for any constants $\eps,\rho$, there exist $(\eps,\rho)$-stable distributions $\cD$ such that each marginal $\cD_r$ is only $(\eps/d,\rho)$-stable. This is a major roadblock for a lower bound construction, and therefore a new idea is required.]{} [We design an *aggregation* technique that does the following. Start with 1D functions $g^1_{j_1}$ and $g^2_{j_2}$ that are hard functions from for $\cD_1$ and $\cD_2$ respectively. Suppose the corresponding distances to monotonicity are $\eps^{(1)}$ and $\eps^{(2)}$. We construct a function $f:[n]^d \mapsto \NN$ that is $\eps^{(1)} + \eps^{(2)}$-far, so we can effectively add their distances. If we can aggregate $\Omega(d)$ 1D functions, each with distance $\eps/d$, then we get a desired hard function. As can be expected, this construction is quite delicate, because we embed violations in many dimensions simultaneouly. Furthermore, we need to argue that this aggregation can produce enough “independent" hard functions, so we get a large enough lower bound (from ). And that is where the hard work lies. ]{} ### Setup and Construction Fix ${\eps}$ and let ${\eps}' = \const{\eps}$. Fix the $({\eps}',\rho)$-stable distribution $\cD$. Since $\cD$ is $({\eps}',\rho)$-stable, for any $\cD'$ with $\|\cD'-\cD\|_{\TV} \leq {\eps}'$, we have $\Delta^*(\cD') \geq \rho\Delta^*(\cD)$. We denote the median BST for $\cD_r$ as $T_r$, $\Delta_r$ as the expected depth w.r.t. $\cD_r$, and $\Delta(\cD) = \sum_{r=1}^d \Delta_r$. The following shows that the median BST is near optimal. \[lem:median\] For any product distribution $\cD = \prod_r \cD_r$, $\Delta(\cD) \leq 5\Delta^*(\cD)$. Fix a coordinate $r$. The depth of a vertex $u$ in $T_r$ is at most $\log_2(1/\mu_{\cD_r}(u))$, so we get $\Delta_r \leq H(\cD_r)$, the Shannon entropy of $\cD_r$. It is also known (cf. Thm 2 in [@Melhorn75]) that $H(\cD_r) \leq \log_23(\Delta^*(\cD_r) + 1)$. To see this, notice that any BST can be converted into a prefix-free ternary code of expected length $(\Delta^*(\cD_r) +1)$, say, over the alphabet ‘left’,‘right’, and ‘stop’. Therefore, if $\Delta^*(\cD_r) \geq 1/2$, we have $\Delta_r \leq 5\Delta^*(\cD_r)$. If $\Delta^*(\cD_r) < 1/2$, then since $\Delta^*(\cD_r) \geq 1 - \Pr[\textrm{root}]$, we get $\mu^* := \mu_{\cD_r}(u^*) > 1/2$ where $u^*$ is the root of the optimal BST $T^*$. But this implies $u^*$ is also the root of $T_r$ by construction of the median BST. Now we can prove via induction. If $p$ and $q$ are the total masses of the nodes in the left and right sub-tree of $T^*$ (and therefore also $T_r$), and $\Delta^*_1$ (resp. $\Delta_1$) and $\Delta^*_2$ (resp. $\Delta_2$)be the expected depths of these subtrees in $T^*$ (resp. $T_r$), then we get, $\Delta^*(\cD_r) = p\Delta^*_1 + q\Delta^*_2 + (1-\mu^*) \leq 5p\Delta_1 + 5q\Delta_2 + (1-\mu^*) \leq 5\Delta_r$. requires the definition of a monotone function and a collection of $\eps$-far from monotone functions with additional properties. The monotone function is $\val(x) := \sum_{r=1}^d 2(2n+1)^r x_r$. The non-monotone functions (which we refer to as “hard" functions) are constructed via aggregation. From , for each dimension $r$ and each level $j \geq 1$ in tree $T_r$, we have a 1D “hard" function ${g^{\scriptscriptstyle r}_{\scriptscriptstyle j}}:[n] \mapsto \NN$. It is useful to abstract out some of the properties of ${g^{\scriptscriptstyle r}_{\scriptscriptstyle j}}$ that were proved in . Let ${L^{r}_{\scriptscriptstyle j}}$ be the nodes in $T_r$ at level $j$. Each level corresponds to a collection of intervals of $[n]$. We use ${L^{r}_{\scriptscriptstyle \geq j}} := \bigcup_{j' \geq j} {L^{r}_{\scriptscriptstyle j'}}$ and ${L^{r}_{\scriptscriptstyle < j}} = \bigcup_{j' < j} {L^{r}_{\scriptscriptstyle j'}}$. We use the shorthand ${\mu^{r}_{\scriptscriptstyle \geq j}}$ to denote $\mu_{\cD_r}({L^{r}_{\scriptscriptstyle \geq j}})$. The following lemma is a restatement of and . \[lem:hard-prop\] Consider ${g^{\scriptscriptstyle r}_{\scriptscriptstyle j}}:[n] \mapsto \NN$, for $j \geq 1$ All violations to monotonicity are contained in intervals corresponding to ${L^{r}_{\scriptscriptstyle j-1}}$, and the distance to monotonicity is at least ${\mu^{r}_{\scriptscriptstyle \geq j}}/2$. Furthermore, any violation $(x,y)$ has ${{\tt lca}}(x,y)$ in ${L^{r}_{\scriptscriptstyle j-1}}$. The aggregation process takes as input a *map* $\psi:[d] \mapsto \{\bot\} \cup \{2,3,4,\ldots\}$. Note that if $\psi(r) \neq \bot$, then $\psi(r) > 1$. Informally, $\psi(r)$, when not equating to $\bot$, tells us the level of $T_r$ whose hard function is to be included in the aggregation. We define ${{\Psi^{-1}}}:= \{r | \psi(r) \neq \bot\}$, the subset of relevant dimensions. Given the map $\psi$, we aggregate the collection of 1D functions $\{{g^{\scriptscriptstyle r}_{\scriptscriptstyle \psi(r)}} | r \in {{\Psi^{-1}}}\}$ into a single hard function for $[n]^d$ as follows. $$\label{eq:gg} \gg(x) := \sum_{r \in {{\Psi^{-1}}}} (2n+1)^r {g^{\scriptscriptstyle r}_{\scriptscriptstyle \psi(r)}}(x_r) + \sum_{r \notin {{\Psi^{-1}}}} 2(2n+1)^r x_r$$ Observe that the latter sum is identical to the corresponding portion in $\val(x)$. The first summand takes the hard function corresponding to the $\psi(r)$th level of $T_r$ for $r\in {{\Psi^{-1}}}$ and aggregates them via multiplying them with a suitable power of $(2n+1)$. \[def:psi\] A map $\psi$ is ***useful*** if the following are true. $\sum_{r \in {{\Psi^{-1}}}} {\mu^{r}_{\scriptscriptstyle \geq \psi(r)}} \in (\eps',1)$ For all $r \in {{\Psi^{-1}}}$, ${\mu^{r}_{\scriptscriptstyle \geq \psi(r)}} \geq \frac{{\mu^{r}_{\scriptscriptstyle \geq \psi(r)\!-\!1}}}{2}$. In plain English, the first point states that total distance of the hard functions picked should be at least ${\eps}'$. The second point is a technicality which is required to argue about the distance of the aggregated function. It states that in each relevant $T_r$, the total mass on the nodes lying in the $\psi(r)$th layer and below shouldn’t be much smaller than the total mass on the nodes lying on the $(\psi(r)\!-\!1)$th layer and below. \[lem:dist-to-mon\] If $\psi$ is useful, $\dist_\cD(\gg,\MON) \geq \eps$. It is convenient to consider restrictions of $\gg$ where all coordinates in $[d] \setminus {{\Psi^{-1}}}$ are fixed. This gives rise to $|{{\Psi^{-1}}}|$-dimensional functions. We argue that each such restriction is $\eps$-far from monotone, which proves the lemma. Abusing notation, we use $\gg$ to refer to an arbitrary such restriction. Fix some $r \in {{\Psi^{-1}}}$. Define the subset $S_r := \{x\in [n]^d: x_s\in {L^{s}_{\scriptscriptstyle < \psi(s)-1}}, ~\forall s\neq r\}$ to be the set of points $x$ with the $s$th coordinate appearing in the first $(\psi(s)-2)$ layers of the tree $T_s$, for all $s\neq r$. We stress that this is well-defined because $\psi(s) \geq 2$ by definition of $\psi$. Note that each $S_r$ is a collection of $r$-lines and the restriction of $\gg$ on each line exactly a multiple of ${g^{\scriptscriptstyle r}_{\scriptscriptstyle \psi(r)}}$. By , all violations to monotonicity in such lines lie in the intervals corresponding to ${L^{r}_{\scriptscriptstyle \geq \psi(r)-1}}$, and the mass of the vertex cover of the violation graph (restricted to the line) is at least ${\mu^{r}_{\scriptscriptstyle \geq \psi(r)}}/2$. Thus the total contribution to distance of $\gg$ from $S_r$ is at least $\frac{{\mu^{r}_{\scriptscriptstyle \geq \psi(r)}}}{2}\cdot \mu_{\cD_{-r}}(\prod_{s \neq r} {L^{s}_{\scriptscriptstyle < \psi(s)-1}})$. What is crucial to note is that the regions of violations in $S_r$ is [*disjoint*]{} from the regions of violation in $S_{r'}$ for $r'\neq r$. Therefore, the contributions to the distance of $\gg$ add up, and this gives $$\begin{aligned} \dist_\cD(\gg,\MON) & \geq & \frac{1}{2} \sum_{r \in {{\Psi^{-1}}}} {\mu^{r}_{\scriptscriptstyle \geq \psi(r)}} \cdot \mu_{\cD_{-r}}\Big(\prod_{s \neq r} {L^{s}_{\scriptscriptstyle < \psi(s)-1}}\Big) \\ & = & \frac{1}{2} \sum_{r \in {{\Psi^{-1}}}} {\mu^{r}_{\scriptscriptstyle \geq \psi(r)}} \prod_{s \neq r} (1 - {\mu^{s}_{\scriptscriptstyle \geq \psi(s)-1}}) \\ & \geq &\frac{1}{2} \sum_{r \in {{\Psi^{-1}}}} {\mu^{r}_{\scriptscriptstyle \geq \psi(r)}} \prod_{s \neq r} (1 - 2{\mu^{s}_{\scriptscriptstyle \geq \psi(s)}}) \ \ \ \ \textrm{(point 2 in def. of useful map)}\end{aligned}$$ We can apply the bound, $\sum_{r \in {{\Psi^{-1}}}} {\mu^{r}_{\scriptscriptstyle \geq \psi(r)}} \in (\eps',1)$, since $\psi$ is useful. We lower bound the product by $\exp(-4\sum_{s \neq r} {\mu^{s}_{\scriptscriptstyle \geq \psi(s)}})$, which by the above bound, is at least $e^{-4}$. So, $\dist_\cD(\gg,\MON) \geq \sum_{r \in {{\Psi^{-1}}}} {\mu^{r}_{\scriptscriptstyle \geq \psi(r)}}/\const \geq \eps'/\const = {\eps}$. \[def:disj\] Two maps $\psi_1, \psi_2$ are *disjoint* if: $\{(r,\psi_1(r)) | r \in \Psi_1^{-1}\}$ and $\{(r,\psi_2(r)) | r \in \Psi_2^{-1}\}$ are disjoint. That is, for every tree $T_r$, $\psi_1$ and $\psi_2$ point to different layers of the tree (or they point to $\bot$). \[lem:disting\] Consider a set of maps $\psi_1, \psi_2, \ldots$ that are all pairwise disjoint. A set of $Q$ queries can distinguish at most $|Q|-1$ of these functions from $\val$. Say a pair $(x,y)$ of queries captures the (unique) tuple $(r,j)$ if the largest coordinate in which $x$ and $y$ differ is $r$, and furthermore ${{\tt lca}}(x_r,y_r)$ in $T_r$ lies in level $(j-1)$. A set $Q$ captures $(r,j)$ if some pair in $Q$ captures $(r,j)$. We first show that if $(x,y)$ distinguishes $\gg$ from $\val$ for some map $\psi$, then $(x,y)$ captures a pair $(r,\psi(r))$ for some $r \in {{\Psi^{-1}}}$. Assume wlog $\val(x) < \val(y)$, and so $\gg(x) > \gg(y)$. Let $a$ be the largest coordinate at which $x$ and $y$ differ; since $\val(x) < \val(y)$, we get $x_a < y_a$. Suppose ${g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(x_a)$ and ${g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(y_a)$ is not a violation. By the construction, this implies that ${g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(y_a) - {g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(x_a) \geq 1$. Furthermore, ${g^{\scriptscriptstyle r}_{\scriptscriptstyle \psi(r)}}$ is always in the range $[1,2n]$ for any $r$. $$\begin{aligned} \gg(y) -\gg(x) & = & (2n+1)^a({g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(y_a) - {g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(x_a)) + \sum_{r < a, r\in {{\Psi^{-1}}}} (2n+1)^r({g^{\scriptscriptstyle r}_{\scriptscriptstyle \psi(r)}}(y_r) - {g^{\scriptscriptstyle r}_{\scriptscriptstyle \psi(r)}}(x_r)) \\ & \geq & (2n+1)^a - (2n)\sum_{r<a} (2n+1)^r \\ & = & (2n+1)^a - (2n)\cdot \frac{(2n+1)^a - 1}{2n} ~~~~ > 0,\end{aligned}$$ So $({g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(x_a),{g^{\scriptscriptstyle a}_{\scriptscriptstyle \psi(a)}}(y_a))$ is a violation. Immediately, we deduce that $\psi(a) \neq \bot$, so $a \in {{\Psi^{-1}}}$. By , ${{\tt lca}}(x_a,y_a)$ lies in level $\psi(a) - 1$ of $T_a$, and hence, $(x,y)$ captures $(a,\psi(a))$. As we prove in , $Q$ queries can capture at most $|Q|-1$ such tuples. The proof is completed by noting the maps $\psi_1,\psi_2,\ldots$ are pairwise disjoint. [claim]{}[qlca]{}\[clm:qlca\] A nonempty set $Q$ can only capture at most $|Q|-1$ tuples $(r,j)$. Proof is by induction on $|Q|$. If $|Q|=2$, then the claim trivially holds. Assume $|Q| > 2$. Let $s$ be the largest dimension such that there are at least two points in $Q$ differing in that dimension. For $c=1$ to $n$, let $Q_c := \{x\in Q: x_s = c\}$. By definition, $Q_c \subset Q$. Reorder the dimensions such that $Q_c$ is non-empty for $c = 1 \ldots q \leq n$. By induction, each $Q_c$ captures at most $|Q_c|-1$ pairs for $1\leq c\leq q$. Consider $(x,y)$ with $x\in Q_c$ and $y\in Q_{c'}$ for $c \neq c'$. The largest coordinate where they differ is exactly $s$. All tuples captured by such pairs is of the form $(s,\ell)$, where $\ell$ is the ${{\tt lca}}$ in $T_s$ of some $c,c' \in \{1\ldots,q\}$. By , the total number of such points is at most $q-1$. Thus, the total number of tuples captured is at most $\sum_{a=1}^q|Q_a| - q + (q-1) = |Q|-1$. ### Constructing the maps Let us go back to the framework of . From and , it suffices to construct a sequence $\psi_1, \psi_2,\ldots$ of pairwise disjoint, useful maps. The number of such maps will exactly be our lower bound. The exact construction is a little tricky, since the conditions of usefulness are somewhat cumbersone. We use the following definition. **Allowed levels:** A level $j$ is allowed w.r.t. dimension $r$ if $j > 1$ and ${\mu^{r}_{\scriptscriptstyle \geq j}} \geq {\mu^{r}_{\scriptscriptstyle \geq j-1}}/2$. This is in lines with point 2 of the usefulness definition. **Level sets $A_r$:** $A_r$ is the set of allowed levels of tree $T_r$. [ It is convenient to define an abstract procedure that constructs these maps. We have a stack $S_r$ for each $r \in [d]$, whose elements are allowed levels. The stack $S_r$ is initialized with $A_r$ in increasing order, that is the head (top entry) of the stack is the least (that is, closest to root) level in $A_r$. In each *round*, we will construct a map $\psi$. Denote the head of $S_r$ by $h_r$. Note that $h_r > 1$ by definition of allowed levels. Maintain a running count initialized to $0$. We go through the stacks in an arbitrary order popping off a *single* element from each stack. In a round, we never touch the same stack more than once. When we pop $S_r$, we set $\psi(r) := h_r$ and add ${\mu^{r}_{\scriptscriptstyle \geq h_r}}$ to the running count. We stop as soon as the running count enters the interval $[\eps',1]$. For all $r$ for which $\psi(r)$ hasn’t been defined, we set $\psi(r) = \bot$. This completes the description of a single map. Observe, by definition of allowed levels and the stopping condition, $\psi$ is useful. When $\sum_{r=1}^d {\mu^{r}_{\scriptscriptstyle \geq h_r}} < \eps'$, we cannot complete the construction. So the procedure terminates, discarding the final map. Let the set of maps constructed be $\PPsi$. By construction, the maps are useful. Furthermore, they are pairwise disjoint, because once a layer is popped out, it never appears again. We now basically show that $|\PPsi|$ is large, using the $(\eps',\rho)$-stability of $\cD$. This proves that the number of hard functions is large. We have to first deal with an annoying corner case of $\cD$. ]{} \[thm:heavy\] If $\sum_r {\mu^{r}_{\scriptscriptstyle \geq 1}} > \rho \Delta(\cD)/12$, then any monotonicity tester requires $\Omega(\rho\Delta^*(\cD))$ queries. We simply apply the hypercube lower bound. Recall the definition of $\theta_r$ described before . Note that ${\mu^{r}_{\scriptscriptstyle \geq 1}}$ is simply the total $\cD_r$-mass of everything in $T_r$ other than the root. By the median property of the $T_r$, $\theta_r$ is ensured to be at least half of this mass, and hence $\theta_r \geq {\mu^{r}_{\scriptscriptstyle \geq 1}}/2$. Combining with , we get a lower bound of $\Omega(\sum_r {\mu^{r}_{\scriptscriptstyle \geq 1}})$, which by assumption, is $\Omega(\rho\Delta(\cD))$. An application of completes the proof. Now we come to the main bound of $|{\PPsi}|$. We need some setup for the proof. The following simple observation is crucial. This follows since $\Exp[Z] = \sum_{k\in \N}\Pr[Z\geq k]$, for any non-negative, integer valued random variable. \[clm:search-depth\] For all $r$, $\sum_{j\geq 1} {\mu^{r}_{\scriptscriptstyle \geq j}} = \Exp_{x\sim \cD_r}[\depth_{T_r}(x)] = \Delta_r$. The following lemma completes the entire lower bound. [lemma]{}[countx]{} \[lem:countX\] Suppose $\sum_r {\mu^{r}_{\scriptscriptstyle \geq 1}} \leq \rho \Delta(\cD)/12$. Then $|{\PPsi}|=\Omega(\rho\Delta(\cD))$. Let $h_r$ denote the head of $S_r$ when the procedure terminates. So, $\sum_{r=1}^d {\mu^{r}_{\scriptscriptstyle \geq h_r}} < \eps'$. For any $\psi \in \PPsi$, $\sum_{r \in {{{\Psi^{-1}}}}} {\mu^{r}_{\scriptscriptstyle \geq \psi(r)}} \leq 1$. Hence, $|{\PPsi}|$ is at least the total sum over popped elements ${\mu^{r}_{\scriptscriptstyle \geq j}}$. Writing this out and expanding out a summation, $$\begin{aligned} |{\PPsi}| \geq \sum_{r \in [d]}\sum_{j < h_r,j\in A_r} {\mu^{r}_{\scriptscriptstyle \geq j}} & = & \sum_{r \in [d]} \Big[\sum_{j = 1}^{h_r-1} {\mu^{r}_{\scriptscriptstyle \geq j}} - {\mu^{r}_{\scriptscriptstyle \geq 1}} - \sum_{1 < j < h_r: j \notin A_r} {\mu^{r}_{\scriptscriptstyle \geq j}} \Big] \end{aligned}$$ Recall that $h_r > 1$ and so the summations are well-defined. For any level $1 < j \notin A_r$, we have ${\mu^{r}_{\scriptscriptstyle \geq j}} < {\mu^{r}_{\scriptscriptstyle \geq j-1}}/2$. Therefore, $\sum_{1 < j < h_r: j \notin A_r} {\mu^{r}_{\scriptscriptstyle \geq j}} < \sum_{j = 1}^{h_r-2} {\mu^{r}_{\scriptscriptstyle \geq j}}/2$. Plugging this bound in and applying the lemma assumption, $$\begin{aligned} \label{eq:toprove} |{\PPsi}| \geq \sum_{r \in [d]} \sum_{j = 1}^{h_r-1} {\mu^{r}_{\scriptscriptstyle \geq j}}/2 - \sum_{r \in [d]} {\mu^{r}_{\scriptscriptstyle \geq 1}} \geq \sum_{r \in [d]} \sum_{j = 1}^{h_r-1} {\mu^{r}_{\scriptscriptstyle \geq j}}/2 - \rho \Delta(\cD)/12\end{aligned}$$ We need to lower bound the double summation above. Observe that the second summation is $\sum_{j\geq 1}{\mu^{r}_{\scriptscriptstyle \geq j}} - ({\mu^{r}_{\scriptscriptstyle \geq h_r}} + {\mu^{r}_{\scriptscriptstyle \geq h_r+1}} + \cdots )$. The first term, by is precisely $\Delta(\cD)$, and by definition of $h_r$, each of the terms in the parenthesis is at most $\eps'$. However, the number of terms in the parenthesis can be quite large, and this doesn’t seem to get any lower bound on the summation. Here’s where stability of $\cD$ saves the day. Construct a distribution $\cD'_r$ be the distribution on $[n]$ as follows. Move the entire probability mass away from ${L^{r}_{\scriptscriptstyle \geq h_r}}$ and distribute it on the ancestral nodes in level $(h_r-1)$ of $T_r$. More precisely, $\mu_{\cD'_r}(u) = 0$ if $u \in {L^{r}_{\scriptscriptstyle \geq h_r}}$, $\mu_{\cD'_r}(u) = \mu_{\cD_r}(u)$ if $u\in {L^{r}_{\scriptscriptstyle < h_r - 1}}$, and $\mu_{\cD'_r}(u) = \sum_{v} \mu_{\cD_r}(v)$ for $u\in {L^{r}_{\scriptscriptstyle h_r-1}}$ where the summation is over children $v$ of $u$ in $T_r$. Letting $\cD' := \prod_r \cD'_i$, we see that $||\cD-\cD'||_{\TV} \leq \sum_{r=1}^d {\mu^{r}_{\scriptscriptstyle \geq h_r}} < \eps'$. Since $\cD$ is $(\eps',\rho)$-stable, we get $\Delta^*(\cD') \geq \rho\Delta^*(\cD)$. Now we can apply on $\cD'_r$ and $T_r$ to get $\EX_{x \sim \cD'_r}[\depth_{T_r}(x)] = \sum_{j\geq 1}\mu_{\cD'_r}({L^{r}_{\scriptscriptstyle \geq j}}) = \sum_{j=1}^{h_r - 1} {\mu^{r}_{\scriptscriptstyle \geq j}}.$ This expected depth is by definition at least $\Delta^*(\cD'_r)$. Therefore, we get a lower bound of $\sum_{r=1}^d \Delta^*(\cD'_r)/2 = \Delta^*(\cD')/2$ on the double summation in . Using the stability of $\cD'$ this is at least $\rho\Delta^*(\cD)/2$. Substituting we get $$\begin{aligned} |{\PPsi}| & \geq & \rho\Delta^*(\cD)/2 - \rho \Delta(\cD)/12 \\ & \geq & \rho\Delta^*(\cD)/2 - 5\rho\Delta^*(\cD)/12 = \Omega(\rho\Delta^*(\cD)) ~~~~~~ \textrm{(by \Lem{median})}\end{aligned}$$ We put it all together to prove the main lower bound, . If $\sum_r {\mu^{r}_{\scriptscriptstyle \geq 1}} > \rho \Delta(\cD)/12$, proves . Otherwise, by we have constructed $\Omega(\rho\Delta^*(\cD))$ pairwise disjoint, useful maps. Each map yields a hard function of distance at least $\eps$ (by ), and these functions satisfy the conditions of , which implies . [0.9]{} [^1]: Microsoft Research, [[email protected]]{} [^2]: Pennsylvania State University, [[email protected]]{}, supported in part by NSF Grants CCF-0964655 and CCF-1320814 [^3]: Sandia National Labs, Livermore, [[email protected]]{} [^4]: Sandia National Labs, Livermore, [[email protected]]{}Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

--- abstract: 'The phase diagram of prolate and oblate particles in the restricted orientations approximation (Zwanzig model) is calculated. Transitions to different inhomogeneous phases (smectic, columnar, oriented, or plastic solid) are studied through minimization of the fundamental measure functional (FMF) of hard parallelepipeds. The study of parallel hard cubes (PHC’s) as a particular case is also included motivated by recent simulations of this system. As a result a rich phase behavior is obtained which include, apart from the usual liquid crystal phases, a very peculiar phase (called here discotic smectic) which was already found in the only existing simulation of the model, and which turns out to be stable because of the restrictions imposed on the orientations. The phase diagram is compared at a qualitative level with simulation results of other anisotropic particle systems.' author: - 'Yuri Martínez-Ratón' title: 'Bulk inhomogeneous phases of anisotropic particles: A fundamental measure functional study of the restricted orientations model' --- Introduction {#Introduccion} ============ Onsager first showed that the isotropic-nematic liquid crystal phase transition occurs in systems of anisotropic particles interacting via hard core repulsions [@Onsager]. He studied a system of hard spherocylinders in the limit of infinite anisometry $\kappa=(L+D)/D\to \infty$ ($\kappa$ is the spherocylinder length to breath ratio) using the second virial form of the free energy, which in this limit is exact for the isotropic phase. The effect that higher virial coefficients have in the isotropic-nematic transition was later studied by Zwanzig, who introduced a model of hard prolate uniaxial parallelepipeds with axes oriented along the three perpendicular directions [@Zwanzig]. This peculiar model, which obviously treats the orientational degrees of freedom in an unrealistic way, has the advantage of being accessible to the calculation of higher virial coefficients up to seventh order in the infinite aspect ratio limit. He showed that including all these virials the isotropic-nematic transition also occur, although the exact value of the coexisting nematic density strongly depends on the order of the approximation. The Padé approximant generated by the truncated cluster expansion provides a much more stable sequence of the parameters which characterize the transition [@Runnels]. This stability leaves little room for doubts regarding the existence of the transition in the model. The virial expansion resumed and expressed in the variable $y=\rho/(1-\rho v)$, with $\rho$ the number density of parallelepipeds and $v$ their volume, converges more rapidly than the traditional expansion in $\rho$, as was shown by Barboy and Gelbart for different hard particle geometries [@Barboy]. Thus, the so-called $y_3$ expansion of the Zwanzig model was applied to the study of the isotropic-nematic transition as well as to the study of properties of its interface [@Moore]. For the latter the authors applied the smoothed density approximation of the free energy functional in the spirit of Tarazona’s weighted density approximation for the fluid of hard spheres [@Tarazona]. The restricted orientations model for hard cylinders was also used to describe the structural properties of molecular fluids near hard walls or confined in a slit. This time the density functional was constructed from the bulk direct correlation function approximated by a linear combination of geometrical functions [@Rickayzen]. Computer simulations of a variety of models of nonspherical hard core particles showed that the excluded volume effects could not only account for the stability of nematics, but also for the existence of liquid crystal inhomogeneous phases such as the smectic [@Strobants_1] and columnar [@Veerman] phases. Particularly the complete phase diagram of freely rotating hard spherocylinders [@Bolhuis], including not only smectic, but also a plastic solid phase and different oriented solid phases was calculated. Several density functional theories, all of them based on weighted or modified weighted density approximations, are able to reproduce reasonably well the isotropic-smectic or nematic-smectic transitions [@Somoza; @Poniewierski; @Velasco] in the whole range of aspect ratios where the smectic is stable, and in some cases, transitions from the isotropic fluid to the plastic or oriented solid phases [@Graf]. In all these approximations the excess free energy is evaluated by integration of the free energy per particle of a reference fluid (typically spheres or hard parallel ellipsoids) evaluated at some weighted or effective density. In some cases, the employed weight is directly the normalized Mayer function between spherocylinders [@Poniewierski; @Velasco]; in others, it is calculated from the knowledge of the bulk correlation function of the reference fluid [@Velasco]. For the latter case, the term proportional to the Mayer function enters into the integrand as a multiplicative factor of the free energy per particle. The hard sphere free energy functional is recovered in both approaches as the limiting case of $L=0$. The fundamental measure theory (FMT) first developed for hard spheres by Rosenfeld [@Rosenfeld_1] was another starting point for constructing a density functional for anisotropic particles. In its general formalism the excess free energy density of the fluid is a function of some weighted densities obtained by convoluting the density profiles with weights which are characteristic functions of the geometry of a single particle whose integrals are the so-called fundamental measures: volume, surface area, and mean radius of the particles. Unfortunately, the Mayer function of two convex anisotropic bodies cannot be decomposed as a finite sum of convolutions of single particle weights [@Rosenfeld_2], which is the keystone for constructing such a functional. Thus, the low density limit of the direct correlation function is no more the Mayer function. In spite of this, Chamoux and Perera have taken advantage of Rosenfeld’s extension of FMT to hard convex bodies by using it to compute the direct correlation function and patching out the low density limit with the exact Mayer function [@Chamoux_1]. In this way they have obtained the equation of state for various convex hard bodies (such as hard ellipsoids, spherocylinders, and cut spheres), have predicted ordered phases and, recently, have study demixing in binary mixtures of rigidly ordered particles [@Chamoux_1; @Chamoux_2]. Following a similar procedure a density functional for anisotropic particles has been proposed which interpolates between the Rosenfeld’s hard sphere functional and Onsager’s functional for elongated rods. The resulting model was tested by calculating the isotropic-nematic transition in systems of hard spherocylinders and hard ellipsoids [@Cinacchi]. Although the authors of this work suggest that the resulting theory can be applied to the study of inhomogeneous systems, the huge computational efforts that their numerical implementations involve is the reason for the absence of any result in this direction. One way to circumvent this difficulty is to reduce the continuous orientational degree of freedom to three discrete orientations (Zwanzig model). Implementing this idea some authors have recently applied the Zwanzig model to the study of interfacial properties of the hard rod fluid interacting with a hard wall or confined in a slit, for a one-component [@vanRoij] and a polydisperse mixture [@Martinez-Raton_1], and also to the study of bulk and interfacial properties of hard platelet binary mixtures [@Harnau]. All these models are based on Onsager’s density functional approximation. The increase of the number of allowed orientations in this functional particularized for hard spherocylinders results in the presence of an artificial nematic-nematic transition in the one component fluid as the authors of Ref. [@vanRoij_1] have shown. This result indicates that certain cares must be taken in the direct extrapolation of the results obtained from this theory. FMF was also constructed for a mixture of parallel hard cubes combining Rosenfeld’s original ideas with a dimensional cross over constraint [@Cuesta]. The latter appears to be very important to describe correctly the structure of inhomogeneous fluids in situations of high confinement and to describe well the structural properties of the solid phase [@Tarazona_1]. The dimensional cross over has been used as an important ingredient to develop a density functional for a binary mixture of hard spheres and needles, assuming that the needles are too thin to interact with each other directly [@Schmidt]. Taking a ternary mixture of parallel hard cubes and scaling each species along one of the three Cartesian axes with the same scaling factor a FMF for the Zwanzig model is obtained. This functional has already been applied to the study of the effect that polydispersity has on the stability of the biaxial phase in a binary mixture of rods and plates [@Martinez-Raton_2] and on the relative stability of the smectic and columnar phases due to the presence of polydispersity [@Martinez-Raton_3]. The FMF for Zwanzig’s model in the homogeneous limit coincide with the scaled particle theory and thus with the so-called $y_3$ expansion which, as pointed out before, first began to be used in Ref. [@Moore] as a model to study the isotropic-nematic phase transitions in fluids of hard parallelepipeds. But this functional, through its minimization, also allows us to calculate inhomogeneous density profiles. This functional has been applied recently to study the isotropic-nematic interface of a binary mixture of hard platelets [@Bier]. Its structural and thermodynamic properties resulting from the FMF minimization show complete wetting by a second nematic. The same phenomenon was found in a binary mixture of hard spherocylinders [@Shundyak]. The phase diagram for Zwanzig’s model including the smectic, columnar, and solid phases has never been carried out, only spinodal instability boundaries have been traced [@Martinez-Raton_3]. The main purpose of this work is to obtain the complete phase diagram for this model and to compare the results with the only existing simulation of the lattice version of the model, which has been carried out for two different aspect ratios [@Casey]. This will test the predictive power of the FMF for anisotropic inhomogeneous phases. As a particular case, the system of parallel hard cubes will be studied. In Ref. [@Martinez-Raton_4] a bifurcation analysis and a Gaussian parametrization of the density profiles were used to calculate the free energy and pressure of the solid phase. Here a free minimization will be performed to calculate not only the solid but also the smectic and columnar phases and compare the obtained results with recent simulations of parallel hard cubes [@Jagla; @Groh]. FMF for Zwanzig model {#FMF} ===================== The FMF for hard parallelepipeds was already described in detail in Ref. [@Cuesta]. A brief summary of the theory will be presented here putting emphasis on its numerical implementation to calculate the equilibrium inhomogeneous phases. A ternary mixture of hard parallelepipeds of cross section $\sigma^2$ and length $L$ with their uniaxial axes pointing to the $x$, $y$, or $z$ directions is described in terms of their density profiles $\rho_{\mu}({\bf r})$ ($\mu=x,y,z$). Following the FMT for hard parallelepipeds in three dimensions the excess free energy density in reduced units can be written as [@Cuesta] $$\begin{aligned} \Phi_{\rm{exc}}({\bf r})=\Phi^{(1)}({\bf r})+\Phi^{(2)}({\bf r})+ \Phi^{(3)}({\bf r}), \label{three}\end{aligned}$$ where the $\Phi^{(k)}$’s are $$\begin{aligned} \Phi^{(1)}&=&-n_0\ln(1-n_3), \label{fi1}\\ \Phi^{(2)}&=&\frac{{\bf n}_1\cdot{\bf n_2}}{1-n_3}, \label{fi2}\\ \Phi^{(3)}&=&\frac{ n_{2x} n_{2y} n_{2z}}{(1-n_3)^2}, \label{fi3}\end{aligned}$$ with weighted densities $$\begin{aligned} n_{\alpha}({\bf r})= \sum_{\mu}\left[\rho_{\mu}\ast \omega^{(\alpha)}_{\mu}\right]({\bf r}), \label{weighted}\end{aligned}$$ i.e., they are sums of convolutions of the density profiles with the following weights: $$\begin{aligned} \omega^{(0)}_{\mu}({\bf r})&=&\frac{1}{8} \prod_{k=1}^3 \delta\left(\frac{\sigma_{\mu}^{k}}{2}-|x_k|\right), \label{w0}\\ \omega^{(3)}_{\mu}({\bf r})&=& \prod_{k=1}^3 \theta\left(\frac{\sigma_{\mu}^{k}}{2}-|x_k|\right), \label{w3}\\ {\omega}_{\mu}^{(1j)}({\bf r})&=& \frac{2\theta\left(\frac{\sigma_{\mu}^{j}}{2}-|x_j|\right)} {\delta\left(\frac{\sigma_{\mu}^{j}}{2}-|x_j|\right)} \omega^{(0)}_{\mu}({\bf r}), \label{w1}\\ {\omega}_{\mu}^{(2j)}({\bf r})&=& \frac{\delta\left(\frac{\sigma_{\mu}^{j}}{2}-|x_j|\right)} {2\theta\left(\frac{\sigma_{\mu}^{j}}{2}-|x_j|\right)} \omega^{(3)}_{\mu}({\bf r}), \label{w2}\end{aligned}$$ where the notation $x_k$ ($k=1,2,3$) for the $x$, $y$, and $z$ coordinates has been employed. The functions $\delta(x)$ and $\theta(x)$ are the usual delta Dirac and Hevisaide functions and $\sigma_{\mu}^{j}=\sigma+(L-\sigma)\delta_{\mu}^j$ with $\delta_{\mu}^j$ the Kronecker delta. The following constraints on the density profiles were imposed: (i) The solid phase has the simple parallelepipedic unit cell with uniaxial symmetry, i.e., the periods in the three spatial directions are $d_{\perp}$ for $x,y$ and $d_{\parallel}$ for $z$. The orientational director is selected parallel to $z$. (ii) The density profile of each species has the form $$\begin{aligned} \rho_{\mu}({\bf r})= \rho \gamma_{\mu}\sum_{{\bf k}={\bf 0}}^{{\bf n}}\alpha^{(\mu)}_{{\bf k}} \prod_{j=1}^3\cos\left( q_jk_jx_j\right), \label{profile}\end{aligned}$$ where $q_j=2\pi/d_j$ is the wave number along the $j$ direction, ${\bf k}=(k_1,k_2,k_3)$ is the vector defined by the reciprocal lattice numbers, and ${\bf n}=(n_1,n_2,n_3)$ is the vector at which the harmonic expansion is truncated. Thus, Eq. (\[profile\]) is the Fourier expansion of the density profiles $\rho_{\mu}({\bf r})$ truncated at some ${\bf n}$. This cutoff is selected in such a way that it guarantees small enough values of $\alpha_{{\bf n}}^{(\mu)}$. The first Fourier amplitudes of all species are fixed to one ($\alpha_{{\bf 0}}^{(\mu)}=1$) and consequently $V_{\rm{cell}}^{-1} \int_{V_{\rm{cell}}} d{\bf r}\rho_{\mu}({\bf r})=\rho \gamma_{\mu}$ with $V_{\rm{cell}}=d_{\perp}^2d_{\parallel}$ the unit cell volume, $\rho$ the mean total density over the unit cell, and $\gamma_{\mu}$ the occupancy probability of species $\mu$ in the unit cell, which obviously fulfills the condition $\sum_{\mu}\gamma_{\mu}=1$. In the plastic solid phase these occupancy probabilities are $1/3$ for each species while they deviate from this value in the oriented solid phase. The uniaxial symmetry also implies that $\gamma_{x}=\gamma_{y}=\gamma_{\perp}$, $\gamma_{z}=\gamma_{\parallel} =1-2\gamma_{\perp}$ and $\rho_x(x,y,z)=\rho_y(y,x,z)$, $\rho_z(x,y,z)=\rho_z(y,x,z)$. Thus, the Fourier amplitudes verify $\alpha^{(x)}_{(k_1,k_2,k_3)} =\alpha^{(y)}_{(k_2,k_1,k_3)}$ and $\alpha^{(z)}_{(k_1,k_2,k_3)}=\alpha^{(z)}_{(k_2,k_1,k_3)}$. The total number of Fourier amplitudes \[except the $(0,0,0)$ term of all species\] is reduced by these symmetries to $N_{\alpha}=(n_{\perp}+1)(n_{\parallel}+1)(3n_{\perp}+4)/2-2$, ($n_1=n_2\equiv n_{\perp}$, $n_3\equiv n_{\parallel}$) independent variables. These variables together with $\gamma_{\perp}$, $q_{\perp}$ and $q_{\parallel}$ span the variable space in which the FMF must be minimized. The density profiles of columnar and smectic phases are obtained from Eq. (\[profile\]) substituting ${\bf n}=(n_{\perp},n_{\perp},0)$ and ${\bf n}=(0,0,n_{\parallel})$. From the definition (\[weighted\]), Eqs. (\[w0\])-(\[w2\]) and the density profiles (\[profile\]), the weighted densities can be easily calculated resulting in $$\begin{aligned} n_{\alpha}({\bf r})&=&\rho\sum_{\mu,{\bf k}}\gamma_{\mu} \alpha^{(\mu)}_{{\bf k}}\chi_{\alpha,{\bf k}}^{(\mu)}\prod_{j=1}^3 \cos \left(q_jk_jx_j\right), \label{promediadas} \\ \chi_{0,{\bf k}}^{(\mu)}&=&\prod_{j=1}^3\phi_0\left(\xi_{j,{\bf k}}^{(\mu)} \right), \\ \chi_{3,{\bf k}}^{(\mu)}&=&v\prod_{j=1}^3\phi_3\left(\xi_{j,{\bf k}}^{(\mu)} \right), \\ \chi_{1j,{\bf k}}^{(\mu)}&=&\sigma_j^{\mu}\frac{\phi_3 \left(\xi_{j,{\bf k}}^{(\mu)}\right)}{\phi_0 \left(\xi_{j,{\bf k}}^{(\mu)}\right)}\chi_{0,{\bf k}}^{(\mu)}, \\ \chi_{2j,{\bf k}}^{(\mu)}&=&\frac{1}{\sigma_j^{\mu}}\frac{\phi_0 \left(\xi_{j,{\bf k}}^{(\mu)}\right)}{\phi_3 \left(\xi_{j,{\bf k}}^{(\mu)}\right)}\chi_{3,{\bf k}}^{(\mu)}, \end{aligned}$$ with $v=L\sigma^2$ the particle volume, $\phi_0(x)=\cos x$, $\phi_3(x)=\sin x/x$, and $\xi_{j,{\bf k}}^{(\mu)}=q_jk_j\sigma_j^{\mu}/2$. The substitution of Eqs. (\[profile\]) and (\[promediadas\]) into the free energy per unit cell $$\begin{aligned} \Phi\equiv\frac{\beta {\cal F}}{V_{\rm{cell}}}&=&V^{-1}_{\rm{cell}} \int_{V_{\rm{cell}}} d{\bf r} \left[\Phi_{\rm{id}}({\bf r})+\Phi_{\rm{exc}}({\bf r})\right], \label{perunit}\\ \Phi_{\rm{id}}({\bf r})&=&\sum_{\mu}\rho_{\mu}({\bf r})\left[ \ln \left(\rho_{\mu}({\bf r})\Lambda_{\mu}^3\right)-1\right],\end{aligned}$$ with $\Phi_{\rm{id}}({\bf r})$ the ideal part of the free energy density, and its minimization with respect to the $N_{\alpha}+3$ variables allows the calculation of the equilibrium free energy and the density profiles of inhomogeneous phases. To characterize the structure and orientational order of these phases the following total density and order parameter profiles will be used: $$\begin{aligned} \rho({\bf r})&=&\sum_{\mu}\rho_{\mu}({\bf r}), \label{total}\\ Q({\bf r})&=&1-\frac{3}{2}\frac{\left[\rho_x({\bf r})+\rho_y({\bf r})\right]} {\rho({\bf r})}. \label{param}\end{aligned}$$ The selection of $Q({\bf r})$ as an order parameter is motivated by its uniaxial symmetry property $Q(x,y,z)=Q(y,x,z)$ and its uniform limit value $Q=1-3\gamma_{\perp}$ ($-1/2\leq Q\leq 1$), which coincides with the usual definition of the nematic order parameter for the Zwanzig model: $Q=0$ ($\gamma_{\perp}=1/3$) for the isotropic phase and $Q=1$ ($\gamma_{\perp}=0$) for the perfectly aligned nematic phase. Although the solid and columnar phases might have local biaxiality \[$\rho_x(x,y,z)\neq \rho_y(x,y,z)$\], the integral over the unit cell of any previously defined biaxial order parameter is always equal to zero as a consequence of the symmetries of the density profiles. Phase diagrams {#PD} ============== The phase diagrams presented in this work were calculated for a set of aspect ratios ranging from $\kappa=0.1$ to $\kappa=10$, corresponding to the aspect ratios of the most anisotropic oblate and prolate parallelepipeds studied here. The volume of all particles (cubes or prolate or oblate parallelepipeds) are fixed to 1 and thus the mean packing fraction $\eta$ is equal to the mean density $\rho$. From the equation $v=L\sigma^2=1$ the parallelepiped edge lengths $L$ and $\sigma$ can be calculated as a function of the aspect ratio $\kappa=L/\sigma$ as $L=\kappa^{2/3}$ and $\sigma=\kappa^{-1/3}$. For each $\kappa$, fixing the mean density $\rho$ and using appropriate initial guesses for the $N_{\alpha}+3$ variables with symmetries corresponding to the smectic, columnar or solid phases, the energy per unit cell (\[perunit\]) was minimized and thus the free energy for each phase was obtained. Varying $\rho$ and repeating the former steps the free-energy branches of the different inhomogeneous phases have been calculated. The common tangent construction allowed the calculation of the coexisting densities between those phases in the case of first order transitions. To evaluate numerically the three dimensional integral of the free energy density (\[perunit\]) a Gauss-Chebyshev quadratures has been employed. Parallel hard cubes {#PHC} ------------------- This subsection is devoted to the study of the parallel hard cube system ($\kappa=1$). The PHC equation of states of the fluid and solid phases as obtained from the FMT and the Monte Carlo simulation results are compared. While the solid phase is very well described with this formalism the exact location of the fluid-solid transition is very poorly estimated. The fundamental reasons of this difference are discussed here through a critical analysis of the fluid equation of state resulting from the FMF. It will be shown that possible modifications of the FMF slightly improve the location of the transition point at the expense of the correct description of the solid branch. In Ref. [@Martinez-Raton_2] the PHC fluid was already studied with the same FMF but using a Gaussian parametrization for the density profile. Through a minimization procedure and also from a bifurcation analysis a second-order fluid-solid transition was found at $\rho=0.3143$ with a lattice period $d=1.3015$ and a fraction of vacancies $\nu=0.3071$ [@Martinez-Raton_2]. Recent simulations on the same system also showed a second-order transition to the solid but with very different transition parameters $\rho=0.48\pm 0.02$ in Ref. [@Jagla] and $\rho=0.533\pm0.010$ in Ref. [@Groh]. No evidence for the vacancies predicted by FMT was found, although the authors recognized that the vacancies might be suppressed by the boundary conditions in the small systems accessible to simulations [@Groh]. The main problem of the FMF for hard cubes is that it recovers in the homogeneous limit the scaled-particle equation of state, which overestimates the pressure calculated from the exact virial expansion up to seventh order. This expansion has a maximum at $\rho\approx 0.6$ and then goes down very quickly to reach negative values [@Swol]. The poorly convergent character of the virial series makes it impossible to construct an equation of state for hard cubes, such as the Carnahan-Starling equation for the hard-sphere fluid, which estimates reasonably well all the known virial coefficients and diverges at close packing. On the other hand, it is well known that the FMF describes accurately the fluid structure in situations of high confinement, including the solid phase near close packing. For example, at high densities the functional recovers the cell theory, which is asymptotically exact when the packing fraction goes to 1, and also compares reasonably well with computer simulations [@Groh]. These nice properties are a consequence of a fundamental restriction, namely, the dimensional cross-over [@Cuesta], imposed in the construction of the FMF. The latter implies that the functional in dimension $D$ reduces to the functional in dimension $D-1$ when the original density profile is constrained to $D-1$ dimensions, i.e., $\rho^{(D)}({\bf r})=\rho^{(D-1)}({\bf r})\delta(x_D)$, where $x_D$ is the coordinate that is eliminated on going from $D$ to $D-1$ dimensions.\ One possible procedure to improve the description of the uniform fluid of hard cubes at the level of the FMF is to follow the same method used in Refs. [@Tarazona_2] and [@Roth], in which the hard-sphere Carnahan-Starling equation of state is imposed through the modification of the third term $\Phi^{(3)}$ \[see Eq. (\[three\])\] of the excess free-energy density while keeping the exact density expansion of the direct correlation function up to first order. Unfortunately the absence of a good equation of state for the PHC fluid with the already mentioned properties makes this procedure less systematic compared to that of hard spheres [@Tarazona_2; @Roth]. Following this purpose the original excess free-energy density for hard cubes (\[three\]) is now substituted by $$\begin{aligned} \Phi_{\rm{exc}}({\bf r})= \sum_{k=1}^3f_k\left(n_3({\bf r})\right)\Phi^{(k)}({\bf r}), \label{substitute}\end{aligned}$$ with the $f_k(n_3)$’s selected in such a way as to keep the correct first order density expansion of the direct correlation function and to obtain the right virial expansion up to the seventh order of the PHC equation of state. As the original FMF for hard cubes gives the third virial coefficient correctly, these conditions imply that $f_{1,2}(n_3)\sim 1+O(n_3^2)$ and $f_3(n_3)\sim 1+O(n_3)$ for small $n_3$. Two further important conditions imposed on the $f_k(n_3)$’s are their limiting behavior when the local packing fraction tends to unity: $\lim_{n_3\to 1}f_{\alpha} (n_3)=1$, which asymptotically guarantees the correct cell-theory limit, and the positive signs of their values, which guarantee the convexity of the fluid free energy. Unfortunately this procedure breaks the dimensional cross-over property, but in principle should describe the fluid-solid transition in hard cubes better. Among all the functions $f_{\alpha}$’s that have been tried, even those which give better results \[the particular case of $f_{1,2}(n_3)=1$\] are far from getting the transition point near the simulation one. In Fig. \[fig1\](a) the scaled-particle equation of state, the improved equation of state $$\begin{aligned} \beta P=\rho+\rho \frac{\partial \Phi_{\rm{exc}}}{\partial \rho}- \Phi_{\rm{exc}},\end{aligned}$$ with $\Phi_{\rm{exc}}$ being the uniform limit of Eq. (\[substitute\]), and finally, the symmetric Padé approximant of the seventh-order virial series are plotted. In the first two curves the bifurcation points are shown. The new bifurcation point calculated from Eq. (\[substitute\]) is located at $\rho=0.3378$, and the period and fraction of vacancies of the solid are $d=1.3249$ and $\nu=0.2143$. As can be seen from Fig. \[fig1\](a), the new equation of state still overestimates the fluid pressure, but to a lesser extent. Although the new functional gets a higher transition density and the fraction of vacancies decreases, there is still disagreement between theory and simulations. The equation of state of the PHC solid calculated from the minimization of the original FMF with respect to the Gaussian density profiles compare very well with simulations for densities $\rho\gtrsim 0.5$, whereas the modified version underestimates the solid pressure. At this point the main conclusion that can be drawn is that the modification of the FMF in order to improve the description of the uniform fluid spoils the good description of the solid phase. As the modification of the FMF was done at the expense of loosing the dimensional cross-over property (and this spoils the good description of highly inhomogeneous phases), and the modified versions do not show too many differences in the prediction of the fluid-solid transition, it is worthless to use them to study nonuniform phases. Setting $q_{\parallel}=q_{\perp}=q=2\pi/d$, $\gamma_{\mu}=1/3$, and $\alpha_{\bf k}^{(\mu)}=\beta_{\bf k}$ in the density profiles (\[profile\]) and minimizing the FMF, Eq. (\[perunit\]), of parallel hard cubes ($\kappa=1$) with respect to the Fourier amplitudes and the wave number $q$, the free energy per unit cell for solid \[${\bf n}=n(1,1,1)$\], columnar \[${\bf n}=n(1,1,0)$\] and smectic \[${\bf n}=n(0,0,1)$\] phases were obtained. The results are shown in Fig. \[fig2\]. From the isotropic liquid at the same density $\rho=0.3134$ three inhomogeneous solutions: solid, columnar, and smectic, bifurcate, with the solid phase being the stable one. While the free energy difference between solid and columnar phases is relatively small, the smectic phase is clearly thermodynamically unfavorable. The number of Fourier amplitudes necessary to describe adequately the density profile increases with the density, and thus the numerical calculations becomes more and more time consuming. Nevertheless, the scenario shown in Fig. \[fig2\], with the solid being the only stable phase, occurs at high densities as the simulations and cell-theory have confirmed [@Groh]. The minimization of the FMF using a Gaussian parametrization of the density profiles of columnar and solid phases shows very similar quantitative results [@Groh]. In fact the equation of state of the parallel hard-cube solid from FMT calculations with this parametrization compares very well at high densities with simulations [@Martinez-Raton_3]. The results presented here are much more accurate than those obtained through the Gaussian parametrization. Prolate parallelepipeds {#PP} ----------------------- This subsection is devoted to study the phase diagram of prolate particles ($\kappa>1$). The results obtained from numerical minimization of the FMF of parallelepipeds with fixed $\kappa=4.5$ are shown in Fig. \[fig3\]. The free energies per unit volume of those phases which are stable in some range of densities are plotted. As can be seen the isotropic phase undergoes a first-order phase transition to the so-called discotic smectic (DSm) phase. This peculiar phase is a layered phase (similar to the smectic phase) but with the long axes of the parallelepipeds lying within the layers. There is no orientational order in the layers, what means that the order parameter $Q(z)$ reaches negative values at the positions of the density peaks. The density and order parameter profiles of the DSm phase at $\rho=0.3$ are plotted in Fig. \[fig4\]. The period in units of the small particle length is $d/\sigma=1.2796$ which means that the particles with long axes perpendicular to the layers (preferentially localized at the center of the interlayer space) intersect approximately three adjacent layers. Simulations of the Zwanzig model with $\kappa=5$ on a lattice showed an I-DSm transition at a density between 0.47 and 0.55 [@Casey]. Although the results were obtained for a lattice spacing of $1/3$ (in units of the shortest particle dimension) the simulations were repeated for values $1/9$ and $1/27$ without changes in the stability of the DSm phase. Thus, the authors concluded that this layered phase may persist in the continuum limit [@Casey]. The difference in the transition density found from FMT ($0.2868$) and from simulations ($\sim 0.5$) can be explained using two arguments: (i) As was already pointed out in Sec. \[FMF\], the FMF in the uniform density limit considerably overestimates the isotropic fluid pressure and thus the theory underestimate the transition densities between homogeneous and inhomogeneous phases. (ii) The transition densities should decrease upon decreasing the lattice spacing in simulations, as the results for the freezing of parallel hard cubes on a lattice (occurring at $\rho=0.568$ for an edge length equal to two lattice spacings, at $\rho=0.402$, for six lattice spacings, and at $\rho=0.314$ for the continuum) illustrate [@Lafuente]. Increasing the mean density further the DSm phase undergoes a first-order transition to the columnar phase as Fig. \[fig3\] shows. The restriction of parallelepiped orientations enhances the columnar phase stability even for prolate particles as a phase diagram, to be described below, will show. This phenomenon can be understood if the Zwanzig model is interpreted as a ternary mixture of particles. Simulations on a binary mixture of parallel spherocylinders with different aspect ratios (specifically 2 and 2.9) show that, instead of a continuous nematic-smectic transition typical of the pure component system, the mixture exhibits a first-order nematic-columnar phase transition [@Stroobants_2]. This result was explained by the poorer packing of rods of different lengths in the smectic phase as compared to that of rods of the same length. Simulations and theory show that one of the most important effects that the aspect ratio polydispersity has on the phase behavior of hard spherocylinders [@Polson] and binary mixtures of oblate and prolate particles [@vanKoij; @Martinez-Raton_3] is the enhancement of the columnar phase stability. All these results show that the columnar phase can be stable even for mixture of particles with different shapes. Although the constituent particles of the Zwanzig model have the same shape, the restriction of their orientations changes strongly its relative packing and thus for some $\kappa$’s enhance the columnar phase stability with respect to other phases. At higher density the columnar phase exhibits a continuous phase transition to an oriented solid phase of prolate parallelepipeds, as shown in Fig. \[fig5\](a). The density and order parameter profiles of the columnar phase at the bifurcation point ($\rho=0.3748$) are shown in Fig. \[fig6\]. The periods of the solid phase along the perpendicular and parallel directions are $d_{\perp}/\sigma=1.2690$ and $d_{\parallel}/L=1.5170$, respectively. From the equation $\rho=(1-\nu) V_{\rm{cell}}^{-1}$ ($V_{\rm{cell}}=d_{\perp}^2d_{\parallel}$ being the unit cell volume), the fraction of vacancies of the solid $\nu$ can be calculated as $0.0845$. The continuous nature of the columnar-oriented solid transition changes to first order at some $\kappa$ between $4$ and $4.5$, as Fig. \[fig5\](b) shows for $\kappa=4$. The order parameter $Q({\bf r})$ is very high in the unit cell except in its borders, where it exhibits small oscillations \[see Fig. \[fig6\](b)\]. These oscillations are a consequence of the microsegregation of species “$x$” and “$y$” in the newly formed solid phase, which is preferentially formed by particles of species “$z$” localized around the position $(x^*,y^*)=(0,0)$. This feature is shown in Fig. \[fig7\], where the sum of the density profiles of species “$x$” and “$y$” \[$\rho_{\perp}({\bf r})=\rho_x({\bf r})+ \rho_y({\bf r})$\] is plotted. While the columnar packing is responsible for the presence of the local maxima at the center of the unit cell, the species “$x$” and “$y$” begin to segregate to the borders of the cell $(\pm 0.5,0)$ and $(0,\pm 0.5)$, respectively (see the four local maxima at these positions) as the new solid phase is formed. The long axes of the perpendicular species lie on the lateral sides of the parallelepipedic unit cell, while their centers of mass are preferentially localized at the centers of these sides.\ The calculation of the free-energy branches for several stable inhomogeneous phases and the phase transitions between them (as it was described for $\kappa=4.5$) has been carried out for 15 values of $\kappa$ (ten of them in the range $1\le\kappa\le 5$ and five of them in the range $5\le\kappa\le 10$). The resulting phase diagram is plotted in Fig. \[fig8\]. The isotropic phase of prolate parallelepipeds with $1\leq \kappa \leq 3.5$ undergoes a continuous phase transition to the plastic solid phase. The transition points are joined with the spinodal line that has been calculated through the divergence of the structure factor. Notwithstanding that a functional minimization was carried out for each $\kappa$ to check the continuous nature of the transitions. The plastic solid is stable for $\kappa\leq 2.5$ up to densities around $0.5$. At these values the numerical minimization turns out to be cumbersome because of the large number of Fourier amplitudes necessary to correctly describe the inhomogeneous profiles. Thus, the high density part of the phase diagram ($\rho \gtrsim 0.5$) has not been calculated with the numerical procedure described above. At higher densities a Gaussian-type parametrization of the density profiles is required, which obviously has a lower degree of accuracy. For $\kappa=2.95$ the plastic solid exhibits a very weak first-order phase transition to the discotic smectic phase (labeled as 1 in Fig. \[fig8\]), and the latter a phase transition to the columnar phase at higher densities. But the most representative region of the phase diagram where the discotic smectic is stable is for $\kappa$ around 4.5 where this layered phase exhibits a first-order phase transition to columnar phase (the shaded area of Fig. \[fig8\] limits the instability region against phase separation between both phases). For $\kappa$ between 4 and 5 the columnar phase undergoes a phase transition (first order for $\kappa=4$ and continuous for other values shown) to the oriented solid phase. The nematic phase begins to be stable for $\kappa >5$ with its stability region bounded below by the first order isotropic-nematic transition and above by a continuous nematic-smectic transition. Finally, the smectic region is bounded above by a continuous transition to the oriented solid phase (see Fig. \[fig8\]). Again the nematic-smectic transition points are joined with spinodal lines and for each $\kappa$ a minimization was carried out to check numerically the continuous character of the transition (the smectic solution begins to be stable right at the spinodal). In Ref. [@Martinez-Raton_5] a bifurcation analysis with the same functional was carried out to study the nature of the nematic-smectic transition. A thermodynamic and mechanical stability analysis showed that the nematic-smectic transition is first order, which is in contradiction with the numerical minimization results presented here. A possible reason that justifies this contradiction could be that the $N$-Sm transition is very weakly first order, so weak that the numerical accuracy used in the functional minimization can not decide about its nature. Another possibility is that the numerical accuracy failure is somewhere in the bifurcation analysis. A careful revision of this analysis is certainly called for in order to settle this point. The available simulation results for freely rotating hard spherocylinders show that the isotropic phase exhibits a transition to the solid phase for $0\leq\kappa\leq 4.1$ (the solid is plastic for $\kappa\leq 1.35$ and oriented for $1.35\leq \kappa\leq 4.1$) while the isotropic-smectic and nematic-smectic transitions begin at $\kappa=4.1$ and $4.7$, respectively [@Bolhuis] \[notice that for hard spherocylinders the length-to-breadth ratio is $\kappa=(L+D)/D$\]. We can see that, despite the different particle geometry and the restricted orientations of the Zwanzig model, the agreement for the threshold $\kappa$ at which spatial instabilities to the solid and smectic phase destabilize the homogeneous phases is rather good. Also the qualitative picture is similar: elongated rods form smectics, and more symmetric particles form solids. The main difference between them is that the Zwanzig phase diagram presents regions where the columnar and discotic smectic phases are stable, a difference due to the restriction of orientations. Oblate parallelepipeds {#OP} ---------------------- The phase diagram of oblate parallelepipeds ($\kappa<1$) is shown in Fig. \[fig9\]. The main differences after comparing the phase diagrams of prolate (Fig. \[fig8\]) and oblate particles are that in the latter: (i) The smectic is no more a stable phase. (ii) The region of columnar phase stability is considerably larger. (iii) The stability region of the plastic solid is reduced (in fact this phase is stable only up to $\kappa^{-1}\approx 2.5$) at the expense of that of the discotic smectic phase. (iv) The transitions to the latter are strongly first order in nature (except for $\kappa^{-1}=4.5$). (v) The oriented solid phase is replaced by a perfectly oriented solid in which “$x$” and “$y$” species are absent. This phase, after scaling in the $z$ direction, is the same as the solid of parallel hard cubes. A solution from the FMF minimization with three dimensional spatial modulations and with $\gamma_{\perp}\neq 0$ has not been found in the parallelepipedic unit cell (the case of face-centered or body-centered cubic unit cells have not been tried here). Finally, similar by to what happens with prolate parallelepipeds, the nematic phase begins to be stable at $\kappa^{-1} \gtrsim 5$. It undergoes a continuous phase transition to the columnar phase (the transition points of Fig. \[fig9\] are joined with the spinodal curve). The parallelepipeds with $\kappa^{-1}=1.5$ exhibit an interesting phase behavior. At low densities the isotropic phase destabilizes with respect to the columnar phase and not with respect to the PS phase. This example shows that the prediction for phase transitions using only the spinodal instability calculations can generate uncertainties about the possible symmetries of the inhomogeneous phases. In fact these calculations do not allow to decide in this example if the new phase is a plastic solid or a columnar phase. Only by a complete minimization of the FMF, could it be concluded that the columnar phase is the stable one. In Fig. \[fig10\] (a) the density profiles of perpendicular \[$\rho_{\perp}(z)$\] and parallel \[$\rho_{\parallel}(z)$\] species are shown for the discotic smectic phase of oblate particles with $\kappa^{-1}=2.5$, while the order parameter profile is shown in Fig. \[fig10\](b). This discotic smectic phase coexists at $\rho=0.4244$ with the plastic solid phase. The period in units of the large parallelepiped edge length is $d/\sigma=1.2142$. The random orientation of the uniaxial axes within the layers is confirmed by the high negative values of the order parameter at the position of the density peak of the perpendicular species. The main difference between the DSm of Fig. \[fig10\] and that of Fig. \[fig7\] is that the “z” species is now localized preferentially not at the center of the interlayer space, but near the smectic layers \[see Fig. \[fig10\](a)\], exhibiting two local maxima at each side of the layer. This effect can be explained by the depletion force that the perpendicular species exerts on the parallel one.\ The $N$-Sm ($N-C$) and the Sm-OS ($C$-POS) transition lines of Figs. \[fig8\] and \[fig9\] converge asymptotically to $\rho=0.3143$, the value of the fluid-solid bifurcation density, as $\kappa\to \infty$ ($\kappa^{-1}\to \infty$). The reason for this is that upon increasing $\kappa$ ($\kappa^{-1}$) the number of rods (plates) with orientation perpendicular to the director becomes vanishing small, and then the system is, after rescaling the $z$ direction, almost equivalent to a system of parallel cubes. Simulations of the Zwanzig model on a $15\times 15\times 15$ lattice with spacing $1/3$ show that oblate parallelepipeds with dimensions $5\times 5\times 1$ undergo a transition to a phase exhibiting a columnar structure [@Casey] at a density somewhere between $0.55<\rho<0.65$. On increasing the system size to $30\times 30\times 30$ the global columnar order disappears, but local correlations persist in the fluid with particle alignment distributed evenly among the three available orientations. In the same work the effect that orientational constraints have on the stability of the inhomogeneous phases was studied. While a system of biaxial $5\times 3\times 1$ “tiles” without orientational constraints (except those inherent to the Zwanzig model) stabilizes in a smectic phase with the shortest edge lengths perpendicular to the layers, the system composed by “tiles” with all their long edge lengths parallel to each other exhibits a phase transition to the smectic phase with these edge lengths perpendicular to the layers, similar to what is found here for uniaxial oblate parallelepipeds (the discotic smectic phase). Simulations of hard cut spheres show that for $\kappa=0.3$ there is an isotropic-solid transition, for $\kappa=0.2$ an isotropic-columnar transition (the isotropic phase might instead be a peculiar “cubatic” phase) and for $\kappa=0.1$ a nematic-columnar one [@Veerman]. From these results it can be concluded that the effect that the degree of particle anisotropy has on the symmetry of the stable phases for both cut spheres and hard parallelepipeds with restricted orientations, is qualitatively similar. Conclusions =========== The goal of this article has been the calculation of the phase diagram of the Zwanzig model for prolate and oblate parallelepipeds centering the attention on the phase transitions to inhomogeneous phases. For this purpose the fundamental measure functional for hard parallelepipeds with restricted orientations has been used. This functional is exactly the same for any particle shape (prolate and oblate depending on $\kappa$), which allows for a unified study of the phase behavior of both kinds of particles. A free minimization of the functional was carried out with the only constraints of choosing a parallelepipedic unit cell and of imposing uniaxial symmetry in the inhomogeneous phases. The latter is justified by uniaxial symmetry of the particles. The degree of approximation to the exact density profiles was controlled by the cutoff imposed on the reciprocal lattice numbers in the Fourier expansion of the density profiles. A system of parallel hard cubes was separately studied, which was motivated by recent simulations on this system [@Jagla; @Groh]. Applying a modified versions of the FMF to improve the description of the PHC liquid, along the same lines as Refs. [@Tarazona_2] and [@Roth], the continuous transition point to the solid phase and the equation of state of the solid were calculated from the divergence of the structure factor and from the functional minimization with respect to Gaussian density profiles. Although the transition density and fraction of vacancies change in the right direction, these results are still far from the simulations. In fact, the solid phase is poorly described by the new functional. The poor convergence of the PHC virial series does not make this procedure as effective as for hard spheres. Further refinement of the method and the proper inclusion of vacancies in simulations of the solid phase will probably improve the agreement between theory and simulations. The original FMF for PHC was minimized to study the relative stability of the smectic, columnar, and solid phases, starting at low densities from the bifurcation point. The solid phase is the only stable phase, followed by the columnar and the smectic (in order of energy stability). At high densities the same behavior is shown from calculations using cell theory, functional minimization with Gaussian density profiles and computer simulations [@Groh]. The system of prolate and oblate parallelepipeds exhibits a very rich phase behavior. Apart from the plastic or oriented solid, smectic, and columnar phases, which are present also in systems of prolate (spherocylinders [@Bolhuis]) and oblate (cut spheres [@Veerman]) particles, a new phase appears: the discotic smectic, the existence of which was confirmed by simulations [@Casey]. The close relation between the particle anisotropy and symmetry of the stable phases (elongated particles form smectics, flattened one form columnars and more isotropic particles form solids) which has been observed in simulations [@Strobants_1; @Bolhuis; @Veerman] and experiments [@Maeda] is confirmed by this simple model. There are two important effects that the restriction of orientations has on the phase diagram topology: (i) The already pointed out stability of the discotic smectic. (ii) The stability of the columnar phase of prolate parallelepipeds for some aspect ratios. 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--- abstract: 'We report phase-resolved optical polarimetric, photometric and spectroscopic observations of  (= EUVE J2115–58.6, Craig 1996) which confirms the system to be a magnetic cataclysmic binary of the polar (AM Herculis) subclass. The optical light curve is sometimes flat and occasionally displays a pronounced bright phase, reminiscent of the self-eclipse of a small accretion spot by the revolving white dwarf, as seen in self-eclipsing polars. Our period search reveals ambiguous results only which can be interpret assuming that the white dwarf is not synchronously rotating with the binary orbit. We find circularly polarized cyclotron radiation with $V/I$ ranging from 0% to $-15\%$ on one occasion, from $-8$% to $+15\%$ on another occasion. Compared with other polars, the self-occulted accretion region of  had an extreme red cyclotron spectrum. In addition, the system has an extreme hard X-ray colour during the ROSAT all-sky survey observation. Both properties suggest a low value of the magnetic field strength, and our best estimate gives $B = 11\pm2$MG. Due to the absence of significant M-star features in our low-resolution spectra we estimate the minimum distance to  to be $d > 250$pc (for an M$5^+$ secondary star).' author: - 'A.D. Schwope' - 'D.A.H. Buckley' - 'D. O’Donoghue' - 'G. Hasinger' - 'J. Trümper' - 'W. Voges' date: 'Received, accepted' title: ': a short-period, asynchronous polar[^1] ' --- 2[He[ii]{}$\lambda$4686]{} 1[He[i]{}$\lambda$4471]{} Introduction ============ EUVEJ2115–58.6 was detected during the EUVE all-sky survey with a countrate of 0.05 cts s$^{-1}$ (Bowyer et al. 1996) and tentatively identified as a magnetic cataclysmic variable by Craig (1996) who identified strong H-Balmer, He[i]{}, He[ii]{} and Ca[ii]{} emission lines. Moderate resolution spectroscopy in the blue wavelength regime was done by Vennes et al. (1996). They observed pronounced radial velocity variations of H$\beta$ and He[ii]{} and determined the orbital period of the system of 110.8min with a possible one-cycle-per-day alias of 102.8min. The source was also detected with the ROSAT-PSPC during the X-ray all-sky survey performed in 1990/1991 at a countrate of 0.380 s$^{-1}$ (galactic coordinates $l^{II}=337\degr$, $b^{II}=-41\degr$). We are currently running a program in order to identify optically bright ROSAT survey sources extracted from the ROSAT All-Sky Survey Bright Source Catalog (1RXS, Voges et al. 1997) at high galactic latitudes ($|b^{II}| > 30\degr$, limiting countrate 0.2 s$^{-1}$). With no certain optical identification at the time,  entered our target list with high priority. When its nature as magnetic CV became clear from a low-resolution spectrum, phase-resolved data were collected in order to study its main characteristics. Later it became clear, that polarimetric and photometric observations has been performed coincidentally from SAAO, too. We present here the combined results of our optical observations obtained over a 70-day basis from South Africa and Chile. Observations and analysis ========================= The ROSAT all-sky survey (RASS) observations --------------------------------------------  was observed during the RASS for a total of 294 sec. The source was scanned 13 times with exposures ranging from 16.2 to 25.9 sec. It showed a modulation of the X-ray flux by 100% reaching a peak countrate of 1.2 s$^{-1}$. The mean survey countrate was $0.38\pm0.19$s$^{-1}$ and the mean hardness ratio HR1$= (H -S)/(H+S) = -0.02\pm0.1$, where $H$ and $S$ are the counts in the ROSAT hard (0.4 – 2.4 keV) and soft (0.1 – 0.4 keV) bands, respectively. Folded over the most likely optical period, the X-ray light curve shows a clear on/off behaviour with length of the X-ray bright phase extending for $\sim$50% of the orbital cycle. With only 102 detected photons the X-ray spectrum of  is not very well constrained. It can be fitted using a single bremsstrahlung component ($kT_{\rm br} > 1$keV) with X-ray spectral flux at 1 keV of 2 photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$. The spectrum is very weakly absorbed, $N_H = 0.4\pm 1.0 \times 10^{20}$cm$^{-2}$, and even compatible with zero absorption. Spectroscopic observations -------------------------- An $R$-band CCD image of the field of  is reproduced in Fig. \[chart\]. Only one possible optical counterpart of the X-ray source lies within the positional error circle of the RASS X-ray observations. A total of 21 low-resolution spectra of this star (12Å  FWHM, integration time 300 sec) and one with intermediate resolution (3Å FWHM, integration time 600 sec) were obtained with the EFOSC2 spectrograph at the ESO/MPG 2.2m-telescope on October 23, 1996, between UT 0:19 and 3:30. The flux-calibrated low-resolution spectra were folded through Johnson $BVR$ filter curves in order to derive broad-band optical light curves. These are suspected to be accurate within $\pm$0.5 mag (absolute) but variability and colours can be determined with much higher accuracy of approximately 0.1 – 0.2 mag. The $B$-band light curve does not show any significant variation, the $V$-band light curve displays marginal variability at a level of 0.2 mag, but the $R$-band light curve shows a pronounced orbital hump with full amplitude of about 0.5 mag centered on HJD 2450379.6231. The optical $R$-band light curve as derived from our low-resolution spectra is shown together with the variation in $V-R$ and the radial velocity variation of the main emission lines in Fig. \[lc\_rv\]. Two more spectra (not shown in Fig. \[lc\_rv\]) were recorded during the preceding faint and bright phases, respectively. Using all the spectra, a photometric period of $\sim$113 min was derived by a period search based on Scargles (1982) algorithm. Mean bright- and faint-phase spectra with low spectral resolution are shown in Fig. \[spec\_low\] and the one spectrum with higher resolution is shown in Fig. \[spec\_hires\].   shows the typical features of a magnetic cataclysmic binary with strong emission lines of the H-Balmer series (including the strong Balmer jump in emission), He[i]{}, He[ ii]{}, and the C[iii]{}/N[iii]{} Bowen blend at 4640/50Å. The lines exhibit a phase-dependent asymmetry and display pronounced radial velocity variations. We have determined the radial velocity of 5 main emission lines (H$\alpha$, H$\beta$, H$\gamma$, He[i]{}5876, He[ ii]{}4686) by fitting single gaussians. No significant difference between the radial velocity variations of the different lines was found. We, therefore, show the average radial velocity of the 5 lines in the lower panel of Fig. \[lc\_rv\]. Although the emission lines are slightly asymmetric, our resolution is not high enough in order to discern between possible (likely) multiple emission components. Although not reflecting the change of the radial velocity variations very well, we used a sine approximation in order to estimate the spectroscopic period. The best fit gives $P_{\rm orb} = 114 \pm 4$min, in agreement with the period derived from the photometric variations of our spectra, and consistent with Vennes et al. (1996) spectroscopic period. Photometry ---------- CCD photometry of  was undertaken at SAAO over 7 nights, from 1996 September 3 to 9. The observations were made on the 0.75-m telescope with the UCT CCD camera, employing a Wright Instruments blue-sensitive EEV CCD chip operated in frame transfer mode. Observations were mostly conducted without a filter, except for those done on 7/8 September, for which alternately a B and I filter were employed. Details of the observations appear in the observing log (Table \[obslog\]), suffice to say that integration times were either 20 or 30 sec for the filterless data, with no dead time between frames. The observations were obtained in both photometric and non-photometric conditions, while seeing was equally as variable, from good (1 arcsec) to poor (2-3 arcsec). The scale of the CCD is 0.37 arcsec $\rm pixel^{-1}$ in normal mode (which was used for the majority of the observations), and twice that value for 2 $\times$ 2 prebinning mode, which was used when the seeing was poor. The field of the UCT CCD on the 0.75-m telescope is $\rm \sim 2.6 \times 1.8~ arcmin^{2}$. After the usual flat-fielding and bias subtraction, batch mode DoPHOT routines (Mateo & Schechter 1989) were used to obtain both PSF profile-fitted and aperture magnitudes. Brighter stars on the frames were used as comparison stars, and differential magnitudes derived. The rms scatter of the corrected comparison stars was typically 0.006 magnitudes for the filterless photometry, and $\sim$0.01 mag for the B & I data. exhibits a large degree of variability, up to $\sim 1.4$ mag in I, 1 mag for filterless (“white-light”), and much less (0.4 mag) for B. The pronounced “hump” seen in the longer wavelength light curves (I & filterless) is not seen in the B light curve (see Fig. \[D1\]). Furthermore, there are substantial night-to-night changes in the light curves, evidenced by a less pronounced hump, or even a double hump on the first night (see Fig. \[E2\]). [llllccccll]{} & & & & & & &\ & & & & &(-2450000) &(h) &(s)\ & & & & & & & &\ \ 3/4 &Sep &96 &W002 &0.75 m &330.387 &2.78 &30 &w.l.\ 4/5 &Sep &96 &W005 &0.75 m &331.436 &4.32 &20 &w.l.\ 6/7 &Sep &96 &W007 &0.75 m &333.420 &3.46 &30 &B\ 7/8 &Sep &96 &W009,W010 &0.75 m &334.467 &3.46 &60,30 &B,I\ 9/10 &Sep &96 &W018 &0.75 m &336.538 &3.52 &20 &w.l.\ & & & & & & & &\ \ 15/16 &Sep &96 &P0342 &1.90 m &342.398 &3.84 &10,180 &w.l.\ 10/11 &Nov &96 &P0398 &1.90 m &398.316 &3.54 &10,180 &w.l.\ 11/12 &Nov &96 &P0399 &1.90 m &399.304 &2.82 &10,180 &w.l.\ & & & & & & & &\ Polarimetry ----------- White-light photopolarimetry of  was undertaken on the SAAO 1.9-m telescope using the UCT Polarimeter (Cropper 1985) on 1997 September 15/16 and November 10/11 and 11/12. The relative faintness of the object precluded any filtered observations. The instrument was run in the so-called “Stokes mode”, simultaneously measuring linear and circular polarization every 180 s, and the intensity every 10 s. Polarization standards were observed at the end of each night in order to derive the instrumental waveplate offsets, which are constant to $\pm \sim 2^{\circ}$ from night to night. Sky measurement were obtained every 15 to 20 min, and the sky background values interpolated with a polynomial spline before subtraction from stellar intensity data arrays. The observations were obtained in photometric conditions with $\sim$ 1 arcsecond seeing, except for the last night (11/12 Nov), for which the seeing was worse (2 arcseconds).  shows clearly detectable circular polarization, ranging from $\sim -15$ to $+15\%$ (Fig. \[circpol\]). The accuracy of our linear polarization measurement was $\sim$3% and within this accuracy no clear detection of the linear polarisation is seen. The various circular polarization curves are quite dissimilar, indicating that large changes in the accretion geometry between the different observations took place. The most simple curve at JD 398 is reminiscent of a one-pole accreting AM Herculis star with extended self-eclipses of the accretion region by the revolving white dwarf. The large positive circular polarization seen for a rather short phase interval at JD 342 then indicates that also a second accretion region on the opposite hemisphere (with opposite circular polarization polarity) became active. The light curves recorded simultaneously displays no pronounced orbital hump, and resemble those seen at JD 330 and 331 (see Fig. \[E2\]). Results and discussion ====================== The polarimetric observations confirm beyond doubt that  is indeed a magnetic cataclysmic binary, probably of AM Herculis type (polar). The borders between polars and intermediate polars (IPs) were blurred by the discovery of soft X-ray emitting polarized IPs like PQ Gem (other ‘soft IPs’ are discussed by Haberl & Motch 1995) on the one hand and asynchronous polars like BY Cam on the other hand. However, the degree of synchronism between the white dwarf and the binary rotation is still an important parameter and we start our analysis therefore by a period search using our photometric, polarimetric and spectroscopic data and compare them with the results of Vennes et al. (1996). Period search ------------- We subjected the filterless and I-band CCD light curves (Figs. \[D1\] and \[E2\]) to a period analysis, using both discrete Fourier transform (DFT) and phase dispersion minimization (PDM) periodograms. The power spectra show no significant features for periods $<$1000 sec, up to the Nyquist frequency. Pronounced power appears at $\sim$ 1.5 mHz (110 min), and at the first and second harmonics, particularly for the DFT, which is more susceptible to non-sinusoidal waveforms than the PDM periodogram. An inspection of the periodograms near the dominant frequency indicates that there is not a “clean” distribution of frequency peaks with the usual 1 cycle d$^{-1}$ alias structure. This is not surprising given the previously mentioned nightly changes to the shape of the light curve. In Fig. \[periodo\] we show the periodograms centered near the dominant frequency at $\sim$0.15 mHz. The strongest peaks in both the DFT and PDM power spectra occurs at a period corresponding to 114.74min (or its alias at 106.27min), although other nearby peaks are only mildly less significant. Pre-whitening by the dominant period removes most of the power at 0.15 mHz (see 4th panel from the top in Fig. \[periodo\]), indicating that the complex period structure is likely an artefact of the data sampling. The highest peak for the prewhitened data occurs at 124min, far away from other period estimates, the second highest peak at 110min is close to the spectroscopic period. After the photopolarimetry runs, we included those intensity data in our period analysis. Some of these data have poor quality due to mediocre seeing, and the consequent loss of light from the small aperture we were forced to use. Inclusion of the polarimetric intensities did not resolve the ambiguity over the photometric period, and the periodogram of all the combined photometry (CCD and polarization intensities) was rather different, with the dominant peak occuring at 98.6 min, and its aliases, none of which coincide with either the spectroscopic or previously determined photometric periods. A periodogram of only the $V/I$ circular polarization data (lowest panel of Fig. \[periodo\]) shows highest power at a period of 131.3min, which is inconsistant with all other period determinations. However, it may be significant that one of the aliases (at 111.8 min) is very close to the purported orbital spectroscopic period (110.8 min) reported by Vennes et al. (1996). The periodograms are clearly affected by the coming and going of the bright phase. This becomes evident if one uses for the period search only those datasets, where an orbital hump is clearly detectable (JD 334, 336, and 379, the times of hump center at these three occasions can be found in Tab. \[hump\_center\_times\]). The corresponding periodogram has power mainly at 109.84min (566 cycles between days 336 and 379). If one tentatively assumes that the accretion geometry is the same when the circular polarization curve looks most simple (JD 398) and when the photometric light curve pattern shows a pronounced hump (JD 334, 346, and 379) and performs a period search for the times of mid-hump at these four occasions, a period of 109.65min emerges as a possible solution (567 cycles between days 336 and 379). The different period estimates can be compared to the spectroscopic period, as derived from the emission line radial velocities by Vennes et al. (1996), which shows two strongly aliased peaks corresponding to periods of 110.8 and 102.8 min. Neither of our estimates coincides exactly with either possible value of the spectroscopic period. On the basis of the present photometric and polarimetric data, it seems that the photometric and spectroscopic periods could be discordant, indicating that the system is asynchronous to a small degree ($\sim$1%). The changing shape of the light curve may indicate a pole-swapping accretion mode. If the far (‘southern’) pole is accreting, selfeclipses by the revolving white dwarf occur, leading to pronounced orbital modulation of the optical light. If the near (‘northern’) pole accretes, the accretion region possibly never becomes eclipsed and orbital photometric modulations are smoothed, also the changes in polarity of $V/I$ supports the accretion region moving from one pole to the other. ---------------- ------------ Date HJD (-2450000) 7/8 Sept 1996 334.5444 9/10 Sept 1996 336.4513 9/10 Sept 1996 336.5282 23 Oct 1996 379.6231 ---------------- ------------ : \[hump\_center\_times\] Times of mid-hump of  at specified dates The difference between the orbital period $P_{\rm orb}$ and the spin period of the white dwarf $P_{\rm wd}$ might be estimated assuming that $P_{\rm orb}$ is represented by the spectroscopic period of 110.8min and that $P_{\rm wd}$ is represented by one of the photometric periods derived for far-pole accretion, 109.84min or 109.65min. Hence, the degree of asynchronism is of the order of $\sim$1%. A unique determination of this quantity requires photometric monitoring of the system over one beat cycle (which is $\sim$7 days). The cyclotron spectrum of the far pole -------------------------------------- As in other selfeclipsing polars (e.g. Schwope et al. 1995), the difference spectrum between the bright and the faint phase can be regarded as the cyclotron spectrum originating from the accreting spot active at that time. This spectrum is shown on a linear scale in the lower panel of Fig. \[spec\_low\] and on a logarithmic scale in Fig. \[cycspec\]. Also included in Fig. \[cycspec\] are suitably scaled cyclotron spectra of other polars which have measured magnetic field strengths. The cyclotron spectrum of  rises steeply towards long wavelength with peak wavelength clearly longward of 8000Å. It does not show any sign of modulation by cyclotron harmonics. Since also no Zeeman lines were observed, no direct measurement of the field strength in  seems to be possible. The colour of the cyclotron spectrum indicates that we are observing the high-harmonic optically thin part of the spectrum which is determined only by the strong frequency-dependence of the cyclotron absorption coefficient . Thus at least an estimate for the field strength can be derived, assuming that the cyclotron spectra of low-field polars in the optically thin regime are similar. In order to test this hypothesis and use it as tool for the field determination of  we synthesized a common cyclotron spectrum from low-field polars with measured field strengths (Fig. \[cycspec\]). The absorption coefficient  in dimensionless frequency units, i.e. normalized to the cyclotron fundamental, is a function of the plasma temperature and the projection angle only (the latter is the angle between the magnetic field and the propagation vectors). The spread of these parameters must not be too large among the different objects for a reliable comparison of their spectra. The plasma temperatures of the different objects are not yet measured. Parameters influencing the plasma temperature are the mass of the white dwarf, the field strength and the specific mass accretion rate. The former two are known to be more or less the same for the objects concerned with here, while the latter is widely unknown. For the time being we assume similar values for the different objects. We used BL Hyi (Schwope et al. 1995), EP Dra (Schwope & Mengel 1997) and RXJ1957-57 (Thomas et al. 1996), all of which have field strengths in the range 12–16 MG. The projection angle at the particular phases of the spectra used in the construction of the common cyclotron spectrum correspond to $\sim$70–80. With the known field strength, the observed spectra were transformed from wavelength to the dimensionless frequency, normalized to the cyclotron fundamental frequency. The observed spectra peak at harmonic numbers 8–11, the optically thick Rayleigh-Jeans component lies in each case in the unobserved infrared spectral regime. The different observed turnover frequencies (from being optically thick to optically thin) reflect different sizes of and densities in the emission regions. The different spectra were then translated to each other by shifting them vertically until they agree at a certain frequency (or harmonic number $m_{H}$). Constraints for the choice of the finally adopted number, $m_{H} = 14$, were sufficient low optical thickness (shifts $m_{H}$ to large values) and sufficient high flux in the observed spectra (shifts $m_{H}$ to small values). The agreement between the different spectra in their usable (optically thin) part is surprisingly good, thus justifying our assumptions on plasma temperature and orientation. We compared the observed cyclotron spectrum of  with those of the other polars by shifting it along both axes of Fig. \[cycspec\] until best agreement was reached. A shift along the abscissa corresponds to a change of the adopted value of the field strength $B$. A shift along the ordinate gives just the normalization of the spectrum. Our best estimate for the field strength thus achieved is 11MG. We believe that the field strength cannot be much in excess of $\sim$13MG, because of the observed steepness of the cyclotron spectrum of . The field strength is probably not much lower than $\sim$9MG. The blue end of the cyclotron spectrum would then correspond to harmonic numbers as high as and in excess of $m_{\rm H} \simeq 20$, which has never been observed in any polar (due to the negligible power radiated in these high harmonics). We thus regard $B =11\pm2$MG as a reasonable estimate of the magnetic field strength. A distance estimate ------------------- Our faint-phase low-resolution spectrum does not show any prominent feature originating from the secondary star. We estimate its contribution to this spectrum to be less than 30% ($\lse 0.55 \times 10^{-16}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$ at 7540Å). On the assumption that it is a main sequence star, which is generally accepted for cataclysmic binaries just below the period gap, we may estimate its distance using Bailey’s (1981) method together with the improved calibration by Ramseyer (1994). The major uncertainty in this procedure comes from our photometric accuracy and the poorly known mass-radius relation of main-sequence stars of late spectral type. Using e.g. either the calibration by Caillault & Patterson (1990) or Neece (1984) we obtain the possible spectral type (mass), scaled K-brightness (using M-dwarf template spectra), surface brighness $S_K$ and stellar radius as (Sp, $K$, $S_K$, $\log(R_2/R_\odot)$) = (M5$^+$, 15.3, 5.3, -0.61) and (M4.5, 16.1, 4.7, -0.73), respectively. With $\log d = ((K-S_K+5)/5 - \log(R_2/R_\odot))$ this yields a lower limit distance estimate of $d > 250$pc for the former, and $d > 350$pc for the latter case. Should the spectral type of the secondary be later than we have assumned the lower limit distance would be smaller.  as X-ray and EUV-emitter ------------------------- Although  is a bright, variable source at X-ray wavelengths, it was missed by previous identification programmes (soft survey: e.g. Beuermann & Burwitz 1995, hard survey: Hasinger et al. 1996) due to its unusual hard X-ray spectrum which places it between the selection boundaries of these previous identification programmes. One may thus speculate about some more low-field polars as counterparts of relatively hard ROSAT survey sources. The X-ray spectral shape as seen with ROSAT (although not well determined) does not make the assumption of a soft blackbody component necessary, unless to the case of all other AM Herculis stars. This makes the system similar to the ‘classical’ intermediate polars (known before ROSAT), which have hard thermal bremsstrahlung spectra only. The EUV-detection on the other hand clearly shows the presence of a soft component. It is not clear, however, if both components orginate from the same accretion spot. Perhaps the system has one IP-like pole emitting predominantly hard X-ray bremsstrahlung which was active during the RASS and a second polar-like accretion region emitting soft and hard X-rays which was active during the EUVE sky survey? With the present very limited observational data we clearly cannot answer these questions and we need more EUV- and X-ray observations with full phase coverage in both modes of accretion. Conclusions ----------- We have found clear and unique evidence for the magnetic nature of the new cataclysmic variable  by the detection of strong and variable circular polarization. In addition, the radial velocity pattern and the shape of the optical light curve (when showing the pronounced hump) suggest an AM Herculis type nature of this object. We found, however, some features which do not fit in the most simple picture of a synchronously rotating polar: (1) there is no consistent photometric and spectroscopic period; (2) the optical lightcurve is sometimes flat and sometimes strongly modulated, which is not related to changes in the mass acrretion rate; (3) the circular polarization curve is not repeatable, showing either only one or both signs of polarization. One possible explanation for these deviations is the presence of a small asynchronism ($\sim$1%) between the orbital and the spin periods of the white dwarf. The origin of such an asynchronism is unknown. Other polars with comparable orbital period (hence, mass accretion rate and thus accretion torque) and field strength (hence, synchronisation torque) are not known to show this behaviour, they are synchronized. There are three more asynchonous polars known, V1500 Cyg (Stockman et al. 1988), BY Cam (e.g. Mouchet et al. 1997), and RXJ1940–10 (Patterson et al. 1995) with orbital periods of 197, 201, and 202min, beat periods between spin and orbital period of $\sim$14, 7.8, and -49.5 days, respectively. All these systems are long-period polars, i.e. they have orbital periods above the 2-3 hour period gap. For V1500 Cyg the reason for the asynchronism was found in its 1975 nova explosion. The mechanism behind for the two other systems are unknown. If confirmed,  would be the first asynchronous polar below the period gap. To confirm it’s nature is a challenge for observers, to understand it’s physics a challenge for theorists. This work was supported by the BMB+F under grant 50 OR 9403 5. 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--- abstract: 'We describe the implementation of the PhotoZ code in the framework of the Astro-WISE package and as part of the Photometric Classification Server of the PanSTARRS pipeline. Both systems allow the automatic measurement of photometric redshifts for the millions of objects being observed in the PanSTARRS project or expected to be observed by future surveys like KIDS, DES or EUCLID.' author: - 'R.P. Saglia' - 'J. Snigula' - 'R. Senger' - 'R. Bender' date: 'Received: date / Accepted: date' subtitle: A photometric redshift code for large datasets title: 'Implementation of PhotoZ under Astro-WISE ' --- Introduction {#Intro} ============ Since the completion of the Sloan Digital Sky Survey (SDSS) [@SDSS] optical astronomy has moved on from the detailed studies of single objects to a phase where catalogues with millions of entries can be produced. This has allowed for detailed statistical studies of entire populations, as well as searches for extremely rare objects. Consequently, astronomers are forced to update their approach to data analysis and to embed their codes in database-supported applications that support efficient automatic procedures and easy administration of the analysis processes. One such case is the measurement of photometric redshifts for the hundreds of millions of galaxies that ongoing or future optical and near infrared photometric surveys will deliver. We are directly involved in three large scale imaging surveys. The Panoramic Survey Telescope and Rapid Response System 1 (PanSTARRS1, see [@Kaiser04]) project started regular operations in May 2010 and is producing a 5 band (grizy) survey of 3/4 of the sky that at the end of the forseen 3 years of observations, will be $\approx 1$ mag deeper than SDSS. Approximately two hundred million galaxies, a similar number of stars, about a million quasars and $\approx 7000$ Type Ia Supernovae will be detected. VIKING (VISTA Kilo-Degree Infrared Galaxy Survey [^1]) is a near-infrared 4 band (ZJHK) survey of 1500 square degrees of extragalactic sky that started in December 2009 at the VISTA telescope. This will be complemented in 5 optical bands (ubgri) by the Kilo Degree Survey (KIDS [^2]) at the VST telescope, to start in October 2011. Finally, the Dark Energy Survey (DES [^3]) will image 5000 square degrees around the southern galactic pole in 4 optical bands (bgri) at the CTIO telescope. Looking to the future, we are participating in the EUCLID [^4] bid. If approved, the satellite will image 20000 square degrees of the extragalactic sky in the optical and NIR channels, providing unprecedented deep photometry for many millions of galaxies. The science driving these projects ranges from Baryonic Acoustic Oscillations and growth of structure, to weak shear, galaxy-galaxy lensing and lensing tomography. All of them rely on the determination of accurate photometric redshifts for extremely large numbers of galaxies. Further science goals, like the detection of high redshift quasars and galaxies, the discovery of very cool stars, or the study of galaxy evolution with cosmic time will also profit from the availability of good photometric redshift and star/galaxy photometric classification. Therefore, in the last few years we have designed and implemented schemes to derive and keep organized photometric redshifts, probability distributions and star/galaxy classification for extremely large datasets. Here we describe two aspects of these efforts; the PhotoZ implementation for Astro-WISE and the PanSTARRS1 Photometric Classification Server. The structure of the paper is as follows. Sect. \[sec:PhotoZ\] considers the algorithm at the core of our implementations and its recent scientific use. Sect. \[sec:Implementation\] discusses the implementation of the code for large data sets. In Sect. \[sec:AWPhotoZ\] we present its Astro-WISE incarnation and give examples of its use and evaluate its performances in Sect. \[sec:AWPerformances\]. Sect. \[sec:PCS\] is dedicated to the implementation of the code for the PanSTARRS1 survey. We draw our conclusions in Sect. \[sec:Conclusions\]. The PhotoZ code: algorithm and science applications {#sec:PhotoZ} =================================================== In the last decade several efficient codes for the determination of photometric redshifts have been developed and a fair summary of these efforts would go well beyond the scope of the present contribution. In short, there are mainly two approaches, one based on empirical methods, the other on template fitting. In the first case one tries to parametrize the low-dimensional surface in color-redshift space that galaxies occupy using low-order polynomials, nearest-neighbor searches or neural networks [@Csabai03; @Collister04]. These codes extract the information directly from the data, given an appropriate training set with spectroscopic information. Template fitting methods work instead with a set of model spectra from observed galaxies and stellar population models [@Padmanabhan05; @Ilbert06; @Mobasher07; @Pello09]. The PhotoZ code that we have implemented under Astro-WISE belongs to the second category and its original incarnation is described in [@Bender01]. The code estimates redshifts $z$ by comparing a set of discrete template SEDs T to the broadband photometry of the (redshifted) galaxies. For each SED the full redshift likelihood function including priors for redshift, absolute luminosity and SED probability is computed using the Bayes’ theorem: $$\label{Bayes} P(z,T|C,M,...) \propto p(C|z,T)p(z,T|M),$$ where $C$ is the vector of measured colors, $M$ the galaxy absolute magnitude, $p(C|z,T)\propto exp(-\chi^2/2)$ is the probability of obtaining a normalized $\chi^2$ for the given dataset with its errors, redshift and template $T$, and $p(z,T|M)$ the prior distribution. This is a product of parametrized functions of the type: $$\label{eq:Priors} p(y)\propto y^nexp\left[-ln(2)\left( \frac{y-\hat{y}}{\sigma_y}\right)^p\right],$$ where the variable $y$ stands for redshift or absolute magnitudes. Typically we use $n=0$, $p=6$ or $8$, and $\hat{y}$ and $\sigma_y$ with appropriate values for mean redshifts and ranges, or mean absolute magnitudes and ranges, which depend on the SED type. The set of galaxy templates is semi-empirical and is chosen to map the color space spanned by the different types of objects at different redshifts. The original set [@Bender01] includes 31 SEDs describing a broad range of galaxy spectral types, from early to late to star-bursting objects. Recently, we added a set of SEDs tailored to fit luminous red galaxies and one SED to represent the average QSO spectrum at redshift $\approx 2$ [@Saglia11]. Furthermore, the method also fits a set of stellar templates, allowing a star/galaxy classification and an estimate of the line-of-sight extinction for stellar objects. The templates cover typically the wavelength range $\lambda=900$ Å up to 25000 Å (with the QSO template covering instead 300-8000 Å) and are sampled with a step typically 10 Åwide (varying from 5 to 20 Å; the QSO SED has $\Delta \lambda=1$ Å). The method has been extensively tested and applied to several photometric catalogues with spectroscopic follow-ups. Given a (deep) photometric dataset covering the wavelength range from the U to the K band, excellent photometric redshifts with $\delta z/(1+z)\sim0.03$ up to $z\approx 5$ with at most a few percent catastrophic failures can be derived for every SED type ([@Gabasch04], [@Feulner05], [@Gabasch08]). When applied to the 5 filter band catalogs $ugriz$ of SDSS [@Greisel11] or $grizy$ of PanSTARRS [@Saglia11], the code delivers $\delta z/(1+z)\sim0.02$ for luminous red galaxies up to redshift $\approx 0.5$. A more detailed description of the scientific merits of PhotoZ goes beyond the scope of this paper, see [@Hildebrandt10] to compare these performances to the ones achieved by other packages. The code is available in Fortran and C++ versions. Implementation for large datasets {#sec:Implementation} ================================= The science applications described in the previous section dealt with some thousand objects and could be managed by simple means, i.e. ascii-based catalogues. In the era of all-sky surveys and/or very deep fields, where millions, if not billions of objects are imaged, this approach is doomed to fail. The support of a database, the automatisation of the procedures and the tools to administrate the testing and analysis of the results become essential ingredients for a successful science project. Therefore, having in mind our participation in the PanSTARRS1 survey and future projects like KIDS, DES and possibly EUCLID (see Introduction), we designed and implemented two packages, PhotoZ for Astro-WISE (see Sect. \[sec:AWPhotoZ\] and \[sec:AWPerformances\]) and the Photometric Classification Server (PCS) for PanSTARRS1 (see Sect. \[sec:PCS\]). PhotoZ for Astro-WISE {#sec:AWPhotoZ} --------------------- We embedded the PhotoZ code in Astro-WISE following the general philosophy of the package. A Python wrapper ([*PhotRedCatalog*]{}) interfaces the Oracle Astro-WISE database to the (Fortran) code, providing the necessary reading, executing and writing calls to construct the ascii input files with the photometry vectors, call the (compiled Fortran) PhotoZ code and transfer the ascii output back into the database. As usual in every Astro-WISE application, each persistent entry created in this last phase allows the backward tracing of the components down to the single raw and calibration frames that went into the production of the photometry used in the process. The option for a posteriori evaluation of the full redshift probability distribution for a list of selected objects is provided. A separate routine ([*PhotRedConfig*]{}) allows the folding of the available spectral energy distributions with the given filter curves on a predefined grid of redshifts to maximize the speed of the PhotoZ code for a given photometric set. Parallelization is obtained by splitting the list of objects to be analysed in smaller chunks, and executing separate calls of PhotoZ on the multiple cluster nodes. Visualization routines give the possibility to plot the best-fitting SED, the best-fitting stellar SED, the datapoints and the redshift probability distribution of selected objects. A schematic description of the structure of the PhotoZ code is given in Fig. \[fig:flowchart\] ![Flowchart of the basic functionality of PhotoZ. Top part: SEDs, the stellar library and the filter curves are retrieved from the data server, the SEDs multiplied with the filter curves to compute the relative fluxes in each band. The results are then again stored on the data servers. Lower part: To create a PhotRedCatalog object, the system retrieves the necessary files, creates an AssociateList of the input SourceLists (i.e. matches the lists in RA and DEC), computes the photometric redshifts, using the information from the PhotRedConfig and finally links the resulting PhotoZ SourceList with the AssociateList.[]{data-label="fig:flowchart"}](AW2b.eps){width="12cm"} Examples and Performances {#sec:AWPerformances} ------------------------- PhotoZ runs under Astro-WISE as implemented at the Munich node on the PanSTARRS cluster, a 175 nodes (each with 2.6GHz 4 CPUs and 6 GB memory, for a total of 700 CPUs) Beowulf machine with 180 TB disk space, attached to a PB robotic storing device, mounted at the Max-Planck Rechenzentrum in Garching. Two servers run the Oracle database. The Munich Astro-WISE node is federated with the central node of Groningen. As an example how the system works, we describe the derivation of the photometric redshifts of galaxies detected in the Medium Deep Field 4 (MDF04) of PanSTARRS1 (see also \[sec:PCS\] and [@Saglia11]) in an Astro-WISE session. In this context below we indicate with “awe$>$” the Python Astro-WISE prompt. For a detailed description [*how to*]{} run the commands discussed below we refer to the Astro-WISE manual [^5]. We first ingest the PanSTARRS1 filter curves: [awe$>$]{} photredfilter = PhotRedFilter( pathname=’PS\_g.filter’) [awe$>$]{} photredfilter.filter=(Filter.mag\_id==’PS\_g g’) [awe$>$]{} ... [awe$>$]{} photredfilter.make() where PS\_g.filter is an ASCII file with two columns, wavelength in Angstroms and the transmission of the PanSTARRS1 g filter at this wavelength. We repeat the process for the filters r, i, z and y. Then we configure the system, specifying the galaxy and stellar libraries (see [@Bender01]): [awe$>$]{} filt = (Filter.name == ’PS\_g’)\[0\] [awe$>$]{} pfg = (PhotRedFilter.filter == filt )\[0\] [awe$>$]{} ... [awe$>$]{} pse = (PhotRedSED.sed\_name == ’mod\_e.sed’)\[0\] [awe$>$]{} ps1 = (PhotRedSED.sed\_name == ’mod\_s210.sed’)\[0\] [awe$>$]{} ... [awe$>$]{} starlib=(PhotRedStarlib.filename==’starlib\_pickles.lis’)\[0\] [awe$>$]{} pc = PhotRedConfig() [awe$>$]{} pc.SEDs=\[pse,ps1,...\] [awe$>$]{} pc.filters=\[pfg,pfr,pfi,pfz,pfy\] [awe$>$]{} pc.starlib=(starlib)\[0\] [awe$>$]{} pc.name=’PanSTARRS1\_MDF04’ [awe$>$]{} pc.make() We now ingest the PanSTARRS1 photometric catalogue into the [*SourceLists*]{} sg, sr, si, sz, sy and generate the photometric redshifts with the commands: [awe$>$]{} pr = PhotRedCatalog() [awe$>$]{} pr.config=pc [awe$>$]{} pr.master=sg [awe$>$]{} pr.sourcelists=\[sg,sr,si,sz,sy\] [awe$>$]{} pr.name=’PanSTARRS1\_MDF04’ [awe$>$]{} pr.make() The results are stored in the pr.associate\_list AssociateList and can be examined through the Oracle database tools and/or the Python awe$>$ prompt. For example, the command [awe$>$]{} pr.plot( 23 ) plots the best-fitting SED, the best-fitting stellar SED, the datapoints and the redshift probability distribution for the objects with identification number 23 in the associate\_list. The derivation of photometric redshifts for the $\approx 350000$ entries in the MDF04 photometric catalogue down to the $r=24$ magnitude takes 3.3 sec if our full PanSTARRS cluster (700 CPUs) is available (i.e. $\approx 150$ objects per second per node). Through the federation mechanism the results can be seen from each Astro-WISE federated node that has the relevant permissions to access the data. We are in the process of optimizing the SEDs and validating the photometric redshifts through available spectroscopic data for the PanSTARRS1 filter set. First results are discussed in [@Saglia11], where a precision of $\approx 0.02(1+z)$ for red luminous galaxies up to redshift $z\approx 0.5$ is achieved. The Photometric Classification Server for PanSTARRS1 {#sec:PCS} ---------------------------------------------------- The Photometric Classification Server (PCS) for PanSTARRS1 provides software tools to perform a photometric star/QSO/galaxy classification, compute photometric redshifts for galaxies and (a subset of) best-fitting temperature, metallicity, gravity and interstellar extinction parameters for stars. A detailed description of the system can be found in [@Saglia08], [@Snigula09] and [@Saglia11]. The code is interfaced to the Published Science Products System (PSPS) database of PanSTARRS1 (see [@Heasley08]), based on Microsoft SQL and inspired in its structure by the SDSS database. The “manual mode” of operations is similar to the one described in Sect. \[sec:AWPhotoZ\]. The user can query the database and run the code off-line through SQL commands and calls to shell scripts, or also through a web interface. This mode is useful when optimizing the SEDs using available datasets with spectroscopic redshifts. The normal mode of operation, however, is fully automatized. The interface to PSPS triggers the analysis of newly produced object catalogues of the static sky on a regular basis. When new entries are found, CAS-like jobs extract them from the database in Hawaii, format them on our PanSTARRS cluster and submit multiple runs of the C++ version of PhotoZ with sub-blocks of data to parallelize the processing. Finally, the resulting photometric redshifts are stored in a local MySQL database and the corresponding table is pulled by the central one in Hawaii. The process to download from the PSPS database the MDF04 catalogue ($\approx 2$ minutes), measure the photometric redshifts on our PanSTARRS cluster ($\approx 30$ sec if 700 nodes are available) and provide the results for pulling by the PSPS database ($\approx 2$ minutes) takes at most 5 minutes. Therefore we expect to sustain the expected regular flow of new photometric data of PanSTARRS1 without problems. The system is open for the implementation of further different approaches to photometric redshifts (see Sect. \[sec:PhotoZ\]). Conclusions {#sec:Conclusions} =========== PhotoZ under Astro-WISE and PCS for PanSTARRS1, the systems to compute accurate photometric redshifts for large datasets described in this paper, are up and running. They are ready to analyse and archive the photometric catalogues with millions of entries that the wide area surveys started recently or starting in the near future will provide. They can be considered as prototypes for the future development of the data analysis schemes of EUCLID [@Laureijs09]. 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--- abstract: 'The use of liquid helium and neon as scintillators for neutrino detection is investigated. Several unique properties of these cryogens make them promising candidates for real-time solar neutrino spectroscopy: large ultraviolet scintillation yields from ionizing radiation, transparency to their own scintillation light, and low levels of radioactive impurities. When neutrinos scatter from electrons in liquid helium or neon, ultraviolet light is emitted. The ultraviolet scintillation light can be efficiently converted to the visible with wavelength shifting films. In this way the neutrino-electron scattering events can be detected by photomultiplier tubes at room temperature. We conclude that the solar neutrino flux from the $\rm p+p\rightarrow e^{+}+d+\nu_{e}$ reaction could be characterized and monitored versus time using a 10 ton mass of liquid helium or neon as a scintillation target.' address: 'Department of Physics, Harvard University, Cambridge, MA 02138, USA' author: - 'D.N. McKinsey and J.M. Doyle' title: 'Liquid Helium and Liquid Neon - Sensitive, Low Background Scintillation Media For The Detection Of Low Energy Neutrinos' --- Introduction ============ The observed deficit in solar neutrino flux at the Earth’s surface is now well established; the neutrino detection rates measured in the Homestake[@Davis68], SAGE/GALLEX[@SAGE; @GALLEX], and Kamiokande/Super-Kamiokande[@Kamiokande; @Super-K] experiments are each significantly less than predicted by the Standard Solar Model (SSM), but taken together are also logically incompatible with any current solar model. Resolution of this problem remains a tantalizing goal. It is plausible that the correct model explaining the observed neutrino detection rates involves flavor oscillation of massive neutrinos. The several scenarios for flavor conversion will most likely be discriminated through measurement of the solar neutrino flux, including temporal variations, at all energies and for all neutrino species. Distortions of the predicted solar neutrino energy spectra could indicate neutrino flavor oscillations, as could daily or seasonal variation of the detected neutrino flux. With these motivations, it is no surprise that real-time detection of neutrinos is rapidly becoming more sophisticated, with many new detectors either in development or recently implemented. One of the most daunting experimental challenges in neutrino observation is the real-time measurement of the full flux of low energy neutrinos from the solar reaction $\rm p+p\rightarrow e^{+}+d+\nu_{e}$. This “pp” reaction is the most intense source of solar neutrinos, and initiates the chain of fusion reactions in the sun. The emitted pp neutrinos range in energy from 0 to 420 keV and have a precisely predicted flux of $\rm 5.94\times 10^{10}\, s^{-1} \, cm^{-2} $ at the Earth[@Bahcall98]. Despite this high flux, the pp neutrinos have proven difficult to characterize in real time; low energy neutrinos yield low energy scattering events, and these are difficult to detect and discriminate from radioactive backgrounds. In order to characterize and monitor the pp neutrino flux, a detector is needed that has a high signal yield for neutrino-induced events, a high rate of such events, and a low background rate from intrinsic radioactivity. We are familiar with several approaches to the real-time detection of pp neutrinos: bolometric detection of helium atoms liberated by rotons from a liquid helium bath (HERON)[@HERON], measurement of electron tracks generated in a pressurized He (HELLAZ) or $\rm CF_{4}$ (SUPER-MuNu) gas-filled time projection chamber[@HELLAZ; @MUNU], and the use of a low energy neutrino absorbing nuclide that follows absorption with a delayed gamma emission(LENS)[@LENS]. Here we propose a detector that uses liquid helium or neon as a scintillation target. This scheme offers the advantages of high scintillation yield, high neutrino detection rate, low intrinsic radioactivity, and simplicity. Experimental Overview ===================== Detection of neutrinos in our proposed experiment is based on neutrino-electron elastic scattering, $\rm \nu_{x} + e^{-} \rightarrow \nu_{x} + e^{-}$, where x = ($\rm e$, $\rm \mu$, $\rm \tau $). For pp neutrinos, the scattered electron can range in energy from 0 to 260 keV. The scattering cross-section for electron neutrinos is about $\rm 1.2\times10^{-45} \, cm^{2}$ (about 4 times larger than for $\rm \mu$ or $\rm \tau$ neutrinos)[@Bahcall]. This small cross-section leads to the need for a large detector. With 10 tons of active scintillator ($\rm 3\times 10^{30}$ electrons), a total solar neutrino scattering rate of roughly 27 per day will occur with about 18 of these from p-p neutrinos (according to the SSM)[@Bahcall98]. This mass of liquid helium (neon) fills a 5.1 (2.6) meter diameter sphere. We have diagrammed our proposed experiment in Figure \[fig:apparatus\]. The design characteristics are similar to those used currently in the Borexino experiment[@BOREXINO], with crucial differences arising from the choice of scintillator and associated cryogenics. A spherical geometry is chosen for conceptual simplicity (a cylindrical volume, for example, could be used instead). In the center of the experiment is an active region (10 tons) of liquid helium or neon. Surrounding the active region is a thin shell of transparent material. On the inner surface of this shell is evaporated a layer of tetraphenyl butadiene (TPB), a wavelength shifting fluor. Around the active (inner) region is a shielding (outer) region filled with either liquid neon or liquid helium. If neon is used as a shielding medium, it should be about 2 meters thick, while if the shielding region is liquid helium, this region should be 5 meters thick. These liquids are held in a large transparent tank (or 2 separate tanks, see below). Surrounding the central tank(s), separated by vacuum, is another transparent tank filled with pure liquid nitrogen. Outside the cryogens, at room temperature, is a large array of low-activity photomultiplier tubes, all facing the interior and fitted with light concentrators. Around the entire assembly is a stainless-steel tank, filled with water. Detection of solar neutrinos is via scintillation originating from neutrino-electron scattering that occurs in the active region. These events cause intense emission of extreme ultraviolet light (EUV), centered at a wavelength of approximately 80 nm[@Stockton72; @Packard70]. This light is absorbed by the TPB waveshifter, causing fluorescence in the blue ($\sim$ 430 nm). The blue light travels through the shield region, through the transparent acrylic walls and liquid nitrogen, and is detected by the photomultipliers at room temperature. Detection electronics are triggered by multiple photomultiplier coincidence, indicating a potential neutrino scattering event. There are several aspects of this geometry that lead to important advantages. EUV light that originates in the active region will hit the TPB film and be converted into blue light, but EUV light that originates outside the active region will simply be absorbed and will not contribute to the background. The liquid nitrogen acts both as black-body radiation shielding and gamma ray shielding, while the tank of deionized water outside the photomultipliers acts as further shielding. The entire experiment will be located deep underground to reduce cosmic ray events. Muon events will be actively vetoed. Vetoing could be done using a set of photomultipliers to detect Cerenkov light in the water tank. Signal ====== A relatively clear model of scintillations in liquid helium and neon can be elucidated from the numerous experimental characterizations of charged-particle-induced scintillation in condensed noble gases[@Stockton72; @Packard70; @Surko70; @Roberts73; @Kubota79; @Hitachi83; @Hab98]. When an energetic charged particle passes through the liquid, numerous ion-electron pairs and excited atoms are created. The ions immediately attract surrounding ground state atoms and form ion clusters. When the ion clusters recombine with electrons, excited diatomic molecules are created. Similarly, the excited atoms react with surrounding ground state atoms, also forming excited diatomic molecules. Fluorescence in condensed noble gases is observed to be almost entirely composed of a wide continuum of EUV light, emitted when these excited diatomic molecules decay to the monoatomic ground state. The energy of emission is less than the difference in energies between the ground state (two separated atoms) and the first atomic excited state for any given noble gas. The scintillation target is thus transparent to its own scintillation light, and a detector based on a condensed noble gas can be built to essentially arbitrary size without signal loss from reabsorption. Liquid helium scintillations have been more quantitatively studied than neon scintillations. It has been found that conversion of electron kinetic energy into prompt scintillation light is highly efficient; about 24% of the energy of an energetic electron is converted into prompt EUV light[@Adams98], corresponding to 15,000 photons per MeV of electron energy. Recent work towards detection of ultracold neutrons trapped in liquid helium[@Doyle94], has resulted in the characterization of efficient wavelength shifting fluors that convert EUV light into blue visible light[@McK97]. This blue light is well matched to the peak sensitivity of available photomultiplier tubes. TPB is the fluor of choice, having a (prompt, $\rm <$ 20 ns) photon-to-photon conversion efficiency from the EUV to the blue of at least 70% (and a total conversion efficiency of 135%)[@McK97; @Mattoni]. The prompt scintillation component from the combined liquid helium-waveshifter system has been measured to have a 20 ns width, allowing the use of coincidence techniques to reduce background[@Hab98]. (In liquid argon and liquid xenon, the prompt ultraviolet photon yield has been measured to be even larger; Doke *et. al.* have measured yields of 40,000 and 42,000 photons/MeV respectively[@Doke90]. This indicates that it is likely that neon has a comparable yield.) Given a scintillation yield of 15,000 photons per MeV, a waveshifting efficiency of 70%, a photomultiplier covering fraction of 70%, and a bialkali photocathode quantum efficiency of 20%, a total photoelectron yield of about 1500 per MeV could be achieved from the prompt component. With this expected photoelectron yield, the energy of a 100 keV neutrino-electron scattering event could be measured with an average of 150 photoelectrons, attaining 16% energy resolution. Liquid neon can be expected to be a similarly fast and efficient scintillation medium, with properties similar to those found in liquid helium. Packard et. al. have found that the electron-excited emission spectrum of liquid neon peaks at 77 nm[@Packard70]. Liquid neon should also have an intense afterpulsing component due to the extreme ultraviolet radiation of triplet molecules. In liquid helium, the lifetime of this slow component has been measured to be 13 seconds[@McK99], close to the radiative lifetime of the ground state triplet molecule[@Chablowski88]. But the theoretically predicted lifetime of ground state triplet neon molecules [@Schneider74] is only 11.9 $\rm \mu s$. In liquid neon, the ground triplet molecular lifetime has been measured to be 2.9 $\rm \mu s$[@Suemoto79]. Intense afterpulsing following neutrino scattering events could be used to positively identify events within the active neon, and could also be added into the prompt signal to improve pulse height resolution. However, our detection scheme does not necessarily require the use of this afterpulsing signal. Cryogenics ========== We describe here the cryogenic and structural requirements for a low energy neutrino detector whose active region is a 10-ton reservoir of liquid helium or neon. We consider three cases. The backgrounds due to construction materials are discussed in section V. **Case A: Liquid neon active region, liquid neon shielding region.** Here the transparent tank holding the shielding and active regions would be constructed of a copper grid and a transparent, low radioactivity material, such as quartz or acrylic. Copper is used to give the tank walls high thermal conductivity and structural rigidity, while the quartz or acrylic allows scintillation light through to the photomultipliers. Given a total surface area of $\rm \pi (6.6\,m)^{2} = 137\,m^{2}$ and a conservatively estimated emissivity[@Pobell] of 1, a total of 270 W is absorbed by the tank walls and routed through a copper heat link to a closed-cycle helium gas refrigerator outside the shielding. If the copper grid covers 20% of the tank surface, has a bulk thermal conductivity of 15 W $\rm cm^{-1}$ $\rm K^{-1}$, and this copper is 10 cm thick, then the power absorbed from 77 K blackbody radiation results in a temperature difference across the tank of no more than 2 degrees. The use of copper to maintain a low thermal gradient is necessary because of the narrow temperature window at which neon is liquid ($\rm 24.5 - 27.1$ K) and the poor thermal conductivity ($\rm \sim 10^{-3}$ W $\rm cm^{-1}$ $\rm K^{-1}$) of liquid neon. The cryogenic constraints on this tank may be relaxed if convection in the liquid neon is found to play an appreciable role in the flow of heat through its volume. The active and shielding regions are separated by a thin ($\sim$ 0.1 mm) shell of transparent plastic or quartz. This shell simply floats in the neon and is held in place by nylon strings connecting the shell to the copper tank. The shell may have small holes in it to allow liquid neon to flow freely between the active and shielding regions. **Case B: Liquid helium active region, liquid neon shielding region.** As in case A, the active and shielding regions are held in a copper grid composite tank. The tank must however be of larger diameter (9.1 m instead of 6.6 m) to accomodate the larger active region. Also, the active and shielding regions must be separated by a vacuum space because of the different temperatures of the liquid neon and liquid helium. The separation of the active and shielding regions must be accomplished with as little material as possible so as to minimize radioactive backgrounds. Appropriate separation may be possible using a 1 mm thick Kevlar-acrylic composite shell, with shielding and active regions held apart using small acrylic pegs. **Case C: Liquid helium active region, liquid helium shielding region.** Liquid helium is not an effective enough gamma ray absorber to protect the active region from copper activity. Therefore the tank must be made from a transparent, low radioactivity material such as acrylic. The heat load from 77 K is large (1430 W), but by cooling the helium through its superfluid transition temperature (2.2 K) to achieve high thermal conductivity, the temperature of the helium may be made constant throughout its volume. The high thermal load on the helium may be handled with a large pumped helium system outside the stainless steel tank. As in Case A, the active and shielding regions may be separated with a thin sheet of plastic or quartz. **General Considerations.** The liquid nitrogen shielding may be held in either a copper grid composite or acrylic tank. The nitrogen should be thick enough (1-2 m) to sufficiently absorb gamma rays from the photomultipliers and stainless tank. Acrylic is a low activity, transparent, strong material. At low temperatures, acrylic remains strong and tough. The yield strength of acrylic increases significantly as temperature is lowered, while the fracture toughness remains roughly constant[@Ward]. Nevertheless, any acrylic containers will have to be designed carefully to avoid unnecessary thermal and mechanical stresses, as the cryogens are of larger scale than is common in low temperature work. Backgrounds =========== Condensed noble gases have an important advantage over organic scintillators: they have no $\rm ^{14}C$ contamination. But among the condensed noble gases, only liquid neon and liquid helium can satisfy the strictest requirements of low radioactive contamination[@Seguinot]. Natural argon is contaminated by the two long-lived isotopes $\rm ^{39}Ar$ and $\rm ^{42}Ar$, and natural krypton contains $\rm ^{85}Kr$ that precludes its use in low background detectors. Liquid xenon would need to be cleaned of Ar and Kr, and double beta decay of $\rm ^{136}Xe$ would have to be addressed. In addition, while liquid xenon has been put to increasing use in searches for dark matter, its high price (at least \$1,000,000 per ton) makes liquid xenon unattractive for use in a large low energy neutrino detector. Helium and neon have no unstable naturally occuring isotopes and therefore no inherent radioactive backgrounds. They do however need to be cleaned of dissolved Ar and Kr, as well as possible low-level contamination by K, U, and Th, but their low boiling temperatures allows for simple and effective solutions to these problems. Distillation can effectively remove argon and krypton, and by passing the helium or neon through a cold trap, the non-noble radioactive contaminants can be frozen out. In neon one remaining possible radioactive contaminant is tritium. If it is found that commercially available neon is contaminated with low levels of tritium, then it can be easily removed by chemical means. Impurities within the helium or neon are therefore not expected to be a significant source of background. Helium and neon are also relatively inexpensive[@neon]. Because liquid helium and neon are easily cleaned of radioactive isotopes, the limiting backgrounds are expected to arise from the various construction materials. Copper (used in cases A and B) has been shown to possess low levels of radioactive impurities[@Avignone86]; an estimate of the activity of copper stored underground for a year[@HERON] gives .02 events $\rm kg^{-1}$ $\rm minute^{-1}$. Possible impurity levels of other necessary materials can be estimated from the results of the BOREXINO[@Alimonti98] and SNO[@SNO] collaborations. It is found[@Polycast] that acrylic is commercially available with U and Th levels of less than $\rm 10^{-13}$ g/g. Photomultiplier assemblies can be constructed with U and Th levels of $\rm 10^{-8}$ g/g. Gamma rays emitted from the copper, acrylic, photomultipliers, stainless steel tank, and heat link will Compton scatter in the nitrogen and shielding regions, producing Cerenkov light that can be detected by the photomultipliers. There will be a significant rate of such events; for example, the BOREXINO group reports a gamma flux of $\rm 2 \times 10^{6}\, day^{-1} \, m^{-2}$ from their photomultiplier assembly. Fortunately, the light yield from gamma Compton scattering events should be relatively small. Cerenkov light should result in no more than 10 photoelectrons per MeV[@SNO], and visible scintillation light should contribute even less. In liquid helium scintillations, the visible light output has been measured to be 500 times less intense than the extreme ultraviolet output[@Stockton72; @Dennis69]. Furthermore, the visible output is concentrated in wavelengths greater than 640 nm, where photocathode responsivities can be chosen to be low. In liquid neon, the visible light emissions are similarly weak, with wavelengths that are shifted even further into the infrared[@Suemoto79]. As a result, the outer neon region, without exposure to an ultraviolet waveshifter, will yield an insignificant amount of visible light from gamma scattering events within its volume. However, even with these effects the high rate of gamma scattering events in the shielding will produce significant background at low photoelectron number. This will therefore set a low energy threshold for neutrino events of roughly 20 keV. This leaves only 10% of solar neutrinos undetected. With a 2 (5) meter thick liquid neon (helium) shielding region, the rate of gammas entering the active volume should be less than 1/day, compared to the predicted 27/day solar neutrino counting rate. Also, gamma rays that penetrate the shielding region will have relatively high energies and are likely to deposit most of their energy in the active region, allowing energy cuts to further reduce background. The background levels arising from events in the shielding regions can be independently tested by running the experiment without any waveshifter. A variety of other effects may help to decrease background counts. The three-dimensional photomultiplier arrangement will allow rough determination of the event location. Events in the active volume will be more evenly spread over the photomultipliers than events in the liquid nitrogen and shielding volume. Also, the light concentrators affixed to the photomultiplier tubes will restrict their immediate field of vision to the active volume. The expected intense ultraviolet afterpulsing from the active liquid neon (see section III) could also provide an important test against background events. Radioactive contamination requirements of the materials separating the active and shield regions are stringent. However, very little of these materials are necessary. If clear plastic is used as a divider between the active and shielding regions, radioactive background from U and Th should be insignificant (given U and Th levels of less than $10^{-13}$ g/g.) However, $\rm ^{14}C$ contamination is a serious issue. In the BOREXINO experiment, $\rm ^{14}C$ levels were demonstrated to be less than $1.9 \times 10^{-18}$ $\rm ^{14}C$/C in organic scintillator synthesized from petroleum[@Alimonti98]. The theoretical estimate for $\rm ^{14}C/C$ in old petroleum is $\sim 5 \times 10^{-21}$, and the higher measured value is presumed to arise during scintillator synthesis or later handling. A $1.9 \times 10^{-18}$ $\rm ^{14}C$/C level in a 100 $\rm \mu m$ thick plastic divider would result in roughly 80 (30) events per day if helium (neon) is used as the active medium. This would obscure the expected 27 neutrino events per day. However, the fact that very little material is required ($\sim$ 10 kg of plastic compared to 100 tons of organic scintillator used in the BOREXINO experiment) suggests it is reasonable to expect that the $\rm ^{14}C$ concentration could be held to an acceptable level. In scheme B, a strong, largely transparent material is needed to separate the liquid helium and liquid neon shielding regions. Because the amount of plastic needed is larger than in cases A and C, a lower level of radioactive impurities is necessary. A second option is to use thin quartz sheet as a substrate. If old silicon is used (older than 50,000 years), then $\rm ^{32}Si$ and $\rm ^{14}C$ are not a problem[@Martoff]. But, of course, $\rm ^{238}U$, $\rm ^{40}K$, $\rm ^{232}Th$, $\rm ^{3}H$ and $\rm ^{22}Na$ must be shown to contribute less than 1 event per day in the energy range of interest. This should be possible because cleanliness levels of less than $10^{-12}$ g/g are routinely achieved in pure Si through zone-refining techniques[@Cabrera84]. By converting this clean Si into silane ($\rm SiH_{4}$) gas, ridding the silane gas of radioactive impurities, and then oxidizing, sufficiently clean $\rm SiO_{2}$ could be produced. Again, the fact that very little quartz is needed makes this contamination level a reasonable requirement. Contamination requirements on the TPB are not so stringent, as only $\rm 0.2 \, mg \, cm^{-2}$ is necessary for efficient wavelength shifting[@McK97]. Muons are another potential source of background. Muons will pass through the experiment at a rate of about $\rm 25 \, day^{-1}\, m^{-2}$ (at Gran Sasso). These prompt events can be eliminated through active vetoing. One way to do this is to detect the Cerenkov radiation in the ultrapure water tank using a second set of photomultipliers[@BOREXINO]. In addition, muons that pass through the active region will produce extremely bright, easily distinguishable scintillation pulses. In the neon experiment, neutrons and radioactive species can be produced by muons stopping in the active volume. With only a small fraction ($\sim$ .008) of muons stopping[@Cassiday73], and with 40% of these stopped muons absorbed by neon nuclei[@Bertin73], a rate of muon radiogenesis of about 0.5 per day follows. Most of these events result in the production of $\rm ^{19}F$, a stable isotope. Prompt muon coincidence rejection and energy cuts will reduce background due to the remaining events (e.g. prompt gammas from neutron absorption, decay of long-lived nuclei) to negligible levels. Muons can also lead to the production of neutrons in the surrounding rock. These neutrons, as well as those emitted from fission products and ($\rm \alpha$, n) reactions, will be moderated and absorbed in the ultrapure water tank, possibly with the help of boric acid dissolved in the water[@HERON], and are not expected to constitute a significant source of background. Conclusion ========== There are several other experimental programs currently underway to develop real time detectors of pp neutrinos. We believe the method described above compares favorably to all of these. However, making exact technical comparisons with HELLAZ, SUPER-MuNu, and LENS is beyond the scope of this paper. Because the HERON experiment also uses a liquid cryogen it is possible to make a few simple comparisons. The HERON program uses liquid helium as a neutrino scattering medium, and bolometers to detect helium atoms liberated by rotons from the liquid helium surface[@HERON]. The possible event rate achievable with HERON is similar to that possible using our proposed scintillation technique with helium as the active scintillator. If liquid neon is used, however, the event rate is 8 times larger for a given active volume. Our design is technically simpler because it requires temperatures of only 27 K (2 K) for liquid neon (helium), while HERON requires 30 mK superfluid helium to avoid roton scattering. HERON has the requirement (not present in our proposed design) that the helium be isotopically pure to avoid $\rm ^{3}He$-roton scattering centers. The added effort and complexity of isotopic purification of 10 tons of helium is significant. A significant technical requirement present in our proposed experiment and not in HERON is the need for large, strong clear plastic tanks at low temperatures. Also, unlike HERON, our proposed experiment relies almost entirely on high purity shielding materials to reduce background, obviating the need for precise event reconstruction for background reduction but requiring additional materials processing. The use of liquid helium or neon as a scintillation medium is a promising method for the detection of low energy neutrinos. First, the background level should be very low because of the extreme cleanliness possible in the active region. All other materials (with higher levels of contamination) can either be well shielded from the active volume or are present in such small amounts that their contribution may be made negligible. Second, the photoelectron output from neutrino scattering events should be high because of the intense extreme ultraviolet scintillation yield. Detection with standard PMTs is made possible by the availability of efficient wavelength shifters. Third, the rate of detected neutrino scattering events will be comparable or larger than those expected in other experimental techniques. Finally, this experiment uses only existing technologies; a small “proof of principle” apparatus could be constructed and tested in relatively little time. Along with the calibration and monitoring of the pp neutrino flux, this detector will be sensitive to other neutrino sources. For example, the relative and absolute intensities of the $\rm ^{7}Be$ and pep solar neutrino lines might be measured using this sort of detector, yielding a good diagnostic test of what happens to neutrinos after they are emitted[@Bahcall96]. Whether these line intensities could be measured over radioactive background (and other neutrino spectra) must be tested by Monte Carlo methods. We conclude that liquid helium and neon are intriguing possible detectors for solar neutrinos. 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--- abstract: 'The classical $R$-matrix structure for the $n$-particle Calogero-Moser models with (type IV) elliptic potentials is investigated. We show there is no momentum independent $R$-matrix (without spectral parameter) when $n\ge4$. The assumption of momentum independence is sufficient to reproduce the dynamical $R$-matrices of Avan and Talon for the type I,II,III degenerations of the elliptic potential. The inclusion of a spectral parameter enables us to find $R$-matrices for the general elliptic potential.' author: - | H. W. Braden Takashi Suzuki\ \ Department of Mathematics and Statistics,\ University of Edinburgh,\ Edinburgh, UK. title: ' $R$-matrices for Elliptic Calogero-Moser Models ' --- =-1.2truecm =-.6in =msbm10 scaled1 ‘=11 =eusm10 scaled 1 =eusm7 scaled 1 =eusm5 scaled 1 == = @\#1[\#1&lt;10 \#1 \#1=10 A\#1=11 B\#1=12 C \#1=13 D\#1=14 E\#1=15 F]{} @\#1[[@@[\#1]{}]{}]{} @@\#1[\#1]{} ‘= ‘=11 =eufm10 scaled 1 =eufm7 scaled 1 =eufm5 scaled 1 == = @\#1[\#1&lt;10 \#1 \#1=10 A\#1=11 B\#1=12 C \#1=13 D\#1=14 E\#1=15 F]{} @\#1[[@@[\#1]{}]{}]{} @@\#1[\#1]{} ‘= ‘=11 =msxm10 scaled 1 =msxm7 scaled 1 =msxm5 scaled 1 =msym10 scaled 1 =msym7 scaled 1 =msym5 scaled 1 == === = @\#1[\#1&lt;10 \#1 \#1=10 A\#1=11 B\#1=12 C \#1=13 D\#1=14 E\#1=15 F]{} @\#1[[@@[\#1]{}]{}]{} @@\#1[\#1]{} ‘= \[section\] \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Definition]{} \[theorem\][Axiom]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Example]{} -9.5cm 11.4cm Edinburgh-/92-93/03 .2cm 11.4cm hep-th/9309033 8.8cm Introduction ============ The Calogero-Moser model [@Cal75; @Moser] is perhaps the paradigm of a completely integrable system of $n$-particles on the line which interact via pairwise potentials $v(q_i-q_j)$. The most general form of the potential in such models [@OlP81] is the (so called type IV) elliptic potential, $v(q)={a\sp2}\wp(a q)$, where $\wp$ is the Weierstrass elliptic function. The various degenerations of this function yield rational (type I, $v(q)=1/{q\sp2}$), hyperbolic (type II, $v(q)=a\sp2/{\sinh(a q)\sp2}$) and trigonometric (type III, $v(q)=a\sp2/{\sin(a q)\sp2}$) potentials. In accord with their importance these models have been studied from many different perspectives. They have a Lax pair formulation, $i {\dot L}= [L,M]$; the ansatz for this Lax pair leads to a study of functional equations[@Cal76; @Ruijs87; @BC90; @BB93]. Further, these models may be expressed as the Hamiltonian reduction [@KKS; @ABT] of integrable flows on the cotangent bundle of symmetric spaces. The quantisation [@OlP83] of these models has also been of some interest. Recently the related (type V) potential $v(q)=1/{q\sp2}+g{q\sp2}$ has been shown to be relevant to the collective field theory of strings[@AJ91]. The proof of the complete integrability of a system given in terms of a Lax pair involves several stages. The first, an immediate consequence of a Lax pair formulation, is the observation that the quantities $\Tr_E {L\sp k}$ are conserved. Here the trace is taken over the representation $E$ of the Lie algebra ${\Got{g}}$ to which the operator $L$ is associated. Another stage is to show these provide enough functionally independent conserved quantities. Finally, and this is perhaps the most tedious step, one must show the quantities are in involution, [*i.e.*]{} $\{\Tr_E {L\sp k},\Tr_E {L\sp m}\}=0$. This step is model dependent. For the Calogero-Moser this stage may be achieved by arguments based on asymptotics[@Moser], inverse scattering[@OlP76] or direct recursion[@W77]. Given an $L$-operator, an alternative approach to proving this Poisson commutativity proceeds via the $R$-matrix[@STS; @FT; @BV90]. An $R$-matrix is an $E\otimes E$ matrix satisfying {LL}=\[R, L1\]-\[[R]{},1L\] . \[rmatrix\] An immediate consequence of the existence of an $R$-matrix is that $$\{\Tr_E {L\sp k},\Tr_E {L\sp m}\}= \Tr_{E\otimes E}\{ {L\sp k}\x {L\sp m}\}= k m \Tr_{E\otimes E} {L\sp {k-1}}\otimes{L\sp {m-1}} \{L\x L\}= 0.$$ The vanishing follows from (\[rmatrix\]) by expressing $\{ L \x L\}$ as a commutator and using the cyclicity of the trace. An $R$-matrix also allows one to canonically construct the matrix $M$ of the Lax pair[@STS]. Recently Avan and Talon[@AT] constructed the $R$-matrices for the Calogero-Moser models and potentials of type I, II, III and V, thus providing this alternate means of proof. The $R$-matrices found by Avan and Talon were dynamical: that is, they depended on the dynamical variables (in their case positions) of the model. In contrast to systems governed by purely numerical $R$-matrices[@FT], dynamical $R$-matrices needn’t satisfy the Yang-Baxter equation and the theory of dynamical $R$-matrices is not well understood[@Maillet; @EEKT; @BB]. One hopes that such concrete examples of dynamical $R$-matrices as are provided by the Calogero-Moser models will aid in the elucidation of this theory. Our present work investigates the $R$-matrix structure of the Calogero-Moser models further. The $R$-matrices of Avan and Talon were constructed on the basis of two assumptions, namely momentum independence and the vanishing of certain terms of the $R$-matrix. These assumptions were found to be consistent with the potentials of type I, II, III, V but did not allow the type IV potential. One might ask what happens if these assumptions are relaxed. We shall show that the second of Avan and Talon’s assumptions actually follows from that of momentum independence (given sufficient particles). We can therefore conclude that no momentum independent $R$-matrix exists for the $L$-operators under consideration. In [@Krich] Krichever enlarged the class of $L$-operators yielding type IV potentials to include a spectral parameter. The usual $L$-operators exactly correspond to those values of the spectral parameter for which the operators are hermitian. We show that the inclusion of the spectral parameter allows us to construct a momentum independent $R$-matrix. The plan of this paper is as follows. In Section 2 we will introduce our notation, the $L$-operators under consideration and the equations to be solved. Section 3 looks at the simplifications resulting from the assumption of momentum independence. Section 4 then shows that for the usual $L$-operator no momentum independent $R$-matrix can be found. Upon introducing a spectral parameter in Section 5, we then exhibit a solution to the corresponding equations. We conclude with a brief discussion. Preliminaries ============= As an alternative to the matrix entry calculations often presented, we give our calculations in terms of a basis of the underlying Lie algebra. Although we will ultimately specialise to the $gl_n$ case, we believe this to be both computationally and conceptually easier. It will also enable us to isolate those features peculiar to $gl_n$. We begin by deriving the relevant equations to be solved and introducing our notation. Let $X_\mu$ denote a Cartan-Weyl basis for the (semi-simple) Lie algebra ${\Got{g}}$ associated with the operator $L$. That is $\{X_\mu\}=\{H_i,E_\alpha\}$, where $\{H_i\}$ is a basis for the Cartan subalgebra ${\Got{h}}$ and $\{E_\alpha\}$ is the set of step operators (labelled by the root system $\Phi$ of ${\Got{g}}$) and $$[H_i,E_\alpha ]=\alpha_i E_\alpha \quad\quad [E_\alpha,E_{-\alpha}]= \alpha\cdot H\quad\quad {\rm and}\quad\quad [E_\alpha,E_\beta ]=N_{\alpha,\beta}E_{\alpha+\beta} \quad{\rm if}\ \alpha+\beta\in\Phi .$$ With $[X_\mu,X_\nu]=c_{\mu \nu}\sp\lambda\ X_\lambda$ defining the structure constants of ${\Got{g}}$, we see for example[^1] that $c_{i\,\alpha}\sp\beta=\delta_{\alpha,\beta}\alpha_i$ and $c_{\alpha\, -\alpha}\sp i =\alpha_i$. Further, we adopt the hermiticity convention ${E_\alpha}\sp\dagger=E_{-\alpha}$ for the representations of ${\Got{g}}$. The structure constants may then be chosen to be real and to have the symmetries: $c_{\alpha\, \beta}\sp\lambda = -c_{\beta\,\alpha}\sp\lambda = c_{\lambda\, -\beta}\sp\alpha= -c_{-\alpha\, -\beta}\sp{-\lambda}$. With this notation at hand we express the $L$ operator as L\_LX\_= pH + i\_w\_E\_where[@Krich] w\_w(q;u)= [(u-q)]{} [e]{}. \[eq:w\] Here $\sigma(x)$ and $\zeta(x)=\sigma\sp\prime(x)/ \sigma(x)$ are the Weierstrass sigma and zeta functions[@WW]. The quantity $u$ in (\[eq:w\]) is known as the spectral parameter and we will only make its appearance explicit when confusion might otherwise arise. It will also be convenient to use the shorthand $f_\alpha$ for a function on ${\Got{h}}$ that takes the value $f(\alpha\cdot q)$ when evaluated at $q$. The functions $w_\alpha$ satisfy the addition formula w\_w\_-w\_w\_= ( z\_-z\_)w\_[+]{}, z\_(u)=[w\_(u)]{}= \_+[12]{}(u) . \[eq:addition\] Clearly $L=L\sp\dagger$ when $w_\alpha=-w_{-\alpha}$ and this is the case usually considered. Requiring hermiticity restricts the spectral parameter with the result that $u\in\{\omega_1,\omega_2,-\omega_1-\omega_2\}$, where $2\omega_{1,2}$ are the periods of the associated elliptic functions. For the Lie algebra $gl_n$, the case of most interest to us, $\Phi=\{e_i-e_j,\ 1\leq i\not= j\leq n\}$ with the $e_i$ an orthonormal basis of ${\Bbb R}\sp n$. If $e_{rs}$ denotes the elementary matrix with $(r,s)-th$ entry one and zero elsewhere, then the $n\times n$ matrix representation $H_i=e_{ii}$ and $E_\alpha=e_{ij}$ when $\alpha=e_i-e_j$ gives the usual representation of $L$. Working with the simple algebra $a_n$ corresponds to the center of mass frame. Let us begin unravelling (\[rmatrix\]). The left hand side becomes {LL}=\_[,]{} { L, L} X\_X\_=i\_[j,]{}\_j w\_(H\_jE\_- E\_H\_j) \[rmrhs\] upon using $\{p_j,w_\alpha\}= \{p_j,\alpha\cdot q\}w_\alpha\sp\prime=\alpha_j w_\alpha\sp\prime$. Turning now to the right hand side of (\[rmatrix\]) we have $R=R\sp{\mu\nu}X_\mu\otimes X_\nu$ and ${R\sp\pi}=R\sp{\nu\mu}X_\mu\otimes X_\nu$. Then $$\begin{aligned} [R, L\otimes 1]-[{R\sp\pi},1\otimes L]&=& R\sp{\mu\nu}([X_\mu,L]\otimes X_\nu-X_\nu\otimes[X_\mu,L])\cr &=& R\sp{\mu\nu}L\sp\lambda ([X_\mu,X_\lambda]\otimes X_\nu-X_\nu\otimes[X_\mu,X_\lambda])\cr &=&( R\sp{\tau\nu}c_{\tau\lambda}\sp\mu L\sp\lambda - R\sp{\tau\mu}c_{\tau\lambda}\sp\nu L\sp\lambda) X_\mu\otimes X_\nu. \nonumber\end{aligned}$$ In terms of the Lie algebra basis, (\[rmatrix\]) then becomes the equation { L, L}= Rc\_L- Rc\_L. \[rmatrixnew\] Observe that (\[rmatrixnew\]) has the structure of a matrix equation, V R +RV =A, for the unknown matrix $R$ in terms of the specified $A\sp{\mu\nu}=\{ L\sp\mu, L\sp\nu \}$ and $V\sp{\mu\nu}= c_{\nu\lambda}\sp\mu L\sp\lambda$. Equation (\[rmatrixnew\]) yields three different equations, depending on the range of $\{\mu,\nu\}$. For $(\mu,\nu)= (i,j),(i,\alpha)$ and $(\alpha,\beta)$ respectively, these are $$\begin{aligned} \label{eq:ij} 0&=& \sum_{\alpha}(R\sp{\alpha j}\alpha_i-R\sp{\alpha i}\alpha_j)w_{-\alpha}\\ \label{eq:ia} -\alpha_i w_\alpha\sp\prime &=& i\alpha\cdot p\ R\sp{\alpha i} +\alpha\cdot R\sp{i}w_\alpha + \sum_{\beta}(\beta_i w_\beta R\sp{-\beta\alpha } + w_{\alpha -\beta} R\sp{\beta i}c_{\beta\ \alpha -\beta}\sp\alpha) \\ \noalign{\hbox{and}}\cr \label{eq:ab} 0&=& \alpha\cdot R\sp{\beta}w_\alpha -\beta\cdot R\sp{\alpha}w_\beta +i ( \alpha\cdot p\ R\sp{\alpha\beta}-\beta\cdot p\ R\sp{\beta\alpha})\\ \nonumber && +\sum_{\gamma}( R\sp{\gamma\beta} c_{\gamma\, \alpha-\gamma}\sp\alpha w_{\alpha-\gamma} -R\sp{\gamma\alpha} c_{\gamma\, \beta-\gamma}\sp\beta w_{\beta-\gamma} )\end{aligned}$$ Here we have introduced the shorthand $\beta\cdot R\sp{\mu}\equiv\sum_{i}\beta_i R\sp{i\mu}$. Equations (\[eq:ij\]-\[eq:ab\]) are the components of (\[rmatrix\]) in our basis. Momentum Independence ===================== We now turn to the solution of equations (\[eq:ij\]-\[eq:ab\]) subject to [ the assumption that the $R$-matrix is independent of momentum]{}. This assumption (introduced in [@AT]) means that R=0,R+R=0R=0=. \[eq:momcons\] The first of these restrictions follows from (\[eq:ia\]) while the remainder come from (\[eq:ab\]). For example, in the matrix components of $gl_n$ introduced earlier, we have $R\sp{\alpha i}=0\Leftrightarrow R\sp{jkii}=0$ and we thus obtain equations (14) of [@AT]. At this stage, equation (\[eq:ij\]) is satisfied identically and the variables remaining are $R\sp{ij},R\sp{i\alpha},R\sp{\alpha\alpha}$ and $R\sp{\alpha\, -\alpha}$. The remaining equations to be solved are $$\begin{aligned} -\alpha_i w_\alpha\sp\prime &=& \alpha\cdot R\sp{i}w_\alpha + \alpha_i R\sp{-\alpha\alpha}w_\alpha -\alpha_i R\sp{\alpha\alpha}w_{-\alpha} \label{eq:ian} \\ \noalign{\hbox{and}}\cr \alpha\cdot R\sp{\beta}w_\alpha -\beta\cdot R\sp{\alpha}w_\beta &=& c_{\alpha\gamma}\sp\beta( R\sp{\alpha\alpha}w_\gamma-R\sp{\beta\beta}w_{-\gamma}) + c_{-\alpha\gamma}\sp\beta( R\sp{-\alpha\alpha}w_\gamma+R\sp{-\beta\beta}w_{\gamma}). \label{eq:abn}\end{aligned}$$ The first term on the right-hand side of (\[eq:abn\]) is nonvanishing only for $\gamma=\beta-\alpha\in\Phi$ while the second term is nonvanishing for $\gamma=\beta+\alpha$. We note that for the simply-laced algebras ($\alpha\cdot\alpha=2,\ \forall\ \alpha\in\Phi$), at most one of the terms on the right-hand of (\[eq:abn\]) can be nonvanishing and we henceforth assume this to be the case. Now, by viewing the root $\gamma=\beta-\alpha$ as being also the sum $\gamma=\beta+(-\alpha)$, we obtain the two equations $$\begin{aligned} \label{eq:g-} \alpha\cdot R\sp{\beta}w_\alpha -\beta\cdot R\sp{\alpha}w_\beta &=& c_{\alpha\gamma}\sp\beta( R\sp{\alpha\alpha}w_\gamma-R\sp{\beta\beta}w_{-\gamma}) \\ \noalign{\hbox{and}}\cr \label{eq:g+} -\alpha\cdot R\sp{\beta}w_{-\alpha} -\beta\cdot R\sp{-\alpha}w_\beta &=& c_{\alpha\gamma}\sp\beta( R\sp{\alpha-\alpha}w_\gamma+R\sp{-\beta\beta}w_{\gamma}).\end{aligned}$$ We shall utilise the consistency of these equations below. Our first observation is that $R\sp{ij}=\eta\,\delta\sp{ij}+P\sp{ij}$ for some constant $\eta$ that we shall later determine and matrix $P\sp{ij}$ orthogonal to the roots, $\alpha\cdot P\sp{j}=0\,\forall\,j$. To see this, view (\[eq:ian\]) as an equation between vectors; thus $\alpha\cdot R\sp{i}$ must be proportional to $\alpha_i$. If we define the constant of proportionality by $\alpha\cdot R\sp{i}=\eta_\alpha\alpha_i$, where $\eta_\alpha$ could in principle depend on $\alpha$, then by linearity $$(\alpha+\beta)\cdot R\sp{i}= \eta_{\alpha+\beta}(\alpha_i+\beta_i)=\eta_\alpha\alpha_i+\eta_\beta\beta_i$$ and so $\eta_{\alpha+\beta}=\eta_\alpha=\eta_\beta\equiv\eta$. Now for each $\alpha$ we have $\sum_i \alpha_iR\sp{ij}=\eta\alpha_j$, and so $R\sp{ij}=\eta\,\delta\sp{ij}+P\sp{ij}$. The matrix $P\sp{ij}$ orthogonal to the roots arises when we have $u(1)$ factors present in ${\Got{g}}$. Thus we have -w\_=w\_+ Rw\_ - Rw\_[-]{}. \[eq:iann\] The assumption of momentum independence leads then to the two equations (\[eq:abn\]) and (\[eq:iann\]). The case $w_\alpha=-w_{-\alpha}$. ================================= Our discussion has so far made no use of the form of $w_\alpha$. For the remainder of this section we will assume that $w_\alpha$ is an odd function, the case usually considered. As we shall see, this results in some quite strong conclusions. The next section, which deals with the inclusion of a spectral parameter, will consider the more general case. First let us show $\eta=0$. Upon subtracting from (\[eq:iann\]) the analogous equation obtained by replacing $\alpha$ with $-\alpha$ and using the second equation of (\[eq:momcons\]), we find =-[12]{}(R+R). \[eq:eta\] Further, upon subtracting (\[eq:g-\]) from (\[eq:g+\]), we obtain $$\begin{aligned} \beta\cdot(R\sp{\alpha}-R\sp{-\alpha})w_\beta &=& c_{\alpha\gamma}\sp\beta( R\sp{\alpha-\alpha}+R\sp{-\beta\beta}-R\sp{\alpha\alpha}-R\sp{\beta\beta}) w_\gamma .\\ \noalign{\hbox{ The same operations applied to the analogous equations based now on $\gamma=(-\alpha)-(-\beta)$}} \noalign{\hbox{ and $-\gamma=\alpha-\beta$ yield }}\cr \beta\cdot(R\sp{\alpha}-R\sp{-\alpha})w_\beta &=& c_{\alpha\gamma}\sp\beta( R\sp{-\alpha-\alpha}+R\sp{-\beta-\beta}-R\sp{-\alpha\alpha}-R\sp{\beta-\beta}) w_\gamma .\end{aligned}$$ Upon comparing these last two equations and using (\[eq:eta\]) together with $R\sp{\alpha\, -\alpha}+R\sp{-\alpha\, \alpha}=0$, we find that $\eta=0$. Therefore $R\sp{ij}=P\sp{ij}$; the choice $P\sp{ij}=0$ corresponds to the middle two equations of Avan and Talon’s second assumption[@AT]. We have now reduced the possible nonzero variables of the $R$-matrix to $R\sp{i\alpha},R\sp{\alpha\alpha}$ and $R\sp{\alpha\, -\alpha}$ subject to R+R=0,R+R=0,R+R=- [w\_w\_]{} \[eq:redcons\] and (R-R)w\_=c\_( -[w\_w\_]{} -[w\_w\_]{} +2 R )w\_-c\_[-]{}( -[w\_w\_]{} +[w\_w\_]{} +2 R)w\_. \[eq:add\] Of course $c_{\alpha\gamma}\sp\beta=0$ unless $\gamma=\beta-\alpha\in\Phi$ (in which case $\alpha\cdot\gamma\not=0$). Our method of solving these equations proceeds as follows. Let us define the quantity $A_{\beta\gamma}$ by A\_(R-R)w\_-c\_[-]{}z\_. Obviously this is closely related to the left hand side of (\[eq:add\]). Our aim will be to show this quantity to be constant, from which we will be able to deduce the remaining equations of Avan and Talon’s second assumption. $ A_{\alpha\beta}=A_{\beta\alpha}=-A_{\beta\gamma}=A_{\gamma-\alpha}= A_{-\beta\gamma}. $ First let us motivate the definition of $A_{\beta\gamma}$ and then derive its symmetries. Suppose $\gamma=\beta-\alpha\in\Phi$ so only the first term of (\[eq:add\]) is nonvanishing. The analogous equation to (\[eq:add\]) for $\beta=(\beta-\alpha)-(-\alpha)$ is (R-R)w\_= c\_[-]{}( [w\_w\_]{} -[w\_w\_]{} +2 R )w\_. \[eq:addp\] Upon adding $w_\beta \times(\ref{eq:add})$ to $w_\gamma\times(\ref{eq:addp})$ and using the symmetries of the structure constants together with the addition formula (\[eq:addition\]) for $w_\alpha$, we obtain $$\alpha\cdot(R\sp{\beta}-R\sp{-\beta})w_\beta+ \alpha\cdot(R\sp{\gamma}-R\sp{-\gamma})w_\gamma= c_{\alpha\gamma}\sp\beta (z_{\gamma}-z_\beta).$$ (For the case at hand, $z_{\gamma}=z_{-\gamma}$.) After substituting $\alpha=\beta-\gamma$ in this expression and making use of the fact $\alpha\cdot(R\sp{\alpha}-R\sp{-\alpha})=0$, which follows from (\[eq:add\]), we obtain (R-R)w\_-c\_[-]{}z\_= (R-R)w\_-c\_z\_. \[eq:aas\] Therefore $A_{\beta\gamma}=A_{\gamma\beta}$. Upon using $\alpha=\beta-\gamma$ the remaining symmetries are similarly shown. Thus to every triangle formed by three roots $\alpha,\beta,\gamma$ we have associated a single constant $A_{\alpha\beta}$ (up to a sign which is taken care of below). Now suppose the root $\alpha$ may be expressed as a sum of two distinct pairs of roots, $\alpha=\beta-\gamma=\beta\sp\prime-\gamma\sp\prime$. This requires that $n\ge4$. For the simply-laced case being considered, we may further assume $\alpha\cdot(\gamma-\gamma\sp\prime)=0$ and that our labelling is such that $\gamma-\gamma\sp\prime\in\Phi$. What then is the relation between $A_{\alpha-\gamma}$ and $A_{\alpha{-\gamma\sp\prime}}$? Using (\[eq:add\]) we see that (-)(R-R)=0. \[eq:cons\] Further $c_{\beta -\gamma}\sp{\alpha}= c_{\beta\sp\prime -\gamma\sp\prime}\sp{\alpha}$, and so $$A_{\alpha-\gamma}=-\gamma\cdot(R\sp{\alpha}-R\sp{-\alpha})w_\alpha -c_{\beta -\gamma}\sp{\alpha}z_\alpha = -\gamma\sp\prime\cdot(R\sp{\alpha}-R\sp{-\alpha})w_\alpha -c_{\beta\sp\prime -\gamma\sp\prime}\sp{\alpha}z_\alpha= A_{\alpha-\gamma\sp\prime} .$$ We have just shown that the constants $A_{\alpha-\gamma}$ associated with a triangle of roots are the same whenever they share a root $\alpha$ as a common edge. Now we can get from one triangle of roots to any other by intermediate root triangles. Therefore the constants $A_{\alpha-\gamma}$ depend on all the roots in the same fashion and we have shown $A_{\alpha\beta}=c_{\beta-\alpha\,\alpha}\sp{\beta}{\cal A} $ for some function ${\cal A}$. On combining this lemma and (\[eq:add\]) we see that (R-R)w\_w\_=c\_( [w\_w\_]{} +[w\_w\_]{} +2 R )w\_w\_= c\_([A]{}+z\_)w\_. \[eq:sol\] Thus ${\cal A}$ determines $R\sp{-\alpha \alpha}$ and (via (\[eq:redcons\])) $R\sp{\alpha \alpha}$, assuming we are given $w_\alpha$ and $z_\alpha$. Further, For $n\ge4$ ${\cal A}$ is a constant and $ R\sp{-\alpha \alpha}$ is a function of $\alpha$ only. Once again, suppose the root $\alpha$ may be expressed as a sum of two distinct pairs of roots, $\alpha=\beta-\gamma=\beta\sp\prime-\gamma\sp\prime$. Comparing (\[eq:sol\]) with the analogous equation in $\beta\sp\prime$, $\gamma\sp\prime$ enables us to show that $$\big( {w_\beta\sp\prime \over w_\beta} -{{z_\beta w_\alpha}\over{w_\gamma w_\beta}} \big) - \big( {w_{\beta\sp\prime}\sp\prime \over w_{\beta\sp\prime}}- {{z_{\beta\sp\prime}w_\alpha}\over{w_{\gamma\sp\prime}w_{\beta\sp\prime}} } \big) = \big( {w_\alpha\over{w_\gamma w_\beta}}- {w_\alpha\over{w_{\gamma\sp\prime}w_{\beta\sp\prime}}} \big){\cal A} ,$$ and so we may solve for ${\cal A}$ explicitly in terms of the roots shown. We have argued however that ${\cal A}$ depends on all of the roots in the same fashion. Therefore ${\cal A}$ is a constant. Having shown that ${\cal A}$ is a constant, let us rewrite (\[eq:sol\]) again, assuming that $\alpha=\beta-\gamma=\beta\sp\prime-\gamma\sp\prime$. Then, $$({\cal A}+z_\beta){{w_\alpha}\over{w_\gamma w_\beta}}- {w_\beta\sp\prime \over w_\beta} = 2 R\sp{-\alpha \alpha}+{w_\alpha\sp\prime \over w_\alpha}= ({\cal A}+z_{\beta\sp\prime}) {{w_\alpha}\over{w_{\gamma\sp\prime}w_{\beta\sp\prime}}}- {w_{\beta\sp\prime}\sp\prime \over w_{\beta\sp\prime}}.$$ The right-hand side of this equation is a function of $\beta$ and $\gamma$ only, while the left is a function of $\beta\sp\prime$ and $\gamma\sp\prime$. Thus both are functions of $\alpha$ only and the remainder of the lemma follows. The final stage of our argument consists of showing there is no constant ${\cal A}$ which makes $2 R\sp{-\alpha \alpha}+{w_\alpha\sp\prime / w_\alpha}$ a function of $\alpha$ only for the elliptic potentials being considered. Take for example[@WW] $w_\alpha=1/sn\,(\alpha\cdot x,k)$. Here $$({\cal A}+z_\beta){{w_\alpha}\over{w_\gamma w_\beta}}- {w_\beta\sp\prime \over w_\beta} = ({\cal A}-{{1+k\sp2}\over2}+{k\sp2\over w_\alpha\sp2 }) {{w_\alpha}\over{w_\gamma w_\beta}}-{w_\alpha\sp\prime \over w_\alpha},$$ and we cannot both have ${\cal A}$ constant and this expression depending only on $\alpha$ unless $k=0$. For $k=0$, which corresponds to the type III degeneration, we find $R\sp{-\alpha \alpha}=-{w_\alpha\sp\prime / w_\alpha}$, $R\sp{\alpha \alpha}=0$ and $$\alpha\cdot(R\sp{\beta}-R\sp{-\beta}) =c_{\gamma\alpha}\sp\beta w_\beta .$$ For $gl_n$, this has a solution $R\sp{i\beta}=w_\beta/2$ when $\beta\cdot e_i \not=0$, and zero otherwise. We have thus obtained the remainder of Avan and Talon’s assumptions together with their solution[@AT]. We have therefore shown If $n\ge4$ there are no momentum independent $R$-matrices for the nondegenerate type IV potential and $w_\alpha=-w_{-\alpha}$. When $n=2,3$ our consistency arguments do not arise. For $n=2$ there is only one root and $n=3$ only one root triangle and solutions in both cases are possible. Inclusion of a Spectral Parameter ================================= Having shown there are no momentum independent $R$-matrices for $L$-operators with $w_\alpha=-w_{-\alpha}$, several possibilities remain. We may for example relax the assumption of momentum independence and solve the full equations (\[eq:ij\]-\[eq:ab\]), or we may look at a broader class of functions $w_\alpha$. We will adopt the latter approach in this note and seek momentum independent $R$-matrices for the the class of $L$-operators (introduced by Krichever) containing a spectral parameter. The generalisation of (\[rmatrix\]) to the situation with spectral parameter is {L(u)L(v)}=\[R(u,v), L(u)1\]-\[[R]{}(u,v),1L(v)\]. \[rmatrixsp\] If $R(u,v)=R\sp{\mu\nu}(u,v)X_\mu\otimes X_\nu$ then ${R\sp\pi}(u,v)$ is defined by ${R\sp\pi}(u,v)=R\sp{\nu\mu}(v,u)X_\mu\otimes X_\nu$. We proceed in the same manner given earlier. The left-hand side of (\[rmatrixsp\]) is {L(u)L(v)}= i\_[j,]{}(\_j w\_(v) H\_jE\_- \_j w\_(u) E\_H\_j), and in terms of our basis (\[rmatrixsp\]) takes the form { L(u), L(v) }= R(u,v)c\_L(u) - R(v,u)c\_L(v) . Again three equations arise, depending on the range of $\{\mu,\nu\}$. (The new possibility $(\mu,\nu)=(\alpha,i)$ yields the same equation as $(\mu,\nu)=(i,\alpha)$ with $u$ and $v$ interchanged.) Once again the assumption that $R(u,v)$ is momentum independent greatly reduces the possible nonzero components of $R(u,v)$. We find R(u,v)=0,R(u,v)+R(v,u)=0R(u,v)=0=. The components to be determined are $R\sp{ij}(u,v),R\sp{i\alpha}(u,v),R\sp{\alpha\alpha}(u,v)$ and $R\sp{\alpha -\alpha}(u,v)$. Again we may argue that $R\sp{ij}(u,v)=\eta(u,v)\,\delta\sp{ij}+P\sp{ij}$, and we arrive at two equations \[eq:spa\] -w\_(v)=(u,v) w\_(v) + R(u,v)w\_(u) - R(v,u)w\_[-]{}(u) and $$\begin{aligned} \label{eq:spb} \alpha\cdot R\sp{\beta}(u,v)w_\alpha(u) -\beta\cdot R\sp{\alpha}(v,u) w_\beta(v)&=& c_{\alpha\gamma}\sp\beta\Bigl( R\sp{\alpha\alpha}(v,u)w_\gamma(v)-R\sp{\beta\beta}(u,v)w_{-\gamma}(u) \Bigl)\phantom{1234} \\ \nonumber &&+ c_{-\alpha\gamma}\sp\beta\Bigl( R\sp{-\alpha\alpha}(v,u)w_\gamma(v)+R\sp{-\beta\beta}(u,v)w_{\gamma}(u) \Bigl).\end{aligned}$$ These equations are the analogues of (\[eq:iann\]) and (\[eq:abn\]) respectively. At this stage we make the ansatz (u,v)=(v-u)+(u)-(v),R(v,u)=0\[eq:ansatza\] and R(u,v)=w\_(v-u) e. \[eq:ansatzb\] For any Lie algebra, this ansatz solves (\[eq:spa\]) and reduces (\[eq:spb\]) to the equation \[eq:reduction\] - =c\_. The consistency conditions exploited in the last section are implicit here in the structure constants $c_{\alpha\beta}\sp\gamma$. Certainly, for the case of $gl_n$ we may solve (\[eq:reduction\]) by setting R(u,v)=[12]{}w\_(v)e\_i = 0 \[eq:ansatzc\] and zero otherwise. The remainder of this section will be devoted to proving some of these assertions. First let us show that (\[eq:spa\]) is satisfied. Now $$R\sp{-\alpha\alpha}(u,v){w_\alpha(u)\over w_\alpha(v)}= {{\sigma(u-\alpha)\sigma(v)\sigma(v-u-\alpha)}\over {\sigma(u)\sigma(v-\alpha)\sigma(\alpha)\sigma(v-u)}} =\zeta(-u)+\zeta(v-\alpha)+\zeta(\alpha)-\zeta(v-u).$$ The exponential factors in our ansatz for $R\sp{-\alpha\alpha}(u,v)$ have been chosen so that there is cancellation leaving only the $\sigma$ factors in the middle term. The final equality makes use of the identity[@WW] $${\sigma(x+y)\sigma(y+z)\sigma(z+x)\over\sigma(x)\sigma(y)\sigma(z)\sigma(x+y+z)} = \zeta(x)+\zeta(y)+\zeta(z)-\zeta(x+y+z).$$ Finally, upon making use of ${w_\alpha\sp\prime(v) /w_\alpha(v)}=\zeta(v)-\zeta(\alpha)-\zeta(v-\alpha)$, we find (\[eq:spa\]) holds for our choice of $\eta$. [ As for (\[eq:spb\]), first observe that $$\begin{aligned} \lefteqn{ R\sp{-\alpha\alpha}(v,u)w_{\alpha+\beta}(v)+ R\sp{-\beta\beta}(u,v)w_{\alpha+\beta}(u) }\\ &=& \Bigl[ -{{\sigma(u-v-\alpha)\sigma(v-\alpha-\beta)}\over {\sigma(\alpha)\sigma(v)}} + {{\sigma(v-u-\beta)\sigma(u-\alpha-\beta)}\over {\sigma(\beta)\sigma(u)}} \Bigl] {{e\sp{\zeta(u)\alpha+\zeta(v)\beta}}\over{\sigma(\alpha+\beta)\sigma(v-u)}}\cr &=& w_{\alpha}(u)w_{\beta}(v).\end{aligned}$$ ]{} To obtain the final equality, we have employed the three-term equation of Weierstrass[@WW §$20\cdot53$], $$\begin{aligned} \sigma(x-y)\sigma(x+y)\sigma(z-t)\sigma(z+t)+ \sigma(y-z)\sigma(y+z)\sigma(x-t)\sigma(x+t)\cr \nonumber +\sigma(z-x)\sigma(z+x)\sigma(y-t)\sigma(y+t)=0.\end{aligned}$$ This observation means that (\[eq:spb\]) reduces to (\[eq:reduction\]). Discussion ========== This paper has further investigated the $R$-matrix structure of the Calogero-Moser models under the assumption of momentum independence. We have shown that for the usual $L$-operator ($L=L\sp\dagger\Leftrightarrow w_\alpha=-w_{-\alpha}$) and nondegenerate type IV potential, no momentum independent $R$-matrix exists whenever $n\ge4$. Indeed our analysis showed that momentum independence actually gives the $R$-matrices of [@AT] for the type I-III potentials when the otherwise arbitrary projection operator $P\sp{ij}$ is chosen to vanish. For $n=2$ and $3$, solutions may however be found for the type IV potential. By enlarging the class of $L$-operators under consideration to the family considered by Krichever, we were able to construct an appropriate spectral parameter-dependent $R$-matrix. This was given by (\[eq:ansatza\],\[eq:ansatzb\],\[eq:ansatzc\]). We have worked throughout in terms of a basis of the Lie algebra. This has the merit of reducing the problem to the two equations (\[eq:spa\],\[eq:spb\]) and highlighting the implicit consistency conditions. The ansatz presented by (\[eq:ansatza\],\[eq:ansatzb\]) is independent of the Lie algebra to which $L$ is associated. While we can certainly find consistent solutions to the resulting equation (\[eq:reduction\]) in the case of $gl_n$ we have not fully examined this equation in the general setting. We must conclude by mentioning the very recent, related work of Sklyanin[@Sklyanin93] which also constructs $R$-matrices for the type IV Calogero-Moser model with spectral parameter. At first glance our solutions are different and we have delayed the written presentation of this work in order to clarify this point. Certainly Sklyanin’s approach is very different from our own. We may easily verify that Sklyanin’s ansatz satisfies our (\[eq:ansatza\],\[eq:ansatzb\]) and so provides a solution. The differences between the solutions has its origin in our respective presentation of the $L$-operators. Sklyanin in fact works with a conjugate of Krichever’s $L$-operator, $U L(u) U\sp{-1}$ where $U_{ij}=e\sp{\zeta(u)x_i}\delta_{ij}$. Once this is observed, our solutions are in fact in agreement, our works providing independent proofs of this fact. 0.5in T.S. acknowledges financial support from both the Daiwa Anglo-Japanese Foundation and Fuju-kai Foundation. H.W.B. thanks A.J.Macfarlane for remarks about the importance of spectral parameter dependent $R$-matrices. [xx]{} F.Calogero, [*Lett. Nuovo Cim.*]{} [**13**]{}, 411 (1975). J.Moser, [*Adv. Math.*]{} [**16**]{}, 1 (1975). M.A.Olshanetsky, A.M.Perelomov, [*Phys. Rep.*]{} [**71**]{}, 313 (1981). F.Calogero, [*Lett. Nuovo Cim.*]{} [**16**]{}, 77 (1976). S. N. M. Ruijsenaars, [*Commun. Math. Phys.*]{} [**110**]{}, 191 (1987). M.Bruschi, F.Calogero, [*SIAM J. Math. Anal.*]{} [**21**]{}, 1019 (1990). H.W.Braden and V.M.Buchstaber. Generalised Integrable Systems and Grassmannians. (In progress) D.Kazhdan, B.Kostant, S.Sternberg, [*Commun. on Pure and Appl. Math.*]{} [**31**]{}, 481 (1978). J.Avan, O.Babelon, M.Talon. Construction of the classical $R$-matrices for the Toda and Calogero models. PAR LPTHE 93-31, June 1993; [hep-th/9306102]{} M. A. Olshanetsky, A. M. Perelomov, [*Phys. Rep.*]{} [**94**]{}, 313 (1983). J.Avan, A.Jevicki, [*Phys. Lett.*]{} [**B 266**]{}, 35 (1991). M. A. Olshanetsky, A. M. Perelomov, [*Inv. Math.*]{} [**37**]{}, 93 (1976). S.Wojciechowski, [*Lett. Nuov. Cim.*]{} [**18**]{}, 103 (1977). M.A.Semenov-Tian-Shansky, [*Funct. Anal. Appl.*]{} [**17**]{}, 259 (1983). L.D.Faddeev, L.A.Takhtajan, [*Hamiltonian methods in the theory of solitons,*]{} Berlin, Heidelberg, New York: Springer 1987. O.Babelon, C.-M.Viallet, [*Phys. Lett.*]{} [**237**]{}, 411 (1990). J.Avan, M.Talon, [*Phys. Lett.*]{} [**B303**]{}, 33 (1993). J.-M. Maillet, [*Nucl. Phys.*]{} [**B269**]{}, 54 (1986). J.C.Eilbeck, V.Z.Enolskii, V.B.Kuznetsov, A.V.Tsiganov. Linear $r$-matrix algebra for classical separable systems. Preprint Univ. of Amsterdam, Math. ser., 93-11; [hep-th/9306155]{} O.Babelon, D.Bernard. The sine-Gordon solitons as a N-body problem. SPhT-93-072; LPTHE-93-40. I. M. Krichever, [*Func. Anal. Appl.*]{} [**14**]{}, 282 (1980). E.T.Whittaker and G.N.Watson, [*A Course of Modern Analysis,*]{} Cambridge University Press 1927. E.K.Sklyanin. Dynamical r-matrices for the elliptic Calogero-Moser Model. LPTHE-93-42; [hep-th/9308060]{} [^1]: Throughout, Roman indices will denote the Cartan subalgebra basis elements while the early Greek indices $\alpha,\beta,\ldots$ will denote the step operators.

--- abstract: 'We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally extends to this compactification and there is a notion of smoothness for such data. We indicate how this weak sort of compactification may be used to develop useful de Rham and crystalline cohomology theories for semistable log schemes over the log point over a field which are not necessarily proper.' author: - '<span style="font-variant:small-caps;">Elmar Grosse-Klönne</span>' title: Compactifications of Log Morphisms --- [**Introduction**]{} Let $X$ be a smooth variety over a field $k$. It is well known that for the study of the cohomology of $X$ — or even for its very definition (e.g. crystalline [@maghreb], rigid [@berco]), or the definition of nice coefficients for it (e.g. integrable connections with regular singularities) — it is often indispensable to take into account also a boundary $D=\overline{X}-X$ of $X$ in a smooth compactification $X\subset\overline{X}$ of $X$. If $D\subset\overline{X}$ is a normal crossing divisor on $\overline{X}$, the cohomology can conveniently be studied in the framework of logarithmic algebraic geometry. On the other hand, log geometry proved also useful to define the cohomology of proper normal crossing varieties $X$ over $k$ which occur as a fibre of a semistable family, or more generally are $d$-semistable ([@fkato]), see [@steen], [@mokr]. In the present paper we attempt to develop a concept in log geometry particularly suitable to treat the mixed situation: given a non-proper $d$-semistable normal crossing variety $X/k$, we want to explain how an open immersion of $X$ into a proper $k$-scheme $\overline{X}$ can be used to investigate the cohomology of $X$, the stress lying on the fact that we avoid the assumption that $\overline{X}$ be $d$-semistable and require a weaker condition instead. Fix a base scheme $W$ for all occuring schemes. Let $T$ be a log scheme. The central definition of this note is that of a [*$T$-log scheme with boundary*]{}: A morphism of log schemes $X\to T$ together with an open log schematically dense embedding of log schemes $i:X\to\overline{X}$. For brevity, we often denote it simply by $(\overline{X},X)$. Morphisms of $T$-log schemes with boundary are defined in an obvious way. There are notions of exact and of boundary exact closed immersions of $T$-log schemes with boundary. The relative logarithmic de Rham complex $\Omega^{\bullet}_{X/T}$ on $X$ extends canonically to a complex $\Omega^{\bullet}_{(\overline{X},X)/T}$ on $\overline{X}$. These definitions are justified by a theory of smoothness for $T$-log schemes with boundary, well suited for cohomology purposes. Roughly, a $T$-log scheme with boundary $(\overline{X},X)$ is said to be weakly smooth if it satisfies a lifting property for morphisms from a nilpotent exact closed immersion of $T$-log schemes with boundary to $(\overline{X},X)$. Weak smoothness implies that $\Omega^{\bullet}_{(\overline{X},X)/T}$ is locally free. $(\overline{X},X)$ is said to be smooth if it is weakly smooth and if for boundary exact closed immersions $(\overline{Y},Y)\to(\overline{V},V)$ of $T$-log schemes with boundary, and morphisms $(\overline{Y},Y)\to(\overline{X},X)$ of $T$-log schemes with boundary, the projections $\overline{X}\overline{\times}_T\overline{V}\to\overline{V}$ lift log étale locally near the image of $\overline{Y}$ in $\overline{X}\overline{\times}_T\overline{V}$ to strict and smooth morphisms of log schemes (see the text for the definition of $\overline{X}\overline{\times}_T\overline{V}$). This definition is of course geared to its application to (crystalline) cohomology. However, our main theorem gives a convenient criterion for smoothness in terms of morphisms of monoids, very similar to Kato’s criterion for usual log smoothness. We emphasize that even if $f:X\to T$ actually extends to a morphism of log schemes $\overline{f}:\overline{X}\to T$, our notion of smoothness is more general: $(\overline{X},X)$ might be smooth as a $T$-log scheme with boundary while $\overline{f}$ is not a log smooth morphism in the usual sense (or even not ideally smooth as defined by Ogus [@ogid]). See for example the discussion at the beginning of Section 3. In this regard, the theme of this paper is that (usual) log smoothness in an ‘interior’ $X\subset\overline{X}$ of a morphism of log schemes $\overline{f}:\overline{X}\to T$ should already ensure that $\overline{f}$ has nice cohomology. (A similar principle underlies the definition of rigid cohomology [@berco].) We hope that our definitions are useful for a definition of log rigid cohomology, in the case of nontrivial log structures on the base; in special cases they already turned out to be so, see [@hkstrat]. Section 1 contains the basic definitions and presents several examples. The main Section is the second one which is devoted to smoothness. The main theorem is the smoothness criterion \[tglatt\]. In Section 3 we discuss the example of semistable $k$-log schemes with boundary (here $T$ is the log point over a field). These are smooth in the sense of Section 2 and we try to demonstrate how they can be used as substitutes for compactifications by usual semistable proper $k$-log schemes. We indicate several applications to de Rham cohomology and crystalline cohomology.\ $T$-log schemes with boundary ============================= We fix a base scheme $W$; all schemes and morphisms of schemes are to be understood over $W$. All morphisms of schemes are quasi-separated. We also assume that all morphisms of schemes are quasi-compact: the only reason for this additional assumption is that it implies the existence of schematic images (=“closed images”) of morphisms: see [@EGA] I, 9.5. We say that an open immmersion $i:X\to\overline{X}$ is schematically dense if $\overline{X}$ coincides with the schematic image of $i$. For the basic notions of log algebraic geometry we refer to K. Kato [@kalo]. Log structures are understood for the étale topology. By abuse of notation, for a scheme $X$ and a morphism of monoids $\alpha:N\to{\cal O}_X(X)$ (where ${\cal O}_X(X)$ is understood multiplicatively), we will denote by $(X,\alpha)$ the log scheme with underlying scheme $X$ whose log structure is associated with the chart $\alpha$. For a log scheme $(X,{\cal N}_X)=(X,{\cal N}_X\to{\cal O}_X)$ we will often just write $X$ if it is clear from the context to which log structure on $X$ we refer, i.e. in those cases the log structure is dropped in our notation. Similarly for morphisms of log schemes. An [*exactification*]{} of a closed immersion of fine log schemes $Y\to X$ is a factorization $Y\stackrel{i}{\to}Z\stackrel{f}{\to}X$ with $i$ an exact closed immersion and $f$ log étale. Recall that a morphism of log schemes $f:(X,{\cal N}_X)\to(Y,{\cal N}_Y)$ is said to be [*strict*]{} if $f^*{\cal N}_Y\to{\cal N}_X$ is an isomorphism. For a monoid $N$ we denote by $N^{\rm gp}$ the associated group. For a finitely generated integral monoid $Q$ we let $$W[Q]=W\times_{{\mbox{\rm Spec}}(\mathbb{Z})}{\mbox{\rm Spec}}(\mathbb{Z}[Q])$$and give it the canonical log structure for which $Q$ is a chart.\ \(i) A morphism of log schemes $f:(X,{\cal N}_X)\to(Y,{\cal N}_{{Y}})$ factors over the [*log schematic image*]{} $(\overline{X},{\cal N}_{\overline{X}})$ of $f$ which is defined as follows: The underlying scheme $\overline{X}$ is the schematic image of the morphism of schemes $X\to Y$ underlying $f$. Let $X\stackrel{i}{\to}\overline{X}\stackrel{\overline{f}}{\to} Y$ be the corresponding morphisms of schemes. The log structure ${\cal N}_{\overline{X}}$ is by definition the image of the natural composite map of log structures $\overline{f}^*{\cal N}_Y\to i_*f^*{\cal N}_Y\to i_*{\cal N}_X$ on $\overline{X}$. Here $i_*$ denotes the functor [*push forward log structure*]{}.\ (ii) A morphism of log schemes $f:(X,{\cal N}_X)\to(Y,{\cal N}_{{Y}})$ is said to be [*log schematically dominant*]{} if $({Y},{\cal N}_{{Y}})$ coincides with the log schematic image of $f$; it is said to be [*log schematically dense*]{} if in addition the underlying morphism of schemes is an open immersion.\ A morphism of log schemes $i:(X,{\cal N}_X)\to(\overline{X},{\cal N}_{\overline{X}})$ is log schematically dense if and only if the underlying morphism of schemes is a schematically dense open immersion and the canonical morphism of log structures ${\cal N}_{\overline{X}}\to i_*{\cal N}_X$ is injective.\ \[pfeil\] Let $(X,{\cal N}_X)$ be a log scheme and $i:X\to\overline{X}$ a schematically dense open immersion of its underlying scheme into another scheme $\overline{X}$. Denote by $i_{*,{\rm sh}}{\cal N}_X$ the [*sheaf theoretic push forward*]{} of the sheaf of monoids ${\cal N}_X$. There exists a unique map $i_{*,{\rm sh}}{\cal N}_X\to (i_*{\cal N}_X)^{\rm gp}$ compatible with the natural maps $i_*{\cal N}_X\to i_{*,{\rm sh}}{\cal N}_X$ and $i_*{\cal N}_X\to(i_*{\cal N}_X)^{\rm gp}$. [Proof:]{} First observe that ${\cal O}_{\overline{X}}\to i_*{\cal O}_X$ is injective, so henceforth we regard ${\cal O}_{\overline{X}}$ as a subsheaf of $i_*{\cal O}_X$. Also note $(i_*{\cal O}_X)^{\times}=i_*({\cal O}_X^{\times})$. It follows that we can view $i_*{\cal N}_X$ as the subsheaf of $i_{*,{\rm sh}}{\cal N}_X$ formed by those sections which map to ${\cal O}_{\overline{X}}$ under the map $\alpha:i_{*,{\rm sh}}{\cal N}_X\to i_*{\cal O}_X$ which we get by functoriality of sheaf theoretic push forward. To prove the lemma it is enough to show that $i_{*,{\rm sh}}{\cal N}_X$ arises from $i_*{\cal N}_X$ by inverting those sections $m$ for which the restrictions $\alpha(m)|_X$ are invertible. But this is the case: Take $m\in i_{*,{\rm sh}}{\cal N}_X$. Since $i_*{\cal O}_X$ arises from ${\cal O}_{\overline{X}}$ by inverting those Sections for which the restrictions to $X$ are invertible, we find $f,g\in{\cal O}_{\overline{X}}$ with $g|_X$ invertible and with $\alpha(m)=g^{-1}f$. We saw $g=\alpha(n)$ for some $n\in i_*{\cal N}_X$. Now $nm\in i_*{\cal N}_X$ and our claim and hence the lemma follows.\ \[feinbild\] The log schematic image $(\overline{X},{\cal N}_{\overline{X}})$ of a morphism of fine log schemes $f:(X,{\cal N}_X)\to(Y,{\cal N}_{{Y}})$ is a fine log scheme. [Proof:]{} The coherence of ${\cal N}_{\overline{X}}$ follows from that of ${\cal N}_{{Y}}$. We have ${\cal N}_{\overline{X}}\subset i_*{\cal N}_X\subset i_{*,{\rm sh}}{\cal N}_X$, for the second inclusion see the proof of Lemma \[pfeil\]. Therefore the integrality of ${\cal N}_X$ implies that of ${\cal N}_{\overline{X}}$.\ A [*log scheme with boundary*]{} is a triple $((X,{\cal N}_X), (\overline{X},{\cal N}_{\overline{X}}),i)$ where $i:(X,{\cal N}_X)\to(\overline{X},{\cal N}_{\overline{X}})$ is a log schematically dense morphism such that $i^*{\cal N}_{\overline{X}}={\cal N}_X$ and $(i_*{\cal N}_X)^{\rm gp}={\cal N}_{\overline{X}}^{\rm gp}$. Let $(T,{\cal N}_T)$ be a log scheme. A $(T,{\cal N}_T)$[*-log scheme with boundary*]{} is a log scheme with boundary $((X,{\cal N}_X), (\overline{X},{\cal N}_{\overline{X}}),i)$ together with a morphism of log schemes $g:(X,{\cal N}_X)\to(T,{\cal N}_T)$.\ We think of $\overline{X}-X$ as a boundary of $X$. We will often drop $i$, $g$ and the log structures from our notation and just speak of the $T$-log scheme with boundary $(\overline{X},X)$. So in the following definition which justifies the whole concept.\ The sheaf of relative differentials of a $T$-log scheme with boundary $(\overline{X},X)$ is defined as follows: Denote by $\tau$ the composite map$$i_{*,{\rm sh}}g^{-1}{\cal N}_T\to i_{*,{\rm sh}}{\cal N}_X\to (i_*{\cal N}_X)^{\rm gp}={\cal N}_{\overline{X}}^{\rm gp}$$where the second arrow is the one from Lemma \[pfeil\]. Let $\Omega^1_{\overline{X}/W}$ be the sheaf of differentials of the morphism of underlying schemes $\overline{X}\to W$. Then $\Omega^1_{(\overline{X},X)/T}$ is the quotient of $$\Omega^1_{\overline{X}/W}\oplus({\cal O}_{\overline{X}}\otimes_{\mathbb{Z}}{\cal N}^{\rm gp}_{\overline{X}})$$ divided by the ${\cal O}_{\overline{X}}$-submodule generated by local sections of the forms $$\begin{aligned} (d\alpha(a),0)-(0,\alpha(a)\otimes a)&\quad\quad\mbox{ with }a\in{\cal N}_{\overline{X}}\notag\\(0,1\otimes a)&\quad\quad\mbox{ with }a\in {\mbox{\rm Im}}(\tau).\notag \end{aligned}$$We define the de Rham complex $\Omega^{\bullet}_{(\overline{X},X)/T}$ by taking exterior powers and the differential as usual.\ \[clabo\] Let $(\overline{X},X)$ be a $T$-log scheme with boundary.\ (1) The restriction $\Omega^1_{(\overline{X},X)/T}|_X$ naturally coincides with the usual sheaf of relative logarithmic differentials of $g:(X,{\cal N}_X)\to(T,{\cal N}_T)$.\ (2) Suppose $g$ extends to a morphism of log schemes $\overline{g}:(\overline{X},{\cal N}_{\overline{X}})\to(T,{\cal N}_T)$. Let us assume the following conditions: (i) : The underlying scheme of $T$ is the spectrum of a field. (ii) : For any étale morphism $\overline{V}\to\overline{X}$ with $\overline{V}$ connected, the scheme $V=\overline{V}\times_{\overline{X}}X$ is also connected. Then $\Omega^1_{(\overline{X},X)/T}$ naturally coincides with the usual sheaf $\Omega^1_{\overline{X}/T}$ of relative logarithmic differentials of $\overline{g}$. [Proof:]{} (1) is immediate. (2) and its proof were suggested by the referee. Write $T={\mbox{\rm Spec}}(k)$. By base change, we may assume that $k$ is separably closed. It suffices to prove that the morphism $\overline{g}^{-1}{\mathcal N}_T\to i_{*,{\rm sh}}g^{-1}{\mathcal N}_T$ is an isomorphism. Let $x$ be a geometric point of $\overline{X}$ and let $\overline{V}$ be the strict Henselization of $\overline{X}$ at $x$. Put $V=\overline{V}\times_{\overline{X}}X$. Then, by the assumption (i), we see that both $\overline{V}$ and $V$ are connected. Hence we have$$(\overline{g}^{-1}{\mathcal N}_T)_x=\Gamma(\overline{V},\overline{g}^{-1}{\mathcal N}_T)=\Gamma(T,{\mathcal N}_T)$$$$(i_{*,{\rm sh}}g^{-1}{\mathcal N}_T)_x=\Gamma(\overline{V},\overline{g}^{-1}{\mathcal N}_T)=\Gamma(T,{\mathcal N}_T)$$and the lemma follows.\ One class of examples where the condition (i) + (ii) of Lemma \[clabo\] (2) holds true are the semistable $T$-log schemes discussed in Section 3; but for them, the conclusion $\Omega^1_{(\overline{X},X)/T}=\Omega^1_{\overline{X}/T}$ (if $g$ extends to $\overline{g}$) is immediate anyway. Undoubtly, if $g$ extends to $\overline{g}$, the conclusion of Lemma \[clabo\] (2) holds under much more general conditions than the stated condition (i) + (ii).\ The following examples will be discussed later on.\ (a) Let $Q,P$ be finitely generated monoids and let $\rho:Q\to P^{\rm gp}$ be a morphism. Let $P'$ be the submonoid of $P^{\rm gp}$ generated by $P$ and ${\mbox{\rm Im}}(\rho)$. Then$$(W[P],W[P'])$$is a $T=W[Q]$-log scheme with boundary.\ (b1) Let $Q=\mathbb{N}$ with generator $t\in Q$. Let $t_1,\ldots,t_r$ be the standard generators of $\mathbb{N}^r$. Let $X=W[\mathbb{N}^r]$, the affine $r$-space over $W$ with the log structure defined by the divisor $V(t_1\cdot\ldots\cdot t_r)$. By means of $t\mapsto t_1\cdot\ldots\cdot t_r$ this is a $T=W[Q]$-log scheme. We compactify ${X}$ by$$\overline{X}=W\times_{{\mbox{\rm Spec}}(\mathbb{Z})}(\times_{{\mbox{\rm Spec}}(\mathbb{Z})}({\mbox{\rm Proj}}(\mathbb{Z}[t_0,t_i])_{1\le i\le r}))=({\bf P}_W^1)^r$$and take for ${\cal N}_{\overline{X}}$ the log structure defined by the normal crossing divisor $$(\overline{X}-X)\cup(\mbox{the closure of }V(t_1\cdot\ldots\cdot t_r)\subset X\mbox{ in }\overline{X}).$$(b2) Let $X$ and $T$ be as in (b1). Another compactifiction of $X$ is projective $r$-space, i.e. $\overline{X}'={\bf P}_W^r$; similarly we take ${\cal N}_{\overline{X'}}$ as the log structure defined by the normal crossing divisor $(\overline{X}'-X)\cup(\mbox{the closure of }V(t_1\cdot\ldots\cdot t_r)\subset X\mbox{ in }\overline{X}')$.\ (c) Let $k$ be a field, $W={\mbox{\rm Spec}}(k)$ and let again $Q=\mathbb{N}$ with generator $t\in Q$. The following type of $S=W[Q]$-log scheme with boundary (which generalizes .(b1) if $W={\mbox{\rm Spec}}(k)$ there) gives rise, by base change $t\mapsto 0$, to the $T$-log schemes with boundary discussed in Section \[semist\] below. Let $\overline{X}$ be a smooth $W$-scheme, $X\subset \overline{X}$ a dense open subscheme, $D=\overline{X}-X$. Let $X\to S$ be a flat morphism, smooth away from the origin. Let $X_0$ be the fibre above the origin, let $\overline{X}_0$ be its schematic closure in $\overline{X}$ and suppose that $D\cup\overline{X}_0$ is a divisor with normal crossings on $\overline{X}$.\ (d) Let $k$ be a field and let $T=({\mbox{\rm Spec}}(k),\mathbb{N}\stackrel{0}{\to}k)$, the standard logarithmic point ([@fkato]). Let $Y$ be a semistable $k$-log scheme in the sense of [@mokr] 2.4.1 or [@fkato]. That is, $Y$ is a fine $T$-log scheme $(Y,{\cal N}_Y)$ satisfying the following conditions. Étale locally on $Y$ there exist integers $i\ge 1$ and charts $\mathbb{N}^i\to{\cal N}_Y(Y)$ for ${\cal N}_Y$ such that\ (i) if on the log scheme $T$ we use the chart $\mathbb{N}\to k, 1\mapsto 0$, the diagonal morphism $\mathbb{N}\stackrel{\delta}{\to}\mathbb{N}^i$ is a chart for the structure morphism of log schemes $Y\to T$, and\ (ii) the induced morphism of schemes $$Y\to{\mbox{\rm Spec}}(k)\times_{{\mbox{\rm Spec}}(k[t])}{\mbox{\rm Spec}}(k[t_1,\ldots,t_i])$$ is smooth in the classical sense. Let $\overline{X}$ be the union of some irreducible components of $Y$ and let $X$ be the open subscheme of $\overline{X}$ which is the complement in $Y$ of the union of all irreducible components not contained in $\overline{X}$. Then $\overline{X}$ inherits a structure of $T$-log scheme, but it is not log smooth over $T$. However, we can view $(\overline{X},X)$ as a $T$-log scheme with boundary (forgetting that the morphism $X\to T$ actually extends to $\overline{X}$): as such it is what we will call [*smooth*]{} below.\ A concrete example for .(c) (see [@hkstrat] for more details). Again let $k$ be a field and let $S=W[\mathbb{N}]$ with generator $q$ of $\mathbb{N}$. Let $Y$ be a semistable $k$-log scheme with set of irreducible components $\{Y_j\}_{j\in R}$ all of which we assume to be smooth. As in [@kalo] p.222/223 we define for every $j\in R$ an invertible ${\cal O}_Y$-module ${\cal F}_j$ as follows: Let ${\cal N}_{Y,j}$ be the subsheaf of the log structure ${\cal N}_Y$ of $Y$ which is the preimage of ${\mbox{\rm Ker}}({\cal O}_Y\to{\cal O}_{Y_j})$. This ${\cal N}_{Y,j}$ is a principal homogeneous space over ${\cal O}_Y^{\times}$, and its associated invertible ${\cal O}_Y$-module is ${\cal F}_j$. Now fix a subset $I\subset R$ with $|I|=i$ and let $L=R-I$. Suppose $M=\cap_{j\in I}Y_j$ is nonempty. Let $$V_M={\mbox{\rm Spec}}({\mbox{\rm Sym}}_{{\cal O}_M}(\oplus({\cal F}_j)_{j\in I}))=\times_M({\mbox{\rm Spec}}({\mbox{\rm Sym}}_{{\cal O}_M}({\cal F}_j)))_{j\in I}.$$By its definition, the affine vector bundle $V_M$ over $M$ comes with a natural coordinate cross, a normal crossing divisor on $V_M$. The intersection of $M$ with all irreducible components of $Y$ not containing $M$ is a normal crossing divisor $D$ on $M$. Let $D_V'\subset V_M$ be its preimage under the structure map $V_M\to M$ and let $D_V\subset V_M$ be the union of $D_V'$ with the natural coordinate cross in $V_M$. Then $D_V$ is a normal crossing divisor on $V_M$. Let ${\cal N}_{V_M}$ be the corresponding log structure on $V_M$. There exists a distinguished element $a\in\Gamma(V_M,{\cal O}_M)$ having $D_V$ as its set of zeros and such that the assignment $q\mapsto a$ defines a morphism of log schemes $V_M\to S$ with the following property: The induced $S$-log scheme $(M,{\cal N}_{V_M}|_M)$ on the zero Section $M\to V_M$ coincides with the $S$-log scheme $(M,{\cal N}_Y|M)$ induced by $Y$. This $a\in\Gamma(V_M,{\cal O}_M)={\mbox{\rm Sym}}_{{\cal O}_M}(\oplus({\cal F}_j)_{j\in I})(M)$ can be described as follows: Denote the image of $q\in{\cal N}_S(S)$ (here ${\cal N}_S$ is the log structure of $S$) under the structure map ${\cal N}_S(S)\to {\cal N}_Y(Y)\to {\cal N}_Y|_M(M)$ again by $q$. Locally on $M$ it can be (non-uniquely) factored as $q=t_0\prod_{j\in I}v_j$ where $v_j$ is a (local) generator of ${\cal F}_j|_M$ and $t_0$ maps to a (local) defining equation $a_0\in{\cal O}_M$ of the divisor $D$ in $M$. Then $a=a_0.(\oplus_{j\in I}v_j)\in{\mbox{\rm Sym}}_{{\cal O}_M}(\oplus_{j\in I}{\cal F}_j)(M)$ is the wanted element, globally well defined. We can view $V_M$ in a canonical way as a (schematically) dense open subscheme of $$P_M=\times_M({\mbox{\rm Proj}}({\mbox{\rm Sym}}_{{\cal O}_M}({\cal O}_M\oplus{\cal F}_j)))_{j\in I}$$by identifying a homogenous section $s\in{\mbox{\rm Sym}}_{{\cal O}_M}({\cal F}_j)$ of degree $n$ with the degree zero section $s/1_{{\cal O}_M}^n$ of ${\mbox{\rm Sym}}_{{\cal O}_M}({\cal O}_M\oplus{\cal F}_j)[1_{{\cal O}_M}^{-1}]$. We give $P_M$ the log structure defined by the normal crossing divisor $(P_M-V_M)\cup \overline{D}_V$, where $\overline{D}_V$ is the closure of $D_V$ in $P_M$. Then $(P_M,V_M)$ is a $S$-log scheme with boundary.\ A [*morphism*]{} of $T$-log schemes with boundary $f:(\overline{X},X)\to(\overline{X}',X')$ is a morphism of log schemes $$f:(\overline{X},{\cal N}_{\overline{X}})\to(\overline{X}',{\cal N}_{\overline{X}'})$$ with $X\subset f^{-1}(X')$ and restricting to a morphism of $T$-log schemes $(X,{\cal N}_{{X}})\to({X}',{\cal N}_{{X}'})$. We have a fully faithful functor from the category of $T$-log schemes to the category of $T$-log schemes with boundary. Namely, take $Y$ to $(Y,Y)$. Beware that $(T,T)$ is [*not*]{} a final object in the category of $T$-log schemes with boundary. We have obvious base change functors for morphisms $W'\to W$ to our underlying base scheme $W$ and everything we develop here behaves well with respect to these base changes. We also have [*base change functors for closed immersions of log schemes*]{} $T'\to T$ as follows: if $(\overline{X},X)$ is a $T$-log scheme with boundary, let $X_{T'}=X\times_TT'$ be the fibre product in the category of log schemes. Define the log scheme $\overline{X}_{T'}$ as the log schematic image of the morphism of log schemes $X_{T'}\to\overline{X}$. Then $(\overline{X}_{T'},X_{T'})$ is a $T'$-log scheme with boundary.\ For the rest of this paper we always assume that the log scheme $T$ is fine. All fibre products of fine log schemes are taken in the category of fine log schemes, unless specified otherwise. A $T$-log scheme with boundary $(\overline{X},X)$ is said to be [*fine*]{} if the log scheme $(\overline{X},{\cal N}_{\overline{X}})$ is fine. In the category of fine $T$-log schemes with boundary, products exist. [Proof:]{} Given fine $T$-log schemes with boundary $(\overline{X}_1,X_1)$ and $(\overline{X}_2,X_2)$, set $$(\overline{X}_1,X_1)\times_T(\overline{X}_2,X_2)=(\overline{X}_1\overline{\times}_T\overline{X}_2,X_1\times_TX_2).$$Here $X_1\times_TX_2$ denotes the fibre product in the category of fine $T$-log schemes, and $\overline{X}_1\overline{\times}_T\overline{X}_2$ is defined as the log schematic image of $X_1\times_TX_2\to \overline{X}_1\times_{W}\overline{X}_2$. (So $\overline{X}_1\overline{\times}_T\overline{X}_2$ depends also on $X_1$ and $X_2$, contrary to what the notation suggests. Note that by the construction [@kalo] 2.7, the scheme underlying $X_1\times_TX_2$ is a subscheme of the scheme theoretic fibre product, hence is a subscheme of the scheme underlying $\overline{X}_1\times_{W}\overline{X}_2$.) That $\overline{X}_1\overline{\times}_T\overline{X}_2$ is fine follows from Lemma \[feinbild\].\ It is to have fibre products why we did not require $X=f^{-1}(X')$ in the definition of morphisms of $T$-log schemes with boundary. If the structural map from the underlying scheme of the log scheme $T$ to $W$ is an isomorphism, one has $(\overline{X},X)\cong(\overline{X},X)\times_T(T,T)$. However, we stress that in contrast to taking the base change with the identity $T\to T$ (cf. .), the operation of taking the fibre product with the $T$-log scheme with boundary $(T,T)$ is non-trivial in general. For example, let $Q=\mathbb{N}$ with generator $q\in Q$, let $T=W[Q]$ and let $U_1, U_2$ be the standard generators of $\mathbb{N}^2$. For $i\in\mathbb{Z}$ let $\overline{X}_i=W[\mathbb{N}^2]$, and let $X_i=W[\mathbb{Z}\oplus\mathbb{N}]$, the open subscheme of $\overline{X}_i$ where $U_1$ is invertible. Define a structure of $T$-log scheme with boundary on $(\overline{X}_i,X_i)$ by sending $q\mapsto U_1^iU_2$. Then $$(\overline{X}_i,X_i)\cong(\overline{X}_i,X_i)\times_T(T,T)\quad\mbox{ if }i\ge0$$$$(\overline{X}_i,X_i)\not\cong(\overline{X}_i,X_i)\times_T(T,T)\quad\mbox{ if }i<0.$$Indeed, $\overline{X}_i\overline{\times}_TT$ is the closure in $W[Q\oplus\mathbb{N}^2]$ of the closed subscheme $V(q-U_1^iU_2)$ of $W[Q\oplus\mathbb{Z}\oplus\mathbb{N}]$. If $i\ge0$ this is the subscheme $V(q-U_1^iU_2)$ of $W[Q\oplus\mathbb{N}^2]$ which maps isomorphically to $W[\mathbb{N}^2]$. If $i<0$ this is the subscheme $V(qU_1^{-i}-U_2)$ of $W[Q\oplus\mathbb{N}^2]$ which does not map isomorphically to $W[\mathbb{N}^2]$.\ Let $(\overline{X},X)$ be a fine $T$-log scheme with boundary. A [*chart*]{} $(Q\to P^{\rm gp}\supset P)$ for $(\overline{X},X)$ over $T$ is a chart $\lambda:P\to\Gamma(\overline{X},{\cal N}_{\overline{X}})$ for $(\overline{X},{\cal N}_{\overline{X}})$, a chart $\sigma:Q\to \Gamma({T},{\cal N}_{{T}})$ for $({T},{\cal N}_{{T}})$ and a morphism $\rho:Q\to P^{\rm gp}$ such that $\lambda^{\rm gp}\circ\rho=\tau\circ\sigma$, where $\tau:\Gamma({T},{\cal N}_{{T}})\to\Gamma({X},{\cal N}_{{X}})\to\Gamma(\overline{X},{\cal N}_{\overline{X}}^{\rm gp})$ is the composite of the structural map with that from Lemma \[pfeil\]. Étale locally on $\overline{X}$, charts for $(\overline{X},X)$ exist. [Proof:]{} (corrected version due to the referee) We may by [@kalo] assume that $(\overline{X},{\cal N}_{\overline{X}})$ has a chart $g:G\to \Gamma(\overline{X},{\cal N}_{\overline{X}})$ and $({T},{\cal N}_{{T}})$ has a chart $\sigma:Q\to \Gamma({T},{\cal N}_{{T}})$. Let $x\in X$ and let ${\cal N}_{\overline{X},\overline{x}}$ be the stalk of ${\cal N}_{\overline{X}}$ at the separable closure $\overline{x}$ of $x$. Let $\varphi$ be the composite $$Q\stackrel{\sigma}{\to}\Gamma({T},{\cal N}_{{T}})\stackrel{\tau}{\to}\Gamma(\overline{X},{\cal N}_{\overline{X}}^{\rm gp})\to{\cal N}_{\overline{X},\overline{x}}^{\rm gp}.$$ Choose generators $q_1,\ldots,q_m$ of $Q$ and elements $x_i, y_i\in {\cal N}_{\overline{X},\overline{x}}$ $(1\le i\le m)$ such that $\varphi(q_i)=x_iy_i^{-1}$. Next, choose elements $a_i, b_i\in G$ and $u_i, v_i\in {\cal O}^{\times}_{\overline{X},\overline{x}}$ $(1\le i\le m)$ satisfying $g(a_i)=x_iu_i$ and $g(b_i)=y_iv_i$: these elements exist because $g$ is a chart. Now let$$f:G^{\rm gp}\oplus Q^{\rm gp}\oplus\mathbb{Z}^m\oplus\mathbb{Z}^m\longrightarrow{\cal N}_{\overline{X},\overline{x}}^{\rm gp}$$be the morphism defined by$$(h,q,(k_i)_{i=1}^m,(l_i)_{i=1}^m)\mapsto g^{\rm gp}(h)\varphi^{\rm gp}(q)\prod_{i=1}^mu_i^{k_i}\prod_{i=1}^mv_i^{l_i},$$and define $P$ by $P=f^{-1}({\cal N}_{\overline{X},\overline{x}})$. Then $f|_P:P\to {\cal N}_{\overline{X},\overline{x}}$ extends to a chart around $\bar{x}$ by [@kalo] 2.10. It remains to prove that the canonical inclusion $Q\to G^{\rm gp}\oplus Q^{\rm gp}\oplus\mathbb{Z}^m\oplus\mathbb{Z}^m, q\mapsto (1,q,0,0)$ actually takes values in $P^{\rm gp}$. Write a given $q\in Q$ as $q=\prod_{i=1}^mq_i^{n_i}$ with $n_i\in\mathbb{N}$. Then we have $$f(q)p=\prod_{i=1}^m(\frac{x_i}{y_i})^{n_i}=\prod_{i=1}^m(\frac{x_iu_i}{y_iv_i}\cdot\frac{v_i}{u_i})^{n_i}=\frac{f((\prod_ia_i^{n_i},0,(0),(n_i)_i))}{f((\prod_ib_i^{n_i},0,(n_i)_i,(0)))}.$$ Put $\alpha=(\prod_ia_i^{n_i},0,(0),(n_i)_i)$ and $\beta=(\prod_ib_i^{n_i},0,(n_i)_i,(0))$. Then we have $\alpha, \beta\in P$ and $f(q\beta)=f(\alpha)$. So $q\beta$ is in $P$ by the definition of $P$ and so $q$ maps to $P^{\rm gp}$.\ Smoothness ========== \(1) A morphism of $T$-log schemes with boundary $(\overline{Y},Y)\to(\overline{X},X)$ is said to be a [*boundary exact closed immersion*]{} if $\overline{Y}\to\overline{X}$ is an exact closed immersion and if for every open neighbourhood $U$ of $Y$ in $X$, there exists an open neighbourhood $\overline{U}$ of $\overline{Y}$ in $\overline{X}$ with $U$ schematically dense in $\overline{U}$.\ (2) A [*first order thickening*]{} of $T$-log schemes with boundary is a morphism $(\overline{L}',L')\to(\overline{L},L)$ such that $\overline{L'}\to\overline{L}$ is an exact closed immersion defined by a square zero ideal in ${\cal O}_{\overline{L}}$.\ (3) A fine $T$-log scheme with boundary $(\overline{X},X)$ is said to be [*weakly smooth*]{} if $\overline{X}$ is locally of finite presentation over $W$ and if the following condition holds: for every first order thickening $\eta:(\overline{L}',L')\to(\overline{L},L)$ and for every morphism $\mu:(\overline{L}',L')\to(\overline{X},X)$ there is étale locally on $\overline{L}$ a morphism $\epsilon:(\overline{L},L)\to(\overline{X},X)$ such that $\mu=\epsilon\circ\eta$.\ (4) A $T$-log scheme with boundary $(\overline{X},X)$ is said to be [*smooth*]{} if it is weakly smooth and satisfies the following property: For all morphisms $(\overline{Y},Y)\to(\overline{X},X)$ and all boundary exact closed immersions $(\overline{Y},Y)\to(\overline{V},V)$ of fine $T$-log schemes with boundary, there exists étale locally on $(\overline{X}\overline{\times}_T\overline{V})$ an exactification $$\overline{Y}\to Z\to(\overline{X}\overline{\times}_T\overline{V})$$of the diagonal embedding $\overline{Y}\to(\overline{X}\overline{\times}_T\overline{V})$ (a morphism of log schemes in the usual sense) such that the projection $Z\to(\overline{X}\overline{\times}_T\overline{V})\to \overline{V}$ is strict and log smooth.\ Recall that by [@kalo] 3.8, ‘strict and log smooth’ is equivalent to ‘strict and smooth on underlying schemes’. A $T$-log scheme $X$ is log smooth if and only if $(X,X)/T$ is weakly smooth. Assume this is the case. Then $(X,X)/T$ satisfies the smoothness condition with respect to test objects $({X},X){\leftarrow}(\overline{Y},Y)\rightarrow({V},V)$ (i.e. for which $\overline{V}=V$), because ${X}\overline{\times}_T{V}\stackrel{p}{\to}{V}$ is clearly log smooth. For general $(\overline{V},V)$ (and log smooth $T$-log schemes $X$) we have at least Theorem \[sglatt\] and Theorem \[tglatt\] below (note that the hypotheses of Proposition \[wecri\] below for $(X,X)/T$ are [*equivalent*]{} to log smoothness of $X/T$, by [@kalo] 3.5 and as worked out in [@fkato]).\ \[smocri\] Let $(\overline{X},X)$ be a weakly smooth $T$-log scheme with boundary and let $T_1\to T$ be an exact closed immersion. Then $(\overline{X}_{T_1},X_{T_1})$ is a weakly smooth $T_1$-log scheme with boundary. [Proof:]{} Let $$(\overline{X}_{T_1},X_{T_1})\stackrel{\mu}{\leftarrow}(\overline{L}',L')\stackrel{\eta}{\to}(\overline{L},L)$$ be a test object over $T_1$. By the weak smoothness of $(\overline{X},X)/T$ we get $\epsilon:(\overline{L},L)\to(\overline{X},X)$ étale locally on $\overline{L}$ such that $\mu=\epsilon\circ\eta$. The restriction $\epsilon|_L:L\to\overline{X}$ goes through $X_{T_1}$; since $L$ is log schematically dense in $\overline{L}$ this implies that $\epsilon$ goes through $(\overline{X}_{T_1},X_{T_1})$ (the schematic image is transitive, [@EGA] I, 9.5.5).\ \[wecri\] Suppose $W$ is locally noetherian. Let $Q$ be a finitely generated integral monoid, let $S=W[Q]$ and let $T\to S$ be an exact closed immersion. Let $(\overline{X},X)$ be a $T$-log scheme with boundary. Suppose that étale locally on $\overline{X}$ there are charts $Q\to P^{\rm gp}\supset P$ for $(\overline{X},X)$ over $T$ as in . such that the following conditions ${\rm (i), (ii)}$ are satisfied: (i) : The kernel and the torsion part of the cokernel of $Q^{\rm gp}\to P^{\rm gp}$ are finite groups of orders invertible on $W$. (ii) : Let $P'$ be the submonoid of $P^{\rm gp}$ generated by $P$ and the image of $Q\to P^{\rm gp}$ and let $W[P]_T$ be the schematic closure of $W[P']\times_ST=W[P']_T$ in $W[P]$. Then $\lambda:\overline{X}\to W[P]_T$ is smooth on underlying schemes. Then $(\overline{X},X)/T$ is weakly smooth. [Proof:]{} (Note that $\lambda$ in (ii) exists by the schematic density of $X\to\overline{X}$.) Let $$(\overline{X},X)\stackrel{\mu}{\leftarrow}(\overline{L}',L')\stackrel{\eta}{\to}(\overline{L},L)$$ be a test object over $T$. Using (i), one can follow the arguments in [@kalo] 3.4 to construct morphisms $(\overline{L},L)\to(W[P],W[P'])$ of $S$-log schemes with boundary. Necessarily $L$ maps in fact to $W[P']_T$. Since $L\to \overline{L}$ is log schematically dense, $\overline{L}$ maps in fact to $W[P]_T$. By (ii) this morphism can be lifted further to a morphism $\overline{L}\to\overline{X}$ inducing $(\overline{L},L)\to(\overline{X},X)$ as desired.\ \[sglatt\] In the situation of Proposition \[wecri\], suppose in addition $S=T$ and $T\to S$ is the identity. Then for every $S$-log scheme with boundary $(\overline{V},V)$, the projection $\overline{X}\overline{\times}_S\overline{V}\stackrel{p}{\to}\overline{V}$ is log smooth. [Proof:]{} We may assume that $(\overline{X},X)$ over $T$ has a chart as described in Proposition \[wecri\] and that $(\overline{V},V)$ over $T$ has a chart $Q\to F^{\rm gp}$, $F\to{\cal N}_{\overline{V}}(\overline{V})$. Our assumptions imply that$$\overline{X}\times_W\overline{V}\to W[P]\times_W\overline{V}$$is smooth on underlying schemes. It is also strict, hence log smooth. Perform the base change with the closed immersion of log schemes$$W[P]\overline{\times}_S\overline{V}\to W[P]\times_W\overline{V}$$to get the log smooth morphism$$\overline{X}\overline{\times}_S\overline{V}\stackrel{}{\to}W[P]\overline{\times}_S\overline{V}$$(by our construction of fibre products, $W[P]\overline{\times}_S\overline{V}$ is the log schematic closure of $W[P']{\times}_S{V}$). Its composite with the projection $$W[P]\overline{\times}_S\overline{V}\stackrel{\beta}{\to}\overline{V}$$is $p$, hence it is enough to show that $\beta$ is log smooth. Now $\beta$ arises by the base change $\overline{V}\to W[F]$ from the projection $$W[P]\overline{\times}_SW[F]\stackrel{\gamma}{\to}W[F]$$so that it is enough to show that $\gamma$ is log smooth. Let $F'$ be the submonoid of $F^{\rm gp}$ generated by $F$ and the image of $Q\to F^{\rm gp}$. Let $(P'\oplus_QF')^{\rm int}$ be the push out of $P'\leftarrow Q\to F'$ in the category of integral monoids, i.e. $(P'\oplus_QF')^{\rm int}={\mbox{\rm Im}}(P'\oplus_QF'\to (P'\oplus_QF')^{\rm gp})$ where $P'\oplus_QF'$ is the push out in the category of monoids. (If $Q$ is generated by a single element then actually $(P'\oplus_QF')^{\rm int}=P'\oplus_QF'$ by [@kalo] 4.1.) Define the finitely generated integral monoid $$R={\mbox{\rm Im}}(P\oplus F\to (P'\oplus_QF')^{\rm int}).$$Then $\gamma$ can be identified with the natural map $W[R]\to W[F]$. That this is log smooth follows from [@kalo] 3.4 once we know that $$a:F^{\rm gp}\to R^{\rm gp}=(P^{\rm gp}\oplus_{Q^{\rm gp}}F^{\rm gp})$$has kernel and torsion part of the cokernel finitely generated of orders invertible on $W$. But this follows from the corresponding facts for $b:Q^{\rm gp}\to P^{\rm gp}$ because we have isomorphisms ${\mbox{\rm Ker}}(b)\cong{\mbox{\rm Ker}}(a)$ and ${\mbox{\rm Coker}}(b)\cong{\mbox{\rm Coker}}(a)$.\ \[tglatt\] In the situation of Proposition \[wecri\], $(\overline{X},X)/T$ is smooth. [Proof:]{} It remains to verify the second condition in the definition of smoothness. Let $(\overline{Y},Y)\to(\overline{X},X)$ and $(\overline{Y},Y)\to(\overline{V},V)$ be test objects. We may assume that $\overline{Y}$ is connected. Remove all irreducible components of $\overline{V}$ not meeting ${\mbox{\rm Im}}(\overline{Y})$ so that we may assume that each open neighbourhood of ${\mbox{\rm Im}}(\overline{Y})$ in $\overline{V}$ is schematically dense. After étale localization we may assume that $(\overline{X},X)$ has a chart $P\to\Gamma(\overline{X},{\cal N}_{\overline{X}}^{\rm gp})$ as in Proposition \[wecri\]. Viewing our test objects as objects over $S$ we can form the fibre product of fine $S$-log schemes with boundary $(W[P]\overline{\times}_S\overline{V},W[P']\times_SV)$. Étale locally on $W[P]\overline{\times}_S\overline{V}$ we find an exactification $$\overline{Y}\stackrel{i}{\to} \tilde{Z}\stackrel{\tilde{g}}{\to} W[P]\overline{\times}_S\overline{V}$$of the diagonal embedding $\overline{Y}\to W[P]\overline{\times}_S\overline{V}$. We may assume that $\tilde{Z}$ is connected. After further étale localization on $\tilde{Z}$ we may also assume that $\tilde{q}=\tilde{p}\circ \tilde{g}:\tilde{Z}\to\overline{V}$ is strict, where $\tilde{p}:(W[P]\overline{\times}_S\overline{V})\to \overline{V}$ is the projection: this follows from the fact that for $y\in \overline{Y}$ the stalks of the log structures ${\cal N}_{\tilde{Z}}$ and $\tilde{q}^*{\cal N}_{\overline{V}}$ at the separable closure of $i(y)$ coincide, because $\overline{Y}\stackrel{i}{\to} \tilde{Z}$ and $\overline{Y}\to \overline{V}$ are exact closed immersions. By Theorem \[sglatt\], $\tilde{p}$ is log smooth. Thus $\tilde{q}$ is also log smooth, hence is smooth on underlying schemes. Let $$\tilde{Z}^0=\tilde{Z}\times_{(W[P]\overline{\times}_S\overline{V})}(W[P']\times_SV),$$an open subscheme of $\tilde{Z}$ containing ${\mbox{\rm Im}}(Y)$. Consider the restriction $\tilde{q}^0:\tilde{Z}^0\to\overline{V}$ of $\tilde{q}$. Since it is smooth on underlying schemes, it maps schematically dominantly to an open neighbourhood of ${\mbox{\rm Im}}(Y)$ in $V$ (here a morphism of schemes ${\cal X}\to{\cal Y}$ is said to be schematically dominant if its schematic image coincides with ${\cal Y}$). It follows that $\tilde{q}^0$ maps schematically dominantly also to $\overline{V}$ because of our assumption on $\overline{V}$ and the fact that $(\overline{Y},Y)\to(\overline{V},V)$ is boundary exact. Thus $\tilde{q}$ is a classically smooth morphism from the connected scheme $\tilde{Z}$ to another scheme $\overline{V}$ such that its restriction to the open subscheme $\tilde{Z}^0$ maps schematically dominantly to $\overline{V}$. This implies that $\tilde{Z}^0$ is schematically dense in $\tilde{Z}$, because (schematically) dominant classically smooth morphisms from a connected scheme induce bijections between the respective sets of irreducible components. It follows that $\tilde{g}$ factors as $$\tilde{Z}\stackrel{g}{\to}(W[P]_T\overline{\times}_T\overline{V})\stackrel{k}{\to}W[P]\overline{\times}_S\overline{V}:$$first as a morphism of underlying schemes because its restriction to the open schematically dense subscheme $\tilde{Z}^0$ factors through $$W[P']_T\times_T{V}=W[P']\times_S{V};$$but then also as a morphism of log schemes, because $k$ is strict. The morphism $g$ is log étale because the composite $\tilde{g}$ with the closed embedding $k$ is log étale. Let $$Z=\tilde{Z}\times_{(W[P]_T\overline{\times}_T\overline{V})}(\overline{X}\overline{\times}_T\overline{V}).$$From the assumption (ii) in Proposition \[wecri\] we deduce that $\overline{X}\overline{\times}_T\overline{V}\to W[P]_T\overline{\times}_T\overline{V}$ is log smooth and strict, hence $Z\to\tilde{Z}$ is log smooth and strict, hence smooth on underlying schemes. Together with the smoothness of $\tilde{q}$ it follows that $Z\to\overline{V}$ is smooth on underlying schemes. Furthermore $Z\to \overline{X}\overline{\times}_T\overline{V}$ is log étale because $g$ is log étale. Finally, $Y\to Z$ is an exact closed immersion because $Z\to\tilde{Z}$ is strict and $Y\to\tilde{Z}$ is an exact closed immersion. The theorem is proven.\ The interest in smoothness as we defined it lies in the following proposition, which enables us to develop nice cohomology theories for $T$-log schemes with boundary.\ \[keykoh\] Let $(\overline{Y},Y)\to(\overline{X}_i,X_i)$ be boundary exact closed immersions into smooth $T$-log schemes with boundary ($i=1,2$). Then there exist étale locally on $(\overline{X}_1\overline{\times}_T\overline{X}_2)$ factorizations$$(\overline{Y},Y)\stackrel{\iota}{\to}(\overline{Z},Z)\stackrel{}{\to}(\overline{X}_1\overline{\times}_T\overline{X}_2,X_1\times_TX_2)$$of the diagonal embedding such that $\iota$ is a boundary exact closed immmersion, the map $\overline{Z}\to\overline{X}_1\overline{\times}_T\overline{X}_2$ is log étale, and the projections $p_i:\overline{Z}\to\overline{X}_i$ are strict and log smooth, hence smooth on underlying schemes. [Proof:]{} By the definition of smoothness we find étale locally exactifications ($i=1,2$)$$\overline{Y}\to \overline{Z}_i\to\overline{X}_1\overline{\times}_T\overline{X}_2$$such that the projections $\overline{Z}_i\to \overline{X}_i$ are strict and log smooth. Let $$\overline{Z}'=\overline{Z}_1\times_{(\overline{X}_1\overline{\times}_T\overline{X}_2)}\overline{Z}_2$$and let $\overline{Y}\to\overline{Z}\to\overline{Z}'$ be an exactification of $\overline{Y}\to\overline{Z}'$. After perhaps étale localization on $\overline{Z}$ as in the proof of Theorem \[tglatt\] we may assume that the projections $\overline{Z}\to\overline{Z}_i$ are strict. Hence the projections $p_i:\overline{Z}\to\overline{X}_i$ are strict and log smooth. This implies that $$Z=p_1^{-1}(X_1)\cap p_2^{-1}(X_2)$$is log schematically dense in $\overline{Z}$. Indeed, it suffices to prove the log schematic density of $Z$ in $p_1^{-1}(X_1)$ and of $p_1^{-1}(X_1)$ in $\overline{Z}$. Both assertions follow from the general fact that for a strict and log smooth (and in particular classically smooth) morphism of log schemes $h:L\to S$ and a log schematically dense open immersion $S'\to S$, also $h^{-1}(S')$ with its pull back log structure from $S'$ is log schematically dense in $L$: this is easy to prove since the question is local for the étale topology and we therefore may assume that $h$ is a relative affine space. The classical smoothness of (say) $p_1$ and the boundary exactness of $(\overline{Y},Y)\to(\overline{X}_1,X_1)$ imply that $(\overline{Y},Y)\to(\overline{Z},Z)$ is boundary exact (for each connnected component $\overline{Z}'$ of $\overline{Z}$ the map $\pi_0(\overline{Z}')\to\pi_0(\overline{X}_1)$ between sets of irreducible components induced by $p_1$ is injective). We are done.\ We make the exactification $Z\to\overline{X}\overline{\times}_T\overline{V}$ in Theorem \[tglatt\] explicit in some examples, underlining the delicacy of the base change argument in Theorem \[tglatt\]. In the following, for free variables $U_1,\ldots,V_1,\ldots$ we denote by $W[U_1,\ldots,V_1^{\pm},\ldots]$ the log scheme $$W[\mathbb{N}\oplus\ldots\oplus\mathbb{Z}\oplus\ldots]$$ with generators $U_1,\ldots$ for $\mathbb{N}\oplus\ldots$ and generators $V_1,\ldots$ for $\mathbb{Z}\oplus\ldots$. For $f\in\mathbb{Z}[U_1,\ldots,V_1^{\pm},\ldots]$ we denote by $W[U_1,\ldots,V_1^{\pm},\ldots]/f$ the exact closed subscheme defined by $f$.\ (a) Let $Q=\mathbb{N}$ with generator $q$. Let $X=W[U_1^{\pm},U_2]\subset\overline{X}=W[U_1,U_2]$. Define $X\to S$ by sending $q\mapsto U_1^{-1}U_2$, thus $(\overline{X},X)$ is a smooth $S$-log scheme with boundary. The self fibre product of $S$-log schemes with boundary is $$(\overline{X}_1,X_1)=(\overline{X},X)\overline{\times}_S(\overline{X},X)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$$$$=(W[U_1,U_2,V_1,V_2]/(V_1U_2-V_2U_1),W[U_1^{\pm},U_2,V_1^{\pm},V_2]/(U_1^{-1}U_2-V_1^{-1}V_2)).$$Note that the projections $q_j:\overline{X}_1\to\overline{X}$ are not flat (the fibres above the respective origins are two dimensional), although they are log smooth. We construct the desired log étale map $Z\stackrel{g}{\to}\overline{X}_1$ according to the procedure in [@kalo], 4.10. Embed $\mathbb{Z}\to\mathbb{Z}^4$ by sending $n\mapsto(n,-n,-n,n)$ and let $H$ be the image of the canonical map $\mathbb{N}^4\to(\mathbb{Z}^4/\mathbb{Z})$. Then $\overline{X}_1=W[H]$. Let $h:(\mathbb{Z}^4/\mathbb{Z})\to\mathbb{Z}^2$ be the map which sends the class of $(n_1,n_2,n_3,n_4)$ to $(n_1+n_3,n_2+n_4)$, and let $K=h^{-1}(\mathbb{N}^2)$. Then $Z=W[K]$ works. More explicitly: We have an isomorphism $K\cong\mathbb{N}^2\oplus\mathbb{Z}$ by sending the class of $(n_1,n_2,n_3,n_4)$ to $(n_1+n_3,n_2+n_4,n_1+n_2)$. Then $$Z=W[S_1,S_2,S_3^{\pm}]$$ and $g$ is given by $U_1\mapsto S_1S_3,\quad U_2\mapsto S_2S_3,\quad V_1\mapsto S_1,\quad V_2\mapsto S_2$. Now consider the base change with $T=W[q]/q\to S$ defined by sending $q\mapsto 0$. For $j=1,2$ let $\overline{X}_{1,j}=\overline{X}_1\times_{\overline{X}}\overline{X}_T$ where in the fibre product we use the $j$-th projection as the structure map for the first factor. Let $\overline{X}_{T,1}=\overline{X}_T\overline{\times}_T\overline{X}_T$. Then we find $\overline{X}_{1,1}=W[U_1,V_1,V_2]/(V_2U_1)$, $\overline{X}_{1,2}=W[U_1,U_2,V_1]/(V_1U_2)$, thus containing $\overline{X}_{T,1}=W[U_1,V_1]$ as a [*proper*]{} subscheme.\ (b) Let $S, X, \overline{X}$ be as in (a), but this time define $X\to S$ by sending $q\mapsto U_1U_2$. Again $(\overline{X},X)$ is smooth. We use the embedding $\mathbb{Z}\to\mathbb{Z}^4$ which sends $n\mapsto(n,n,-n,-n)$, to define $H={\mbox{\rm Im}}(\mathbb{N}^4\to(\mathbb{Z}^4/\mathbb{Z}))$. Let $h:(\mathbb{Z}^4/\mathbb{Z})\to\mathbb{Z}^2$ be the map which sends the class of $(n_1,n_2,n_3,n_4)$ to $(n_1+n_3,n_2+n_4)$, and let $K=h^{-1}(\mathbb{N}^2)$. We have an isomorphism $K\cong\mathbb{N}^2\oplus\mathbb{Z}$ by sending the class of $(n_1,n_2,n_3,n_4)$ to $(n_1+n_3,n_2+n_4,n_1-n_2)$. We thus find $$\overline{X}_1=W[H]=W[U_1,U_2,V_1,V_2]/(U_1U_2-V_1V_2),$$ $Z=W[S_1,S_2,S_3^{\pm}]$ and $g:Z\to\overline{X}_1$ is given by $U_1\mapsto S_1S_3,\quad U_2\mapsto S_2S_3^{-1},\quad V_1\mapsto S_1,\quad V_2\mapsto S_2$. Note that in this case the projections $q_j:\overline{X}_1\to\overline{X}$ are flat. Now consider the base change with $T=W[q]/q\to S$ defined by sending $q\mapsto 0$. Then, in contrast to (a), we find $\overline{X}_{1,1}=\overline{X}_{1,2}=\overline{X}_{T,1}$ (with $\overline{X}_{1,1}, \overline{X}_{1,2}, \overline{X}_{T,1}$ as in (a)).\ (c) Using the criterion \[tglatt\] one checks that the log schemes with boundary mentioned in .(b)–(d) and . are smooth. In fact, the example (a) just discussed is a special case of . (b) or .. Example (b) (or rather its base change with $T=W[q]/q\to S$ as above) is a special case of . (d).\ $\Omega^1_{(\overline{X},X)/T}$ is locally free of finite rank if $(\overline{X},X)$ is weakly smooth over $T$. [Proof:]{} The same as in the classical case.\ Semistable log schemes with boundary {#semist} ==================================== In this Section $k$ is a field, $Q=\mathbb{N}$ with generator $q$ and $T=({\mbox{\rm Spec}}(k),Q\stackrel{0}{\to}k)$. Definitions ----------- A [*standard semistable $T$-log scheme with boundary*]{} is a $T$-log scheme with boundary isomorphic to:$$(\overline{X},X)=({\mbox{\rm Spec}}(\frac{k[t_1,\ldots,t_{i_2}]}{(t_1,\ldots,t_{i_1})}),{\mbox{\rm Spec}}(\frac{k[t_1,\ldots,t_{i_1},t_{i_1+1}^{\pm},\ldots,t_{i_2}^{\pm}]}{(t_1,\ldots,t_{i_1})}))$$for some integers $1\le i_1\le i_2$ such that $$P=\mathbb{N}^{i_2}\to{\cal N}_{\overline{X}}(\overline{X}),\quad 1_i\mapsto t_i\quad\mbox{ for }1\le i\le i_2$$ $$Q=\mathbb{N}\to P^{\rm gp}=\mathbb{Z}^{i_2},\quad q\mapsto (1_1,\ldots,1_{i_1},r_{i_1+1},\ldots,r_{i_2})$$with some $r_{j}\in\mathbb{Z}$ for $i_1+1\le j\le i_2$ is a chart in the sense of .. A [*semistable $T$-log scheme with boundary*]{} is a $T$-log scheme with boundary $(\overline{Y},Y)$ such that étale locally on $\overline{Y}$ there exist morphisms $(\overline{Y},Y)\to(\overline{X},X)$ to standard semistable $T$-log schemes with boundary such that $\overline{Y}\to\overline{X}$ is strict and log smooth, and $Y={\overline Y}\times_{\overline{X}}X$. Note that $Y$ is then a semistable $k$-log scheme in the usual sense defined in .(d). A [*normal crossing variety over $k$*]{} is a $k$-scheme which étale locally admits smooth morphisms to the underlying schemes of semistable $k$-log schemes. Following [@fkato] we say that a log structure ${\cal N}_{\overline{Y}}$ on a normal crossing variety $\overline{Y}$ over $k$ is [*of embedding type*]{} if étale locally on $\overline{Y}$ the log scheme $(\overline{Y},{\cal N}_{\overline{Y}})$ is isomorphic to a semistable $k$-log scheme. (The point is that we do not require a [*global*]{} structure map of log schemes $(\overline{Y},{\cal N}_{\overline{Y}})\to T$.)\ Let us discuss for a moment the standard semistable $T$-log schemes with boundary $(\overline{X},X)$. If in the above definition $r_j\ge 0$ for all $j$, then $f:X\to T$ actually extends to a (non log smooth in general) usual morphism of log schemes $\overline{f}:\overline{X}\to T$. If even $r_j=0$ for all $j$ then $\overline{f}$ is nothing but a semistable $k$-log scheme with an additional horizontal divisor not interfering with the structure map of log structures; in particular it is log smooth. If at least $r_j\in\{0,1\}$ for all $j$ the morphism $\overline{f}$ is ideally smooth in the sense of Ogus [@ogid]. Examples with $r_j=1$ for all $j$ are those in .(d). The concept of semistable $T$-log schemes with boundary helps us to also understand the cases with local numbers $r_j\notin\{0,1\}$: Any $(\overline{Y},Y)$ semistable $T$-log scheme with boundary is smooth, by Theorem \[tglatt\], and as we will see below this implies analogs of classical results for their cohomology. Examples of semistable $T$-log schemes with boundary with local numbers $r_j$ possibly not in $\{0,1\}$ are those in . or those from \[Grothe\] below. Or think of a flat family of varieties over ${\mbox{\rm Spec}}(k[q])$ with smooth general fibre and whose reduced subscheme of the special fibre is a normal crossing variety, but where some components of the special fibre may have multiplicities $>1$: then unions of irreducible components of this special fibre with multiplicity $=1$ are semistable $T$-log schemes with boundary. One more big class of examples with local numbers $r_j$ possibly not in $\{0,1\}$ is obtained by the following lemma, which follows from computations with local coordinates: Let $Y\to\overline{Y}$ be an embedding of $k$-schemes which étale locally looks like the underlying embedding of $k$-schemes of a semistable $T$-log scheme with boundary (i.e. for each geometric point $y$ of $\overline{Y}$ there is a semistable $T$-log scheme with boundary which on underlying schemes looks like $Y\to\overline{Y}$ around $y$). Suppose ${\cal N}_{\overline{Y}}$ is a log structure of embedding type on $\overline{Y}$ such that $(Y,{\cal N}_{\overline{Y}}|_Y)$ is a semistable $k$-log scheme (for an appropriate structure morphism to $T$). Then $((\overline{Y},{\cal N}_{\overline{Y}}),Y)$ is a semistable $T$-log scheme with boundary. Fumiharu Kato in [@fkato] has worked out precise criteria for these two properties of normal crossing varieties over $k$ — to admit a log structure of embedding type, resp. to admit a log structure of semistable type. Now suppose we are given a semistable $T$-log scheme $Y$. An “optimal” compactification would be a dense open embedding into a proper semistable $k$-log scheme in the classical sense, or at least into an ideally smooth proper $k$-log scheme; however, advocating the main idea of this paper, a dense open embedding $Y\to\overline{Y}$ into a log scheme $\overline{Y}$ such that $(\overline{Y},Y)$ is a proper semistable $T$-log scheme with boundary is also very useful, and this might be easier to find, or (more importantly) be naturally at hand in particular situations.\ De Rham cohomology ------------------ Here we assume ${\mbox{\rm char}}(k)=0$. Let $Z$ be a smooth $k$-scheme and let $V$ be a normal crossing divisor on $Z$. Suppose there exists a flat morphism $f:(Z-V)\to{\mbox{\rm Spec}}(k[q])$, smooth above $q\ne 0$ and with semistable fibre $X$ above the origin $q=0$. Let $\overline{X}$ be the closure of $X$ in $Z$ and suppose also that $\overline{X}\cup V$ is a normal crossing divisor on $Z$. Endow $Z$ with the log structure defined by $\overline{X}\cup V$ and endow all subschemes of $Z$ with the induced log structure (we will suppress mentioning of this log structure in our notation). Then $(\overline{X},X)$ is a semistable $T$-log scheme with boundary. Let $D=\overline{X}\cap V=\overline{X}-X$ and let $\overline{X}=\cup_{1\le i\le a}\overline{X}_i$ be the decomposition into irreducible components in a fixed ordering and suppose that each $\overline{X}_i$ is classically smooth. Let $\Omega_{X/T}^{\bullet}$ be the relative logarithmic de Rham complex.\ \[Grothe\] The restriction map$$R\Gamma(\overline{X},\Omega_{(\overline{X},X)/T}^{\bullet})\to R\Gamma(X,\Omega_{X/T}^{\bullet})$$is an isomorphism. [Proof:]{} We use a technique of Steenbrink [@steen] to reduce to a standard fact. Let $\Omega_Z^{\bullet}$ be the de Rham complex over $k$ on $Z$ with logarithmic poles along $\overline{X}\cup V$. Note that ${\mbox{\rm dlog}}(f^*(q))\in\Gamma(Z-V,\Omega_Z^{1})$ extends uniquely to a global Section $\theta\in\Gamma(Z,\Omega_Z^{1})$. Let $\Omega_{Z,V}^{\bullet}$ be the de Rham complex on $Z$ with logarithmic poles only along $V$; thus $\Omega_{Z,V}^{\bullet}$ is a [*subcomplex*]{} of $\Omega_Z^{\bullet}$. Define the vertical weight filtration on $\Omega_Z^{\bullet}$ by$$P_j\Omega^i_Z={\mbox{\rm Im}}(\Omega^j_Z\otimes\Omega^{i-j}_{Z,V}\to \Omega^i_Z).$$For $j\ge 1$ let $\overline{X}^j$ be the disjoint sum of all $\cap_{i\in I}\overline{X}_i$ where $I$ runs through the subsets of $\{1,\ldots,a\}$ with $j$ elements. Let $\tau_j:\overline{X}^j\to \overline{X}$ be the canonical map and let $\Omega_{\overline{X}^j}^{\bullet}$ be the de Rham complex on $\overline{X}^j$ with logarithmic poles along $\overline{X}^j\cap \tau_{j}^{-1}(D)$. Then we have isomorphisms of complexes$$(*)\quad\quad {\rm res}:{\mbox{\rm Gr}}_j\Omega_Z^{\bullet}\cong \tau_{j,*}\Omega_{\overline{X}^j}^{\bullet}[-j],$$characterized as follows: Let $x_1,\ldots,x_d$ be local coordinates on $Z$ such that $x_i$ for $1\le i\le a\le d$ is a local coordinate for $\overline{X}_i$. If $$\omega=\alpha\wedge{\mbox{\rm dlog}}(x_{i_1})\wedge\ldots \wedge{\mbox{\rm dlog}}(x_{i_j})\in P_j\Omega_Z^{\bullet}$$ with $i_1<\ldots<i_j<a$, then ${\rm res}$ sends the class of $\omega$ to the class of $\alpha$. Now let $$A^{pq}=\Omega_Z^{p+q+1}/P_q\Omega_Z^{p+q+1},\quad\quad\quad P_jA^{pq}=P_{2q+j+1}\Omega_Z^{p+q+1}/P_q\Omega_Z^{p+q+1}.$$Using the differentials $d':A^{pq}\to A^{p+1,q},\quad\omega\mapsto d\omega$ and $d'':A^{pq}\to A^{p,q+1},\quad\omega\mapsto \theta\wedge\omega$ we get a filtered double complex $A^{\bullet\bullet}$. We claim that $$0\to\frac{\Omega^p_Z\otimes{\cal O}_{\overline{X}}}{(\Omega^{p-1}_Z\otimes{\cal O}_{\overline{X}})\wedge\theta}\stackrel{\wedge\theta}{\to}A^{p0}\stackrel{\wedge\theta}{\to}A^{p1}\stackrel{\wedge\theta}{\to}\ldots$$is exact. Indeed, it is enough to show that for all $p$, all $j\ge 2$ the sequences$${\mbox{\rm Gr}}_{j-1}\Omega^{p-1}_Z\stackrel{\wedge\theta}{\to}{\mbox{\rm Gr}}_{j}\Omega^{p}_Z\stackrel{\wedge\theta}{\to}{\mbox{\rm Gr}}_{j+1}\Omega^{p+1}_Z\stackrel{\wedge\theta}{\to}\ldots$$$$0\to P_0\Omega^{p-1}_Z/{\cal J}_{\overline{X}}.\Omega^{p-1}_Z\stackrel{\wedge\theta}{\to}{\mbox{\rm Gr}}_{1}\Omega^{p}_Z\stackrel{\wedge\theta}{\to}{\mbox{\rm Gr}}_{1}\Omega^{p+1}_Z\stackrel{\wedge\theta}{\to}\ldots$$are exact, where ${\cal J}_{\overline{X}}={\mbox{\rm Ker}}({\cal O}_{Z}\to{\cal O}_{\overline{X}})$. This follows from $(*)$ and the exactness of $$0\to P_0\Omega^{p}_Z/{\cal J}_{\overline{X}}.\Omega^{p}_Z\to\tau_{1,*}\Omega^p_{\overline{X}^1}\to\tau_{2,*}\Omega^p_{\overline{X}^2}\to\ldots.$$The claim follows. It implies that the maps$$\Omega_{(\overline{X},X)/T}^{p}=\frac{\Omega^p_Z\otimes{\cal O}_{\overline{X}}}{(\Omega^{p-1}_Z\otimes{\cal O}_{\overline{X}})\wedge\theta}\to A^{p0}\subset A^p,\quad\omega\mapsto (-1)^p\theta\wedge\omega$$define a quasi-isomorphism $\Omega_{(\overline{X},X)/T}^{\bullet}\to A^{\bullet}$, hence a spectral sequence$$E_1^{-r,q+r}=H^q(\overline{X},{\mbox{\rm Gr}}_r A^{\bullet})\Longrightarrow H^q(\overline{X},\Omega_{(\overline{X},X)/T}^{\bullet}).$$Now we can of course repeat all this on $Z-V$ instead of $Z$, and restriction from $Z$ to $Z-V$ gives a canonical morphism between the respective spectral sequences. That this is an isomorphism can be checked on the initial terms, and using the isomorphism $(*)$ this boils down to proving that the restriction maps$$H^p(\overline{X}^j,\Omega_{\overline{X}^j}^{\bullet})\to H^p(X^j,\Omega_{\overline{X}^j}^{\bullet})$$are isomorphisms where we set $X^j=\overline{X}^j\cap\tau_j^{-1}(X)$. But this is well known. The proof is finished.\ Now assume ${\overline X}$ is proper. Similar to the classical Hodge theory, the Hodge filtration on $$H^p(\overline{X},\Omega_{(\overline{X},X)/T}^{\bullet})=H^p(X,\Omega_{X/T}^{\bullet})$$ obtained by stupidly filtering $\Omega_{(\overline{X},X)/T}^{\bullet}$ should be meaningful. Another application of Proposition \[Grothe\] might be a Poincaré duality theorem. Suppose the underlying scheme of $\overline{X}$ is of pure dimension $d$. Let ${\cal I}_{D}={\mbox{\rm Ker}}({\cal O}_{\overline{X}}\to{\cal O}_{D})$ and define the de Rham cohomology with compact support of $(\overline{X},X)/T$ as $$R\Gamma(\overline{X},{\cal I}_D\otimes\Omega_{(\overline{X},X)/T}^{\bullet}).$$It is a natural question to ask if this is dual to $R\Gamma(\overline{X},\Omega_{(\overline{X},X)/T}^{\bullet})=R\Gamma(X,\Omega_{X/T}^{\bullet})$. The key would be as usual the construction of a trace map $H^d(\overline{X},{\cal I}_D\otimes\Omega_{(\overline{X},X)/T}^d)\to k$.\ Another application of semistable $T$-log schemes with boundary is the possibility to define the notion of [*regular singularities*]{} of a given integrable log connection on a semistable $T$-log scheme $X$, provided we have an embedding $X\to\overline{X}$ such that $(\overline{X},X)$ is a proper semistable $T$-log scheme with boundary.\ Here is an application of the construction in . to the de Rham cohomology of certain semistable $k$-log schemes (a simplified variant of the application given in [@hkstrat]; in fact, the present paper formalizes and generalizes a key construction from [@hkstrat]). In . assume that ${\mbox{\rm char}}(k)=0$ and that $M$ is the intersection of [*all*]{} irreducible components of $Y$. Recall that we constructed a morphism of log schemes $V_M\to S=({\mbox{\rm Spec}}(k[q]),1\mapsto q)$. For $k$-valued points $\alpha\to S$ (with pull back log structure) let $V_{M}^{\alpha}=V_M\times_S{\alpha}$. Using the $S$-log scheme with boundary $(P_M,V_M)$ one can show that the derived category objects $R\Gamma(V_{M}^{\alpha},\Omega^{\bullet}_{V_M^{\alpha}/\alpha})$ (with $\Omega^{\bullet}_{V_M^{\alpha}/\alpha}$ the relative logarithmic de Rham complex; if $\alpha\ne 0$ this is the classical one) are canonically isomorphic for varying $\alpha$. Namely, the canonical restriction maps $$R\Gamma(P_{M},\Omega^{\bullet}_{(P_M,V_M)/S})\to R\Gamma(V_{M}^{\alpha},\Omega^{\bullet}_{V_M^{\alpha}/\alpha})$$are isomorphisms for all $\alpha$.\ Crystalline cohomology ---------------------- Let $\tilde{S}$ be a scheme such that ${\cal O}_{\tilde{S}}$ is killed by a non-zero integer, $I\subset{\cal O}_{\tilde{S}}$ a quasi-coherent ideal with DP-structure $\gamma$ on it, and let $\tilde{\cal L}$ be a fine log structure on $\tilde{S}$. Let $(S,{\cal L})$ be an exact closed log subscheme of $(\tilde{S},\tilde{\cal L})$ defined by a sub-DP-ideal of $I$ and let $f:(X,{\cal N})\to(S,{\cal L})$ be a log smooth and integral morphism of log schemes. An important reason why log crystalline cohomology of $(X,{\cal N})$ over $(\tilde{S},\tilde{\cal L})$ works well is that locally on $X$ there exist smooth and integral, hence flat morphisms $\tilde{f}:(\tilde{X},\tilde{\cal N})\to(\tilde{S},\tilde{\cal L})$ with $f=\tilde{f}\times_{(\tilde{S},\tilde{\cal L})}(S,{\cal L})$. This implies that the crystalline complex of $X/\tilde{S}$ (with respect to any embedding system) is flat over ${\cal O}_{\tilde{S}}$, see [@hyoka] 2.22, and on this property many fundamental theorems rely. Now let $W$ be a discrete valuation ring of mixed characteristic $(0,p)$ with maximal ideal generated by $p$. For $n\in \mathbb{N}$ let $W_n=W/(p^n)$, $k=W_1$ and $K_0={\mbox{\rm Quot}}(W)$, and let $T_n$ be the exact closed log subscheme of $S=W[Q]$ defined by the ideal $(p^n,q)$ (abusing previous notation we now take ${\mbox{\rm Spec}}(W)$ as the base scheme $W$ of .). Thus $T=T_1$. We will often view $T$-log schemes with boundary as $T_n$-log schemes with boundary for $n\in \mathbb{N}$.\ \[loclif\] Let $(\overline{Y},Y)/T$ be a semistable $T$-log scheme with boundary. Then there exist étale locally on $\overline{Y}$ smooth $T_n$-log schemes with boundary $(\overline{Y}_n,Y_n)$ such that $(\overline{Y},Y)=(\overline{Y}_n,Y_n)\overline{\times}_{T_n}T$, the closed immersion $(\overline{Y},Y)\to(\overline{Y}_n,Y_n)$ is boundary exact, and such that $\Omega^1_{(\overline{Y}_n,Y_n)/T_n}$ is flat over ${\cal O}_{T_n}$ and commutes with base changes $T_m\to T_n$ for $m\le n$. [Proof:]{} We may suppose that there is a strict and log smooth morphism $$h:(\overline{Y},Y)\to (\overline{X},X)=({\mbox{\rm Spec}}(\frac{k[t_1,\ldots,t_{i_2}]}{(t_1,\ldots,t_{i_1})}),{\mbox{\rm Spec}}(\frac{k[t_1,\ldots,t_{i_1},t_{i_1+1}^{\pm},\ldots,t_{i_2}^{\pm}]}{(t_1,\ldots,t_{i_1})}))$$for some integers $1\le i_1\le i_2$ such that $P=\mathbb{N}^{i_2}$ is a chart for $(\overline{X},X)$ sending $1_i\mapsto t_i$ for $1\le i\le i_2$ and such that the structure map is given by $$Q=\mathbb{N}\to P^{\rm gp}=\mathbb{Z}^{i_2},\quad q\mapsto (1_1,\ldots,1_{i_1},r_{i_1+1},\ldots,r_{i_2})$$with some $r_{j}\in\mathbb{Z}$ for $i_1+1\le j\le i_2$. We lift $(\overline{X},X)$ to$$(\overline{X}_n,X_n)=({\mbox{\rm Spec}}(\frac{W_n[t_1,\ldots,t_{i_2}]}{(t_1,\ldots,t_{i_1})}),{\mbox{\rm Spec}}(\frac{W_n[t_1,\ldots,t_{i_1},t_{i_1+1}^{\pm},\ldots,t_{i_2}^{\pm}]}{(t_1,\ldots,t_{i_1})}))$$using the same formulas for the log structure maps. Local liftings of $h$ to $(\overline{X}_n,X_n)$ result from the classical theory, since ‘strict and log smooth’ is equivalent to ‘smooth on underlying schemes’.\ \[kriskey\] Let $n\in\mathbb{N}$ and let $(\overline{Y},Y)\to(\overline{X}_i,X_i)$ be boundary exact closed immersions into smooth $T_n$-log schemes with boundary ($i=1,2$). Then there exist étale locally on $(\overline{X}_1\overline{\times}_{T_n}\overline{X}_2)$ factorizations of the diagonal embedding $$(\overline{Y},Y)\stackrel{\iota}{\to}(\overline{Z},Z)\stackrel{}{\to}(\overline{X}_1\overline{\times}_{T_n}\overline{X}_2,X_1\times_{T_n}X_2)$$with $\iota$ a boundary exact closed immmersion, the map $\overline{Z}\to\overline{X}_1\overline{\times}_{T_n}\overline{X}_2$ log étale, the projections $p_i:\overline{Z}\to\overline{X}_i$ strict and log smooth, and with the following property: Let $\overline{D}_{12}$ (resp. $\overline{D}_i$) denote the ${\rm DP}$ envelopes of (the underlying scheme morphism of) $\overline{Y}\to\overline{Z}$ (resp. of $\overline{Y}\to\overline{X}_i$), and let $q_i:\overline{D}_{12}\to\overline{D}_i$ be the canonical projections. Then there exist $u_{i1},\ldots,u_{im_i}\in {\cal O}_{\overline{D}_{12}}$ for $i=1$ and $i=2$ such that $du_{i1},\ldots,du_{im_i}$ form a basis of $\Omega^1_{\overline{Z}/\overline{X}_i}$ and such that the assignments $U_{ij}^{[k]}\mapsto u_{ij}^{[k]}$ ($k\in\mathbb{N}$) induce isomorphisms$$q_i^{-1}{\cal O}_{\overline{D}_i}\langle U_{i1},\ldots,U_{im_i}\rangle\cong{\cal O}_{\overline{D}_{12}}$$where on the left hand side we mean the ${\rm DP}$ envelope of the free polynomial ring. Lemma \[kriskey\] follows from Proposition \[keykoh\], and the same proofs give variants of Proposition \[keykoh\] and Lemma \[kriskey\] for more than two embeddings $(\overline{Y},Y)\to(\overline{X}_i,X_i)$ (and hence with products with more than two factors). As in [@kalo] one shows that the DP envelopes of $(\overline{Y},Y)$ in chosen exactifications of these products (e.g. the DP envelope $\overline{D}_{12}$ in Lemma \[kriskey\]) are independent of the chosen exactifications. For a given semistable $T$-log scheme with boundary $(\overline{Y},Y)$ we now define its crystalline cohomology relative to $T_n$ by the standard method (cf. [@hyoka] 2.18): Choose an open covering $\overline{Y}=\cup_{\overline{U}\in\overline{\cal U}}\overline{U}$ and for each $(\overline{U},U=Y\cap\overline{U})$ a lift $(\overline{U}_n,U_n)$ as in Lemma \[loclif\]. Taking products we get a simplicial $T_n$-log scheme with boundary $(\overline{U}^{\bullet}_n,U^{\bullet}_n)$ which is an embedding system for $(\overline{Y},Y)$ over $T_n$. Let $\overline{D}_n^{\bullet}$ be the DP envelope of $(\overline{Y},Y)$ in $(\overline{U}_n^{\bullet},U^{\bullet}_n)$, i.e. the simplicial scheme formed by the DP envelopes of local exactifications of $(\overline{Y},Y)\to(\overline{U}_n^{\bullet},U^{\bullet}_n)$ as in Lemma \[kriskey\]. Then we set$$R\Gamma_{\rm crys}((\overline{Y},Y)/T_n)=R\Gamma(\overline{D}_n^{\bullet},\Omega^{\bullet}_{(\overline{U}_n,U_n)/T_n}\otimes{\cal O}_{\overline{D}_n^{\bullet}}).$$That this definition is independent of the chosen embedding follows from Lemma \[kriskey\] and the DP Poincaré lemma.\ \(a) For $m\le n$ we have $$R\Gamma_{\rm crys}((\overline{Y},Y)/T_m)\cong R\Gamma_{\rm crys}((\overline{Y},Y)/T_n)\otimes^{\mathbb{L}}_{W_n}W_m.$$(b) If $\overline{Y}$ is proper over $k$, the cohomology of $$R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}((\overline{Y},Y)/T_n)$$ (resp. of $R\Gamma_{\rm crys}((\overline{Y},Y)/T_n)$) is finitely generated over $W$ (resp. over $W_n$). [Proof:]{} Just as in [@hyoka] 2.22 one deduces from Lemmata \[loclif\] and \[kriskey\] that $\Omega^{\bullet}_{(\overline{U}_n,U_n)/T_n}\otimes{\cal O}_{\overline{D}_n^{\bullet}}$ is a $W_n$-flat sheaf complex on $\overline{D}_n^{\bullet}$ and this implies (a). If $\overline{Y}$ is proper over $k$ it follows that $R\Gamma_{\rm crys}((\overline{Y},Y)/T_1)=R\Gamma(\overline{Y},\Omega^{\bullet}_{(\overline{Y},Y)/T_1})$ has finite dimensional cohomology over $k$ since each $\Omega^{j}_{(\overline{Y},Y)/T_1}$ is coherent. Together with (a) we conclude as in the classical case.\ Ogus [@ogcon] and Shiho [@shiho] have defined logarithmic convergent cohomology in great generality and “in crystalline spirit”. Here we content ourselves with the following definition. Let $E$ be a fine $T$-log scheme. Let $T_{\infty}$ be the formal log scheme $({\mbox{\rm Spf}}(W),1\mapsto 0)$. Choose an exact closed immersion $E\to G$ into a log smooth formal $T_{\infty}$-log scheme $G$ topologically of finite type over $W$. Associated to $G$ is a $K_0$-rigid space $G_{K_0}$ together with a specialization map ${\rm sp}$ to the special fibre of $G$. The preimage ${\rm sp}^{-1}(E)=]E[_G$ of the embedded $E$, the tube of $E$, is an admissible open subspace of $G_{K_0}$. The logarithmic de Rham complex $\Omega^{\bullet}_{G/T_{\infty}}$ on $G$ gives rise, tensored with $\mathbb{Q}$, to a sheaf complex $\Omega^{\bullet}_{G_{K_0}/T_{\infty,K_0}}$ on $G_{K_0}$ and we set$$R\Gamma_{\rm conv}(E/T_{\infty})=R\Gamma(]E[_G,\Omega^{\bullet}_{G_{K_0}/T_{\infty,K_0}}),$$an object in the derived category of $K_0$-vector spaces. If there are embeddings $E\to G$ as above only locally on $E$, one works with embedding systems.\ Now let $Y$ be a semistable $k$-log scheme with smooth irreducible components and let $M$ be the intersection of some of its irreducible components. Endow $M$ with the structure of $T$-log scheme induced from $Y$. Note that $M$ is not log smooth over $T$ (unless $Y$ has only a single irreducible component) and its usual log crystalline cohomology is pathological; it does not provide a canonical integral lattice in the log convergent cohomology of $M$, as we will now construct one by another method. In . we constructed a $S_1$-log scheme with boundary $(P_M,V_M)$ where $S_1$ is the exact closed log subscheme of $S$ defined by the ideal $(p)$. Perform the base change with the exact closed subscheme $T$ of $S_1$ defined by the ideal $(q)$ to get $(P_M^0,V_M^0)=(P_M\times_{S_1}T,V_M\times_{S_1}T)$. This is a semistable $T$-log scheme with boundary as defined above.\ \[crisconv\] There exists a canonical isomorphism $$R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}(({P_M^0},V_M^0)/T_n)\otimes_{W}K_0\cong R\Gamma_{\rm conv}(M/T_{\infty}).$$In particular, if $M$ is proper, each $R^j\Gamma_{\rm conv}(M/T_{\infty})$ is finite dimensional.\ [Proof:]{} [*Step 1:*]{} The map is$$R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}(({P_M^0},V_M^0)/T_n)\otimes_{W}K_0\to R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}(({V_M^0},V_M^0)/T_n)\otimes_{W}K_0$$$$=R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}(V_M^0/T_n)\otimes_{W}K_0\stackrel{(i)}{\cong}R\Gamma_{\rm conv}(V_M^0/T_{\infty})\to R\Gamma_{\rm conv}(M/T_{\infty})$$where the left hand side in $(i)$ is the usual log crystalline cohomology of $V_M^0/T_n$ and the isomorphism $(i)$ holds by log smoothness of $V_M^0/T$. That this map is an ismorphism can be checked locally.\ [*Step 2:*]{} We may therefore assume that there exists a smooth (in the classical sense) affine connected ${\mbox{\rm Spec}}(W)$-scheme $\tilde{\cal M}={\mbox{\rm Spec}}(\tilde{B})$ lifting $M$ and that the invertible sheaves ${\cal F}_j|_M$ on $M$ are trivial (notation from .); let $v_j$ be a generator of ${\cal F}_j|_M$. Furthermore we may assume that the divisor $D$ on $M$ (the intersection of $M$ with all irreducible components of $Y$ not containing $M$) lifts to a (relative ${\mbox{\rm Spec}}(W)$) normal crosssings divisor $\tilde{\cal D}$ on $\tilde{\cal M}$. Let $$\tilde{\cal V}_{\cal M}={\mbox{\rm Spec}}(\tilde{B}[x_j]_{j\in I})$$$$\tilde{\cal P}_{\cal M}=\times_{\tilde{\cal M}}({\mbox{\rm Proj}}(\tilde{B}[y_j,x_j]_{j\in I}).$$Identifying the free variable $x_j$ with a lift of $v_j$ we view $\tilde{\cal V}_{\cal M}$ as a lift of $V_M$; identifying moreover the free variable $y_j$ with a lift of $1_{{\cal O}_M}$ we view $\tilde{\cal P}_{\cal M}$ as a lift of $P_M$; identifying a homogenous element $s\in\tilde{B}[x_j]_{j\in I}$ of degree $n$ with the degree zero element $s/y_j^n$ of $\tilde{B}[y_j^{\pm},x_j]$ we view $\tilde{\cal V}_{\cal M}$ as an open subscheme of $\tilde{\cal P}_{\cal M}$. As in . we factor the distinguished element $a\in{\mbox{\rm Sym}}_{{\cal O}_M}(\oplus({\cal F}_j)_{j\in I})(M)$ as $a=a_0.(\oplus_{j\in I}v_j)$ with defining equation $a_0\in{\cal O}_M$ of the divisor $D$ in $M$. Lift $a_0$ to a defining equation $\tilde{a}_0\in \tilde{B}$ of $\tilde{\cal D}$ in $\tilde{\cal M}$. This $\tilde{a}_0$ also defines a normal crossing divisor $\tilde{\cal D}_{\tilde{\cal V}}$ on $\tilde{\cal V}_{\cal M}$. Set $\tilde{a}=\tilde{a}_0\prod_{j\in I}x_j\in \tilde{B}[x_j]_{j\in I}$ and consider the following normal crossing divisor on $\tilde{\cal P}_{\cal M}$: the union of $\tilde{\cal P}_{\cal M}-\tilde{\cal V}_{\cal M}$ with the closure (in $\tilde{\cal P}_{\cal M}$) of the zero set of $\tilde{a}$ (in $\tilde{\cal V}_{\cal M}$). It defines a log structure on $\tilde{\cal P}_{\cal M}$. Define a morphism $\tilde{\cal V}_{\cal M}\to S$ by sending $q\mapsto \tilde{a}$. We have constructed a lift of the $S_1$-log scheme with boundary $(P_M,V_M)$ to a $S$-log scheme with boundary $(\tilde{\cal P}_{\cal M},\tilde{\cal V}_{\cal M})$. Moreover, if we denote by $\tilde{\cal T}_{\infty}$ the exact closed log subscheme of $S$ defined by the ideal $(q)$, then the $\tilde{\cal T}_{\infty}$-log scheme with boundary $(\tilde{\cal P}_{\cal M}^0,\tilde{\cal V}_{\cal M}^0)=(\tilde{\cal P}_{\cal M}\times_S\tilde{\cal T}_{\infty},\tilde{\cal V}_{\cal M}\times_S\tilde{\cal T}_{\infty})$ is a lift of the $T$-log scheme with boundary $(P_M^0,V_M^0)$.\ [*Step 3:*]{} Denote by ${\cal P}_{\cal M}^0$ (resp. ${\cal V}_{\cal M}^0$, resp. ${\cal M}$, resp. ${\cal D}_{\cal V}$) the $p$-adic completions of $\tilde{\cal P}_{\cal M}^0$ (resp. of $\tilde{\cal V}_{\cal M}^0$, resp. of $\tilde{\cal M}$, resp. of $\tilde{\cal D}_{\tilde{\cal V}}$). Denote by ${\cal P}_{{\cal M},n}^0$ (resp. ${\cal V}_{{\cal M},n}^0$, resp. ${\cal M}_n$, resp. ${\cal D}_{{\cal V},n}$) the reduction modulo $p^n$. Let $\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}$ be the $p$-adic completion of the de Rham complex of the $\tilde{\cal T}_{\infty}$-log scheme with boundary $(\tilde{\cal P}_{\cal M}^0,\tilde{\cal V}_{\cal M}^0)$. Its reduction $\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes(\mathbb{Z}/p^n)$ modulo $p^n$ is the de Rham complex $\Omega^{\bullet}_{{\cal P}_{{\cal M},n}^0/T_n}$ of the ${T}_{n}$-log scheme with boundary $({\cal P}_{{\cal M},n}^0,\tilde{\cal V}_{{\cal M},n}^0)$. Observe that the differentials on $\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}$ pass to differentials on $\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes_{{\cal O}_{{\cal P}_{\cal M}^0}}{\cal O}_{\cal M}$ where we use the zero section ${\cal M}\to{\cal P}_{\cal M}^0$. Let $\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes\mathbb{Q}$ be the complex on the rigid space ${\cal P}_{{\cal M},K_0}^0$ obtained by tensoring with $K_0$ the sections of $\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}$ over open affine pieces of ${\cal P}_{\cal M}^0$. Similarly define $\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes_{{\cal O}_{{\cal P}_{\cal M}^0}}{\cal O}_{\cal M}\otimes\mathbb{Q}$. By definition we have $$R\Gamma_{\rm conv}(M/T_{\infty})=R\Gamma(]M[_{{\cal P}_{\cal M}},\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes\mathbb{Q}),$$ $$R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}(({P_M^0},V_M^0)/T_n)\otimes_{W}K_0=R\lim_{\stackrel{\leftarrow}{n}}R\Gamma({\cal P}_{{\cal M},n}^0,\Omega^{\bullet}_{{\cal P}_{{\cal M},n}^0/T_n})\otimes_{W}K_0.$$In view of$$R\Gamma(]M[_{{\cal P}_{\cal M}},\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes_{{\cal O}_{{\cal P}_{\cal M}^0}}{\cal O}_{\cal M}\otimes\mathbb{Q})=R\lim_{\stackrel{\leftarrow}{n}}R\Gamma({\cal P}_{{\cal M},n}^0,\Omega^{\bullet}_{{\cal P}_{{\cal M},n}^0/T_n}\otimes_{{\cal O}_{{\cal P}_{{\cal M},n}^0}}{\cal O}_{{\cal M}_n})\otimes_{W}K_0$$ it is therefore enough to show that the maps$$f_n:R\Gamma({\cal P}_{{\cal M},n}^0,\Omega^{\bullet}_{{\cal P}_{{\cal M},n}^0/T_n})\to R\Gamma({\cal P}_{{\cal M},n}^0,\Omega^{\bullet}_{{\cal P}_{{\cal M},n}^0/T_n}\otimes_{{\cal O}_{{\cal P}_{{\cal M},n}^0}}{\cal O}_{{\cal M}_n})$$ $$g:R\Gamma(]M[_{{\cal P}_{\cal M}},\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes\mathbb{Q})\to R\Gamma(]M[_{{\cal P}_{\cal M}},\Omega^{\bullet}_{{\cal P}_{\cal M}^0/T_{\infty}}\otimes_{{\cal O}_{{\cal P}_{\cal M}^0}}{\cal O}_{\cal M}\otimes\mathbb{Q})$$are isomorphisms.\ [*Step 4:*]{} Let ${\cal D}_{{\cal V},n}=\cup_{l\in L}{\cal D}_{n,l}$ be the decomposition of ${\cal D}_{{\cal V},n}$ into irreducible components. Let ${\cal E}'_{n}$ be the closed subscheme of ${\cal V}_{{\cal M},n}^0$ defined by $\prod_{j\in I}x_j\in \Gamma({\cal V}_{{\cal M},n}^0,{\cal O}_{{\cal V}_{{\cal M},n}^0})$ and let ${\cal E}_{n}$ be the closure of ${\cal E}'_{n}$ in ${\cal P}_{{\cal M},n}^0$. Let ${\cal E}_{n}=\cup_{j\in I}{\cal E}_{n,j}$ be its decomposition into irreducible components. For a pair $P=(P_I,P_L)$ of subsets $P_I\subset I$ and $P_L\subset L$ let$${\cal G}_P=(\cap_{j\in P_I}{\cal E}_{n,j})\cap(\cap_{l\in P_L}{\cal D}_{n,l}),$$so we drop reference to $n$ in our notation, for convenience. Also for convenience we denote the sheaf complex $\Omega^{\bullet}_{{\cal P}_{{\cal M},n}^0/T_n}$ on ${\cal P}_{{\cal M},n}^0$ simply by $\Omega^{\bullet}$. For two pairs $P, P'$ as above with $P_I\cup P_L\ne \emptyset$, with $P_I\subset P'_I$ and $P_L=P'_{L}$ consider the canonical map $$w_{P,P'}:\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_P}\to\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P'}}$$of sheaf complexes on ${\cal P}_{{\cal M},n}^0$. We claim that the map $R\Gamma({\cal P}_{{\cal M},n}^0,w_{P,P'})$ induced by $w_{P,P'}$ in cohomology is an isomorphism. For this we may of course even assume $P_I'=P_I\cup\{j_0\}$ for some $j_0\in I$, $j_0\notin P_I$. In the ${\cal O}_{{\cal G}_{P'}}$-module $\Omega^1\otimes{\cal O}_{{\cal G}_{P'}}$ we fix a complement $N$ of the submodule generated by (the class of) ${\mbox{\rm dlog}}(x_{j_0})\in\Gamma({\cal P}_{{\cal M},n}^0,\Omega^1\otimes{\cal O}_{{\cal G}_{P'}})$ as follows. We use the identification$$\frac{(\Omega^1_{\tilde{\cal M}}(\log(\tilde{\cal D}))\otimes{\cal O}_{{\cal G}_{P'}})\oplus(\oplus_{j\in I}{\cal O}_{{\cal G}_{P'}}.{\mbox{\rm dlog}}(x_j))}{{\cal O}_{{\cal G}_{P'}}.{\mbox{\rm dlog}}(\tilde{a})}=\Omega^1\otimes{\cal O}_{{\cal G}_{P'}}$$(with $\Omega^1_{\tilde{\cal M}}(\log(\tilde{\cal D}))$ the differential module of $(\tilde{\cal M},(\mbox{log str. def. by }\tilde{\cal D}))\to({\mbox{\rm Spec}}(W),\mbox{triv.})$). If $P_L\ne\emptyset$ we may assume that we can factor our $\tilde{a}_0\in\tilde{B}$ from above as $\tilde{a}_0=\tilde{a}_0'h$ with $h\in\tilde{B}$ whose zero set in $\tilde{\cal M}={\mbox{\rm Spec}}(\tilde{B})$ reduces modulo $p^n$ to an irreducible component of $\cup_{l\in P_L}{\cal D}_{l,n}$. We may assume that the ${\cal O}_{\tilde{\cal M}}$-submodule of $\Omega^1_{\tilde{\cal M}}(\log(\tilde{\cal D}))$ generated by ${\mbox{\rm dlog}}(h)$ admits a complement $N'$. Then we get the isomorphism $$(N'\otimes{\cal O}_{{\cal G}_{P'}})\oplus(\oplus_{j\in I}{\cal O}_{{\cal G}_{P'}}.{\mbox{\rm dlog}}(x_j))\cong \Omega^1\otimes{\cal O}_{{\cal G}_{P'}}$$(use ${\mbox{\rm dlog}}(\tilde{a})={\mbox{\rm dlog}}(h)+{\mbox{\rm dlog}}({\tilde{a}}/{h})$). If there exists $j'\in P_I$ we get the isomorphism$$(\Omega^1_{\tilde{\cal M}}(\log(\tilde{\cal D}))\otimes{\cal O}_{{\cal G}_{P'}})\oplus(\oplus_{j\in I-\{j'\}}{\cal O}_{{\cal G}_{P'}}.{\mbox{\rm dlog}}(x_j))\cong\Omega^1\otimes{\cal O}_{{\cal G}_{P'}}$$(use ${\mbox{\rm dlog}}(\tilde{a})={\mbox{\rm dlog}}(x_{j'})+{\mbox{\rm dlog}}({\tilde{a}}/{x_{j'}})$). In both cases, dropping the $j_0$-summand in the left hand side we get $N$ as desired. We see that the ${\cal O}_{{\cal G}_{P'}}$-subalgebra $N^{\bullet}$ of $\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P'}}$ generated by $N$ is stable for the differential $d$, and that we have $$\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P'}}=N^{\bullet}\otimes_{W_n}C^{\bullet}$$ as complexes, where $C^{\bullet}$ is the complex $C^0=W_n$, $C^1=W_n.{\mbox{\rm dlog}}(x_{j_0})$ (here ${\mbox{\rm dlog}}(x_{j_0})$ is nothing but a symbol), $C^m=0$ for $m\ne 0,1$, and zero differential. Let ${\cal R}={\mbox{\rm Proj}}(W_n[y_{j_0},x_{j_0}])$. We have a canonical map ${\cal G}_P\to{\cal R}$. Let $D^{\bullet}$ be the ${\cal O}_{\cal R}$-subalgebra of $\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P}}$ generated by ${\mbox{\rm dlog}}(x_{j_0})\in\Gamma({\cal P}_{{\cal M},n}^0,\Omega^1\otimes{\cal O}_{{\cal G}_{P}})$. It is stable for the differential $d$, and we find$$\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P}}=N^{\bullet}\otimes_{W_n}D^{\bullet}$$as complexes, where $N^{\bullet}$ is mapped to $\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P}}$ via the natural map (and section of $w_{P,P'}$)$$\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P'}}\to\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{P}}$$ induced by the structure map ${\cal E}_{n,j_0}\to{\cal M}_n$. This map also induces a map $C^{\bullet}\to D^{\bullet}$, and it is enough to show that the latter induces isomorphisms in cohomology. But $$H^m({\cal P}_{{\cal M},n}^0,D^{\bullet})\cong H^m({\bf P}^1_{W_n},\Omega_{{\bf P}^1_{W_n}}^{\bullet}(\log\{0,\infty\})),$$which is $W_n$ if $0\le m\le 1$ and zero otherwise, because of the degeneration of the Hodge spectral sequence ([@kalo] 4.12) and $\Omega^1_{{\bf P}^1}(\log\{0,\infty\})\cong{\cal O}_{{\bf P}^1}$. So $C^{\bullet}$ and $D^{\bullet}$ have the same cohomology.\ [*Step 5:*]{} We now show that $f_n$ is an isomorphism. Let ${\cal F}_I=\cup_{j\in I}{\cal E}_{n,j}$, let ${\cal F}_L=\cup_{l\in L}{\cal D}_{n,l}={\cal D}_{{\cal V},n}$ and ${\cal F}_{I,L}={\cal F}_I\cap{\cal F}_L$. All the following tensor products are taken over ${\cal O}_{{\cal P}_{{\cal M},n}^0}$. We will show that in$$\Omega^{\bullet}=\Omega^{\bullet}\otimes {\cal O}_{{\cal P}_{{\cal M},n}^0}\stackrel{\alpha}{\longrightarrow}\Omega^{\bullet}\otimes {\cal O}_{{\cal F}_I}\stackrel{\beta}{\longrightarrow}\Omega^{\bullet}\otimes {\cal O}_{{\cal G}_{(I,\emptyset)}}=\Omega^{\bullet}\otimes{\cal O}_{{\cal M}_n}$$both $\alpha$ and $\beta$ induce isomorphisms in cohomology. The exact sequences $$0\longrightarrow {\cal O}_{{\cal P}_{{\cal M},n}^0}\longrightarrow {\cal O}_{{\cal F}_J}\oplus {\cal O}_{{\cal F}_L}\longrightarrow {\cal O}_{{\cal F}_{I,L}}\longrightarrow0$$$$0\longrightarrow{\cal O}_{{\cal F}_{I}}\longrightarrow{\cal O}_{{\cal F}_{I}}\oplus {\cal O}_{{\cal F}_{I,L}} \longrightarrow {\cal O}_{{\cal F}_{I,L}}\longrightarrow0$$ show that, to prove that $\alpha$ induces cohomology isomorphisms, it is enough to prove that $\Omega^{\bullet}\otimes{\cal O}_{{\cal F}_L}\to\Omega^{\bullet}\otimes {\cal O}_{{\cal F}_{I,L}}$ induces cohomology isomorphisms. To see this, it is enough to show that both $\Omega^{\bullet}\otimes {\cal O}_{{\cal F}_L}\stackrel{\gamma}{\to}\Omega^{\bullet}\otimes {\cal O}_{{\cal F}_L\cap {\cal G}_{(I,\emptyset)}}$ and $\Omega^{\bullet}\otimes {\cal O}_{{\cal F}_{I,L}}\stackrel{\delta}{\to}\Omega^{\bullet}\otimes {\cal O}_{{\cal F}_L\cap {\cal G}_{(I,\emptyset)}}$ induce cohomology isomorphisms. Consider the exact sequence$$\begin{gathered} 0\longrightarrow {\cal O}_{{\cal F}_L}\longrightarrow\oplus_{l\in L}{\cal O}_{{\cal G}_{(\emptyset,\{l\})}}\longrightarrow\oplus_{\stackrel{L'\subset L}{|L'|=2}}{\cal O}_{{\cal G}_{(\emptyset,L')}}\longrightarrow\ldots\longrightarrow{\cal O}_{{\cal G}_{(\emptyset,L)}} \longrightarrow 0\tag{$*$}\end{gathered}$$Comparison of the exact sequences $(*)\otimes\Omega^{\bullet}$ and $(*)\otimes {\cal O}_{{\cal F}_L\cap {\cal G}_{(I,\emptyset)}}\otimes\Omega^{\bullet}$ shows that to prove that $\gamma$ induces cohomology isomorphisms, it is enough to show this for $\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{(\emptyset,L')}}\to\Omega^{\bullet}\otimes{\cal O}_{{\cal G}_{(I,L')}}$ for all $\emptyset\ne L'\subset L$; but this has been done in Step 2. Comparison of $(*)\otimes {\cal O}_{{\cal F}_{I,L}}\otimes\Omega^{\bullet}$ and $(*)\otimes {\cal O}_{{\cal F}_L\cap {\cal G}_{(I,\emptyset)}}\otimes\Omega^{\bullet}$ shows that to prove that $\delta$ induces cohomology isomorphisms, it is enough to show this for $\Omega^{\bullet}\otimes{\cal O}_{{\cal F}_I\cap{\cal G}_{(\emptyset,L')}}\stackrel{\epsilon_G}{\to}\Omega^{\bullet}\otimes {\cal O}_{{\cal G}_{(I,L')}}$ for all $\emptyset\ne L'\subset L$. Consider the exact sequence $$\begin{gathered} 0\longrightarrow {\cal O}_{{\cal F}_I}\longrightarrow\oplus_{j\in I}{\cal O}_{{\cal G}_{(\{j\},\emptyset)}}\longrightarrow\oplus_{\stackrel{I'\subset I}{|I'|=2}}{\cal O}_{{\cal G}_{(I',\emptyset)}}\longrightarrow\ldots\longrightarrow{\cal O}_{{\cal G}_{(I,\emptyset)}} \longrightarrow 0\tag{$**$}\end{gathered}$$ The exact sequence $(**)\otimes {\cal O}_{{\cal F}_I\cap{\cal G}_{(\emptyset,L')}}\otimes\Omega^{\bullet}$ shows that to prove that $\epsilon_G$ induces cohomology isomorphisms, it is enough to show this for $\Omega^{\bullet}\otimes {\cal O}_{{\cal G}_{(I',L')}}\to\Omega^{\bullet}\otimes {\cal O}_{{\cal G}_{(I,L')}}$ for all $\emptyset\ne I'\subset {I}$; but this has been done in Step 2. The exact sequence $(**)\otimes\Omega^{\bullet}$ shows that to prove that $\beta$ induces cohomology isomorphisms, it is enough to show this for $\Omega^{\bullet}\otimes {\cal O}_{{\cal G}_{(I',\emptyset)}}\to\Omega^{\bullet}\otimes {\cal O}_{{\cal G}_{(I,\emptyset)}}$ for all $\emptyset\ne I'\subset {I}$; but this has been done in Step 2. The proof that $f_n$ is an isomorphism is complete. The proof that $g$ is an isomorphism is essentially the same: While Step 4 above boiled down to $H^m({\bf P}^1_{W_n},\Omega_{{\bf P}^1_{W_n}}^{\bullet}(\log\{0,\infty\}))=W_n$ if $0\le m\le 1$, and $=0$ for other $m$, one now uses $H^m({\bf D}^0_{K_0},\Omega_{{\bf D}^0_{K_0}}^{\bullet}(\log\{0\}))=K_0$ if $0\le m\le 1$, and $=0$ for other $m$ (here ${\bf D}^0_{K_0}$ is the open unit disk over $K_0$). The formal reasoning from Step 5 is then the same. The theorem is proven.\ Also unions $H$ of irreducible components of $Y$ are not log smooth over $T$ (unless $H=Y$) and their usual log crystalline cohomology is not useful. However, if $H^{\heartsuit}$ denotes the complement in $H$ of the intersection of $H$ with the closure of $Y-H$ in $Y$, then $(H,H^{\heartsuit})$ is a semistable $T$-log scheme with boundary. There is natural map$$h:R\Gamma_{\rm conv}(H/T)\longrightarrow R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}((H,H^{\heartsuit})/T_n)\otimes_{W}K_0,$$constructed as follows. We say a $T_{\infty}$-log scheme is strictly semistable if all its irreducible components are smooth $W$-schemes and if étale locally it is the central fibre of a morphism ${\mbox{\rm Spec}}(W[t_1,\ldots,t_{n}])\to {\mbox{\rm Spec}}(W[t]),\quad t\mapsto t_1\cdots t_{m}$ (some $n\ge m\ge1$), with the log structures defined by the vanishing locus of $t$ resp. of $t_1\cdots t_{m}$. We find an étale cover $Y=\{Y_i\}_{i\in I}$ of $Y$ and for each $i\in I$ a semistable $T_{\infty}$-log scheme ${\mathcal Y}_i$ together with an isomorphism ${\mathcal Y}_i\otimes_Wk\cong Y_i$. Taking suitable blowing ups in the products of these ${\mathcal Y}_i$ (a standard procedure, compare for example [@mokr]) we get an embedding system for $Y$ over $T_{\infty}$ where a typical local piece $Y_J=\prod_Y(Y_i)_{i\in J}$ of $Y$ is exactly embedded as $Y_J\to {\mathcal Y}_J$ with ${\mathcal Y}_J$ a semistable $T_{\infty}$-log scheme and such there is a closed subscheme ${\mathcal H}_J$ of ${\mathcal Y}_J$, the union of some of its irreducible components, such that $H\times_Y Y_J={\mathcal H}_J\times_{{\mathcal Y}_J}Y$. Now ${\mathcal Y}_J$ is log smooth over $T_{\infty}$, hence its $p$-adic completion $\widehat{\mathcal Y}_J$ may be used to compute $R\Gamma_{\rm conv}(H\times_Y Y_J/T)$. On the other hand, let ${\mathcal H}_J^{\heartsuit}\subset {\mathcal H}_J$ be the open subscheme which is the complemet in ${\mathcal Y}_J$ of all irreducible components of ${\mathcal Y}_J$ which are not fully contained in ${\mathcal H}_J$. Then $({\mathcal H}_J,{\mathcal H}_J^{\heartsuit})$ is a smooth $T_{\infty}$-log scheme with boundary, hence its $p$-adic completion may be used to compute $R\lim_{\stackrel{\leftarrow}{n}}R\Gamma_{\rm crys}((H\times_Y Y_J,H^{\heartsuit}\times_Y Y_J)/T_n)\otimes_{W}K_0$. By the proof of [@berfi] Proposition 1.9 there is a natural map from the structure sheaf of the tube $]H\times_Y Y_J[_{\widehat{\mathcal Y}_J}$ to the structure sheaf of the $p$-adically completed DP envelope, tensored with $\mathbb{Q}$, of $H\times_Y Y_J$ in ${\mathcal H}_J$. It induces a map between our de Rham complexes in question, hence we get $h$. By the same local argument which showed the isomorphy of the map $g$ in the proof of Theorem \[crisconv\] we see that $h$ is an isomorphism; the work on local lifts of $P_M^0$ there is replaced by work on local lifts of $Y$ here. In particular, if $H$ is proper, each $R^j\Gamma_{\rm conv}(H/T_{\infty})$ is finite dimensional.\ Suppose $k$ is perfect. Then there is a canonical Frobenius endomorphism on the log scheme $T_n$ (cf. [@hyoka] 3.1): The canonical lift of the $p$-power map on $k$ to an endomorphism of $W_n$, together with the endomorphism of the log structure which on the standard chart $\mathbb{N}$ is multiplication with $p$. We can also define a Frobenius endomorphism on $R\Gamma_{\rm crys}((\overline{Y},Y)/T_n)$ for a semistable $T$-log scheme with boundary $(\overline{Y},Y)$, because we can define a Frobenius endomorphism on the embedding system used in ., compatible with that on $T_n$. Namely, on a standard $T_n$-log scheme with boundary $(\overline{X}_n,X_n)$ as occurs in the proof of Lemma \[loclif\] we act on the underlying scheme by the Frobenius on $W_n$ and by $t_i\mapsto t_i^p$ (all $i$), and on the log structure we act by the unique compatible map which on our standard chart $\mathbb{N}^{i_2}$ is multiplication with $p$. Then we lift these endomorphisms further (using the lifting property of classical smoothness) to Frobenius lifts of our $\overline{Y}$-covering and hence to the embedding system.\ We finish with perspectives on possible further developments.\ (1) Mokrane [@maghreb] defines the crystalline cohomology of a classically smooth $k$-scheme $U$ as the log crystalline cohomology with poles in $D$ of a smooth compactification $X$ of $U$ with $D=X-U$ a normal crossing divisor. This is a cohomology theory with the usual good properties (finitely generated, Poincaré duality, mixed if $k$ is finite). He shows that under assumptions on resolutions of singularities, this cohomology theory indeed only depends on $U$. We suggest a similar approach to define the crystalline cohomology of a semistable $k$-log scheme $U$: Compactify it (if possible) into a proper semistable $T$-log scheme with boundary $(X,U)$ and take the crystalline cohomology of $(X,U)$. Similarly, classical rigid cohomology as defined by Berthelot [@berco] works with compactifications. Also here, to define log versions it might be useful to work with log schemes with boundary to avoid hypotheses on existence of compactifications by genuine log morphisms.\ (2) We restricted our treatment of crystalline cohomology to that of semistable $T$-log schemes with boundary $(\overline{Y},Y)$ relative to $T_n$. For deformations of $T=T_1$ other then $T_n$ — for example, $({\mbox{\rm Spec}}(W_n), 1\mapsto p)$ — we have at present no suitable analogs of Lemma \[loclif\]. However, such analogs also seem to lack in idealized log geometry: for an ideally log smooth $T$-log scheme (like the union of some irreducible components of a semistable $k$-log scheme in the usual sense), there seems to be in general no lift to a flat and ideally log smooth $({\mbox{\rm Spec}}(W_n), 1\mapsto p)$-log scheme. Some more foundational concepts need to be found. Let us nevertheless propose some tentative definitions of crystalline cohomology for more general fine log schemes $T$ and more general $T$-log schemes with boundary (without claiming any results). Suppose that $p$ is nilpotent in ${\cal O}_W$ and let $(I,\delta)$ be a quasicoherent DP ideal in ${\cal O}_W$. All DP structures on ideals in ${\cal O}_W$-algebras are required to be compatible with $\delta$. Let $T_0$ be a closed subscheme of $T$ and let $\gamma$ be a DP structure on the ideal of $T_0$ in $T$. Let $(\overline{X},{X})$ be a ${T}$-log scheme with boundary, and let $\overline{X}_0$ be the closure in $\overline{X}$ of its locally closed subscheme ${X}\times_{T}{T}_0$. We say $\gamma$ extends to $(\overline{X},{X})$ if there is a DP structure $\alpha$ on the ideal of $\overline{X}_0$ in $\overline{X}$, such that the structure map ${X}\to{T}$ is a DP morphism (if $\alpha$ exists, it is unique, because ${\cal O}_{\overline{X}}\to i_*{\cal O}_{X}$ is injective). Then we say $(\overline{X},{X})$ is a $\gamma$-${T}$-log scheme with boundary. For a $\gamma$-${T}$-log scheme $(\overline{X},{X})$ we can define the crystalline site and the crystalline cohomology of $(\overline{X},{X})$ over $T$ as in the case of usual log schemes. [*Example:*]{} Let $\underline{T}_0\subset \underline{T}$ be a closed immersion. Suppose ${T}$ is the DP envelope of $\underline{T}_0$ in $\underline{T}$ and ${T}_0\subset {T}$ is the closed subscheme defined by its DP ideal; we have $T_0=\underline{T}_0$ if $\delta$ extends to $\underline{T}$. Now if $(\underline{\overline{X}},\underline{X})$ is a $\underline{T}$-log scheme with boundary, we obtain a $\gamma$-${T}$-log scheme with boundary $(\overline{X},X)$ by taking as $\overline{X}$ the DP envelope of the schematic closure of the subscheme $\underline{X}\times_{\underline{T}}\underline{T}_0$ of $\overline{\underline{X}}$.\ [abcdefgh]{} , Cohomologie rigide et cohomologie rigide à supports propres, Première partie, Prépublication IRMAR 96-03, Université de Rennes, 1996. , Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. [**128**]{} (1997), 329-377 , Eléments de géométrie algébrique, Publ. Math. Inst. Hautes Etudes Sci. [**4**]{}, [**8**]{}, [**11**]{}, [**17**]{}, [**20**]{}, [**24**]{}, [**28**]{}, [**32**]{} (1960-67). , Frobenius and monodromy operators in rigid analysis, and Drinfel’d’s symmetric space, preprint. , Semi-stable reduction and crystalline cohomology with logarithmic poles, Asterisque 223 (1994), 221–268. , Log smooth deformation theory, Tohoku Math. J. [**48**]{} (1996), 317–354. , Logarithmic structures of Fontaine-Illusie, Algebraic Analysis, Geometry and Number Theory (Baltimore, MD, 1989), 191–224. , La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Math. J. [**72**]{} (1993), 301–337. , Cohomologie cristalline des variétés ouvertes, Maghreb Math. Rev. [**2**]{} (1993), 161–175. , Logarithmic de Rham cohomology, preprint. , $F$-crystals on schemes with constant log structure. Special issue in honour of Frans Oort, Compositio Math. [**97**]{} (1995), 187–225. , Crystalline fundamental groups. II. Log convergent and rigid cohomology, J. Math. Sci. Univ. Tokyo [**9**]{} (2002), 1–163. , Limits of Hodge structures, Invent. Math. [**31**]{} (1976), 229–257. <span style="font-variant:small-caps;">Mathematisches Institut der Universität Münster\ Einsteinstrasse 62, 48149 Münster, Germany</span>\ *E-mail address*: [email protected]

--- abstract: 'Motivated by advances in observational searches for sub-parsec supermassive black hole binaries (SBHBs) made in the past few years we develop a semi-analytic model to describe spectral emission line signatures of these systems. The goal of this study is to aid the interpretation of spectroscopic searches for binaries and help test one of the leading models of binary accretion flows in the literature: SBHB in a circumbinary disk. In this work we present the methodology and a comparison of the preliminary model with the data. We model SBHB accretion flows as a set of three accretion disks: two mini-disks that are gravitationally bound to the individual black holes and a circumbinary disk. Given a physically motivated parameter space occupied by sub-parsec SBHBs, we calculate a synthetic database of nearly 15 million broad optical emission line profiles and explore the dependence of the profile shapes on characteristic properties of SBHBs. We find that the modeled profiles show distinct statistical properties as a function of the semi-major axis, mass ratio, eccentricity of the binary, and the degree of alignment of the triple disk system. This suggests that the broad emission line profiles from SBHB systems can in principle be used to infer the distribution of these parameters and as such merit further investigation. Calculated profiles are more morphologically heterogeneous than the broad emission lines in observed SBHB candidates and we discuss improved treatment of radiative transfer effects which will allow direct statistical comparison of the two groups.' author: - Khai Nguyen and Tamara Bogdanović bibliography: - 'apj-jour.bib' - 'smbh.bib' title: 'Emission Signatures from Sub-parsec Binary Supermassive Black Holes I: Diagnostic Power of Broad Emission Lines' --- \#1[\#1]{} \#1[[\#1]{}]{} \#1[\#1]{} \#1[[\#1]{}]{} \#1[\#1]{} \#1[[\#1]{}]{} Introduction {#sec:intro} ============ The past ten years have marked a period of active research on supermassive black hole (SBH) pairs and binaries spearheaded by theoretical studies which have investigated how black holes grow, form pairs and interact with their environment. Interest in them has been driven by a realization that SBHs play an important role in evolution of their host galaxies [@ferrarese00; @gebhardt00; @tremaine02] and also, by intention to understand the parent population of merging binaries which are the prime targets for the long anticipated space-based gravitational wave (GW) observatories. We refer to dual SBHs at large separations as [*pairs*]{} and to the gravitationally bound SBHs as [*binaries*]{}, hereafter. Theoretical studies have established that evolution of SBH pairs from kiloparsec to smaller scales is determined by gravitational interactions of individual black holes with their environment [@bbr80; @mayer13]. These include interaction of the SBHs with their own wakes of stars and gas [a.k.a., dynamical friction; @chandra43; @ostriker99; @mm01; @escala04] and scattering of the SBHs by massive gas clouds and spiral arms produced by local and global dynamical instabilities during the merger [@fiacconi13; @roskar14]. During these interactions the SBHs exchange orbital energy and angular momentum with the ambient medium and can in principle grow though accretion [@escala04; @escala05; @dotti06; @dotti07; @dotti09; @callegari11; @khan12; @chapon13]. These factors determine the SBH dynamics and whether they evolve to smaller separations to form a gravitationally bound binary. For example, @callegari09 [@callegari11] find that SBH pairs with mass ratios $q<0.1$ are unlikely to form binaries within a Hubble time at any redshift. On the other hand SBH pairs with initially unequal masses can evolve to be more equal-mass, through preferential accretion onto a smaller SBH. It is therefore likely that SBH pairs with $q\gtrsim0.1$ form a parent population of bound binaries at smaller separations. Gravitationally bound binary forms at the point when the amount of gas and stars enclosed within its orbit becomes comparable to the total mass of the two black holes. For a wide range of host properties and SBH masses this happens at orbital separations $\lesssim 10$ pc [@mayer07; @dotti07; @khan12]. The subsequent rate of binary orbital evolution depends on the nature of gravitational interactions that it experiences and is still an area of active research often abbreviated as [*the last parsec problem*]{}. The name refers to a possible slow-down in the orbital evolution of the parsec-scale supermassive black hole binaries (SBHBs) caused by inefficient interactions with stars [@mm01] and gas [@escala05]. If present, a consequence of this effect would be that a significant fraction of SBHBs in the universe should reside at orbital separations of $\sim 1$pc. Several recent theoretical studies that focus on the evolution of binaries in predominantly stellar backgrounds however report that evolution of binaries to much smaller scales continues unhindered [@berczik06; @preto11; @khan11; @khan12a; @khan13], although the agreement about the leading physical mechanism responsible for the evolution is still not universal [@vasiliev14]. SBH binaries in predominantly gaseous environments have also been a topic of a number of theoretical studies [@an05; @macfadyen08; @bogdanovic08; @bogdanovic11; @cuadra09; @haiman09; @hayasaki09; @roedig12; @shi12; @noble12; @kocsis12b; @kocsis12a; @dorazio13; @farris14; @rafikov16]. They find that binary torques can truncate sufficiently cold circumbinary disks and create an inner low density cavity by evacuating the gas from the central portion of the disk [see @lp79 and references above]. SBHs in this phase can accrete by capturing gas from the inner rim of the circumbinary disk and can in this way maintain mini-disks bound to individual holes. As the binary orbit decays, the inner rim of the circumbinary disk follows it inward until the timescale for orbital decay by gravitational radiation becomes shorter than the viscous timescale[^1] of the disk [@an05]. At that point, the rapid loss of orbital energy and angular momentum through gravitational radiation cause the binary to detach from the circumbinary disk and to accelerate towards coalescence. Through its dependence on the viscous time scale, orbital evolution of a gravitationally bound SBHB in the circumbinary disk depends on thermodynamic properties of the disk. These are uncertain, as they are still prohibitively computationally expensive to model from first principles and are unconstrained by observations. More specifically, the thermodynamics of the disk is determined by the binary dynamics and also the presence of magnetic field and radiative heating and cooling of the gas. While the role of magnetic field in circumbinary disks has been explored in some simulations [@giaco12; @shi12; @noble12; @farris12; @gold14], a fully consistent calculation of radiative processes is still beyond computational reach. Consequently, current theoretical models can be formulated as parameter studies, where difficult-to-model processes are parametrized in some fashion, but cannot uniquely predict the properties of the circumbinary regions or the emission signatures of SBHBs. The circumbinary disk model is therefore an appealing theoretical concept that must be tested through observations. Along similar lines, observations of the orbital properties of SBHBs are key to understanding binary evolution. This is because the frequency of binaries as a function of their orbital separation is directly related to the rate at which binaries evolve towards coalescence. Theoretical models predict that the exchange of angular momentum with the ambient medium is likely to result in SBHB orbits with eccentricities $\gtrsim 0.1$, with the exact value depending on whether gravitationally bound SBHs evolve in mostly stellar or gas rich environments [@roedig11; @sesana11; @khb15]. Known semi-major axis and eccentricity distributions would therefore provide a direct test for a large body of theoretical models. Our understanding of spin magnitudes and orientations in binary SBHs also relies on theoretical considerations. Interest in this topic was triggered by the prediction of numerical relativity that coalescence of SBHs with certain spin configurations can lead to the ejection of a newly formed SBH from its host galaxy. This effect arises due to the asymmetry in emission of GWs in the final stages of an SBH merger and can lead to a GW kick of up to $\sim5000\,{\rm km\,s^{-1}}$ [@campanelli07; @lousto11]. Several subsequent theoretical studies found that accretion and gravitational torques can act to align the spin axes of SBHs evolving in gas rich environments and in such way minimize the GW recoil [@bogdanovic07; @dotti10; @dotti13; @sorathia13; @miller13][^2]. Mutual SBH spin alignment is not expected in gas poor environments, geometrically thick, turbulent and magnetically dominated disks [@fragile05; @fragile07; @mckinney13], allowing a possibility that runaway SBHs and empty nest galaxies may exist. Hence, if observations can independently provide an insight into the geometry of circumbinary disks and spin properties of SBHBs, they would be an important probe of the alignment hypothesis. In this work, which constitutes part I of a series, we develop a preliminary semi-analytic model of an SBHB in circumbinary disk and use it to calculate a database of broad, optical emission line profiles associated with binary systems. Our analysis indicates that such profiles can in principle be used to infer statistical distribution of SBHB parameters, making modeling of binary spectroscopic signatures a worthwhile task. The comparison of the preliminary model with the data from observed SBHB candidates however indicates that further improvements to the model are necessary before the synthetic profiles can be used to interpret the observations. These steps will be presented in subsequent papers. This paper is organized as follows: Section \[sec:observations\] describes the status of ongoing spectroscopic searches for SBHBs, Section \[sec:calculation\] and the Appendix outline the semi-analytic model used in calculation of SBHB emission line signatures, and Section \[sec:results\] gives the description of results. We discuss the validity of assumptions and implications of our results in Section \[sec:discussion\] and conclude in Section \[sec:conclusions\]. Status of Spectroscopic Searches for SBHBs {#sec:observations} ========================================== Key characteristic of gravitationally bound SBHBs is that they are observationally elusive and expected to be intrinsically rare. Theorists estimate that a fraction $<10^{-3}$ of active galactic nuclei (AGNs) at redshift $z<0.7$ may host SBHBs [@volonteri09]. This result implies that any observational search for SBHBs must involve a large sample of AGNs and that observational technique used in the search needs to distinguish signatures of binaries from those of AGNs powered by single SBHs. Spectroscopic searches rely on the detection of the Doppler-shift in the emission line spectrum of an SBHB candidate that arise as a consequence of the binary orbital motion. This approach is reminiscent of a well established technique for detection of the single- and double-line spectroscopic binary stars. In both classes of spectroscopic binaries, the lines are expected to oscillate about their local rest frame wavelength on the orbital time scale of a system. In the context of the binary model, the spectral emission lines are assumed to be associated with the gas accretion disks that are gravitationally bound to the individual SBHs [@gaskell83; @gaskell96; @bogdanovic08]. Given the velocity of the bound gas the emission line profiles from the SBH mini-disks are expected to be Doppler-broadened, similar to the emission lines originating in the broad line regions (BLRs) of AGNs. Moreover, several theoretical studies have shown that in unequal mass binaries accretion occurs preferentially onto the lower mass object [@al96; @gr00; @hayasaki07], rendering it potentially more luminous than the primary. If so, this indicates that some fraction of SBHBs may appear as the single-line spectroscopic binaries. This realization lead to a discovery of a number of SBHB candidates based on the criterion that the culprit sources exhibit broad optical lines offset with respect to the rest frame of the host galaxy [@bogdanovic09a; @dotti09a; @bl09; @tang09; @decarli10; @barrows11; @tsal11; @tsai13][^3]. Because this effect is also expected to arise in the case of a recoiling SBH receding from its host galaxy, the same approach has been used to flag candidates of that type [@komossa08a; @shields09; @civano10; @robinson10; @lusso14]. The key advantage of the method is its simplicity, as the spectra that exhibit emission lines shifted relative to the galaxy rest frame are relatively straightforward to select from large archival data sets, such as the Sloan Digital Sky Survey (SDSS). Its main complication however is that the Doppler shift signature is not unique to these two physical scenarios and complementary observations are needed in order to determine the nature of the observed candidates [e.g., @popovic12; @bogdanovic14]. To address this ambiguity a new generation of spectroscopic searches has been designed to monitor the offset of the broad emission line profiles over multiple epochs and target sources in which modulations in the offset are consistent with the binary orbital motion [@eracleous12; @bon12; @decarli13; @shen13; @ju13; @liu13; @li16]. For example, @eracleous12 searched for $z<0.7$ SDSS quasars whose broad $H\beta$ lines are offset by $\gtrsim 1000\;{\rm km\,s^{-1}}$ and selected 88 quasars for observational followup from the initial group of $\sim 16,000$ objects. After the second and third epoch of observations of this sample, statistically significant changes in the velocity offset were found in 14 [@eracleous12] and 9 objects [@mathes14], respectively, in broad agreement with theoretical predictions for frequency of SBHBs [@volonteri09]. Description of the Model {#sec:calculation} ======================== Emission line profiles from SBHB in circumbinary disk {#sec:cbd} ----------------------------------------------------- Motivated by theoretical models described in the literature and ongoing observations we consider the sub-parsec binaries with mass ratios in the range $0.1\leq q \leq 1$, where $q=M_2/M_1$. The orbital separation and period of such binaries can be expressed in terms of the spectroscopically determined velocity offset, which is their key observable property. If the measured velocity offset can be attributed to the motion of the secondary SBH, as indicated by the accretion rate inversion found in theoretical studies of SBHBs (see Section \[sec:observations\]), then the projected velocity of the secondary, $u_2$, is related to its true orbital speed, $v_{\rm orb2}$, as $u_2$ = $v_{\rm orb2}\,\sin{i}\,|\sin{\phi}|$. Here $i$ is the inclination of the orbital axis of the binary relative to observer’s the line of sight ($i=0$ is face-on) and $\phi$ is the orbital phase at the time of the observation ($\phi = 0$ corresponds to conjunction). Note that the expression for $u_2$ applies to circular orbits, an assumption which we use to obtain illustrative estimates but in our model calculations actually consider both circular and eccentric orbits. Following @eracleous12 we express the period and orbital separation in terms of the total mass $M_8 = (M_1 + M_2)/10^8\,M_\odot$ and the projected velocity of the secondary, $u_{2,3} = u_2/10^3\,{\rm km\,s^{-1}}$ as $$\begin{aligned} a=\frac{0.11\,M_8}{(1+q)^2\,u_{2,3}^2} \left(\frac{\sin{i}}{\sin{45^\circ}} \frac{|\sin{\phi}|}{\sin{45^\circ}}\right)^2\,{\rm pc}\\ P=\frac{332\,M_8}{(1+q)^3\,u_{2,3}^3} \left(\frac{\sin{i}}{\sin{45^\circ}} \frac{|\sin{\phi}|}{\sin{45^\circ}}\right)^3\,{\rm yr}\end{aligned}$$ If the measured velocity offset is instead associated with the primary SBH, the above expressions can be written in terms of the projected velocity of the primary, $u_{1,3} = q\,u_{2,3}$ where $u_{1,3} = u_1/10^3\,{\rm km\,s^{-1}}$. In the expressions above we choose $i=\phi=45^{\circ}$ and discuss the parameter values used in our model calculations in Section \[sec:params\]. The accretion flow is described as a set of three circular accretion disks: two mini-disks that are gravitationally bound to the individual SBHs and a circumbinary disk. The three disks are modeled as independent BLRs, where the extent of the two mini-disks, as well as the central opening in the circumbinary disk are constrained by the size of the binary orbit and are subject to tidal truncation by the binary SBH [@paczynski77; @lp79; @sepinsky07]. In this model both accreting SBHs can shine as AGNs and illuminate their own mini-disk as well as the two other disks in the system. We assume that the bolometric luminosity of each AGN correlates with the accretion rate onto its SBH and that photoionization by the AGNs gives rise to the broad, low-ionization optical emission lines just like in “ordinary" BLRs [@csd89; @csd90]. The emissivity of each disk can then be evaluated as a function of the accretion rate onto the SBHs and the disk size. We utilize the published measurements of accretion rates from simulations of SBHBs [@al96; @gr00; @hayasaki07; @roedig11; @farris14] in order to establish the relative bolometric luminosities of the two AGN in a binary and emissivity of each disk component. Any assumptions about the mutual orientation of the two mini-disks and circumbinary disk are relaxed and they are allowed to assume arbitrary orientations relative to the observer. [cl]{} \[table:parameters\] $q$ & 1 , 9/11 , 2/3 , 3/7 , 1/3 , 1/10\ $a/M$ & $5\times 10^3$ , $10^4$ , $5 \times 10^4$, $10^5$, $10^6$\ $e$ & 0.0 , 0.5\ $f$ & $0^{\circ}$, $72^{\circ}$, $144^{\circ}$, $216^{\circ}$, $288^{\circ}$\ $R_{\rm in1}/M_1$, $R_{\rm in2}/M_2$ & 500 , 1000\ $R_{\rm out3}$ & $3a$\ $i$ & $5^{\circ}$, $55^{\circ}$, $105^{\circ}$, $155^{\circ}$\ $\phi$ & $0^{\circ}$, $36^{\circ}$, $108^{\circ}$, $180^{\circ}$, $242^{\circ}$, $324^{\circ}$\ $\theta_1$, $\theta_2$ & $0^{\circ}$, $30^{\circ}$, $60^{\circ}, 105^{\circ}$, $135^{\circ}$, $165^{\circ} $\ $\phi_1$, $\phi_2$ & $0^{\circ}$, $25^{\circ}$, $60^{\circ}$, $185^{\circ}$, $210^{\circ}$, $235^{\circ}$\ $h_1/M_1$, $h_2/M_2$ & 10\ $\sigma/{\rm km\,s^{-1}}$ & 850 \[table:parameters\] We follow the line profile calculations described in @chen89, @chen89b and @eracleous95 to obtain an emission line profile from each Keplerian, circular, relativistic thin disk in the weak-field approximation. Such line profiles are characteristic of rotating disks and resemble the persistent, double-peaked Balmer lines found in about 10 – 20% of broad-line radio galaxies and about 3% of all active galaxies [@eh94; @eh03; @strateva03]. More generally, several works have demonstrated that disk models of this type can be used to describe emission from BLRs of most AGNs when additional radiative transfer effects of the disk atmosphere on the emission line profiles are accounted for [@cm96; @mc97; @flohic12; @chajet13]. We adopt this approach because of its broad applicability, considering the baseline model first and addressing the mentioned radiative transfer effects in Paper II of the series (see also Section \[secsec:approx\] for more discussion). The emission line fluxes from the three disks are evaluated and summed following the steps presented in the Appendices \[sec:appendixsingle\] and \[appendixbinary\]. The main objective of this calculation is to obtain the final expression for the flux density in the observer’s frame as a function of parameters defined in the reference frame of the binary. Using this approach we calculate a database of profiles by drawing from a parameter space that describes different configurations of SBHBs and their associated circumbinary regions. For a somewhat different approach see also the work by @simic16 who model the SBHB accretion flow as either one or two BLRs, each of which contributes a Gaussian emission line profile. In this work we focus on the H$\beta$ emission line profiles, the second line of the hydrogen Balmer series, but note that this calculation is applicable to all permitted, low-ionization broad emission line profiles. The broad emission lines of particular interest are H$\alpha~\lambda6563$, H$\beta\;\lambda4861$, and Mg II $\;\lambda2798$ because they are prominent and relatively easy to identify in AGN spectra. These low-ionization lines have been established as reliable tracers of dense gas in BLRs and are presently used in spectroscopic searches for SBHBs at low (H$\alpha$, z $<$ 0.4) and high redshift (MgII, z $<$ 2.5). ![Illustration of SBHB geometry for face-on orientation of the orbit. SBHB center of mass marks the origin of the coordinate system which $z$-axis points in the direction of the orbital angular momentum of the binary (out of the page). The $x$-axis points towards the pericenter of the primary SBH orbit and is parallel to the orbital semi-major axis of the binary, $a=(a_1+a_2)$. The orbital phase $f$ is measured counter-clockwise from the $x$-axis to the instantaneous location of secondary SBH. The mini-disks and circumbinary disk are not shown. See text for definition of other variables.[]{data-label="fig:binary"}](f1.eps){width=".4\textwidth"} Parameters of the model {#sec:params} ----------------------- Table \[table:parameters\] summarizes parameter choices for the modeled configurations of SBHB systems. The parameters encode the intrinsic properties of the binary, such as the orbital semi-major axis, eccentricity, the alignment of the triple disk system, as well as the orientation of the SBHB with respect to the distant observer’s line of sight. The sample includes 2,545,200 realizations of binaries on circular orbits and 12,273,000 on elliptical orbits, for a total of 14,818,200 configurations. We describe our parameter choices below and present the details of profile calculation in the Appendices. - *SBHB mass ratio, $q$* – Simulations of galaxy mergers that follow pairing of their massive black holes find that SBH pairs with mass ratios $q<0.1$ are unlikely to form gravitationally bound binaries within a Hubble time at any redshift [@callegari09; @callegari11]. They also find that SBH pairs with initially unequal masses can evolve to be more equal-mass, through preferential accretion onto a smaller SBH. Motivated by these results we choose six values of $q$ in the range $0.1-1$ to represent the mass ratio of the binary. - *Semi-major axis, $a$* – To describe orbital separations of gravitationally bound binaries we chose five values of $a$ ranging from $5000\,M$ to $10^6\, M$, where we use the mass of the binary $M\equiv GM/c^2 = 1.48\times10^{13}\,{\rm cm}\, (M/10^8\,M_\odot)$ as a measure of length in geometric units, where $G=c=1$. For example, for the total mass of the binary of $10^8 M_{\odot}$ this range of semi-major axes corresponds to binary separations $\sim 0.02 - 5$pc. - *Orbital eccentricity, $e$* – Theoretical models that follow evolution of the orbital eccentricity of SBHBs in circumbinary disks suggest that the exchange of angular momentum between them drives a steady increase in binary eccentricity which saturates in the range $0.6-0.8$ [@an05; @cuadra09; @roedig11]. For the purposes of this calculation we choose two values of eccentricity, $e=0.0$ and 0.5, to model SBHBs on both circular and elliptical orbits. - *Orbital phase, $f$* – Five values of the orbital phase are chosen to describe orbital evolution of SBHBs. $f$ is measured from the positive $x$-axis to the instantaneous location of the secondary SBH in counter-clockwise direction, as illustrated in Figure \[fig:binary\]. ![Outer radii of the primary and secondary mini-disks as a function of $a$ and $e$. Lines mark SBHB mass ratios $q = 1/10$ (thin, red lines) , 1/3 (dotted, blue) and 1 (solid, black) based on the model of @paczynski77. For unequal mass ratios the top line marks the size of the larger (primary) mini-disk.[]{data-label="fig:Rout"}](f2.eps){width=".42\textwidth"} - *SBHB accretion rate ratio, $\dot{m}$* – In the context of this model we assume that the emissivity of each broad emission line region is a function of the AGN luminosity and the disk size. In order to establish the relative bolometric luminosities of the two AGN we compile from the literature the values of $\dot{m} = \dot{M}_2/\dot{M}_1$ and parametrize it as a linear function of $q$ for SBHBs on circular and eccentric orbits. $$\begin{aligned} \label{eq:mscale} \dot{m}\simeq \begin{cases} 5.5-4.5q & e=0.0 \\ 1.5-0.5q & e=0.5 \end{cases}\end{aligned}$$ Here $\dot{M}_1$ and $\dot{M}_2$ are the accretion rates onto the primary and secondary SBH, respectively. The two relations capture two key results observed in hydrodynamic simulations of prograde SBHBs [rotating in the same sense as the circumbinary disk; @hayasaki07; @roedig11]: (1) in unequal mass binaries accretion occurs preferentially onto the smaller of the SBHs and (2) the inversion of accretion rates is more severe for SBHBs on circular orbits. This trend has also been captured by other simulations [@al96; @farris14] and models motivated by them [@gerosa15; @young15]. - *Size of the broad line regions, $R_{\rm {in}\it{i}}$ and $R_{\rm {out}\it{i}}$* – Each disk in the triple disk system has an associated BLR defined by a pair of inner and outer radii. In the case of the mini-disks we choose two different values for the BLR inner radius $R_{\rm {in}\it{i}}=500M_{i}$ and $1000M_{i}$, where $i=1,2$ mark the BLR around the primary and secondary SBH, respectively. These choices are motivated by characteristic values for the inner radius of the BLRs in AGNs powered by single SBHs, which emission lines are well modeled by the emission from a Keplerian disk [for e.g., @eh94; @eh03]. The outer radii are naturally determined by the tidal torques of the binary and do not extend beyond the Roche lobes of their SBHs. We follow the approach described by @paczynski77 to estimate the average values of $R_{\rm out1}$ and $R_{\rm out2}$ based on the binary separation $a$, and mass ratio $q$ (see Figure \[fig:Rout\]). We define the size of the circumbinary disk BLR in terms of the SBHB semi-major axis, $R_{\rm in3}=2a$ and $R_{\rm out3}=3a$. The value of the inner radius is directly motivated by theory and simulations which show that SBHB torques create a low density hole with radius about $2a$ in the center of the circumbinary disk [@lp79; @an05; @macfadyen08]. The value of the outer radius of the circumbinary disk BLR is poorly constrained and for the purposes of this calculation we adopt $R_{\rm out3}=3a$. Note that the BLR sizes assumed in this work are consistent with the plausible range empirically derived for low redshift AGN by @kaspi05. - *Emissivity of the broad line regions, $\epsilon_{i}$* – Each disk in the system is further characterized by the emissivity of the BLR, which arises due to the illumination by the two AGNs. For example, the emissivity of the mini-disk around the primary SBH can be expressed as $\epsilon_{1} = \epsilon_{11} + \epsilon_{12}$, where $\epsilon_{11}$ and $\epsilon_{12}$ correspond to the components of emissivity due to the illumination by its own AGN and the AGN associated with the secondary SBH, respectively. The emissivity of each mini disk associated with its own AGN is described as a power law in radius, with the power law index $p=3$, such that $\epsilon_{11} \propto \epsilon_{22} \propto R^{-p}$ [@csd89]. The component of emissivity associated with the companion AGN ($\epsilon_{12}$ and $\epsilon_{21}$) is calculated as a function of its distance and orientation of the mini-disk (Appendix \[appendixbinary\]). The emissivity of the circumbinary disk, $\epsilon_{3}$, is calculated as a sum of emissivities due to the two off-center AGN associated with the primary and secondary SBHs. - *Orientation of the observer relative to the binary orbit, $i$ and $\phi$* – We choose four values of the inclination angle, $i$, to describe the orientation of the observer’s line of sight relative to the vector of orbital angular momentum of the SBHB. For example, $i=0^\circ$ represents the clockwise binary seen face-on and values $i>90^\circ$ represent counter-clockwise binaries. Furthermore, we select six values of the azimuthal angle $\phi$ measured in the binary orbital plane, from the positive $x$-axis to the projection of the observer’s line of sight, in counter-clockwise direction. For circular SBHBs varying the true anomaly $f$ is equivalent to varying the azimuthal orientation of the observer and in this case we adopt a single nominal value of $\phi=0^\circ$ in calculation of the emission line profiles. However, in the case of eccentric SBHBs we explore a full range of $f$ and $\phi$ angles. - *Orientation of the mini disks, $\theta_i$ and $\phi_i$* – We relax assumptions about the orientation of the mini disks with respect to the binary orbit in order to study how profile shapes depend on it. We choose six values of the polar angle ($\theta_i$) and azimuthal angle ($\phi_i$) to describe the orientation of each mini disk with respect to the vector of orbital angular momentum of the binary. For example, when $\theta_1=\theta_2=0^{\circ}$, both mini-disks are coplanar with the SBHB orbit and for $\theta_i>90^\circ$, the gas in the mini-disks exhibits retrograde motion relative to the circumbinary disk. The azimuthal angles $\phi_i$ are measured in the binary orbital plane, from the positive $x$-axis to the projection of the rotation axis of the mini-disk, in counter-clockwise direction. The circumbinary disk is assumed to always be coplanar and in co-rotation with the binary orbit. ![Illustration of profile shapes represented in the emission line database. Total flux (black line) is a sum of components contributed by the primary (red), secondary (blue) and circumbinary disk (green). Flux is shown in arbitrary units against wavelength (bottom $x$-axis) and corresponding velocity offset relative to the binary center of mass (top $x$-axis). Pink vertical line at $4860.09 \AA$ marks the rest wavelength of the H$\beta$ emission line.[]{data-label="fig:diversity"}](f3.eps){width="45.00000%"} Note that some of the model parameters described above are actually not free parameters, because they are constrained by the relevant physical processes and can be expressed in terms of the properties of the binary (this is the case with $\dot{m}$, $R_{\rm out1}$, $R_{\rm out2}$ and $R_{\rm in3}$). The calculation of the emission line profiles requires definition of two additional parameters which have a lesser impact on their shape. Specifically, motivated by the X-ray studies of the broad iron line reverberation [see review by @uttley14], we assume that the central source of the continuum radiation associated with each SBH is compact and has spatial extent of $h_i=10M_i$. Similarly, we describe the broadening of the emission line profiles due to the random (turbulent) motion of the gas in each disk as $\sigma=850\,{\rm km\,s^{-1}}$. We discuss implications of our parameter choices in Section \[sec:discussion\]. Results {#sec:results} ======= In this section we draw attention to unique features of the modeled population of profiles (Section \[sec:features\]) and characterize their shapes in terms of commonly used statistical distribution functions (Section \[sec:stats\]). We then investigate whether the complex, composite profile shapes preserve any dependence on the parameters of the underlying SBHB model (Section \[sec:dependence\]). [cccccccc]{} & $5\times10^3 $ & 15.01 & 40.15 & 35.84 & 8.94 & 0.06 & 0.00\ & $10^4 $ & 28.16 & 54.34 & 17.05 & 0.45 & 0.00 & 0.00\ & $5\times10^4 $ & 77.25 & 22.74 & 0.01 & 0.00 & 0.00 & 0.00\ & $10^5 $ & 87.07 & 12.93 & 0.00 & 0.00 & 0.00 & 0.00\ & $10^6 $ & 100.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\ & $5\times10^3$ & 5.85 & 14.21 & 50.05 & 22.82 & 6.51 & 0.57\ & $10^4$ & 9.03 & 42.36 & 44.05 & 4.39 & 0.17 & 0.00\ & $5\times10^4$ & 37.84 & 61.75 & 0.41 & 0.00 & 0.00 & 0.00\ & $10^5 $ & 63.97 & 35.98 & 0.05 & 0.00 & 0.00 & 0.00\ & $10^6 $ & 100.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \[tab:peaks\] {width="75.00000%"} Characteristic features of the modeled emission line profiles {#sec:features} ------------------------------------------------------------- The most striking property of the modeled emission line profiles is that they can have multiple peaks and their appearance can vary significantly over time, due to the orbital motion of the binary and the resulting variable illumination of the three disks by the two AGNs. Each disk in the triple disk system can give rise to either a single-peaked or a double-peaked profile, depending on the size of its emission region and its orientation with respect to the observer’s line of sight. Generally, the larger the BLR, the more likely is the disk to produce a single-peaked profile. This is because the bulk of the emission is contributed by the outer regions on the disk characterized by lower rotational velocities. Similarly, the lower the inclination of the disk with respect to the observer, the more likely it is that the observed profile is single-peaked since the gas velocity along the line of sight is low. Since in our model we account for a range of BLR sizes and inclinations, the composite emission line profiles can display anywhere from 1 to 6 peaks. Figure \[fig:diversity\] illustrates the diversity of shapes encountered in the profile database, calculated for different binary configurations. Each profile includes contribution from the primary and secondary mini-disks and the circumbinary disk. Individual profiles are broadened by rotational motion and random motion of the gas in the disk. Because the gas in the mini-disks has higher rotational velocity and is closer to the sources of continuum radiation, the emission line profiles contributed by the mini-disks often appear broader and stronger relative to the emission from the circumbinary disk. In Table \[tab:peaks\] we show the percentage of profiles characterized by a given number of peaks as a function of the orbital separation and eccentricity of the SBHB. One readily identifiable trend is that majority of profiles tend to have 1–3 peaks. The profiles with 5 and 6 peaks are relatively rare and entirely absent from SBHB systems with large orbital separations. This can be understood because SBHBs on tight orbits are characterized by compact mini-disks with high orbital velocities about the binary center of mass, both of which give rise to broad and multi-peaked lines in the wavelength space. Another trend is that SBHBs on eccentric orbits tend to have profiles with a higher number of peaks relative to the circular binaries with the same semi-major axis. This is because eccentric SBHBs sample a wider range of orbital velocities, allowing for a larger wavelength offset of individual components in the composite profile. As mentioned in Section \[sec:cbd\], we assume that both accreting SBHs can shine as AGNs and illuminate all three disks in the system. In this setup, both mini-disks are illuminated by their central AGN as well as the off-center companion AGN. The illumination of the circumbinary disk by the two AGNs is always off-center. In Figure \[fig:mutual\] we show the effect of illumination of the primary mini-disk by the two AGNs (similar effect is present for the secondary mini-disk). The sequence of profiles at the top and bottom are created for two different SBHB configurations, arbitrarily chosen for illustration. The illumination by the secondary AGN resembles an off-center hot spot on the accretion disk surface. Figure \[fig:mutual\] shows that contribution to the total flux from such a hotpot sensitively depends on the alignment of the primary mini-disk with the binary orbit. Namely, when the two are coplanar ($\theta_1 = 0^{\circ}$), illumination due to the secondary AGN is negligible because of the small incidence angle of its photons on the mini-disk (first panel of Figure \[fig:mutual\]). When the mini-disk and the binary orbit are misaligned even by a small amount, the illumination by the secondary AGN can make a significant contribution to the line flux (second panel of Figure \[fig:mutual\]). In the case of close binaries with highly misaligned mini-disks we find that this effect can increase the line flux up to several times (last panel). Depending on whether the hotspot moves away or toward the observer (as indicated by the azimuthal angle $\phi$) this extra flux may appear in the blue or the red wing of the emission line giving rise to an asymmetric profile (third and fourth panels). Therefore, the effect of illumination by a dual AGN can in principle be an indicator of the orbital alignment of the triple disk system, if it can be identified in the observed emission line profiles of candidate SBHB systems. One more characteristic feature of the emission line profiles contributed by the triple disk system in our model is that the shape of a profile can change significantly over one orbital period of the binary. The centroids of the emission line profiles contributed by the mini-disks oscillate about the rest wavelength due to the orbital motion of the SBHs in the way similar to the spectroscopic stellar binaries. The emission from the circumbinary disk, which is anchored to the binary center of mass, is on the other hand centered on the rest wavelength of the system. As a result, a combination of the SBHB orbital motion and rotation of gas within each disk can produce complex and distinct features in SBHB systems relative to emission lines from stellar binaries. {width="85.00000%"} Figure \[fig:evolution\] shows time evolution of a profile associated with an SBHB system in which profiles from both mini-disks are double peaked and asymmetric due to relativistic Doppler boosting (i.e., exhibit a higher blue shoulder). The same effect is also noticeable in the composite profile for all orbital phases except $f= 216^{\circ}$, when the blue and the red shoulder of the profile become comparable. At $f=216^{\circ}$ the red wings of the two mini-disk profiles line up in wavelength giving rise to a relatively strong red peak in the composite profile. Statistical properties of emission line profiles {#sec:stats} ------------------------------------------------ The unique features of modeled emission line profiles associated with SBHB systems point to an intriguing possibility that, if it is possible to identify them in the observed SBHB candidates, these markers can be used to learn about the properties of the SBHBs. We analyze the trends in the modeled population of profiles by characterizing their shapes in terms of several commonly used distribution functions. These include the location of the centroid (C), asymmetry index (AI), kurtosis index (KI), full width at half and quarter maximum (FWHM and FWQM), peak shift (PS), and centroid shift (CS). We use the following definitions: $$\begin{aligned} F &=&\sum_{i} F_{\lambda, i}\\ C &=&\frac{1}{F}\sum_{i} \lambda_i F_{\lambda, i}\\ \sigma^2 &=&\frac{1}{F}\sum_{i}(\lambda_i-C)^2 F_{\lambda, i}\\ \rm{AI} &=&\frac{1}{F\sigma^3}\sum_{i}(\lambda_i-C)^3 F_{\lambda, i} \label{eq_ai}\\ \rm{AIP} &=&(C-\lambda_m)/\sigma\label{eq_aip}\\ \rm{KI} &=&\frac{1}{F\sigma^4}\sum_{i}(\lambda_i-C)^4 F_{\lambda, i}\label{eq_ci}\\ \rm{FWHM} &=& \left[\lambda_r(1/2)-\lambda_b(1/2)\right]\frac{c}{\lambda_0}\label{eq_fwhm}\\ \rm{FWQM} &=& [\lambda_r(1/4)-\lambda_b(1/4)]\frac{c}{\lambda_0}\label{eq_fwqm}\\ \rm{PS} &=& (\lambda_p-\lambda_0)\frac{c}{\lambda_0}\label{eq_ps}\\ \rm{CS} &=&(C-\lambda_0)\frac{c}{\lambda_0}\label{eq_cs}\end{aligned}$$ where $F_{\lambda, i}$ is the profile flux density at wavelength $\lambda_i$. The profile flux is normalized by the maximum flux measured at the peak wavelength, $\lambda_{p}$, so that $F_{\lambda}(\lambda_p)\equiv \max(F_{\lambda, i})=1$. $\lambda_b(x)$ and $\lambda_r(x)$ indicate the wavelength in the blue wing or the red wing of the profile, respectively, where the normalized flux drops to some level, $x$. $\lambda_0$ is the rest wavelength of the emission line and $\lambda_m$ is the median wavelength that divides profile into a half, so that 50% of the flux lies to the left and to the right of it. The location of the profile centroid, $C$, is calculated as the flux weighted mean wavelength. We use two measures to characterize the asymmetry of the profiles: the asymmetry index (AI) and the Pearson skewness coefficient (AIP). The positive values of AI and AIP indicate profiles skewed toward short wavelengths (i.e., blue-leaning profiles) and the negative values indicate red-leaning profiles. However, AI and AIP calculated for the same profile sometimes have opposite signs, as they provide different measures of the profile asymmetry. Specifically, AI sensitively depends on the low intensity features in the profile wings, while AIP diagnoses the asymmetry in the bulk of the profile. We use the kurtosis index (KI), calculated as the fourth moment of the flux distribution, to evaluate the “boxiness" of the profiles. By definition, the values of KI are always positive. Smaller values correspond to boxier profiles and larger values indicate cuspy profiles, with the top narrower than bottom. In addition, the relevant line widths, peak and centroid shifts are measured in units of velocity, as defined in equations \[eq\_fwhm\] – \[eq\_cs\]. In calculation of all these statistical properties we adopt a cutoff at $F_c=0.01$ to mimic some fiducial level of spectral noise (but do not introduce actual fluctuations due to noise to the profiles). With “noise" subtracted from the profile, we rescale the flux above the cutoff so that the maximum flux measured at the peak wavelength has the value of 1.0. We investigate the dependance of the distribution functions, characterizing the modeled profile shapes, on the value of $F_c$ in the Appendix \[sec:appendixcutoff\]. {width=".85\textwidth"} {width=".85\textwidth"} {width=".75\textwidth"} We use statistical properties defined in equations \[eq\_ai\] – \[eq\_cs\] to construct a multi-dimensional parameter space of the emission line profiles and investigate their distribution as a function of the underlying SBHB parameters. In the remainder of the paper we visualize the multivariate distribution of profiles with 2-dimensional maps, which represent different projections through this parameter space. For example, in Figures \[fig:phys1c\] and \[fig:phys1e\] we plot maps of AIP and PS values for profiles calculated for circular and eccentric binary configurations, respectively. The color marks the number density of profiles and indicates which portions of the parameter space are favored by the modeled profiles. The top left map in Figures \[fig:phys1c\] and \[fig:phys1e\] illustrates that AIP-PS distributions appear similar in the overall shape, with the eccentric sample having a wider range of the peak velocity shifts. This difference can be attributed to a wider range of orbital velocities sampled by eccentric binaries with the same semi-major axes. This topological similarity in the distribution of profiles from circular and eccentric SBHBs is present throughout the parameter space. Given the overlap, we plot only the distribution maps for the eccentric SBHBs in the rest of the paper and discuss any differences between the circular and eccentric samples in the text. Inspection of the top left panels in Figures \[fig:phys1c\] and \[fig:phys1e\] reveals that a significant fraction of profiles are fairly symmetric (${\rm AIP} \approx 0$) and likely to exhibit the maximum peak at wavelengths shorter than the rest wavelength (${\rm PS}<0\,{\rm km\,s^{-1}}$). The latter is a consequence of the relativistic Doppler boosting, which for each individual disk preferentially boosts the blue shoulder of its emission line profile, creating an effect which is also noticeable in the composite profile. Another feature worth noting is that in both the circular and eccentric samples, the profiles that exhibit the strongest peak at shorter wavelengths are also preferentially blue-leaning and vice versa. In the next section, we inspect the remainder of the profile parameter space for similar trends and consider their relationship with the physical properties of the SBHB. In Figure \[fig:stat1e\] we show examples of line profiles from different parts of the parameter space of Figure \[fig:phys1e\], marked in the footprint of the map in the central panel. The shapes include profiles that exhibit symmetry (7 and 8), strong asymmetry (2 and 5), and large velocity offsets of the strongest peak (4 and 7). As discussed in Section \[sec:features\], the offset of the dominant peak towards longer wavelengths (evident in profiles 4 and 5) can occur in our model only under a specific circumstance: as a consequence of the illumination of a mini-disk by the companion AGN, when the hotspot is moving away from the observer. Inspection of profile 4 shows that the mini-disk with a strong hot spot is that around the secondary SBH (traced by the blue line) and around the primary SBH in profile 5 (traced by the red line). Moreover, profile 1 exemplifies the scenario where secondary illumination by the companion AGN dramatically boosts the blue wing of the profile from the primary mini-disk in configuration where the hotspot is moving towards the observer. Dependence of profiles on the physical parameters of the binary {#sec:dependence} --------------------------------------------------------------- In this Section we investigate how the properties of modeled profiles vary as a function of the SBHB parameters, such as the alignment of the triple disk system, binary mass ratio, orbital separation, and inclination of the binary relative to the observer. We illustrate this dependence in the remainder of the panels in Figures \[fig:phys1c\] and \[fig:phys1e\] where we show subsets of profiles associated with the specific value of SBHB parameter. These show that profiles from SBHBs with wide orbital separations ($a=10^6\,M$) tend to be very symmetric and concentrate in the center of the AIP-PS parameter space, while close binaries ($a=5000\,M$) have a much wider footprint. By implication, this means that only profile 8 shown in Figure \[fig:stat1e\] can be produced by systems with large orbital separations. Similarly, any SBHB configurations where the mini-disks are co-planar with the binary orbit (and circumbinary disk, by assumption) are characterized by symmetric profiles with AIP$\approx 0$ with dominant peak shifted towards the blue part of the spectrum. The misaligned systems on the other hand are equally likely to be blue-leaning as well as red-leaning and reside in the range $-0.4\lesssim{\rm AIP} \lesssim 0.4$. Therefore, profiles 1, 2, 3, 4, 5 and 6 cannot correspond to SBHBs with coplanar disks. As discussed in the previous paragraph, profiles 1, 4 and 5 also show strong contribution due to illumination by the companion AGNs, which is indeed expected to be most pronounced for configurations with misaligned disks. More generally, we find that the effect of illumination by the companion AGN is the main reason for difference between the AIP-PS distribution of profiles from coplanar and misaligned SBHB systems shown in Figure \[fig:phys1e\]. On the other hand profiles associated with SBHB systems with different mass ratios ($q=1/10$ and 1) and different orientations of the binary orbit relative to the observer’s line of sight ($\theta=5^\circ$ and $105^\circ$) show significant overlap in their distributions. Together, these plots indicate that the most important SBHB parameters that determine the degree of asymmetry and the position of the dominant peak in the emission line profile are the intrinsic alignment of the triple disk system and the orbital semi-major axis. {width=".85\textwidth"} {width=".75\textwidth"} In the remaining figures we show four more projections of the multi-dimensional parameter space of the modeled emission line profiles including AI-FWHM (Figures \[fig:phys2e\] and \[fig:stat2e\]), FWQM-CS (Figures \[fig:phys3e\] and \[fig:stat3e\]), AI-KI (Figures \[fig:phys4e\] and \[fig:stat4e\]) and AIP-CS maps (Figures \[fig:phys5e\] and \[fig:stat5e\]). While more maps (i.e., parameter combinations) can in principle be constructed for this parameter space we focus on those that show distinct statistical distributions for any given SBHB property. Figure \[fig:phys2e\] shows AI-FWHM maps for the eccentric sample of binaries, where we used formulation of the asymmetry index defined in equation \[eq\_ai\]. The figure illustrates that profiles in the synthetic database have a wide range of FWHM values that extend to $28,000\,{\rm km\,s^{-1}}$ for eccentric sample of SBHBs. In comparison, the circular sample of binaries (not shown) is characterized by somewhat narrower profiles and ${\rm FWHM} < 23,000\,{\rm km\,s^{-1}}$. This difference can again be attributed to a wider range of orbital velocities sampled by eccentric binaries. Similar to Figure \[fig:phys1c\] this map shows that SBHB systems with coplanar disks and SBHBs on wide orbits tend to produce symmetric profiles with ${\rm AI} \approx 0$, distinct from misaligned systems and close binaries. Furthermore, Figure \[fig:phys2e\] shows that low mass ratio systems ($q=1/10$) and those in which SBHB orbit is close to face-on orientation relative to the observer ($\theta=5^\circ$) also occupy a narrow range of $-1\gtrsim {\rm AI} \gtrsim 1$, relative to the footprint of the entire distribution. This means that a combination of AIP-PS and AI-FWHM maps can in principle be used to break the degeneracy between SBHBs with aligned disks or large $a$ and SBHBs with low inclination or low values of $q$. Figure \[fig:stat2e\] shows the characteristic profile shapes occupying the AI-FWHM parameter space. Panels 2 and 6 illustrate the ability of AI to diagnose asymmetry in the low intensity features in profile wings even when the bulk of the profile is symmetric. By the same token, profile 5 has a lower value of AI than profile 6. This makes AI an useful diagnostic whenever the spectral noise level can be accurately determined and low intensity features clearly isolated. The profiles in panels 4 and 8 of Figure \[fig:stat2e\] have ${\rm AI} \approx 0$ showing that AI does not diagnose the asymmetry in the bulk of the profile. This tendency is the opposite from the AIP index, which makes them complementary diagnostics. Considering this in the context of the distributions discussed in the previous paragraph indicates that the low mass ratio SBHB systems and those in which SBHB orbit is close to face-on orientation relative to the observer can produce emission lines which are asymmetric in the bulk of the profile but show no significant asymmetry in the low intensity wings. {width=".85\textwidth"} {width=".75\textwidth"} Figure \[fig:phys3e\] shows FWQM-CS maps calculated for emission line profiles from eccentric SBHB systems. Modeled profiles exhibit $|{\rm CS}| < 4,000\,{\rm km\,s^{-1}}$ and can have a broad base with ${\rm FWQM} < 30,000\,{\rm km\,s^{-1}}$. This figure illustrates that the location of the centroid is a strong function of $a$ in the sense that profiles from close binaries ($a=5000\,M$) can have a significantly wider range of CS values relative to the wide binaries ($a=10^6\,M$). Similarly the width of the profiles is most strongly affected by $q$ and $i$, where for low mass ratio binaries and nearly face-on systems ${\rm FWQM} < 25,000\,{\rm km\,s^{-1}}$, lower than the entire sample of profiles. Figure \[fig:stat3e\] illustrates the diversity of profile shapes encountered in FWQM-CS parameter space. Interpreted together with Figure \[fig:phys3e\] it shows that the profiles 1, 2 and 3 must be produced by SBHB systems which satisfy either of these conditions: $q>1/10$, $\theta>5^\circ$ or $a\ll10^6\,M$. Similarly, profiles 1, 3, 4, 5, 7 and 8 cannot be associated with SBHB systems in which all disks are coplanar nor with wide binaries. Furthermore, profiles 5 and 7 cannot be associated with SBHB systems with $q=1$ and represent configurations where flux contributed by the primary mini-disk dominates over all other components. {width=".85\textwidth"} {width=".75\textwidth"} Figures \[fig:phys4e\] and \[fig:stat4e\] show AI-KI maps and examples of the emission line profiles associated with eccentric SBHBs, respectively. A large fraction of profiles clusters around low values of AI and KI indicating a large number of symmetric and boxy shapes (see profiles 1 and 7). The AI-KI maps illustrate a strong dependence of the profile shapes on the alignment of the triple disk system, where aligned systems give rise to very boxy profiles with symmetric wings. Similarly, the asymmetry of the low intensity features in the profile wings (profiles 3 and 5) is a sensitive function of $a$ and $i$ but is less sensitive to $q$, because distributions for different values of the SBHB mass ratio overlap to a significant degree. {width=".8\textwidth"} {width=".8\textwidth"} The pair of Figures \[fig:phys5e\] and \[fig:stat5e\] show the AIP-CS projection of the parameter space and the profile shapes in it, respectively. Similar to previous maps, the statistical distributions are a strong function of $a$, followed by the degree of triple disk alignment marked by the angles $\theta_1$ and $\theta_2$. This implies that AIP-CS maps can be used as a relatively sensitive diagnostic for these properties. The statistical distributions as a function of $q$ and $i$ are also distinct, so AIP-CS may also be used to constrain these parameters, although with a somewhat larger degree of degeneracy. The profile shapes shown in Figure \[fig:stat5e\] are drawn from the rim of the AIP-CS distribution and are representative of shapes associated with close SBHBs, those with $q\approx 1$ (with the exception of 1 and 5) and large inclination. Profiles in panels 1, 4, 5 and 8 exhibit asymmetry due to one strongly dominant peak produced by illumination of one mini-disk by the companion AGN. As discussed in Section \[sec:features\], this effect produces strong contribution to the flux of the composite profile in close binaries where the illuminated mini-disk is not aligned with the SBHB orbit, consistent with the binary properties shown in the AIP-CS maps. Profiles 2, 3, 6 and 7 also have a very pronounced, dominant peak which in their case arises due to incidental alignment of constituent profiles, rather than illumination by the other AGN. In this section we analyzed the dependance of the modeled profile shapes on the key parameters describing SBHB and triple disk configurations. For convenience, we summarize the most important results below. - The shapes of modeled emission line profiles are a sensitive function of the binary orbital separation. Compared to systems with small orbital semi-major axis, line profiles of wide SBHBs are more symmetric and occupy a relatively narrow range of values in terms of boxiness, peak and centroid shifts, and FWHM. - Similarly, modeled profile shapes are a sensitive function of the degree of alignment in the triple disk system. Line profiles associated with SBHB systems with nearly coplanar disks tend to be symmetric, boxy, and weakly affected by the secondary illumination from the companion AGN relative to the misaligned systems. - The synthetic emission line profiles tend to be less sensitive (or more degenerate with respect) to the SBHB mass ratio according to a majority of statistical distributions calculated in this work. The exception is the AIP-CS parameter space in which the low and high $q$ systems trace distinct correlations. - Besides the SBHB mass ratio, the modeled profile shapes show a comparable degree of degeneracy with respect to the binary orbital inclination relative to the observer. Compared to systems with high orbital inclination, line profiles of low inclination systems tend to be more symmetric, especially in the extended profile wings, and have somewhat lower values of FWHM. Similar to the SBHB mass ratio, the low and high inclination systems trace distinct correlations in the AIP-CS parameter space. Discussion {#sec:discussion} ========== Do modeled broad emission line profiles carry an imprint of SBHB parameters? ---------------------------------------------------------------------------- The ultimate goal of this investigation is to investigate whether the broad emission line profiles, commonly used in spectroscopic observational searchers to select SBHB candidates, can be used to decode the properties of bona fide SBHBs. In this work we make a step in this direction by first examining whether modeled broad emission line profile shapes convey any information about the parameters of SBHBs and their BLRs. If so, further development of this and similar models is of interest, as well as a comparison of such models with the data. The answer to this question is not obvious a priori: while composite profiles are potentially rich in information, the properties of SBHBs may be difficult to extract because of the complex emission geometry of multiple accretion disks. In practice, this means that any model designed to represent such systems must depend on a number of parameters and so do calculated emission line profiles (listed in Table \[table:parameters\] and Section \[sec:params\] for model presented here). Because of the dependence of profiles on multiple parameters and their degeneracy, it is unlikely that a unique match between a model and an observed SBHB can be achieved by attempting to fit the observed profile with arbitrary parameter combinations. This argues for an approach based on statistical distributions as a more promising way to analyze profile shapes. In this approach observed profiles can be matched to the modeled database based on their values of AIP, KI, FWHM, PS and CS. Each observed profile would map into a subset of modeled profiles with similar statistical properties that represent different modeled SBHB configurations. This correspondence of one observed profile and multiple SBHB configurations is a direct manifestation of degeneracy of the SBHB parameters. As a result, one could make a statement about the likelihood that the observed profile corresponds to some given SBHB configuration. If instead of one, a temporal sequence of observed profiles from the same SBHB is available for comparison with the modeled database, this could further help to reduce degeneracy. Our results indicate that the modeled profiles show distinct statistical properties as a function of the semi-major axis, mass ratio, eccentricity of the binary, and the degree of alignment of the triple disk system. In our model the SBHB systems on eccentric orbits are more likely to produce broader emission line profiles and complex profiles with multiple peaks relative to the circular cases. Thus, an observed profile compared to the synthetic database can be assigned a finite probability in the context of this model that it originates with the circular or eccentric SBHB based on its shape (see however the discussion below). Furthermore, mini-disks in smaller separation binaries which are misaligned with the binary orbital plane are subject to strong illumination by both AGNs in the system. As a consequence of the off-center illumination, such systems give rise to very asymmetric profiles that can exhibit significant peak or centroid velocity shifts. This is the dominant reason why all statistical distributions shown in this work are sensitive functions of parameters that control orbital separation and disk alignment. Indeed, in our model these two features of SBHB systems are most easily discernible based on profile shapes. In comparison, the effects of the binary mass ratio and SBHB orientation relative to a distant observer on profiles shapes are most discernible in the AIP-CS distribution (for both $q$ and $i$) and AI-KI distribution (for $i$ and to a lesser degree $q$). It is interesting to note that SBHBs with low $q$ or nearly face-on orbits ($i\approx 0^\circ$) tend to show a significant degree of correlation between the Pearson’s skewness coefficient and the peak or centroid velocity shifts (Figures \[fig:phys1e\] and \[fig:phys5e\]), where the AIP-CS correlation is more pronounced. This implies that such SBHB systems give rise to specific asymmetric profiles. As the offset of the dominant peak increases, the profile becomes more asymmetric resulting in red leaning profiles with the strongest peak shifted towards red or the blue leaning profiles with the strongest peak shifted towards blue. Visual inspection of such profiles indicates that their shapes tend to be strongly affected by the off-center illumination of the primary mini-disk, which dominates the flux in the composite profile. This can be understood because in our model the emission from the primary mini-disk typically dominates over that from the secondary and circumbinary disks for the smallest values of $q$. Even so, the composite emission line profile does not default to a fairly symmetric double peaked profile from an accretion disk about a single SBH, precisely because of the illumination by the secondary AGN. It follows that the most characteristic features of the modeled profiles presented here are a direct consequence of the presence of multiple BLRs (giving rise to profiles with multiple peaks) and their illumination by two AGN, both of which are an inherent property of the SBHB model. The distinct statistical distributions suggest that SBHB properties are indeed imprinted in the population of modeled profiles, albeit with some degeneracy, which for any given SBHB parameter can be statistically quantified. Based on this we conclude that models of broad emission line profiles from SBHBs in circumbinary disks can have predictive power and as such merit further investigation. Can modeled emission line profiles be compared with the observed profiles from SBHB candidates? ----------------------------------------------------------------------------------------------- The next relevant question is whether the modeled emission line profiles presented here can be directly compared with those from spectroscopically selected SBHB candidates. We make several such comparisons below but note that they do not constitute a proof that the observed candidates are indeed SBH binaries. Examination of the observed emission line profiles from SBHB candidates presented by @eracleous12, @decarli13, @liu13 and @li16, shows that these profiles can be asymmetric and offset but are usually quite smooth and characterized by one or two peaks, unlike some of the profiles in our database with complex structure and up to 6 peaks. Admittedly, the fraction of profiles with such high number of peaks is relatively small in our database and they are more common for eccentric binaries. On the other hand, the modeled single and double-peaked profiles are most common in SBHB configurations with semi-major axes $a\geq 5\times 10^4\,M$, which for a $10^8\,M_\odot$ binary translates to $\geq 0.25$pc. Therefore, if comparison between the observed and modeled profiles is made at the face value, it would favor moderately wide bound binaries. Note that both AIP-PS and AIP-CS correlations have been reported in spectroscopically targeted SBHB candidates and have not been found in a control sample of matching AGN [@eracleous12; @runnoe15]. A qualitative comparison of the observed sample in Figure 18 of @runnoe15 with our modeled AIP-PS distributions in Figures \[fig:phys1c\] and \[fig:phys1e\] shows that they cover a similar range of AIP values but that the observed profiles tend to have peak velocity shifts in a narrow range between $-4000\lesssim {\rm PS} \lesssim 3000\,{\rm km\,s^{-1}}$. In the context of the SBHB model this disfavors configurations of SBHBs with either the smallest or the widest orbital separations and favors moderately wide binaries and binaries with misaligned disks. Another correlation identified in the sample of observed SBHBs by the same authors is between the third moment of the flux distribution of profiles[^4], $\mu_3$, and FWHM. Namely, @eracleous12 report that the values of $\mu_3$ in the observed emission line profiles tend to decrease with increasing FWHM. This correlation is not seemingly present in our modeled sample regardless of the adopted parameter cut. As noted before however, the value of AI (and that of the related parameter $\mu_3$) sensitively depends on the noise level, which in observed profiles is very likely different from the fiducial noise level we adopt in our calculations of AI. We will take this difference into account in future work, when we carry out a more detailed comparison between the observed and modeled samples. Along similar lines, the FWHM measured by @eracleous12 in their sample of 88 candidates reaches up to $18,000\,{\rm km\,s^{-1}}$. Our synthetic profiles are however characterized by values of FWHM as high as $23,000\,{\rm km\,s^{-1}}$ for the circular and $28,000\,{\rm km\,s^{-1}}$ for the eccentric sample of binaries. The modeled profiles are therefore inherently wider than those observed, regardless of the SBHB parameter cut. The tendency of modeled profiles to exhibit richer and more diverse structure can to some degree be ascribed to their dependence on the semi-major axis, as discussed at the beginning of this section, or perhaps a larger degree of “smoothing" in real profiles due to either the presence of noise or a larger velocity dispersion of the emitting gas on average. On the other hand a mismatch in the range of measured FWHM between the two populations cannot be trivially explained. The FWHM measured in modeled profiles is a function of the orbital velocity of the gas in each disk and the orbital velocity of the binary, both of which are inherent characteristics of SBHB systems. If anything, increasing the velocity dispersion of the gas ($\sigma$) in our model, in order to produce smoother profiles, would result in even wider profiles and more tension between the observed and modeled samples. A qualitative comparison therefore highlights some intriguing similarities and also points to differences between the two samples. The former motivate further development of models of broad emission line profiles from SBHB systems, given their potential to interpret profiles from observed bona fide SBHBs. The latter may arise either due to a true difference between the two samples of profiles or because physical processes giving rise to the broad optical emission line profiles were not entirely captured by our model. It is therefore important before attempting more detailed comparisons to examine the impact of any simplifying assumptions made in the current model. Simplifying assumptions and their implications {#secsec:approx} ---------------------------------------------- Perhaps the most important physical mechanism that can significantly modify the appearance of the spectrum and emission lines is the radiative feedback from the binary AGN, capable of driving winds and outflows from the circumbinary region. Several recent simulations of SBHBs in circumbinary disks indicate that despite strong binary torques, accretion into the central cavity continues more-less unhindered relative to the single SBH case [@dorazio13; @farris14; @shi15]. This point is of particular interest because AGN feedback from an accreting binary SBH can considerably change the structure, thermodynamic and ionization properties of the circumbinary region. In this work, we assume that the emissivity of each broad line region arises due to photoionization by the two AGNs but neglect the effects of radiation pressure on the dynamics and optical depth of the emitting gas. In order to address this, in Paper II we will calculate the profiles of low-ionization emission lines by generalizing models that account for radiative transfer effects through a disk wind [@cm96; @mc97; @flohic12; @chajet13]. These works have demonstrated that disk models that reproduce relatively rare double-peaked emitters can also describe emission from BLRs of most AGNs once these effects are accounted for, pointing to their broad applicability. The key effect of the accretion disk wind is to modify the shape of a broad emission line profile. This occurs because the radiation pressure from the central AGN lifts-off the low density gas from the surface of the disk and launches it along streamlines above the disk. The photons (in this case H$\beta$) escaping from a single accretion disk encounter increased optical depth through the emission layer and as a consequence, the peaks of an initially double peaked profile move closer and eventually merge, producing a narrower single peaked profile. Comparisons of such single peaked disk-wind model profiles with emission lines from a set of SDSS quasars show that observed lines are consistent with moderately large optical depth in the disk wind and indicate that most AGNs may be subject to this type of feedback [@flohic12]. It is therefore reasonable to assume that if SBHs in a binary can accrete at rates comparable to the general population of AGNs, they are likely to produce similar effects. The reprocessing of the H$\beta$ photons through the accretion disk wind may indeed produce smoother and narrower profiles in better general agreement with the observed sample of SBHB candidates and AGNs in general. However, the same effect may also “wash out" some of the characteristic features encountered in our modeled profiles, thus weakening their dependence on the properties of the binary. We will assess the diagnostic power from broad emission line profiles affected by the accretion disk wind and associated with SBHBs in circumbinary disks in Paper II. In addition to the accretion disk wind the emissivity can also be modified by shocks, impacts of streams from the circumbinary disk onto the accretion disks around the individual SBHs [@roedig14b] and by the presence of overdense lumps that may form in the inner region of the circumbinary disk [@farris14]. These features have been predicted by some theoretical models and simulations and if indeed present in binary accretion flows, they would increase the complexity of the emission line profiles by creating hot spots and localized regions of high emissivity. The presence, persistance and exact emission properties of these features however sensitively depend on thermodynamics of the SBHB accretion flow, which remains to be understood and at the present cannot be derived from first principles. We do not account for contribution to the emissivity of the broad lines from shocks and overdensities but note that they can be added to the existing model should that be necessary. Another approximation used in our model is that the two mini-disks, as well as the circumbinary disk are circular in shape. In this scenario, the outer edges of the mini-disks and the inner edge of the circumbinary disk are determined by SBHB tidal forces and are not free parameters of the model (see Section \[sec:params\]). Simulations however show that the mini-disks and the circumbinary disk can exhibit varying degrees of eccentricity as a consequence of tidal deformation by the binary, an effect which is most pronounced for comparable-mass binaries [@farris14]. From the stand point of our semi-analytic model this implies that additional parameters may be required in order to describe the geometry of the emission regions around SBHBs, leading to additional degrees of freedom in profile shapes. The distinct property of an elliptical accretion disk is that it can naturally give rise to double peaked emission line profiles in which the red peak is stronger than the blue, a feature that cannot be reproduced by a circular disk [@eracleous95]. In our model this type of asymmetry is present in less than 50% of the profiles and it arises in two ways: either through summation of individual profiles which results in a stronger red peak (see for example panels 6 and 7 in Figure \[fig:stat5e\]) or due to illumination by the companion AGN (panels 4 and 5 in Figure \[fig:stat5e\]). We therefore reproduce such an effect even though we only consider circular BLRs in our model. If our model accounted for elliptical disks the database may contain a larger fraction of profiles with the dominant red peak (reflected in the positive value of the peak velocity shift) but at the expense of a number of additional parameters. An additional assumption used in our model is that of prograde binaries. Namely, motivated by theoretical works described in Section \[sec:intro\] we assume that the SBHB and the circumbinary disk are coplanar and rotate in the same sense. At the same time, the mini-disks are allowed to assume arbitrary orientation (and sense of rotation) relative to the SBHB orbit. A circumbinary disk with an arbitrary orientation relative to the SBHB orbit would however still produce a single- or a double-peaked profile that is centered on the SBHB rest frame, similar to the profiles shown in this work. Since the total flux of the composite profile is dominated by the primary and secondary mini-disks, the assumption about co-planarity of the circumbinary disk should not strongly affect our results. Note however that simulations of retrograde SBHBs in circumbinary disks show a different dependence of SBH accretion rates on orbital eccentricity [@roedig14] from that assumed in Equation \[eq:mscale\] of this work. This is another ingredient that can in principle be added to the model, if counter-rotating SBHB configurations are of interest. Because we evaluate Doppler boosting and gravitational redshift in the weak field limit and neglect bending of light (see Appendix \[sec:appendixsingle\]) we can only faithfully calculate the emission line profiles that arise in configurations in which the photons are emitted far from the immediate environment of black holes (i.e., at distances larger than tens of Schwarzschild radii) and in which they do not travel on grazing trajectories over the SBHs. Both of these requirements are satisfied in our model given the assumed sizes of emission regions and the fact that we do not allow for edge-on configurations characterized by the disk inclination angles in the range $80 - 100^{\circ}$. Along similar lines, we do not account for lensing of one mini-disk by the companion SBH when the two SBHs are in conjunction. Such configurations are expected to be rare and short lived and should not significantly affect the overall statistical distribution of the emission line profiles. The parameter values in Table \[table:parameters\] are chosen so as to provide a relatively uniform but not necessarily dense coverage of the SBHB parameter space. This can be seen in the middle top panel of Figure \[fig:phys3e\], where “branches” at ${\rm FWQM} \approx 22,000\,{\rm km\,s^{-1}}$ and $27,000\,{\rm km\,s^{-1}}$ carry an imprint of the underlying SBHB parameter choices, most likely that of the binary orbital inclination. Because of the extent of the parameter space, the number of sampled configurations quickly adds up to nearly 15 million, even with a handful of choices per parameter. While this rate of sampling may be acceptable for surveying the properties of emission line profiles, a denser coverage can be obtained for sub-regions of the parameter space. It is worth noting that other physical processes can potentially mimic the emission signatures of SBHBs discussed here. These include but are not limited to the recoiling SBHs [@blecha16] and local and global instabilities in single SBH accretion disks that can give rise to transient bright spots and spiral arms [@flohic08; @lewis10]. In that sense, the model described in this paper can be used to interpret observed emission line profiles in the context of the SBHB model but cannot be used to prove that they originate with veritable SBHB systems. For example, profiles of SBHB candidates observed in multiple epochs can be compared against the synthetic database individually, in order to determine the likelihood distribution for underlying SBHB parameters for each profile independently. The entire time series of observed profiles can also be compared against the time series of matching modeled profiles as an added consistency check for the inferred SBHB parameters. We defer this type of analysis to future work. Conclusions and Future Prospects {#sec:conclusions} ================================ This work is motivated by advances in observational searches for SBHBs made in the past few years which are represented by better designed, multi-wavelength and multi-year observational campaigns. Observational challenges notwithstanding, spectroscopic searches for SBHBs seem capable of delivering statistically significant sample of binary candidates and their first results are broadly consistent with theoretical predictions. While selection of a well defined sample of SBHBs remains a principal goal in this research field, an equally important and timely consideration is what can be learned once such sample is available. In this context we develop a model to calculate the broad emission line profiles from SBHBs in circumbinary disks guided by a wealth of theoretical results in the literature. In this work, which constitutes part I of the series, we consider whether complex, composite emission line profiles from SBHB systems can be used as a diagnostic of the binary properties. We use the SBHB model to calculate a database of 14.8 million emission line profiles arising from a triple disk system associated with the binary. In this setup each disk acts as a BLR and contributes emission line flux resulting in a broad composite profile. We analyze the modeled emission line profiles in terms of the commonly used statistical distribution functions in order to determine their dependence on the underlying binary parameters. We find that the modeled profiles show distinct properties as a function of the binary semi-major axis, eccentricity, mass ratio, alignment of the triple disk system and orientation relative to the observer. The most characteristic features of modeled profiles are a direct consequence of the presence of multiple BLRs and their illumination by two accretion powered SBHs, both of which are a unique property of the SBHB model. Thus, models of broad emission line profiles from SBHBs in circumbinary disks have predictive power and can in principle be used to infer distribution of these parameters in real binaries. We identify some intriguing similarities between the observed SBHB candidates and our synthetic profiles. Both groups exhibit correlation between the Pearson skewness coefficient and the peak or centroid velocity shift. Initial comparison of the two samples at the face value favors SBHB candidates which are moderately wide binaries with misaligned disks. On the other hand, the database of modeled profiles contains more diverse profile morphologies and on average wider profiles than the observed sample of SBHB candidates or a general population of AGNs. This suggests that not all relevant physical phenomena are fully captured by our model, a question that given the potential of this and similar models merits further investigation. The leading contender for a physical mechanism that can modify the appearance of the emission lines is radiative feedback from the binary AGN, capable of driving winds and outflows in the circumbinary region. We will investigate the importance of this mechanism in future work, where we will calculate the emission line profiles by taking into account the radiative transfer effects through a disk wind. More specifically, we will re-evaluate the diagnostic power of broad emission lines and carry out a direct comparison of the observed and modeled profile samples. We conclude by noting that the emission signatures discussed here may not be unique to SBHB systems and that there is a possibility that they can be mimicked by other physical processes, driven by local and global instabilities in single SBH accretion disks. The model described in this work is therefore a promising tool that can be used to interpret the observed emission line profiles in the context of the SBHB model but should not be considered a conclusive test of binarity. We are grateful to Michael Eracleous and Jessie Runnoe for their insightful and useful comments and thank the anonymous referee for a thoughtful report. This research was supported in part by the National Science Foundation under Grant No. NSF AST-1211677 and by the National Aeronautics and Space Administration under Grant No. NNX15AK84G issued through the Astrophysics Theory Program. T.B. acknowledges support from the Research Corporation for Science Advancement through a Cottrell Scholar Award. T.B. is a member of the MAGNA project (<http://www.issibern.ch/teams/agnactivity/Home.html>) supported by the International Space Science Institute (ISSI) in Bern, Switzerland. Numerical simulations presented in this paper were performed using the high-performance computing cluster PACE, administered by the Office of Information and Technology at the Georgia Institute of Technology. \[appendix\] A: Broad Emission Line Profiles from Circular Keplerian Disk {#sec:appendixsingle} ============================================================ We describe each disk in the triple disk system as a circular Keplerian, geometrically thin accretion disk in the weak-field approximation as outlined by @chen89, @chen89b and @eracleous95. Specifically, we use implementation that assumes optically thin emission from the skin of the disk [equation 19 in @chen89] and neglect bending of light in gravitational field of an SBH [encoded in equation 8 of both @chen89b; @eracleous95]. We first outline the key elements of this model (hereafter referred to as the [*single disk model*]{}) and then describe modifications we made in order to calculate emission line profiles from triple disk systems. In the single disk model the flux of the broad emission line profile measured in the observer’s frame can be expressed as an integral over the surface of the emitting disk defined in terms of the properties in the disk frame: $$\label{eq:flux1} F(\nu_{\rm obs})=\frac{M^2\nu_0}{d^2} \int\limits_{0}^{2\pi} \int\limits_{\xi_{\rm in}}^{\xi_{\rm out}} I(\xi,\nu_{\rm turb}) D_{\rm rot}^3 \left( 1-\frac{2}{\xi} \right) ^ {-\frac{1}{2}} \xi\, d\xi\, d\varphi$$ where $M$ is the mass of the central object, $\nu_0$ is the rest frequency of the emission line, $d$ is the distance from the center of the disk to the observer, $\xi=r/M$ is the radius in the disk in dimensionless units, $\xi_{\rm in}$ and $\xi_{\rm out}$ are the inner and outer edge of the emission region, respectively, and $\varphi$ is the azimuthal angle measured in the plane of the disk (note that $\varphi$ is different from $\phi$ defined in Section \[sec:params\]). The geometry of such a disk is illustrated in Figure \[fig:single\]. $I(\xi,\nu_{\rm turb})$ is the specific intensity of light emitted at radius $\xi$ and frequency $\nu_{\rm turb}$ $$\label{eq:intensity1} I(\xi,\nu_{\rm turb})= \frac{1}{4\pi} \frac{\epsilon(\xi)}{(2\pi)^{1/2} \sigma} \, \exp[{-(\nu_{\rm turb}-\nu_0)^2/2\sigma^2}]$$ where $\epsilon(\xi)$ is the disk emissivity as a function of radius. In the single disk model $\epsilon = \epsilon_{0} \cdot \xi^{-p}$ represents the emissivity of the disk illuminated by a single, central source. The emissivity constant, $\epsilon_{0}$, is proportional to the luminosity of the photoionizing source, which we assume is powered by accretion onto an SBH. For the purposes of this calculation we therefore assume that $\epsilon_{0}\propto \dot{M}$. Geometric arguments, as well as photoionization calculations, indicate that $p \approx 3$ is a reasonable value for the emissivity index [@csd89] and we adopt it in our calculations. In the next section we extend this formalism to account for illumination of the disk by two AGNs, associated with two SBHs as illustrated in Figure \[fig:single\]. {width=".5\textwidth"} The frequency of photons emerging locally from the disk ($\nu_{\rm turb}$) is shifted by turbulent motion of gas and assumed to have a Gaussian distribution about the rest frequency $\nu_0$ [@chen89b]. We adopt a characteristic velocity dispersion of the gas due to turbulent motion of $\sigma \simeq 850 \, \rm km\,s^{-1}$, which corresponds to a characteristic frequency shift of $\Delta \nu/ \nu_0 = 850\, {\rm km\, s^{-1}} / c \simeq 2.8 \times 10^{-3}$, where $c$ is the speed of light. Our choice of the characteristic velocity dispersion is motivated by the values inferred from radio-loud AGN with double-peaked emission lines, which have been successfully modeled with either the circular or elliptical disk models and turbulent broadening in the range of $\sim 600-3000 \, \rm km\,s^{-1}$ [@eh94; @eh03; @strateva03]. Several additional effects can impact the frequency of emitted photons, including the rotational motion of gas in the disk, relativistic Doppler boosting, and gravitational redshift. Classically, the Doppler factor associated with the motion of the gas in the disk can be expressed as $$\label{eq:dopfactor3} D_{\rm cl}=\frac{1}{1- \boldsymbol{v}_{\rm rot} \boldsymbol{\cdot \hat{s}}} = (1+\xi^{-1/2}\sin i \sin \varphi)^{-1}$$ where $\boldsymbol{v}_{\rm rot} = \xi^{-1/2}\, \boldsymbol{\hat{e}_{\phi}}$ is the velocity vector of a given surface element of the disk in units of the speed of light, $\boldsymbol{\hat{s}}$ is the unit vector along the line of sight of the observer, and $i$ is the inclination of the disk with respect to the observers line of sight, as illustrated in Figure \[fig:single\]. The two relativistic effects can be accounted for by adding terms for the special relativistic beaming, $D_{\rm sr}=\sqrt{1-v_{\rm rot}^2}$, and general relativistic gravitational redshift, $D_{\rm gr}=\sqrt{1-2/\xi}$, yielding $$\label{eq:dopfactor1} D_{\rm rot} = \left(1-\frac{1}{\xi}\right)^{1/2} \left(1-\frac{2}{\xi}\right)^{1/2} D_{\rm cl} \simeq \left(1-\frac{3}{\xi} \right)^{1/2} D_{\rm cl}$$ where we used the weak-field approximation (valid when $\xi \gg 1$) to obtain the final expression. For an emission element in the disk located at $\xi=500$ the shift in the frequency of emitted light due to the relativistic effects can be estimated as $D_{\rm rot} \simeq 0.99699 D_{\rm cl}$. For the $H\beta$ transition this amounts to nearly 15Å, an offset that is in principle detectable given the spectral resolution of optical surveys and therefore should be accounted for in the model. The relativistic Doppler factor can then be expressed as $$\label{eq:dopfactor2} D_{\rm rot}=\nu_{\rm obs}/\nu_{\rm turb}=(1-3/\xi)^{1/2}(1+\xi^{-1/2}\sin i \sin \varphi)^{-1}.$$ where $\nu_{\rm obs}$ marks the frequency of the photon measured by the observer. Our derivation of equation \[eq:flux1\] departs from that of @chen89b and @eracleous95 because it does not include relativistic bending of light and is thus applicable under two conditions. The first is that the photons are emitted by a gas element far away from the black hole ($\xi \gg 1$). The second is that emitted photons do not travel on “grazing" orbits over the black hole. Our calculation satisfies both by having the $H\beta$ photons emerge from the radii in the disk $\xi \geq 500$ and by eliminating edge-on configurations characterized by inclination angles of the disk in the range $80 - 100^{\circ}$. B: Broad Emission Line Profiles From a Triple Disk System {#appendixbinary} ========================================================= In this section we describe modifications to the single disk model introduced in order to calculate the emission line profiles from the triple disk system associated with an SBHB. This is accomplished in three steps in which we: (a) define the orientation of the three disks relative to the orbital plane of the SBHB and relative to a distant observer (b) evaluate the emissivity of each disk illuminated by the two AGNs and (c) sum the three components of flux to calculate the composite emission line profile in the frame of the binary. Geometry of the triple disk system ---------------------------------- In order to determine the mutual orientation of the three disks and the binary orbit, as well as their relative orientation with respect to the observer’s line of sight, we define three coordinate systems, each anchored to the center of its resident disk as in Figure \[fig:single\]. The coordinate system associated with the circumbinary disk coincides with that associated with the binary orbit and has the origin in the SBHB center of mass. We refer to it as the [*binary*]{} or [*SBHB frame*]{} in the rest of the text. In order to distinguish among the properties calculated in these reference frames we introduce subscripts where “1" and “2" correspond to the primary and secondary mini-disks, and “3" to the circumbinary disk, respectively. Furthermore, because we carry out the calculation of flux in dimensionless, geometric units (as shown in the previous section), the subscripts also indicate that distances are measured in units of $M_1$ and $M_2$ in the frames of the primary and secondary mini-disks, and $M=M_1+M_2$ in the frame of the binary. Figure \[fig:binary\] illustrates the coordinate system anchored to the binary orbital plane where the SBHB center of mass marks the origin and the $z$-axis points in the direction of the orbital angular momentum of the binary, directed out of the page. The $x$-axis points towards the pericenter of the primary SBH orbit and is parallel to the orbital semi-major axis of the binary, $a=(a_1+a_2)$. We describe the orientation of the two SBHs in the orbital plane of the binary as a vector pointing from the primary to the secondary SBH (see Figure \[fig:binary\]) $$\label{eq:ell} \boldsymbol{l}= l\; \boldsymbol{\hat{e}}_r = l(\cos{f} \, \boldsymbol{\hat{x}}+ \sin f \: \boldsymbol{\hat{y}})$$ where $l=a(1-e^2)/(1 - e \cos f)$ is the separation of the AGNs, $e$ is the orbital eccentricity, $f$ is the orbital phase of the SBHB measured counter-clockwise from the $x$-axis to the instantaneous location of the secondary SBH, and $\boldsymbol{\hat{e}}_r$ is the unit vector parallel to $\boldsymbol{l}$. We define the orientation of the observer in the SBHB frame with a vector $\boldsymbol{\hat{s}}=\sin i \cos\phi \,\boldsymbol{\hat{x}} + \sin i \sin\phi \,\boldsymbol{\hat{y}} + \cos i \,\boldsymbol{\hat{z}}$. The inclination angle, $i$, describes the orientation of the observer’s line of sight relative to the vector of orbital angular momentum of the SBHB. For example, the inclination angle $i=0^\circ$ represents the clockwise binary seen face-on and values $i>90^\circ$ represent counter-clockwise binaries. The azimuthal angle $\phi$ is measured in the binary orbital plane, from the positive $x$-axis to the projection of the observer’s line of sight, in counter-clockwise direction. For circular SBHBs varying the orbital phase $f$ is equivalent to varying the azimuthal orientation of the observer and in this case we adopt a single nominal value of $\phi=0^\circ$ in calculation of the emission line profiles. This is however not the case for the eccentric binaries, in which case $f$ and $\phi$ take independent values. We define the orientation of the primary mini-disk by specifying the orientation of its rotation axis (given by the unit vector of the disk angular momentum, $\boldsymbol{\hat{L}_1}$) in terms of the polar and azimuthal angles $\theta_1$ and $\phi_1$ measured in the SBHB frame: $$\boldsymbol{\hat{L}_1}= \sin\theta_1 \cos\phi_1 \;\boldsymbol{\hat{x}} + \sin\theta_1 \sin\phi_1 \;\boldsymbol{\hat{y}} + \cos\theta_1 \;\boldsymbol{\hat{z}} \;\; .$$ Equivalently, we use $\theta_2$, $\phi_2$ to specify the orientation of the secondary disk, given by the unit vector of the disk angular momentum, $\boldsymbol{\hat{L}_2}$. As described in Section \[sec:params\] mini-disks are coplanar with the SBHB orbit when $\theta_1=\theta_2=0^{\circ}$ and the gas in the mini-disks exhibits retrograde motion relative to the SBHB when $\theta_i>90^\circ$. The azimuthal angles $\phi_1$ and $\phi_2$ are measured in the binary orbital plane, from the positive $x$-axis to the projections of the rotation axes of the mini-disks, in counter-clockwise direction. With known orientations of the mini-disks and the observer in the SBHB frame we can evaluate the inclinations of the mini-disks with respect to the observer’s line of sight $$\label{eq:i1} \cos i_1= \boldsymbol{\hat{L}_1} \cdot \boldsymbol{\hat{s}} = \sin i \cos\phi \sin\theta_1 \cos\phi_1 + \sin i \sin\phi \sin\theta_1 \sin\phi_1 + \cos i \cos\theta_1$$ The orientation of the secondary AGN relative to the primary mini-disk can be calculated as follows $$\begin{aligned} &&\sin \theta_{M2} \cos\phi_{M2}= \boldsymbol{\hat{e}_r}\cdot \boldsymbol{\hat{x}_1}= \boldsymbol{\hat{e}_r} \cdot \left( \frac{\boldsymbol{\hat{L}_1} \times \boldsymbol{\hat{s}}}{\sin i_1} \times \boldsymbol{\hat{L}_1} \right)\\ &&\sin \theta_{M2} \sin\phi_{M2}= \boldsymbol{\hat{e}_r} \cdot \boldsymbol{\hat{y}_1}= \boldsymbol{\hat{e}_r} \cdot \frac{\boldsymbol{\hat{L}_1} \times \boldsymbol{\hat{s}}}{\sin i_1}\\ &&\cos \theta_{M2}= \boldsymbol{\hat{e}}_r \cdot \boldsymbol{\hat{L}_1} = \sin \theta_1 \cos \phi_1 \cos f + \sin \theta_1 \sin \phi_1 \sin f= \sin\theta_1 \cos (\phi_1 - f)\end{aligned}$$ where $\theta_{M2}$ and $\phi_{M2}$ are the spherical polar coordinates describing the location of the secondary AGN in the primary mini-disk frame. Equivalent expressions can be written for the secondary mini-disk by replacing subscript “1" with “2" $$\begin{aligned} &&\cos i_2= \boldsymbol{\hat{L}_2} \cdot \boldsymbol{\hat{s}}\\ &&\sin \theta_{M1} \sin\phi_{M1}=- \boldsymbol{\hat{e}_r } \cdot \boldsymbol{\hat{y}_2}\\ &&\cos \theta_{M1}= - \boldsymbol{\hat{e}_r} \cdot \boldsymbol{\hat{L}_2}\end{aligned}$$ where the minus signs in the last two equations encode the opposition of the primary and secondary SBHs relative to the center of mass. In all configurations we assume that the circumbinary disk is co-planar with the SBHB orbit and co-rotating with it and therefore $\theta_3\equiv 0^\circ$, $\phi_3 \equiv \phi$ and $i_3 \equiv i$ (i.e., the circumbinary disk frame is coincident with the SBHB frame). These relationships allow us to define the emissivity of surface elements in each of the three disks in the system. Calculation of disk emissivities -------------------------------- We assign emissivities $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ to the primary, secondary and circumbinary disk, respectively. The emissivity of a gas element located at $(\xi_1,\varphi_1)$ in the mini-disk of the primary SBH can be expressed as a sum of components due to the illumination by its own AGN ($\epsilon_{11}$) and the AGN associated with the secondary SBH ($\epsilon_{12}$) $$\label{eq:emiss1} \epsilon_1(\xi_1,\varphi_1)= \epsilon_{11} + \epsilon_{12}= \epsilon_{10}\frac{h_1}{\xi_1^3} + \epsilon_{20}\frac{H \left(\cos i_1 \cos \theta_{M2}\right)\, \left|l_1 \left| \cos\theta_{M2} \right|-q\, h_2\right|}{\left[\xi_1^2 + l_1^2 - 2\xi_1 l_1 \left(\boldsymbol{\hat{\xi}_1} \cdot \boldsymbol{\hat{e}_r} \right) \right]^{3/2}}$$ where $h_1$ and $h_2$ denote the sizes of the sources of continuum radiation associated with the two SBHs. As mentioned earlier, the subscripts indicate that $h_1$ and $h_2$ are dimensionless quantities in units of $M_1$ and $M_2$, respectively, and $l_1$ is the separation of the two AGN in units of $M_1$. Motivated by the X-ray studies of the broad iron line reverberation [@uttley14], we assume that the sources of continuum radiation are compact and have spatial extents of $h_1=10$ and $h_2= 10$. Note that the term $q\, h_2$ in equation \[eq:emiss1\] represents conversion of $h_2$ into the units of $M_1$, for consistency with the rest of the properties calculated in the frame of the primary mini-disk, where $q=M_2/M_1$ is the SBH mass ratio. The second term of equation \[eq:emiss1\] captures the effect of the off-center illumination of the primary BLR by the secondary AGN. Figure \[fig:single\] illustrates that this effect can be seen by a distant observer only when the side of the mini-disk illuminated by the off-center AGN is facing the observer. Alternatively, whenever the secondary AGN is blocked by the primary mini-disk (from the observer’s point of view) this effect will be absent. We describe these outcomes with the Heaviside step function, $H(\cos i_1 \cos \theta_{M2})$, which takes value “1" whenever the secondary AGN belongs to the same half plane with the observer ($\cos i_1 \cos \theta_{M2}\geq 0$) and value “0" otherwise. As discussed in Section \[sec:appendixsingle\], the emissivity constants $\epsilon_{10}$ and $\epsilon_{20}$ are directly proportional to the luminosity of the two AGNs, which we assume are powered by accretion onto the SBHs. Therefore, we express the ratio of the two constants as $\dot{m} = \epsilon_{20}/\epsilon_{10} = \dot{M_2}/\dot{M_1}$. Assuming relative scaling such that $\epsilon_{10} = 1$ implies $\epsilon_{20} = \dot{m}$ and equation \[eq:emiss1\] becomes $$\begin{aligned} \label{eq:emiss2} \begin{split} \epsilon_1(\xi_1,\varphi_1)=\frac{10}{\xi_1^3} + \dot{m}\frac{H(\cos i_1 \cos \theta_{M2})\; |l_1|\cos\theta_{M2}|-10q |}{\left[\xi_1^2 + l_1^2 - 2\xi_1 l_1 \sin\theta_{M2}\left(\cos\varphi_1\cos\phi_{M2}+\sin\varphi_1\sin\phi_{M2}\right) \right]^{3/2}} \end{split}\end{aligned}$$ The vertical brackets in equations \[eq:emiss1\] and \[eq:emiss2\] denote absolute values of the relevant quantities. Similarly, the emissivity of the secondary mini-disk can be expressed as: $$\label{eq:emiss3} \epsilon_2(\xi_2,\varphi_2)= \dot{m}\,\frac{10}{\xi_2^3} + \frac{H(\cos i_2 \cos \theta_{M1})\, \left| l_2 \left| \cos\theta_{M1}\right| -10/q\right|} {\left[\xi_2^2 + l_2^2 + 2\xi_2 l_2 \sin\theta_{M1}\left(\cos\varphi_2\cos\phi_{M1}+\sin\varphi_2\sin\phi_{M1}\right) \right]^{3/2}}$$ In the case of the circumbinary disk the illumination by both AGNs is off-center. We express its emissivity as that of a single accretion disk which center resides at the center of mass of the binary $$\label{eq:emiss4} \epsilon_3(\xi_3,\varphi_3)= \frac{10}{1+q}\frac{1}{(\xi_3^2+l_{31}^2 + 2\,\xi_3 \,l_{31}\cos \varphi_3)^{3/2}}+ \dot{m}\, \frac{10\,q}{1+q}\frac{1}{(\xi_3^2 + l_{32}^2 - 2\,\xi_3\, l_{32} \cos \varphi_3)^{3/2}}$$ Because the coordinate system associated with the circumbinary disk coincides with the reference frame of the binary, all distances in equation \[eq:emiss4\] are normalized by the total mass of the binary $M$. Therefore, $\xi_3=r_3/M$ is the dimensionless radial distance of the gas element to the SBHB center of mass and $l_{31}= q/(1+q) (l/M)$ and $l_{32}= 1/(1+q) (l/M)$ are the dimensionless distances from the center of mass to the primary and secondary SBHs, respectively. Total flux of the composite emission line profile ------------------------------------------------- With known emissivities the flux from each disk in the SBHB system can be calculated as an integral over the surface area, according to equation \[eq:flux1\]. Before summing the fluxes to calculate the composite emission line profile we need to account for the Doppler shift of the photons emitted by the primary and secondary mini-disks due to their orbital motion. Because the circumbinary disk is at rest with respect to the SBHB center of mass we apply no shift to its emission. The emitted composite profile is therefore calculated in the reference frame of the binary. Because the orbital velocities of SBHBs considered in this work are non-relativistic, the Doppler shifts associated with the orbital motion can be evaluated in classical limit. We therefore neglect the effect of relativistic boosting in this case as well as the gravitational redshift and lensing of photons that may arise in configurations when the two SBHs are in conjunction (i.e., lined up along the observer’s line of sight). In classical limit, the Doppler shift associated with the orbital motion of the secondary mini-disk is $D_{\rm orb2}=1/\left(1-\boldsymbol{v_{\rm orb2}} \cdot \boldsymbol{ \hat{s}}\right)$ and $$\frac{\Delta \nu_{\rm orb2}}{\nu_0}= \frac{\nu_{\rm obs}-\nu_0}{\nu_0}= \frac{1}{1-{\boldsymbol{v_{\rm orb2}}}\cdot {\boldsymbol{\hat{s}}}} - 1 \simeq {\boldsymbol{v_{\rm orb2}}}\cdot {\boldsymbol{\hat{s}}}$$ Here $\boldsymbol{v_{\rm orb2}}$ is the velocity vector of the secondary SBH measured in the frame of the binary in units of $c$ and $\boldsymbol{\hat{s}}$ describes the orientation of the observer in the frame of the SBHB as defined earlier. Let $\boldsymbol{v_{\rm orb}}=\boldsymbol{v_{\rm orb2}}-\boldsymbol{v_{\rm orb1}}$ be the relative velocity vector of the two SBHs, as in Figure \[fig:binary\], and $$\begin{aligned} &&v_{\rm orb}^2=\left(\frac{2}{l}-\frac{1}{a}\right)\\ &&\boldsymbol{v_{\rm orb}}=\dot{\boldsymbol{l}}= v_r \,\boldsymbol{\hat{e}_r}+v_f\,\boldsymbol{\hat{e}_{\perp}}=\dot{l}\,\boldsymbol{\hat{e}_r}+l\dot{f}\,\boldsymbol{\hat{e}_{\perp}}\\ &&\boldsymbol{v_{\rm orb}}=\left[\frac{1}{a(1-e^2)}\right]^{1/2} \left[-e \sin f \,\boldsymbol{\hat{e}_r} + (1- e \cos f) \, \boldsymbol{\hat{e}_{\perp}} \right]\end{aligned}$$ where $l$ and $a$ are in units of $M$ and $\boldsymbol{\hat{e}_r}=(\cos f \,\boldsymbol{\hat{x}} + \sin f \,\boldsymbol{\hat{y}})$ and $\boldsymbol{\hat{e}_{\perp}}=(-\sin f\, \boldsymbol{\hat{x}} + \cos f \,\boldsymbol{\hat{y}})$ are the unit vectors parallel and perpendicular to $\boldsymbol{l}$, respectively. By conservation of momentum, ${\boldsymbol{v_{\rm orb2}}} = {\boldsymbol{v_{\rm orb}}} /(1+q) $, yielding the Doppler shifts for emission from the secondary and primary mini-disks $$\begin{aligned} \label{eq:orb1} \Delta \nu_{\rm orb2} = \frac{\nu_0}{1+q}\left[\frac{1}{a(1-e^2)}\right]^{1/2}[- \sin f \sin i \cos\phi + (\cos f - e) \sin i \sin\phi] \end{aligned}$$ $$\label{eq:orb2} \Delta \nu_{\rm orb1}= -q\, \Delta \nu_{\rm orb2}$$ Because we are interested in the value of the total flux in some arbitrary normalized units (as opposed to the absolute units) the distance $d$ from the observer to the SBHB system can be omitted because it is the same for all three disks (see equation \[eq:flux1\]). Given the choice of dimensionless units employed in our calculation of emissivities in equations \[eq:emiss2\]–\[eq:emiss4\], the components of flux associated with the primary, secondary and circumbinary disk are proportional to $M_1^2$, $M_2^2$ and $M^2$, respectively. Therefore, in the expression for the total flux the relative contributions from each disk should be scaled in terms of the SBHB mass ratio $q$ as $$F_{\rm tot} = \frac{1}{(1+q)^2}\, F_1 + \frac{q^2}{(1+q)^2}\, F_2 + F_3$$ C: Dependence of Statistical Distribution Functions on $F_c$ {#sec:appendixcutoff} ============================================================ {width=".8\textwidth"} In this section we investigate the dependence of the distribution functions, characterizing the modeled profile shapes, on the value of $F_c$, a cutoff used to mimic some fiducial level of spectral noise. As noted in Section \[sec:stats\] we adopt $F_c = 0.01$ in calculation of statistical properties presented in this work but do not introduce actual fluctuations due to noise to the profiles. Figure \[fig:cutoff\] illustrates how different noise levels impact the line profile shapes, where in addition to $F_c = 0.01$ we examine the values of 0.1 and 0.2. With “noise" subtracted from the profile, we rescale the flux above this cutoff so that the maximum flux measured at the peak wavelength always has the value of 1.0. One apparent consequence of the higher level of noise is that it can mask low intensity features present in the profile wings and hence, affect its statistical properties. The middle and right panel of Figure \[fig:cutoff\] show that when the noise conceals the low intensity feature between 4900 and $5200\AA$ the profile centroid (marked by the green vertical line) changes from $C=4842\AA$ to $4830\AA$ to $4828\AA$ for $F_c = 0.01$, 0.1 and 0.2, respectively. The sensitivity to the level of noise is particularly pronounced for higher order distribution functions that depend on the term $(\lambda_i - C)^n$, where index $n$ represents the order (see Section \[sec:stats\]). For example, the value of AI, which is proportional to the third moment, indicates that the profile changes from asymmetric (${\rm AI}=2.21$ at $F_c=0.01$) to relatively symmetric (${\rm AI}=0.35$ at $F_c=0.1$) with increasing $F_c$. Similarly, the value of KI, which is proportional to the fourth moment, indicates a transition from a cuspy (${\rm KI}=9.75$ at $F_c=0.01$) to a more boxy profile (${\rm KI}=2.02$ at $F_c=0.1$). This behavior of higher order distribution functions is illustrated in Figure \[fig:ai\_noise\] which shows the AI-KI maps associated with eccentric SBHB systems and calculated for different values of $F_c$. They show that the overall footprint of the 2-dimensional distribution increases for the higher levels of noise while at the same time the average profile (traced by the blue and green colors) becomes more boxy. The value of the Pearson skewness coefficient, AIP, on the other hand exhibits a weak dependance on $F_c$. For the profile in Figure \[fig:cutoff\] for example, ${\rm AIP}=0.21$ at $F_c=0.01$ and ${\rm AIP}=0.19$ at $F_c=0.1$. As discussed earlier, the AI and AIP provide different measures of the profile asymmetry. This is because the AI sensitively depends on the low intensity features in the profile wings and AIP diagnoses the asymmetry in the bulk of the profile. This property of AIP is captured in Figure \[fig:aip\_noise\], which shows the AIP-PS maps associated with eccentric SBHB systems. The map footprint and distribution of values in different panels show little change as both AIP and PS are weak functions of $F_c$. {width=".7\textwidth"} {width=".75\textwidth"} [^1]: The time scale on which the angular momentum is transported outwards through the disk. [^2]: See however @lodato13 for a different view. [^3]: In an alternative approach anomalous line ratios have been used to flag SBHB candidates with perturbed BLRs [@montuori11; @montuori12]. [^4]: Related to the properties calculated in this work as $\mu_3 = \sigma^3 {\rm AI}$.

--- abstract: 'In this work we implement the Minimal Geometric Deformation method to obtain the isotropic sector and the decoupler matter content of any anisotropic solution of the Einstein field equations with cosmological constant in $2+1$ dimensional space–times. We obtain that the solutions of both sectors can be expressed analytically in terms of the metric functions of the original anisotropic solutions instead of formal integral as in its $3+1$ counterpart. As a particular example we study a regular black hole solution and we show that, depending on the sign of the cosmological constant, the solutions correspond to regular black holes violating the null energy condition or to a non–regular black hole without exotic hair. The exotic/non–exotic and the regular/non–regular black hole dualities are discussed.' address: 'Yachay Tech University, School of Physical Sciences & Nanotechnology, 100119-Urcuquí, Ecuador\' author: - 'Ernesto Contreras [[^1] [^2]]{}' title: 'Gravitational decoupling in $2+1$ dimensional space–times with cosmological term' --- Introduction {#intro} ============ The interest in the Minimal Geometric Deformation (MGD) [@randall1999a; @randall1999b; @antoniadis1990; @antoniadis1998; @ovalle2008; @ovalle2009; @ovalle2010; @casadio2012; @ovalle2013; @ovalle2013a; @casadio2014; @casadio2015; @ovalle2015; @casadio2015b; @ovalle2016; @cavalcanti2016; @casadio2016a; @ovalle2017; @rocha2017a; @rocha2017b; @casadio2017a; @ovalle2018; @estrada2018; @ovalle2018a; @lasheras2018; @gabbanelli2018; @sharif2018; @fernandez2018; @fernandez2018b; @contreras2018; @estrada; @contreras2018a; @morales] as a powerful method to decouple the Einstein field equations to obtain new solutions [@ovalle2017; @ovalle2018; @estrada2018; @ovalle2018a; @lasheras2018; @gabbanelli2018; @sharif2018; @contreras2018; @contreras2018a; @rincon2018; @ovalleplb; @contreras2018c; @tello2018; @tello2018a; @tello2018b] has considerably increased. Among the main applications of the method we find studies of local anisotropies in spherically symmetric systems [@lasheras2018; @gabbanelli2018; @tello2018; @tello2018a; @tello2018b], hairy black holes [@ovalle2018a] and new anisotropic solutions in $2+1$ dimensional space–times [@contreras2018; @rincon2018]. The method has been extended to solve the inverse problem [@contreras2018a], namely, given any anisotropic solution of the Einstein field equations it is possible to recover the isotropic sector and the decoupler matter content which, after gravitational interaction, led to the anisotropic configuration. In that work, it was found that, for an anisotropic solution with exotic matter sector (negative energy density), the free parameters involved in the MGD can be setted such that both the isotropic and the decoupler sectors satisfy all the energy conditions. It was the first time that a kind of exotic/non–exotic matter was found using the method. As another extension of MGD and the inverse problem, in Ref. [@contreras2018c] the method have been studied considering Einstein’s equations with cosmological constant and implemented in a polytropic black hole which is a solution with a matter content satisfying all the energy conditions. The main finding there was that the isotropic sector is deeply linked with the appearance of exotic matter, although it can be located inside the horizon. In this sense, this work shows again how the apparition of exotic matter seems unavoidable but one could, in principle, control the energy conditions by tuning the isotropy/anisotropy of a black hole solution. The MGD-decoupling have been implemented in $2+1$ circularly symmetric and static space–times obtaining that both the isotropic and the anisotropic sector fulfil Einstein field equations in contrast to the cases studied in $3+1$ dimensions, where the anisotropic sector satisfies “quasi-Einstein” field equations. In this sense, the isotropic and the decoupler sector leads to a pair of new solution of Einstein’s equations, one for each source. In this work we study MGD in $2+1$ circularly symmetric and static space–times with different purposes and interests. First, as a difference to the previous work (see Ref. [@contreras2018]), we consider $2+1$ Einstein’s equation with cosmological constant. This is because, as the BTZ is a vacuum solution of this configuration, the set of equations coming from MGD method could serve as the starting point to extend interior $2+1$ solutions to anisotropic domains taking into account suitable matching conditions between the compact objects and a BTZ vacuum. Second, we study the inverse MGD problem to explore, among other aspects, the exotic/non–exotic matter content duality previously reported in the $3+1$ dimensional case [@contreras2018a; @contreras2018c]. As we shall see later, the inverse method leads to more tractable expressions to deal with because they correspond to exact analytic instead to formal equations as previously reported [@contreras2018a; @contreras2018c]. This work is organized as follows. In the next section we briefly review the MGD-decoupling method. Then, in section \[iso\], we obtain the isotropic sector and the decoupler matter content considering a regular black hole as anisotropic solution. In section \[ec\] we study the energy conditions to explore the apparition of exotic mater in the solutions and some final comments and conclusion are in the last section. Einstein Equations in $2+1$ space–time dimensions {#mgd} ================================================= In a previous work we considered the MGD-method with cosmological constant [@contreras2018c]. In this work study the $2+1$ dimensional case. Let us consider the Einstein field equations $$\begin{aligned} \label{einsorig} R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa^{2}T_{\mu\nu}^{tot},\end{aligned}$$ and assume that the total energy-momentum tensor is given by $$\begin{aligned} \label{total} T_{\mu\nu}^{(tot)}=T_{\mu\nu}^{(m)}+\theta_{\mu\nu},\end{aligned}$$ As usual, the energy–momentum tensor for a perfect fluid is given by $T^{\mu(m)}_{\nu}=diag(-\rho,p,p)$ and the decoupler matter content reads $\theta^{\mu}_{\nu}=diag(-\rho^{\theta},p_{r}^{\theta},p_{\perp}^{\theta})$. In what follows, we shall assume circularly symmetric space–times with a line element parametrized as $$\begin{aligned} \label{le} ds^{2}=-e^{\nu}dt^{2}+e^{\lambda}dr^{2}+r^{2}d\phi^{2},\end{aligned}$$ where $\nu$ and $\lambda$ are functions of the radial coordinate $r$ only. Considering Eq. (\[le\]) as a solution of the Einstein Field Equations, we obtain $$\begin{aligned} \kappa ^2 \tilde{\rho}&=&-\Lambda+\frac{e^{-\lambda} \lambda'}{2 r}\label{eins1}\\ \kappa ^2 \tilde{p}_{r}&=&\Lambda+\frac{e^{-\lambda} \nu '}{2 r}\label{eins2}\\ \kappa ^2 \tilde{p}_{\perp}&=&\Lambda+\frac{1}{4} e^{-\lambda} \left(-\lambda ' \nu '+2 \nu ''+\nu '^2\right)\label{eins3}\end{aligned}$$ where the prime denotes derivation with respect to the radial coordinate and we have defined $$\begin{aligned} \tilde{\rho}&=&\rho+\rho^{\theta}\label{rot}\\ \tilde{p}_{r}&=&p+p_{r}^{\theta}\label{prt}\\ \tilde{p}_{\perp}&=&p+p_{\perp}^{\theta}.\label{ppt}\end{aligned}$$ The next step consists in decoupling the Einstein Field Equations (\[eins1\]), (\[eins2\]) and (\[eins3\]) by implementing the minimal deformation $$\begin{aligned} \label{def} e^{-\lambda}&=&\mu +\alpha f.\end{aligned}$$ As usual, Eq. [\[def\]]{} leads to two sets of differential equations: one describing an isotropic system sourced by the conserved energy–momentum tensor of a perfect fluid $T^{\mu(m)}_{\nu}$ an the other set corresponding to Einstein field equations sourced by $\theta_{\mu\nu}$. Now, as in a previous work [@contreras2018c], we interpret the cosmological constant as an isotropic fluid, so we include the $\Lambda$–term in the isotropic sector and we obtain $$\begin{aligned} \kappa ^2\rho &=& \frac{-2 \Lambda r-\mu '}{2 r}\label{iso1}\\ \kappa ^2 p&=& \frac{2 \Lambda r+\mu \nu '}{2 r}\label{iso2}\\ \kappa ^2 p&=&\frac{4 \Lambda +\mu ' \nu '+\mu \left(2 \nu ''+\nu '^2\right)}{4 } ,\label{iso3}\end{aligned}$$ for the perfect fluid and $$\begin{aligned} \kappa ^2\rho^{\theta}&=&-\frac{f'}{2 r}\label{aniso1}\\ \kappa^{2} p_{r}^{\theta}&=&\frac{f \nu '}{2 r}\label{aniso2}\\ \kappa^{2} p_{\perp}^{\theta}&=&\frac{f' \nu '+f \left(2 \nu ''+\nu '^2\right)}{4},\label{aniso3}\end{aligned}$$ for the anisotropic system [^3]. Note that that the addition of the cosmological constant only affects the isotropic sector because Eqs. (\[aniso1\]), (\[aniso2\]) and (\[aniso3\]) remain unchanged. For the inverse problem we shall apply the same strategy followed in reference [@contreras2018a], namely, we implement the constraint $$\begin{aligned} \label{cons} \tilde{p}_{\perp}-\tilde{p}_{r}=\alpha(p^{\theta}_{\perp}-p^{\theta}_{r})\end{aligned}$$ It is remarkable that, unlike the $3+1$ case, the solution for $f(r)$ obtained from the constraint (\[cons\]) is an exact expression in terms of $\nu,\lambda$, instead of a combination of formal integrals as found in [@contreras2018a]. In fact, $$\begin{aligned} \label{f} f=\frac{c_1 r^2 e^{-\nu }}{\nu '^2}+\frac{e^{-\lambda}}{\alpha }\end{aligned}$$ where $c_1$ is a constant of integration. Now, from Eq. (\[def\]), we obtain $$\begin{aligned} \label{mu} \mu=-\frac{\alpha c_1 r^2 e^{-\nu}}{\nu '^2}\end{aligned}$$ Note that somehow, the constant $c_1$ controls the geometric deformation: if $c_1\to0$ there is not deformation at all. The matter content of the isotropic sector reads $$\begin{aligned} \rho&=&-\Lambda-\frac{\alpha c_1 e^{-\nu} \left(2 r \nu ''+\nu ' \left(r \nu '-2\right)\right)}{2 \nu '^3}\\ p&=&\Lambda -\frac{\alpha c_1 r e^{-\nu}}{2 \nu '},\end{aligned}$$ and the decoupler matter content satisfies $$\begin{aligned} \rho^{\theta}&=&\frac{1}{2} \left(\frac{c_1 e^{-\nu} \left(2 r \nu ''+\nu ' \left(r \nu '-2\right)\right)}{\nu '^3}+\frac{e^{-\lambda} \lambda '}{\alpha r}\right)\\ p_{r}^{\theta}&=&\frac{c_1 r e^{-\nu}}{2 \nu '}+\frac{e^{-\lambda} \nu '}{2 \alpha r}\\ p_{\perp}^{\theta}&=&\frac{1}{4} \left(\frac{2 c_1 r e^{-\nu}}{\nu '}+\frac{e^{-\lambda} \left(-\lambda '\nu '+2 \nu ''+\nu '^2\right)}{\alpha }\right)\end{aligned}$$ At this point, some comments are in order. First, Eqs. (\[iso1\]), (\[iso2\]) and (\[iso3\]) correspond to Einstein field equations with cosmological constant for a perfect fluid. Second, Eqs. (\[aniso1\]), (\[aniso2\]) and (\[aniso3\]) corresponds to Einstein field equations without cosmological constant and anisotropic decoupler fluid. Note that this set of equations together with those of the isotropic sector allows us to decouple Einstein’s equation with cosmological constant in $2+1$ dimensional space–times for any anisotropic fluid. What is more, the above expressions can be used to extend isotropic solutions embedded in a BTZ vacuum to anisotropic domains after the implementation of suitable matching conditions. Finally, that the inverse problem leads to exact analytical expressions entails the “isotropization” of a broader set of systems than its $3+1$ dimensional counterpart which is given in terms of formal integrals. In this sense, the inverse problem in $2+1$ dimensions can be implemented starting from any anisotropic solutions at difference to the $3+1$ case where depending on the particular form of the anisotropic solution, the inverse problem would yield to formal instead exact analytical solutions. In the next section we shall implement the inverse problem using a well known regular and circularly symmetric black hole solution as anisotropic system. Isotropic sector of a regular black hole in 2+1 dimensions {#iso} ========================================================== In this section we shall illustrate the inverse MGD problem using as anisotropic solution a regular black hole metric. The reason to consider a regular black hole solution is twofold: to explore the conditions for the apparition of exotic matter and to study if the MGD inverse problem can affect the regularity of the solution. Let us consider the regular black hole solution [@garcia2017] with metric functions $$\begin{aligned} e^\nu&=& -M-\Lambda r^2-q^{2}\log\left(a^2+r^2\right)\\ e^{-\lambda}&=&-M-\Lambda r^2-q^{2}\log \left(a^2+r^2\right)\end{aligned}$$ where $M$, $\Lambda$, $a$ and $q$ are free parameters. This geometry is sustained by a matter content given by $$\begin{aligned} \tilde{\rho}&=&\frac{q^2}{8\pi\left(a^2+r^2\right)}\\ \tilde{p}_{r}&=&-\frac{q^2}{8\pi\left(a^2+r^2\right)}\\ \tilde{p}_{\perp}&=&\frac{q^2 \left(r^2-a^2\right)}{8\pi\left(a^2+r^2\right)^2}\end{aligned}$$ This solution corresponds to a black hole whenever $ -M-\Lambda r^2-q^{2}\log\left(a^2+r^2\right)=0$ leads to two real roots (or one real root in the extreme case) for some values of the parameters $\{M,\Lambda,q,a\}$ [@garcia2017]. What is more, the solution is regular everywhere, which can be deduced from the invariants $$\begin{aligned} R&=&\frac{2 q^2 \left(3 a^2+r^2\right)}{\left(a^2+r^2\right)^2}+6 \Lambda\\ Ricc=\mathcal{K}&=&\frac{4 q^4 \left(3 a^4+2 a^2 r^2+r^4\right)}{\left(a^2+r^2\right)^4}\nonumber\\ &&+\frac{8 \Lambda q^2 \left(3 a^2+r^2\right)}{\left(a^2+r^2\right)^2} +12 \Lambda ^2,\end{aligned}$$ where $R$, $Ricc$ and $\mathcal{K}$ correspond to the Ricci, Ricci squared and Kretschmann scalar respectively. From now on we shall apply the inverse MGD problem to obtain the isotropic generator and the decoupler matter content associated with this regular black hole solution. From Eq. (\[f\]), the decoupler function $f$ reads $$\begin{aligned} f&=&-\frac{c_1 \left(a^2+r^2\right)^2 \left(q^2 \log \left(a^2+r^2\right)+M+\Lambda r^2\right)}{4 \left(\Lambda \left(a^2+r^2\right)+q^2\right)^2}\nonumber\\ &&-\frac{q^2 \log \left(a^2+r^2\right)+M+\Lambda r^2}{\alpha }\end{aligned}$$ Now, from Eq. (\[mu\]), the radial metric function of the isotropic sector is given by $$\begin{aligned} \mu=\frac{\alpha c_1 \left(a^2+r^2\right)^2 \left(q^2 \log \left(a^2+r^2\right)+M+\Lambda r^2\right)}{4 \left(\Lambda \left(a^2+r^2\right)+q^2\right)^2}\end{aligned}$$ Replacing the above result in the set of isotropic Einstein equations, Eqs. (\[iso1\]), (\[iso2\]) and (\[iso3\]), the perfect fluid reads $$\begin{aligned} &&\rho= -\frac{\alpha c_1 a^{2}_{r}\left(2 \Lambda q^2 \left(a^{2}_{r}+r^2\right)+\Lambda ^2 a_{r}^2+2 M q^2+q^4\right)}{32\pi \lambda_{r}^3}\nonumber\\ &&-\frac{2 \alpha c_1 q^4 a^{2}_{r} \log a^{2}_{r}+4 \Lambda \lambda_{r}^3}{ 32\pi\lambda_{r}^3}\\ &&p=\frac{\alpha c_1 \left(a^2+r^2\right)}{32\pi \left(\Lambda \left(a^2+r^2\right)+q^2\right)}+\frac{\Lambda }{8\pi}\end{aligned}$$ where $$\begin{aligned} a_{r}^{2}&:=&a^{2}+r^2\\ \Lambda_{r}&:=&\Lambda a^{2}_{r}+q^2\end{aligned}$$ At this point some comments are in order. First, note that, as in the $3+1$ case, the inverse problem does not modify the position of the killing horizon. In fact, the horizon appears whenever $-M-\Lambda r^2-q^{2}\log\left(a^2+r^2\right)=0$ which, as discussed above, leads to one or two real roots for the black hole solution. Second, the regularity of the solution depends on the positivity of the parameter $\Lambda$. In fact, the invariants $$\begin{aligned} R&=&-\alpha c_{1}a_{r}^{2}\left(\frac{ \left(2 M q^2+3 q^4\right)}{2 \Lambda_{r}^3} -\frac{2 q^4 \log a_{r}^{2}}{2 \Lambda_{r}^3}\right.\nonumber\\ &&\left.-\frac{2 \Lambda q^2 \left(3 a^2+4 r^2\right)}{2 \Lambda_{r}^3}-\frac{3 \Lambda ^2 a_{r}^{3}}{2 \Lambda_{r}^3}\right)\\ Ricc=\mathcal{K}&=&\frac{\alpha ^2 c_1^2 a_{r}^4 \left(\mathcal{F}+2 \Lambda _r^4\right)}{4 \Lambda _r^6}\end{aligned}$$ where $$\begin{aligned} \mathcal{F}=\left(2 q^4 \log a_{r}^{2}+2 \Lambda q^2 a_{r}^{2}+ \Lambda ^2 a_{r}^4+2 M q^2+q^4\right)^2,\end{aligned}$$ reveal that the solution is regular everywhere whenever $\Lambda_{r}=\Lambda \left(a^2+r^2\right)+q^2\ne 0$, which can be satisfied if $\Lambda>0$. In particular, from the horizon condition we can obtain the condition for $\Lambda>0$ in terms of the other parameters as $$\begin{aligned} \Lambda=-\frac{M+q^{2}\log(a^{2}+r_{H}^{2})}{r_{H}^{2}}\end{aligned}$$ from where it must be imposed that $r_{H}^{2}+a^{2}<1$ and $q^{2}|\log(a^{2}+r_{H}^{2})|>M$, with $r_{H}$ the horizon radius. In this case, the solution corresponds to a regular isotropic black hole solution. Note that, in the case $\Lambda<0$ the solution has a critical radius $r_{c}$ as frequently found in the application of MGD. This result would lead to a naked singularity for $r_{c}>r_{H}$ or to a non regular black hole solution for $r_{c}<r_{H}$. In the last case, we say that the isotropic sector of the regular black hole corresponds to a non-regular black hole solution. Now we focus our attention into the decoupler sector. In this case, the metric functions are $\{\nu,f\}$ and the decoupler matter content reads $$\begin{aligned} \rho^{\theta}&=&\frac{c_1 a_r^2 \left(2 \Lambda q^2 \left(a_{r}^2+r^{2}\right)+\Lambda ^2 a_r^4+2 M q^2+q^4\right)}{4 \Lambda _r^3}\nonumber\\ &&+\frac{q^2}{\alpha a_r^2} +\frac{c_1 q^4 a_r^2 \log a_r^2}{2 \Lambda _r^3}+\frac{\Lambda }{\alpha }\\ p_{r}^{\theta}&=&-\frac{c_1 a_r^2}{4 \Lambda _r}-\frac{q^2}{\alpha a_r^2}+\frac{\Lambda }{\alpha }\\ p_{\perp}^{\theta}&=&\frac{q^2 \left(r^2-a^2\right)}{\alpha a_r^4}-\frac{c_1 a_r^2}{4 \Lambda _r}-\frac{\Lambda }{\alpha}\end{aligned}$$ The above solution corresponds to an anisotropic regular black hole solution for $\Lambda>0$. In fact, the solution has a killing horizon when $-M-\Lambda r^2-q^{2}\log\left(a^2+r^2\right)=0$. The invariants are given by $$\begin{aligned} R&=&\frac{\mathcal{H}_{1}}{\left(a^2+r^2\right)^2 \left(\Lambda \left(a^2+r^2\right)+q^2\right)^3}\\ Ricc&=&\frac{\mathcal{H}_{2}}{\left(a^2+r^2\right)^4 \left(\Lambda \left(a^2+r^2\right)+q^2\right)^6}\\ \mathcal{K}&=&\frac{\mathcal{H}_{3}}{\left(a^2+r^2\right)^4 \left(\Lambda \left(a^2+r^2\right)+q^2\right)^6}\end{aligned}$$ where $\mathcal{H}_{1}$, $\mathcal{H}_{2}$ and $\mathcal{H}_{3}$ are (too long) regular functions in terms of polynomials of $r$ and $\log(a^{2}+r^{2})$. Note that, as discussed above, the regularity of the solution depends on the sign of $\Lambda$. More precisely, for $\Lambda<0$ the solution is the one of a non–regular black hole solution. It is worth mentioning that the solutions obtained here could be considered as “hairy” black holes and that the nature of such a hair fields depends on the energy conditions that we shall discuss in what follows. Energy conditions {#ec} ================= In this section we study the energy conditions of the obtained solution for the distinct cases outlined before. Case I. $\Lambda>0$ ------------------- As previously discussed, this case corresponds to regular black hole solutions for the isotropic and decoupler sector with the horizon radius located at $-M-\Lambda r_{H}^{2}-q^{2}\log(a^{2}+r_{H}^{2})=0$. Given the nature of the solution, a numerical analysis is mandatory. However, setting suitable values of $M,\Lambda,q,a$ to obtain real horizons and $\Lambda>0$, we infer that the behaviour of the energy density can be written as $$\begin{aligned} &&\lim\limits_{r\to0}\rho=A\alpha c_1-B\\ &&\lim\limits_{r\to\infty}\rho=-C \alpha c_1-B\end{aligned}$$ with $A$, $B$, $C$ real and positive numbers. In particular, for $\Lambda=2$, $M=1$, $q=1$ and $a= 0.1$ we obtain $$\begin{aligned} &&\lim\limits_{r\to0}\rho=0.000578336 \alpha c_1-0.0795775\\ &&\lim\limits_{r\to\infty}\rho=-0.00497359 \alpha c_1-0.0795775\end{aligned}$$ Note that the apparition of exotic matter is unavoidable. In fact, if $\alpha c_1$ is a positive (negative) quantity such that $\lim\limits_{r\to0}\rho>0\ (\lim\limits_{r\to0}\rho<0)$, it is obtained that $\lim\limits_{r\to\infty}\rho>0\ (\lim\limits_{r\to\infty}\rho<0)$ necessarily. In figure \[fig1\] we show the profile of the energy density for some values of $\alpha c_{1}$. ![\[fig1\] Energy density for for $\alpha c_{1}=-200$ (black solid line), $\alpha c_{1}=-140$ (dashed blue line) $\alpha c_{1}=400$ (short dashed red line) and $\alpha c_{1}=500$ (dotted green line). ](fig1){width="\linewidth"} For the decoupler sector we obtain that the negative values for the energy density can be avoided with a suitable choice of the parameter. Without loss of generality, let us set $\Lambda=2$, $M=1$, $q=1$ and $a= 0.1$ to ensure that the horizon radius is real and positive. Now $$\begin{aligned} \lim\limits_{r\to0}\rho^{\theta}&=&\frac{102.}{\alpha }-0.0145352 c_1\\ \lim\limits_{r\to\infty}\rho^{\theta}&=&\frac{2}{\alpha }+\frac{c_1}{8}\end{aligned}$$ Note that for $\alpha$ and $c_{1}$ positive values, the exotic matter content can be avoided whenever $c_{1}\le\frac{7017.46}{\alpha }$. In figure \[fig2\] we show the profile of $\rho^{\theta}$ for $\alpha=1$ and some values for $c_{1}$. ![\[fig2\] Energy density for for $c_{1}=1$ (black solid line), $c_{1}=10$ (dashed blue line) $c_{1}=50$ (short dashed red line) and $c_{1}=100$ (dotted green line). ](fig2){width="\linewidth"} We would like to conclude this section by emphasizing that for $\Lambda>0$ the exotic matter can be partially avoided. In fact, while the perfect fluid solution contain negative energy density for suitable values of $M,q,\Lambda,a$, the apparition of exotic matter in the decoupler sector can be circumvented for certain values of the parameters involved. In the next section we shall study the energy conditions for $\Lambda<0$. Case II. $\Lambda<0$ -------------------- In this case, $\Lambda<0$ as long as $r_{H}^{2}+a^{2}<1$ and$q^{2}|\log(a^{2}+r_{H}^{2})|>M$ or $r^{2}+a^{2}>1$. For example, we can choose the values $M=2$, $q=1$, $a=1$ from where $\Lambda=-0.902359$. For these values, the horizon radius is located at $r_{H}=2$ and the critical radius is at $r_{c}=0.328946$ such that the solution corresponds to a non–regular black hole solution. For this particular choosing of the parameters and for $\alpha c_{1}>0$, the asymptotic behaviour of the energy density is given by $$\begin{aligned} &&\lim\limits_{r\to r_{c}}\rho=\infty\\ &&\lim\limits_{r\to\infty}\rho=0.0110235 \alpha c_1+0.0359037.\end{aligned}$$ In figure in figure \[fig3\] we show the profile of $\rho$ as a function of the radial coordinate for different values of $\alpha c_{1}>0$. ![\[fig3\] Energy density for for $\alpha c_{1}=0.1$ (black solid line), $\alpha c_{1}=0.2$ (dashed blue line) $\alpha c_{1}=0.5$ (short dashed red line) and $\alpha c_{1}=1$ (dotted green line). ](fig3){width="\linewidth"} Now, let us turn out our attention in the decoupler sector. For $M=2$, $q=1$, $a=1$ and $\Lambda=-0.902359$ we obtain $$\begin{aligned} \lim\limits_{r\to\infty}\rho^{\theta}=-\frac{0.902359}{\alpha }-0.277051 c_1\end{aligned}$$ so that for $\alpha<0$, $c_{1}<0$ the energy density reach a positive value asymptotically . In particular for $\alpha=-1$ we obtain that $$\begin{aligned} \lim\limits_{r\to r_{c}}\rho^{\theta}\to\infty\end{aligned}$$ and we obtain that the exotic matter can be avoided. In figure \[fig4\] we show the profile of the energy density $\rho^{\theta}$ for $\alpha=-1$ and different values of $c_{1}$. ![\[fig4\] Energy density for for $c_{1}=-1$ (black solid line), $c_{1}=-2$ (dashed blue line) $c_{1}=-3$ (short dashed red line) and $c_{1}=-4$ (dotted green line). ](fig3){width="\linewidth"} At his point a couple of comments are in order. First, note that in both cases (isotropic and decoupler sector) the exotic content can be avoided. Second, for suitable choices of the parameters, the solution corresponds to a non–regular black hole containing a non–vanishing critical radius. In this sense, we conclude that although the exotic/non–exotic duality can be circumvented for $\Lambda<0$, the nature of the black hole solution of the isotropic and decoupler sector leads to a kind of regular/non–regular duality. Conclusions {#remarks} =========== In this work we have extended the Minimal Geometric Deformation method in $2+1$ dimensional space–times to decouple the Einstein field equations including cosmological constant. We obtained that the isotropic sector obeys Einstein’s equation with cosmological constant but the decoupler part consists in a system without cosmological term. In this sense, we can combine any $2+1$ isotropic, static and circularly symmetric interior solution of the Einstein field equations with cosmological constant embedded in a BTZ vacuum with certain decoupler matter solution and suitable matching conditions to obtain new anisotropic interior solutions in the three dimensional realm. We showed that the inverse problem leads to exact analytical solutions for the decoupling and the isotropic metric in terms of the original anisotropic solution instead to formal integrals obtained in the $3+1$ counterpart. The scope of this result to obtain analytical solutions is broad. Indeed, it can be implemented taking into account any anisotropic solution as the starting point because it does not involve formal integrals as the $3+1$ case. As a particular example we implemented the inverse problem to a regular $2+1$ black hole solution. We obtain that for a positive cosmological constant the isotropic sector corresponds to a regular isotropic black hole in presence of a “exotic” hair (negative energy density), and the decoupler sector is a regular anisotropic black hole which, under certain circumstances, can be surrounded by a matter content with positive energy density so that the apparition of exotic matter can be avoided. For negative cosmological constant both the isotropic and the decoupler sector corresponds to non–regular black hole solution where the existence of exotic hair can be avoided with a suitable choice of the free parameters. It is worth mentioning that on one hand the non–regular black hole solution is singular in a non–vanishing radius as often occur in the implementation of the Minimal Geometric Deformation protocol. On the other hand, the exotic matter can be avoided but the price that it has to be paid is that the the solutions are not regular anymore. In this sense, the kind of exotic/non-exotic matter duality appearing in previous works transmute to a regular/non-regular duality in the cases where the exotic content can be avoided. Acknowledgement =============== The author would like to acknowledge Pedro Bargueño for fruitful discussions. L. Randal, R. Sundrum. Phys. Rev. Lett. [**83**]{}, 3370 (1999). L. Randal, R. Sundrum. Phys. Rev. Lett. [**83**]{}, 4690 (1999). I. 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--- abstract: 'The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu’s metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki’s proof of Kollár’s injectivity theorem. Moreover we investigate the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics.' address: 'Kagoshima University, 1-21-35 Koorimoto, Kagoshima 890-0065, Japan.' author: - 'SHIN-ICHI MATSUMURA' title: | A Nadel vanishing theorem\ for metrics with minimal singularities\ on big line bundles --- [^1] Introduction ============ The Kodaira vanishing theorem plays an important role when we approach certain fundamental problems of algebraic geometry and the theory of several complex variables (for example, asymptotics of linear systems, extension problems of holomorphic sections, the minimal model program, and so on). By using multiplier ideal sheaves associated to singular metrics, this theorem can be generalized to the Nadel vanishing theorem, which can be seen as an analytic analogue of the Kawamata-Viehweg vanishing theorem of algebraic geometry ([@Kaw82], [@Vie82]). In this paper, we study singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu’s metrics) and a Nadel vanishing theorem for them from the view point of complex (differential) geometry. \[Nad\] Let $F$ be a big line bundle on a smooth projective variety $X$ and $h$ be a $($singular$)$ metric on $F$ with strictly positive curvature. Then we have $$H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(h)})= 0 \quad \text{for}\ \text{any}\ i >0.$$ Here ${\mathcal{I}(h)}$ denotes the multiplier ideal sheaf of the $($singular$)$ metric $h$ and $K_{X}$ denotes the canonical bundle of $X$. We mainly treat metrics with minimal singularities $h_{\min}$ and Siu’s metrics $h_{\rm{Siu}}$ (see Section 2, 3 for the definition). These metrics satisfy many important properties (for example they admit an analytic Zariski decomposition), thus several authors study these metrics (see [@DEL00], [@Dem], [@Pau12], [@Siu98]). The main purpose of this paper is to establish a Nadel vanishing theorem for $h_{\min}$ and $h_{\rm{Siu}}$. When we investigate the cohomology groups with coefficients in $K_{X} \otimes F \otimes {\mathcal{I}(h_{\min})}$ and $K_{X} \otimes F \otimes {\mathcal{I}(h_{\rm{Siu}})}$, we encounter the following difficulties:\ (1) $h_{\min}$ and $h_{\rm{Siu}}$ may have non-algebraic (transcendental) singularities.\ (2) $h_{\min}$ and $h_{\rm{Siu}}$ do not have strictly positive curvature except the trivial case. The proof of Theorem \[Nad\] heavily depends on the assumption that the curvature of $h$ is strictly" positive. Under this assumption, we can construct solutions of the ${\overline{\partial}}$-equation with $L^{2}$-estimates, which implies Theorem \[Nad\] (see [@Dem82]). In fact, the theorem fails without this assumption even if the curvature of $h$ is semi-positive. Nevertheless, we can expect that all higher cohomology groups with coefficients in $K_{X} \otimes F \otimes {\mathcal{I}(h_{\min})}$ and $K_{X} \otimes F \otimes {\mathcal{I}(h_{\rm{Siu}})}$ vanish from the special characteristics of $h_{\min}$ and $h_{\rm{Siu}}$. This is because, for a big line bundle $F$ we have already known $$\begin{aligned} H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(\|F \|)}) = 0 \quad \text{for}\ \text{any}\ i >0, \end{aligned}$$ where ${\mathcal{I}(\|F \|)}$ is the asymptotic multiplier ideal sheaf of $F$ (see [@DEL00] for the precise definition). The multiplier ideal sheaves ${\mathcal{I}(h_{{{\rm{min}}}})}$ and ${\mathcal{I}(h_{\rm{Siu}})}$ can be seen as an analytic counterpart of ${\mathcal{I}(\|F \|)}$. The asymptotic multiplier ideal sheaf ${\mathcal{I}(\|F \|)}$ does not always coincide with $ {\mathcal{I}(h_{\min})}$, but it is conjectured that the equality ${\mathcal{I}(\|F \|)} = {\mathcal{I}(h_{\min})}$ holds if $F$ is big or more generally abundant (that is, the numerical dimension $\nu(F)$ agrees with the Kodaira dimension $\kappa(F)$). Hence it is interesting to study a Nadel vanishing theorem for ${\mathcal{I}(h_{{{\rm{min}}}})}$ and ${\mathcal{I}(h_{\rm{Siu}})}$. This is a natural problem, however it has been an open problem for a long time. In this paper, we give an affirmative answer for this problem, which is the main result of this paper. \[m-min\] Let $F$ be a big line bundle on a smooth projective variety $X$ and $h_{\min}$ be a metric with minimal singularities on $F$. Then we have $$H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(h_{\min})}) = 0 \quad {\text{for}}\ {\text{any}}\ i >0.$$ This theorem follows from Theorem \[main\], which says that all higher cohomology groups with coefficients in $K_{X} \otimes F \otimes {\mathcal{I}(h)}$ vanish if the singular metric $h$ is less singular than some singular metric with strictly positive curvature. From Theorem \[main\], we can also obtain the same conclusion for $h_{\rm{Siu}}$ (see Corollary \[main-co\]). These results are closely related to the openness conjecture of Demailly-Koll$\rm{\acute{a}}$r in [@DK01], which is a conjecture on singularities of plurisubharmonic (psh for short) functions. Indeed, if $F$ is a big line bundle and $h_{\min}$ has algebraic singularities, we can easily check ${\mathcal{I}(\|F \|)} = {\mathcal{I}(h_{\min})}$. However we emphasize that $h_{\min}$ does not always have algebraic singularities (see Section 4). For the proof of the main result, we need to take a transcendental approach since we need to overcome difficulties (1), (2). The proof is based on a combination of the technique to solve the ${\overline{\partial}}$-equation and Enoki’s proof of Kollár’s injectivity theorem for semi-positive line bundles (see [@Eno90], [@Kol86]). The strategy of the proof can be divided into four steps as follows: In Step 1, we approximate the metric $h_{\min}$ by singular metrics $\{ h_{{\varepsilon}} \}_{{\varepsilon}>0}$ that are smooth on a Zariski open set. Then we can represent a given cohomology class by the associated harmonic form $u_{{\varepsilon}}$ with respect to $h_{{\varepsilon}}$ on the Zariski open set. In Step 2, by using a suitable (holomorphic) section $s$ of some positive multiple $F^{m}$, we show that the norm of $D^{''*} su_{{\varepsilon}}$ converges to zero as letting ${\varepsilon}$ go to zero, where $D^{''*} $ is the adjoint operator of ${\overline{\partial}}$. Unfortunately the curvature of $h_{{\varepsilon}}$ is not semi-positive. For this reason, we need to generalize Enoki’s technique for Kollár’s injectivity theorem by applying the theory of harmonic integrals. In Step 3, we solve the ${\overline{\partial}}$-equation. By considering $su_{{\varepsilon}}$ instead of $u_{{\varepsilon}}$, we can construct a solution $\beta_{{\varepsilon}}$ of the ${\overline{\partial}}$-equation ${\overline{\partial}}\beta_{{\varepsilon}} = su_{{\varepsilon}}$. Moreover we can show that the $L^{2}$-norm $\| \beta_{{\varepsilon}} \|$ is uniformly bounded from the special characteristics of $h_{\min}$. In Step 4, we investigate the limit of $u_{{\varepsilon}}$. The above arguments yield $$\| su_{{\varepsilon}} \| ^{2} = {{\langle\!\langle}}su_{{\varepsilon}}, {\overline{\partial}}\beta_{{\varepsilon}} {{\rangle\!\rangle}}\leq \| D^{''*} su_{{\varepsilon}}\| \| \beta_{{\varepsilon}} \| \to 0 \quad {\text{as }} {\varepsilon}\to 0.$$ From this convergence we prove that $u_{{\varepsilon}}$ converges to zero in a suitable sense, which completes the proof. This paper is divided into four sections. In Section 2, we recall the fundamental results that are often used in this paper. In Section 3, we give a proof of the main result. In Section 4, we give examples that tell us that metrics with minimal singularities do not always have algebraic singularities. This paper is a revised version of our preprint [@Mat13-A]. Ten months after we finish writing our preprint, Guan and Zhou announced that they solved the strong openness conjecture in [@GZ13]. Although their celebrated result implies the main theorem, we believe that it is worth to display our techniques. This is because, our view point is quite different from that of them and our techniques seem to have some applications. In fact, we can give an interesting generalization of the injectivity theorem by applying our techniques (see [@Mat13-B]). Acknowledgment {#acknowledgment .unnumbered} -------------- The author is supported by the Grant-in-Aid for Young Scientists (B) $\sharp$25800051 from JSPS. Preliminaries ============= In this section, we recall the fundamental results needed later. Refer to [@Dem], [@Dem-book], [@Fuj12-A] and [@Laz] for more details. Singular Metrics and the Nadel Vanishing Theorem ------------------------------------------------ Throughout this subsection, let $X$ be a compact complex manifold and $F$ be a line bundle on $X$. First we recall the definitions and properties of singular metrics and their multiplier ideal sheaves. \[s-met\] (1) A (hermitian) metric $h$ on $F$ is called a [*singular metric*]{}, if for a local trivialization $\theta : F|_{U} \cong U \times \mathbb{C}$ and a local section $\xi$ of $F$ on an open set $U \subset X$, there exists an $L^{1}_{\rm{loc}}$-function $\varphi$ on $U$ such that $$|\xi|_{h} = |\theta (\xi)| e^{- \varphi}.$$ Here $\varphi$ is called the local *weight* of $h$ with respect to the trivialization.\ (2) A (singular) metric $h$ on $F$ is said to have [*algebraic*]{} (resp. [*analytic*]{}) [*singularities*]{}, if there exists an ideal sheaf $\mathcal{I} \subset \mathcal{O}_{X}$ such that a local weight $ \varphi$ of $h$ can be locally written as $$\varphi = \frac{c}{2} \log \big( |f_{1}|^{2} + |f_{2}|^{2} + \cdots + |f_{k}|^{2}\big) +v,$$ where $c \in \mathbb{Q}_{>0}$ (resp. $c \in \mathbb{R}_{>0}$), $f_{1}, \dots f_{k}$ are local generators of $\mathcal{I}$ and $v$ is a smooth function.\ (3) The [*curvature current*]{} $\sqrt{-1} \Theta_{h}(F)$ associated to $h$ is defined by $$\sqrt{-1} \Theta_{h}(F) = {dd^c}\varphi,$$ where $ \varphi$ is a local weight of $h$. In this paper, we simply abbreviate the singular metric (resp. the curvature current) to the metric (resp. the curvature). The Levi form ${dd^c}\varphi$ is taken in the sense of distributions. Note that curvatures are $(1,1)$-currents but not always smooth $(1,1)$-forms. The curvature $\sqrt{-1} \Theta_{h}(F)$ is said to be [*positive*]{} if $\sqrt{-1} \Theta_{h}(F) \geq 0$ in the sense of currents. If $\sqrt{-1} \Theta_{h}(F) \geq \omega$ for some hermitian form $\omega$ on $X$, then $h$ is said to have [*strictly positive*]{} curvature. Let $h$ be a metric on $F$ such that $\sqrt{-1} \Theta_{h}(F) \geq \gamma$ for some smooth $(1,1)$-form $\gamma$ on $X$. Then the ideal sheaf ${\mathcal{I}(h)}$ defined to be $${\mathcal{I}(h)}(U):= \{f \in \mathcal{O}_{X}(U)\mid |f|e^{-\varphi} \in L^{2}_{\rm{loc}}(U) \}$$ for an open set $U \subset X$ is called the *multiplier ideal sheaf* associated to $h$. The original Nadel vanishing theorem (Theorem \[Nad\]) asserts that all higher cohomology groups with coefficients in $K_{X} \otimes F \otimes {\mathcal{I}(h)}$ vanish if $h$ has strictly positive curvature. Our motivation is that one can obtain a Nadel vanishing theorem for metrics with minimal singularities $h_{{{\rm{min}}}}$. When $F$ is not an ample line bundle, the metric $h_{{{\rm{min}}}}$ never has strictly positive curvature. Nevertheless, as we stated in Section 1, we can obtain a Nadel vanishing theorem for $h_{{{\rm{min}}}}$ and $h_{\rm{Siu}}$. Let us recall the definition of metrics with minimal singularities. Let $h_{1}$ and $h_{2}$ be metrics on $F$ with positive curvature. The metric $h_{1}$ is said to be [*less singular*]{} than $h_{2}$ if $h_{1} \leq C h_{2}$ for some positive constant $C>0$. When $F$ is pseudo-effective (that is, $F$ admits a metric with positive curvature), we can construct a metric $h_{\min}$ on $F$ with the following properties:\ $\bullet$ $h_{\min}$ has semi-positive curvature.\ $\bullet$ $h_{\min}$ is less singular than any metric on $F$ with positive curvature.\ Such a metric is uniquely determined up to equivalence of singularities (see [@Dem (6.4) Theorem] for more details). Metrics with minimal singularities do not always have analytic singularities. Indeed, if $h_{{{\rm{min}}}}$ has analytic singularities, then $F$ admits a birational Zariski decomposition (see Section 4). However, there exist big line bundles that do not admit a birational Zariski decomposition. Equisingular Approximations --------------------------- In the proof of the main result, we apply the equisingular approximation to a given metric. In this subsection, we reformulate [@DPS01 Theorem 2.3.] with our notation and give further remarks. $($[@DPS01 Theorem 2.3.]$).$ \[equi\] Let $X$ be a compact Kähler manifold and $F$ be a line bundle with a metric $h$ with $($semi-$)$positive curvature. Then there exist metrics $\{h_{{\varepsilon}} \}_{1\gg {\varepsilon}>0}$ on $F$ with the following properties:\ $(a)$ $h_{{\varepsilon}}$ is smooth on $X \setminus Z_{{\varepsilon}}$, where $Z_{{\varepsilon}}$ is a subvariety on $X$.\ $(b)$ $h_{{\varepsilon}_{2}} \leq h_{{\varepsilon}_{1}} \leq h$ holds for any $0< {\varepsilon}_{1} < {\varepsilon}_{2} $.\ $(c)$ ${\mathcal{I}(h)}= {\mathcal{I}(h_{{\varepsilon}})}$.\ $(d)$ $\sqrt{-1} \Theta_{h_{{\varepsilon}}}(F) \geq -{\varepsilon}\omega$.\ Moreover, if the set $\{x \in X \mid \nu(\varphi, x) >0 \}$ is contained in a subvariety $Z$, then we can add the property that $Z_{{\varepsilon}} $ is contained in $Z$ for any ${\varepsilon}> 0$. Here $\nu(\varphi, x)$ denotes the Lelong number at $x$ of a weight $\varphi$ of $h$. Fix a smooth metric $g$ on $F$. Then there exists an $L^{1}$-function $\varphi$ on $X$ with $h=g e^{- \varphi}$. By applying [@DPS01 Theorem 2.3.] to $\varphi$, we can obtain quasi-psh functions $\varphi _{\nu}$ with equisingularities. For a given ${\varepsilon}>0$, by taking a large $\nu=\nu({\varepsilon})$, we define $h_{{\varepsilon}}$ by $h_{{\varepsilon}}:=g e^{- \varphi_{\nu({\varepsilon})}}$. Then the metric $h_{{\varepsilon}}$ satisfies properties (a), (b), (c), (d). The latter conclusion follows from the proof. We will see this fact shortly, by using the notation in [@DPS01]. In their proof, they locally approximate $\varphi$ by $\varphi_{{\varepsilon}, \nu, j}$ with logarithmic pole. By inequality (2.5) in [@DPS01], the Lelong number of $\varphi_{{\varepsilon}, \nu, j}$ is less than or equal to that of $\varphi$. Hence $\varphi_{{\varepsilon}, \nu, j}$ is smooth on $X \setminus Z$ since $\varphi_{{\varepsilon}, \nu, j}$ has a logarithmic pole. Then $\varphi _{\nu}$ is obtained from Richberg’s regularization of the supremum of these functions (see around (2.5) and (2.7)). Since the supremum is continuous on $X \setminus Z$, we obtain the latter conclusion. Theory of Harmonic Integrals ---------------------------- For our purpose, we recall some facts on the theory of harmonic integrals. Throughout this subsection, let $Y$ be a (not necessarily compact) complex manifold with a hermitian form $\omega$ and $E$ be a line bundle on $Y$ with a smooth metric $h$. For $E$-valued $(p,q)$-forms $u$, $v$, the point-wise inner product $\langle u, v\rangle _{h, \omega}$ can be defined. Further, the (global) inner product ${{\langle\!\langle}}u, v {{\rangle\!\rangle}}_{h, \omega}$ can also be defined by $${{\langle\!\langle}}u, v {{\rangle\!\rangle}}_{h, \omega}:= \int_{Y} \langle u, v\rangle _{h, \omega}\ \omega^{n}.$$ Then the $L^{2}$-space of the $E$-valued $(p, q)$-forms is defined as follows: $$L_{(2)}^{p, q}(Y, E)_{h, \omega}:= \{u \mid u \text{ is an }E \text{-valued\ } (p, q)\text{-form with } \|u \|_{h, \omega}< \infty \}.$$ The connection $D_{h}$ on $E$ is determined by the holomorphic structure of $E$ and the hermitian metric $h$, which is called the Chern connection. The Chern connection $D_{h}$ can be written as $D_{h} = D'_{h} + D''_{h}$ with the $(1,0)$-connection $D'_{h}$ and the $(0,1)$-connection $D''_{h}$. (The $(0,1)$-connection $D''_{h}$ agrees with ${\overline{\partial}}$.) The connections $D'_{h}$ and $D''_{h}$ can be seen as a closed and densely defined operator on $L_{(2)}^{p, q}(Y, E)_{h, \omega}$. If $\omega$ is a *complete* metric on $Y$, then the formal adjoints $D^{'*}_{h}$ and $D^{''*}_{h}$ coincide with the Hilbert space adjoints (see [@Dem-book (3,2) Theorem in Chapter 8]). \[Nak\] Let $ \omega$ be a complete Kähler metric on $Y$ of dimension $n$. Assume that $\sqrt{-1}\Theta_{h}(E) \geq - C \omega$ for some positive constant $C>0$. Then for every $u \in L_{(2)}^{n, i}(Y, E)_{h, \omega}$ with $u \in {\rm{Dom}}D_{h}^{''*} \cap {\rm{Dom}} \overline{\partial}$, the following equality holds$:$ $$\| D_{h}^{''*}u \|_{h, \omega}^{2} + \|\overline{\partial} u \|_{h, \omega}^{2} = \| D_{h}^{'*}u \|_{h, \omega}^{2} + {{\langle\!\langle}}\sqrt{-1}\Theta_{h}(E)\Lambda_{\omega} u, u {{\rangle\!\rangle}}_{h, \omega}.$$ Here $\Lambda_{\omega}$ denotes the adjoint operator of the wedge product $\omega \wedge \cdot$. This proposition can be obtained from Nakano’s identity and the density lemma. Since $\omega$ is a Kähler form, we have $$\Delta^{''} = \Delta^{''} + [\sqrt{-1}\Theta_{h}(E), \Lambda_{\omega}]$$ by Nakano’s identity. Here $\Delta^{'}$ (resp. $\Delta^{''}$) is the Laplacian operator defined by $\Delta^{'}:= D_{h}^{'}D_{h}^{'*} +D_{h}^{'*} D_{h}^{'}$ (resp. $\Delta^{''}:= D_{h}^{''}D_{h}^{''*} +D_{h}^{''*} D_{h}^{''}$) and \[$\cdot$, $\cdot$\] is the graded Lie bracket. Therefore if $u$ is smooth and compactly supported, the equality in the proposition holds. For a given $u$, there exist smooth and compactly supported $E$-valued $(n,i)$-forms $\{u_{k}\}_{k=1}^{\infty}$ such that $u_{k} \to u$, $D_{h}^{''*} u_{k} \to D_{h}^{''*}u$ and $\overline{\partial}u_{k} \to \overline{\partial}u$ in $L^{n,\bullet}_{(2)}(Y, E)_{h, \omega}$ as $k \to \infty$ since $\omega$ is a complete metric (see [@Dem-book Chapter 8]). By the assumption of $\sqrt{-1}\Theta(E) \geq - C \omega$, the second term of the right hand side $${{\langle\!\langle}}\sqrt{-1}\Theta_{h}(E)\Lambda_{\omega} u_{k}, u_{k} {{\rangle\!\rangle}}_{h, \omega} = \int_{Y} \langle \sqrt{-1}\Theta_{h}(E)\Lambda_{\omega} u_{k}, u_{k} \rangle_{h, \omega}\ \omega^{n}.$$ is bounded below. Therefore we obtain the conclusion by Lebesgue’s convergence theorem. Topics Related to the Main Problem ---------------------------------- It is worth mentioning that the main result of this paper is related to the openness conjecture of Demailly-Kollár on singularities of psh functions ([@DK01 5.3 Remark]). When $h_{\min}$ is a metric with minimal singularities on a big line bundle $F$, we have $$\mathcal{I}_{+}(h_{{{\rm{min}}}}) \subset \mathcal{I}(\| F \|) \subset \mathcal{I}(h_{{{\rm{min}}}}),$$ where $\mathcal{I}_{+}(h_{{{\rm{min}}}})$ is the [*upper semi-continuous regularization*]{} of the multiplier ideal sheaf of $h_{\min}$ (see [@DEL00]). Further all higher cohomology groups with coefficients in $K_{X}\otimes F \otimes \mathcal{I}_{+}(h_{{{\rm{min}}}})$ vanish. Indeed, it is easy to check that there exists a metric $h$ on $F$ such that the curvature of $h$ is strictly positive and ${\mathcal{I}(h)}=\mathcal{I}_{+}(h_{{{\rm{min}}}})$. However the main problem of this paper has been an open problem even if $F$ is big since it is more difficult to handle ${\mathcal{I}(h_{\min})}$. It is an interesting question to ask whether or not $\mathcal{I}_{+}(\varphi)$ agrees with $\mathcal{I}(\varphi)$ for a psh function $\varphi$, which was first posed in [@DEL00]. It is a natural question, but it has been an open problem. It is known that $\mathcal{I}_{+}(\varphi) = {\mathcal{I}(\varphi)}$ holds when $\varphi$ has analytic singularities, but unfortunately $h_{\min}$ does not always have analytic singularities (see Section 4). This question is also related to the openness conjecture of Demailly-Kollár. Recently, Berndtsson announced that this conjecture was affirmatively solved in [@Ber13]. However the openness conjecture does not, at least directly, imply that $\mathcal{I}_{+}(\varphi) = {\mathcal{I}(\varphi)}$ holds. See [@Kim10 Proposition 3.2] for relations between this question and the openness conjecture. Main results ============ In this section, we prove Theorem \[main\] and Corollary \[main-co\], which lead to Theorem \[m-min\]. Proof of the Main Theorem ------------------------- In this subsection, we give a proof of Theorem \[main\]. \[main\] Let $F$ be a big line bundle on a smooth projective variety $X$ and $h$ be a $($singular$)$ metric on $F$ with $($semi-$)$positive curvature. Assume that there exist a $($non-zero$)$ section $s$ of some positive multiple $F^{m}$ of $F$ and a $($singular$)$ metric $h_{{{\rm{big}}}}$ on $F$ satisfying the following conditions$:$\ $(1)$ The metric $h_{{{\rm{big}}}}$ has strictly positive curvature.\ $(2)$ The metric $h$ is less singular than $h_{{{\rm{big}}}}$.\ $(3)$ The point-wise norm $|s|_{h^m_{{{\rm{big}}}}}$ of $s$ with respect to $h^{m}_{{{\rm{big}}}}$ is bounded on $X$.\ Then we have $$H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(h_{})}) = 0 \quad {\text{for}}\ {\text{any}}\ i >0.$$ *Proof of Theorem \[main\].*   The idea of the proof is as follows: We observe the case where $h$ is smooth on $X$ (or some Zariski open set of $X$). Take an arbitrary cohomology class $\{u \} \in H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(h)})$ represented by the $F$-valued $(n, i)$-form $u$ such that the norm $\|u \|_{h}$ is bounded and $u$ is harmonic. If the norm $\| u \|_{h_{{{\rm{big}}}}}$ is bounded, we can construct a solution $\gamma \in L_{(2)}^{n, i-1}(X, F)_{h}$ of the ${\overline{\partial}}$-equation $ {\overline{\partial}}\gamma = u$ by assumptions (1), (2), which completes the proof. However we can never expect that $\|u \|_{{{\rm{big}}}}$ is bounded since $h_{{{\rm{big}}}}$ is more singular than $h$. Now we consider the ${\overline{\partial}}$-equation ${\overline{\partial}}\beta = su$ instead of $ {\overline{\partial}}\gamma = u$. Then the norm $\|su\|_{h_{{{\rm{big}}}}^{m}h}$ is bounded since the singularities of $h_{{{\rm{big}}}}$ are canceled by $s$ thanks to assumption (3). Further the metric $h_{{{\rm{big}}}}^{m}h$ has strictly positive curvature. Therefore we can obtain a solution $\beta \in L_{(2)}^{n, i-1}(X, F^{m+1})_{h^{m+1}}$ of the ${\overline{\partial}}$-equation ${\overline{\partial}}\beta = su$. On the other hand, $su$ can be shown to be harmonic since $u$ is harmonic and the curvature of $h$ is semi-positive. It follows from Enoki’s technique for the injectivity theorem. From these arguments, we can conclude that $u$ is zero. Unfortunately $h$ may not smooth even on Zariski open sets of $X$. In Step 1, we approximate the metric $h$ by metrics $\{ h_{{\varepsilon}} \}_{{\varepsilon}>0}$ that are smooth on a Zariski open set. Then we must investigate the asymptotic behavior of the harmonic forms that represent a given cohomology class since the positivity of the curvature of $h_{{\varepsilon}}$ can be lost. In this step, we fix the notation to apply the theory of harmonic integrals. At the end of this step, we give the sketch of the proof.\ [**[Step 1 (The equisingular approximation of $h$)]{}**]{}\ We can take a Kähler form $\omega$ on $X$ such that $\sqrt{-1} \Theta_{h_{{{\rm{big}}}}}(F) \geq \omega$ since the curvature of $h_{{{\rm{big}}}}$ is strictly positive. For the proof we apply the theory of harmonic integrals, but $h$ may not be smooth. For this reason, we need to approximate $h$ by metrics $\{ h_{{\varepsilon}} \}_{{\varepsilon}>0}$ that are smooth on a Zariski open set. By Theorem \[equi\], we can obtain metrics $\{ h_{{\varepsilon}} \}_{{\varepsilon}>0}$ on $F$ with the following properties:\ $(a)$ $h_{{\varepsilon}}$ is smooth on $X \setminus Z_{{\varepsilon}}$, where $Z_{{\varepsilon}}$ is a subvariety on $X$.\ $(b)$ $h_{{\varepsilon}_{2}} \leq h_{{\varepsilon}_{1}} \leq h$ holds for any $0< {\varepsilon}_{1} < {\varepsilon}_{2} $.\ $(c)$ ${\mathcal{I}(h)}= {\mathcal{I}(h_{{\varepsilon}})}$.\ $(d)$ $\sqrt{-1} \Theta_{h_{{\varepsilon}}}(F) \geq -{\varepsilon}\omega$.\ By assumptions (2), (3), the point-wise norm $|s|_{h^{m}}$ is also bounded on $X$, thus the set $\{x \in X \mid \nu(\varphi, x)>0\}$ is contained in the subvariety $Z:=\{z \in X \mid s(x)=0 \}$. Here $\nu(\varphi, x)$ denotes the Lelong number at $x \in X$ of a weight $\varphi$ of $h$. This implies that $h_{{\varepsilon}}$ is smooth on $X \setminus Z$ (see the latter conclusion of Theorem \[equi\]). Hence we can add a stronger property than property $(a)$, namely\ $(e)$ $h_{{\varepsilon}}$ is smooth on $Y:=X \setminus Z$, where Z is a subvariety independent of ${\varepsilon}$. Now we construct a complete Kähler metric on $Y$ with suitable potential function. Take a quasi-psh function $\psi$ on $X$ such that $\psi$ is smooth on $Y$ and $\psi$ has a logarithmic pole along $Z$. Since quasi-psh functions are upper semi-continuous, we may assume $\psi \leq -e$. Then we define the function $\varphi$ and the $(1,1)$-form ${\widetilde{\omega}}$ on $Y$ by $$\varphi:=\frac{1}{\log(-\psi)}\quad \text{and}\quad {\widetilde{\omega}}:= \ell \omega + {dd^c}\varphi,$$ where $\ell$ is a positive number. By taking a sufficiently large $\ell >0$, we can easily see that the $(1,1)$-form ${\widetilde{\omega}}$ satisfies the following properties:\ (A) ${\widetilde{\omega}}$ is a complete Kähler metric on $Y$.\ (B) $\varphi$ is bounded on $X$.\ (C) ${\widetilde{\omega}}\geq \omega $.\ Indeed, properties (B), (C) are obvious. Property (A) follows from straightforward computations (see [@Fuj12-A Lemma 3.1]). In the proof, we consider the harmonic forms on $Y$ with respect to $h_{{\varepsilon}}$ and ${\widetilde{\omega}}$. Let $L_{(2)}^{n, i}(Y, F)_{h_{{\varepsilon}}, {\widetilde{\omega}}}$ be the space of the square integrable $F$-valued $(n,i)$-forms $\alpha$ with respect to the inner product defined by $$\|\alpha \|^{2}_{h_{\varepsilon}, {\widetilde{\omega}}}:= \int_{Y} |\alpha |^{2}_{h_{{\varepsilon}}, {\widetilde{\omega}}}\ {\widetilde{\omega}}^{n}.$$ Then we have the orthogonal decomposition $$L_{(2)}^{n, i}(Y, F)_{h_{{\varepsilon}}, {\widetilde{\omega}}} = {\rm{Im}} {\overline{\partial}}\oplus \mathcal{H}^{n, i}(F)_{h_{{\varepsilon}}, {\widetilde{\omega}}} \oplus {\rm{Im}} D^{''*}_{h_{{\varepsilon}}}.$$ Here the operators $D^{'*}_{h_{{\varepsilon}}}$ $D^{''*}_{h_{{\varepsilon}}}$ are the Hilbert space adjoints of $D^{'}_{h_{{\varepsilon}}}$ and ${\overline{\partial}}$. (Since ${\widetilde{\omega}}$ is complete, these coincide with the closed extensions of the formal adjoints in the sense of distributions.) The space $\mathcal{H}^{n, i}(F)_{h_{{\varepsilon}}, {\widetilde{\omega}}}$ denotes the space of the harmonic forms with respect to $h_{{\varepsilon}}$ and ${\widetilde{\omega}}$, namely $$\mathcal{H}^{n, i}(F)_{h_{{\varepsilon}}, {\widetilde{\omega}}}= \{\alpha \mid \alpha \text{ is an } F\text{-valued } (n,i)\text{-form such that } {\overline{\partial}}\alpha= D^{''*}_{h_{{\varepsilon}}}\alpha=0 \}.$$ Harmonic forms are smooth by the regularization theorem for elliptic operators. These facts may be known to specialists. The precise proof can be found in [@Fuj12-A Claim 1]. From property (C) we have the inequality $|\beta|^{2}_{{\widetilde{\omega}}}\ {\widetilde{\omega}}^{n} \leq |\beta|^{2}_{\omega}\ \omega^{n}$ for any $(n, i)$-form $\beta$. From this inequality and property (b) of $h_{{\varepsilon}}$, we obtain $$\label{ine} \|\alpha \|_{h_{{\varepsilon}}, {\widetilde{\omega}}} \leq \|\alpha \|_{h_{{\varepsilon}}, \omega} \leq \|\alpha \|_{h, \omega}$$ for any $F$-valued $(n,i)$-form $\alpha$. This inequality is often used in the proof. Take an arbitrary cohomology class $\{u \} \in H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(h)})$ represented by an $F$-valued $(n, i)$-form $u$ with $\|u \|_{h, \omega} < \infty$. Our goal is to show that the cohomology class $\{u \}$ is actually zero. By inequality (\[ine\]), we know $u \in L_{(2)}^{n, i}(Y, F)_{h_{{\varepsilon}}, {\widetilde{\omega}}}$ for any ${\varepsilon}> 0$. By the orthogonal decomposition above, there exist $u_{{\varepsilon}} \in \mathcal{H}^{n, i}(F)_{h_{{\varepsilon}}, {\widetilde{\omega}}}$ and $v_{{\varepsilon}} \in L_{(2)}^{n,i-1}(Y, F)_{h_{{\varepsilon}}, {\widetilde{\omega}}}$ such that $$u=u_{{\varepsilon}}+{\overline{\partial}}v_{{\varepsilon}}.$$ Notice that the component of ${\rm{Im}} D^{''*}_{h_{{\varepsilon}}}$ is zero since $u$ is ${\overline{\partial}}$-closed. The strategy of the proof is as follows: In Step 2, we show $$\|D^{''*}_{h^{m+1}_{{\varepsilon}}} s u_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \to 0 \quad {\text{as }} {\varepsilon}\to 0.$$ This step can be regarded as a generalization of Enoki’s proof of Kollár’s injectivity theorem. In Step 3, we construct a solution $\beta_{{\varepsilon}}$ of the ${\overline{\partial}}$-equation ${\overline{\partial}}\beta_{{\varepsilon}} = s u_{{\varepsilon}}$ such that the norm $\| \beta_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ is uniformly bounded. By these steps, we know $$\| su_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} ^{2} = {{\langle\!\langle}}su_{{\varepsilon}}, {\overline{\partial}}\beta_{{\varepsilon}} {{\rangle\!\rangle}}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \leq \| D^{''*}_{h^{m+1}_{{\varepsilon}}} su_{{\varepsilon}}\|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \| \beta_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \to 0 \quad {\text{as }} {\varepsilon}\to 0.$$ In Step 4, we show that $u_{{\varepsilon}}$ converges to zero in a suitable sense. This completes the proof.\ [**[Step 2 (A generalization of Enoki’s proof of the injectivity theorem)]{}**]{}\ The aim of this step is to prove the following claim. \[D”\] The norm $\|D^{''*}_{h^{m+1}_{{\varepsilon}}} s u_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ converges to zero as letting ${\varepsilon}$ go to zero. *Proof of Claim \[D”\].*   From the definition of $u_{{\varepsilon}}$ and inequality (\[ine\]), we have $$\label{ine2} \|u_{{\varepsilon}} \|_{h_{{\varepsilon}}, {\widetilde{\omega}}} \leq \|u \|_{h_{{\varepsilon}}, {\widetilde{\omega}}} \leq \|u \|_{h, \omega}.$$ In the proof, these inequalities play an important role. By applying Proposition \[Nak\] to $u_{{\varepsilon}}$, we obtain $$\label{B-eq} 0 = {{\langle\!\langle}}\sqrt{-1}\Theta_{h_{{\varepsilon}}}(F) \Lambda_{{\widetilde{\omega}}} u_{{\varepsilon}}, u_{{\varepsilon}} {{\rangle\!\rangle}}_{h_{{\varepsilon}}, {\widetilde{\omega}}} + \|D^{'*}_{h_{{\varepsilon}}}u_{{\varepsilon}} \|^{2}_{h_{{\varepsilon}}, {\widetilde{\omega}}}.$$ Note that the left hand side is zero since $u_{{\varepsilon}}$ is harmonic. Let $A_{{\varepsilon}}$ be the first term and $B_{{\varepsilon}}$ the second term of the right hand side of equality (\[B-eq\]). From now on, we will show that the first term $A_{{\varepsilon}}$ and the second term $B_{{\varepsilon}}$ converge to zero. For simplicity, we put $$g_{{\varepsilon}}:= \langle \sqrt{-1}\Theta_{h_{{\varepsilon}}}(F) \Lambda_{{\widetilde{\omega}}} u_{{\varepsilon}}, u_{{\varepsilon}} \rangle_{h_{{\varepsilon}}, {\widetilde{\omega}}}.$$ Then we can easily see that there exists a positive constant $C$ such that $$\label{ine3} g_{{\varepsilon}} \geq -{\varepsilon}C |u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}}, {\widetilde{\omega}}}.$$ Indeed, let $\lambda_{1}^{{\varepsilon}} \leq \lambda_{2}^{{\varepsilon}} \leq \dots \leq \lambda_{n}^{{\varepsilon}} $ be the eigenvalues of $\sqrt{-1}\Theta_{h_{{\varepsilon}}}(F)$ with respect to ${\widetilde{\omega}}$. Then for any point $y \in Y$, there exists a local coordinates $(z_{1}, z_{2}, \dots, z_{n})$ centered at $y$ such that $$\begin{aligned} \sqrt{-1}\Theta_{h_{{\varepsilon}}}(F) = \sum_{j=1}^{n} \lambda_{j}^{{\varepsilon}} dz_{j} \wedge d\overline{z_{j}} \quad \text{and} \quad {\widetilde{\omega}}= \sum_{j=1}^{n} dz_{j} \wedge d\overline{z_{j}} \quad {\rm{ at}}\ y. \end{aligned}$$ When we locally write $u_{{\varepsilon}}$ as $u_{{\varepsilon}} =\sum_{|K|=i} f_{K}^{{\varepsilon}}\ dz_{1}\wedge \dots \wedge dz_{n} \wedge d\overline{z}_{K}$, by straightforward computations we have $$g_{{\varepsilon}}= \sum_{|K|=i} \Big{(} \sum_{j \in K} \lambda_{j}^{{\varepsilon}} \Big{)} |f_{K}^{{\varepsilon}}|^{2}_{h_{{\varepsilon}}}.$$ On the other hand, from property (C) of ${\widetilde{\omega}}$ and property (d) of $h_{{\varepsilon}}$, we have $\sqrt{-1}\Theta_{h_{{\varepsilon}}}(F) \geq -{\varepsilon}\omega \geq -{\varepsilon}{\widetilde{\omega}}$, which implies that $\lambda_{j}^{{\varepsilon}} \geq -{\varepsilon}$. Therefore we obtain inequality (\[ine3\]). From inequalities (\[ine2\]), (\[ine3\]) and equality (\[B-eq\]), we obtain $$\begin{aligned} 0 \geq A_{{\varepsilon}} &= \int_{Y} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n} \\ & \geq -{\varepsilon}C \int_{Y} |u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}}, {\widetilde{\omega}}}\ {\widetilde{\omega}}^{n}\\ & \geq -{\varepsilon}C \|u \|^{2}_{h, \omega}. \end{aligned}$$ Hence $A_{{\varepsilon}}$ converges to zero. By equality (\[B-eq\]), $B_{{\varepsilon}}$ also converges to zero. We will apply Proposition \[Nak\] to $su_{{\varepsilon}}$. We first see that the norm $\|su_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ is bounded. By assumptions (2), (3), the point-wise norm $|s|_{h^{m}}$ with respect to $h^{m}$ is bounded. Further, we have $|s|_{h_{{\varepsilon}}^{m}} \leq |s|_{h^{m}}$ from property (b) of $h_{{\varepsilon}}$, thus we obtain $$\|s u_{{\varepsilon}} \|_{h_{{\varepsilon}}^{m+1}, {\widetilde{\omega}}} \leq \sup_{X} |s|_{h_{{\varepsilon}}^{m}} \|u_{{\varepsilon}} \|_{h_{{\varepsilon}}, {\widetilde{\omega}}} \leq \sup_{X} |s|_{h^{m}} \|u \|_{h, \omega} < \infty.$$ By applying Proposition \[Nak\] to $su_{{\varepsilon}}$, we obtain $$\label{B-eq2} \|D^{''*}_{h_{{\varepsilon}}^{m+1}} su_{{\varepsilon}} \|^{2}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} = {{\langle\!\langle}}\sqrt{-1}\Theta_{h^{m+1}_{{\varepsilon}}}(F^{m+1}) \Lambda_{{\widetilde{\omega}}} su_{{\varepsilon}}, su_{{\varepsilon}} {{\rangle\!\rangle}}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} + \|D^{'*}_{h_{{\varepsilon}}^{m+1}}su_{{\varepsilon}} \|^{2}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}.$$ Here we used ${\overline{\partial}}s u_{{\varepsilon}}=0$. First we see that the second term of the right hand side converges to zero. Since $s$ is a holomorphic $(0, 0)$-form, we can check the equality $D^{'*}_{h^{m+1}_{{\varepsilon}}}su_{{\varepsilon}} = s D^{'*}_{h_{{\varepsilon}}}u_{{\varepsilon}}$, which yields $$\|D^{'*}_{h_{{\varepsilon}}^{m+1}}su_{{\varepsilon}} \|^{2}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \leq \sup_{X}|s|^{2}_{h_{{\varepsilon}}^{m}} \int_{Y} |D^{'*}_{h_{{\varepsilon}}}u_{{\varepsilon}} |^{2}_{h_{{\varepsilon}}, {\widetilde{\omega}}}\ {\widetilde{\omega}}^{n} \leq \sup_{X}|s|^{2}_{h^{m}} B_{{\varepsilon}}.$$ Since $B_{{\varepsilon}}$ converges to zero, the norm $\|D^{'*}_{h^{m+1}_{{\varepsilon}}}su_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ also converges to zero. For the proof of the claim, it remains to show that the first term of the right hand side of equation (\[B-eq2\]) converges to zero. For this purpose, we investigate $A_{{\varepsilon}}$ in details. By the definition of $A_{{\varepsilon}}$, we have $$A_{{\varepsilon}}= \int_{\{ g_{{\varepsilon}} \geq 0 \}} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n} + \int_{\{ g_{{\varepsilon}} \leq 0 \}} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n}.$$ Let $A^{+}_{{\varepsilon}}$ be the first term and $A^{-}_{{\varepsilon}}$ be the second term of the right hand side. Then $A^{+}_{{\varepsilon}}$ and $A^{-}_{{\varepsilon}}$ converge to zero. Indeed, by simple computations and inequalities (\[ine2\]), (\[ine3\]), we have $$\begin{aligned} 0 \geq A^{-}_{{\varepsilon}} &\geq -{\varepsilon}C \int_{\{ g_{{\varepsilon}} \leq 0 \}}|u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}}, {\widetilde{\omega}}}\ {\widetilde{\omega}}^{n} \\ &\geq -{\varepsilon}C \int_{Y}|u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}}, {\widetilde{\omega}}}\ {\widetilde{\omega}}^{n} \\ &\geq -{\varepsilon}C \| u \|^{2} _{h, \omega}. \end{aligned}$$ It implies that $A^{+}_{{\varepsilon}}$ and $A^{-}_{{\varepsilon}}$ go to zero since $A_{{\varepsilon}} = A^{+}_{{\varepsilon}} + A^{-}_{{\varepsilon}}$ goes to zero. Now we have $$\begin{aligned} \langle \sqrt{-1}\Theta_{h^{m+1}_{{\varepsilon}}}(F^{m+1}) \Lambda_{{\widetilde{\omega}}} su_{{\varepsilon}}, su_{{\varepsilon}} \rangle_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} &= (m+1)|s|^{2}_{h^{m}_{{\varepsilon}}}\langle \sqrt{-1}\Theta_{h_{{\varepsilon}}}(F) \Lambda_{{\widetilde{\omega}}} u_{{\varepsilon}}, u_{{\varepsilon}} \rangle_{h_{{\varepsilon}}, {\widetilde{\omega}}} \\ &= (m+1)|s|^{2}_{h^{m}_{{\varepsilon}}}\ g_{{\varepsilon}}.\end{aligned}$$ Therefore we obtain $$\begin{aligned} {{\langle\!\langle}}\sqrt{-1}\Theta_{h^{m+1}_{{\varepsilon}}}(F^{m+1}) \Lambda_{{\widetilde{\omega}}} su_{{\varepsilon}}, su_{{\varepsilon}} {{\rangle\!\rangle}}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} &= (m+1)\int_{Y} |s|^{2}_{h^{m}_{{\varepsilon}}} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n} \\ &= (m+1) \Big\{ \int_{\{ g_{{\varepsilon}} \geq 0 \}} |s|^{2}_{h^{m}_{{\varepsilon}}} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n} + \int_{\{ g_{{\varepsilon}} \leq 0 \}} |s|^{2}_{h^{m}_{{\varepsilon}}} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n} \Big\}.\end{aligned}$$ Let $C^{+}_{{\varepsilon}}$ be the first term and $C^{-}_{{\varepsilon}}$ be the second term of the right hand side. Then we have $$\begin{aligned} 0 \leq C^{+}_{{\varepsilon}} &\leq (m+1)\sup_{X}|s|^{2}_{h^{m}_{{\varepsilon}}} \int_{\{ g_{{\varepsilon}} \geq 0 \}} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n} \\ & \leq (m+1)\sup_{X}|s|^{2}_{h^{m}}\ A^{+}_{{\varepsilon}}.\end{aligned}$$ On the other hand, we have $$\begin{aligned} 0 \geq C^{-}_{{\varepsilon}} &\geq (m+1) \sup_{X}|s|^{2}_{h^{m}_{{\varepsilon}}} \int_{\{ g_{{\varepsilon}} \leq 0 \}} g_{{\varepsilon}}\ {\widetilde{\omega}}^{n} \\ &\geq -{\varepsilon}C (m+1) \sup_{X}|s|^{2}_{h^{m}_{{\varepsilon}}} \int_{\{ g_{{\varepsilon}} \leq 0 \}} |u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}}, \omega}\ {\widetilde{\omega}}^{n} \\ &\geq -{\varepsilon}C (m+1) \sup_{X}|s|^{2}_{h^{m}_{{\varepsilon}}} \int_{Y} |u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}}, \omega}\ {\widetilde{\omega}}^{n} \\ &\geq -{\varepsilon}C (m+1) \sup_{X}|s|^{2}_{h^{m}} \|u \|^{2}_{h, \omega}. \end{aligned}$$ By these inequalities, we know that $C^{+}_{{\varepsilon}}$ and $C^{-}_{{\varepsilon}}$ converge to zero. Thus we obtain the conclusion of Claim \[D”\] from equality (\[B-eq2\]).\ [**[Step 3 (Solutions of the ${\overline{\partial}}$-equation with uniformly bounded $L^{2}$-norms)]{}**]{}\ A positive multiple $s^{k}$ of the section $s$ also satisfies assumption (3), thus we may assume that $m$ is greater than $\ell$. Under this additional assumption, we prove the following claim. \[sol\] For every ${\varepsilon}>0$, there exists an $F^{m}$-valued $(n, i-1)$-form $\beta_{{\varepsilon}}$ such that $${\overline{\partial}}\beta_{{\varepsilon}} = su_{{\varepsilon}} \quad \text{and} \quad \ \| \beta_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \text{ is uniformly bounded.}$$ *Proof of Claim \[sol\].*   To construct solutions of the ${\overline{\partial}}$-equation $ {\overline{\partial}}\beta_{{\varepsilon}} = su_{{\varepsilon}} $, we apply [@Dem82 Théorèm 4.1]. For this purpose, we consider the new metric $H_{{\varepsilon}}$ on $F^{m+1}$ defined by $H_{{\varepsilon}}:= h^{m}_{{{\rm{big}}}}h_{{\varepsilon}} e^{-\varphi}$. First we compute the curvature of $H_{{\varepsilon}}$. We have $$\begin{aligned} \sqrt{-1}\Theta_{H_{{\varepsilon}}}(F^{m+1}) &= m \sqrt{-1} \Theta_{h_{{{\rm{big}}}}}(F)+ \sqrt{-1}\Theta_{h_{{\varepsilon}}}(F)+ {dd^c}\varphi \\ &\geq m \omega - {\varepsilon}\omega + {dd^c}\varphi. \end{aligned}$$ By the definition of ${\widetilde{\omega}}$ and the additional assumption of $m>\ell$, we obtain $\sqrt{-1}\Theta_{H_{{\varepsilon}}}(F^{m+1}) \geq {\widetilde{\omega}}$. Now we consider the norm of $su_{{\varepsilon}}$ with respect to $H_{{\varepsilon}}$. By assumption (3), the point-wise norm $|s|_{h^{m}_{big}}$ is bounded on $X$. Further, since $\varphi$ is bounded on $X$, there exist positive constants $C_{1}$ and $C_{2}$ such that $C_{1} \leq e^{-\varphi} \leq C_{2}$ on $X$. Then by inequality (\[ine2\]) we obtain $$\begin{aligned} \|su_{{\varepsilon}} \|_{H_{{\varepsilon}}, {\widetilde{\omega}}} &\leq C_{2} \sup_{X} |s|_{h^{m}_{{{\rm{big}}}}} \|u_{{\varepsilon}} \|_{h_{{\varepsilon}}, {\widetilde{\omega}}} \\ &\leq C_{2} \sup_{X} |s|_{h^{m}_{{{\rm{big}}}}} \|u \|_{{h}, \omega}.\end{aligned}$$ In particular, the norm $\|su_{{\varepsilon}} \|_{H_{{\varepsilon}}, {\widetilde{\omega}}} $ is bounded. Moreover, the right hand side does not depend on ${\varepsilon}$. By applying [@Dem82 Théorèm 4.1], we can find a solution $\beta_{{\varepsilon}}$ of the ${\overline{\partial}}$-equation $ {\overline{\partial}}\beta_{{\varepsilon}} = su_{{\varepsilon}}$ with $$\|\beta_{{\varepsilon}} \|^{2}_{H_{{\varepsilon}}, {\widetilde{\omega}}} \leq \frac{1}{i}\|su_{{\varepsilon}} \|^{2}_{H_{{\varepsilon}}, {\widetilde{\omega}}}.$$ Since $h$ is less singular than $h_{{{\rm{big}}}}$, there exists a positive constant $C_{3} > 0$ such that $h_{{\varepsilon}} \leq h \leq C_{3} h_{{{\rm{big}}}}$. Then we can easily see $$C^{-m}_{3} C_{1} \|\beta_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \leq \|\beta_{{\varepsilon}} \|_{H_{{\varepsilon}}, {\widetilde{\omega}}}.$$ Since the right hand side can be estimated by the constant independent of ${\varepsilon}$, the norm $\|\beta_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ is uniformly bounded. This completes the proof.\ [**[Step 4 (The limit of harmonic forms)]{}**]{}\ In this step, we investigate the limit of $u_{{\varepsilon}}$ and complete the proof of Theorem \[main\]. First we prove the following claim. \[converge\] The norm $\| s u_{{\varepsilon}}\|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ converges to zero as ${\varepsilon}\to \infty$. *Proof of Claim \[converge\].*   Take $\beta_{{\varepsilon}} \in L_{(2)}^{n, i-1} (F^{m+1})_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ satisfying the properties in Claim \[sol\]. Then straightforward computations yield $$\begin{aligned} \| s u_{{\varepsilon}}\|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}^{2} &={{\langle\!\langle}}s u_{{\varepsilon}}, {\overline{\partial}}\beta_{{\varepsilon}} {{\rangle\!\rangle}}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \\ &= {{\langle\!\langle}}D^{''*}_{h^{m+1}_{{\varepsilon}}}s u_{{\varepsilon}},\beta_{{\varepsilon}} {{\rangle\!\rangle}}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \\ &\leq \|D^{''*}_{h^{m+1}_{{\varepsilon}}} s u_{{\varepsilon}}\| _{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} \|\beta_{{\varepsilon}} \|_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}. \end{aligned}$$ The norm of $\beta_{{\varepsilon}}$ is uniformly bounded by Claim \[sol\]. On the other hand, the norm $\|D^{''*}_{h^{m+1}_{{\varepsilon}}} s u_{{\varepsilon}}\| _{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}}$ converges to zero by Claim \[D”\]. Therefore the norm $\| s u_{{\varepsilon}}\|_{h_{{\varepsilon}}, {\widetilde{\omega}}}$ also converges to zero. From now on, we fix a small positive number ${\varepsilon}_{0}>0$. Then for any positive number ${\varepsilon}$ with $0< {\varepsilon}< {\varepsilon}_{0}$, by property (b) of $h_{{\varepsilon}}$, we obtain $$\|u_{{\varepsilon}} \|_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}} \leq \|u_{{\varepsilon}} \|_{h_{{\varepsilon}}, {\widetilde{\omega}}} \leq \|u \|_{h, \omega}.$$ These inequalities say that the norms of $\{ u_{{\varepsilon}} \}_{{\varepsilon}>0}$ with respect to $h_{{\varepsilon}_{0}}$ are uniformly bounded. Therefore there exists a subsequence of $\{ u_{{\varepsilon}} \}_{{\varepsilon}>0}$ that converges to $\alpha \in L_{(2)}^{n, i}(Y, F)_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}$ with respect to the weak $L^{2}$-topology. For simplicity, we denote this subsequence by the same notation $\{ u_{{\varepsilon}} \}_{{\varepsilon}>0}$. Then we prove the following claim. \[zero\] The weak limit $\alpha$ of $\{ u_{{\varepsilon}} \}_{{\varepsilon}>0}$ in $\in L_{(2)}^{n, i}(Y, F)_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}$ is zero. *Proof of Claim \[zero\].*   For every positive number $\delta>0$, we define the subset $A_{\delta}$ of $Y$ by $A_{\delta}:= \{x \in Y \mid |s|^{2}_{h^{m}_{{\varepsilon}_{0}}} > \delta \}$. Since a weight of $h_{{\varepsilon}_{0}}$ is upper semi-continuous, $|s|^{2}_{h^{m}_{{\varepsilon}_{0}}}$ is lower semi-continuous. Hence $A_{\delta}$ is an open set of $Y$. We estimate the norm of $u_{{\varepsilon}}$ on $A_{\delta}$. By easy computations, we have $$\begin{aligned} \| s u_{{\varepsilon}} \|^{2}_{h^{m+1}_{{\varepsilon}}, {\widetilde{\omega}}} &\geq \| s u_{{\varepsilon}} \|^{2}_{h^{m+1}_{{\varepsilon}_{0}}, {\widetilde{\omega}}} \\ &\geq \int_{A_{\delta}} |s|^{2}_{ h^{m}_{{\varepsilon}_{0}} } |u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}\ {\widetilde{\omega}}^{n} \\ &\geq \delta \int_{A_{\delta}} |u_{{\varepsilon}}|^{2}_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}\ {\widetilde{\omega}}^{n} \geq 0\end{aligned}$$ for any $\delta>0$. Note that $u_{{\varepsilon}} |_{A_{\delta}}$ converges to $\alpha |_{A_{\delta}}$ with respect to the weak $L^{2}$ topology in $ L_{(2)}^{n, i}(A_{\delta}, F)_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}$, where $u_{{\varepsilon}} |_{A_{\delta}}$ (resp. $\alpha |_{A_{\delta}}$) is the restriction of $u_{{\varepsilon}}$ (resp. $\alpha$) to $A_{\delta}$. Indeed, for every $\gamma \in L_{(2)}^{n, i}(A_{\delta}, F)_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}$, the inner product ${{\langle\!\langle}}u_{{\varepsilon}} |_{A_{\delta}}, \gamma {{\rangle\!\rangle}}_{A_{\delta}} = {{\langle\!\langle}}u_{{\varepsilon}}, \widetilde{\gamma} {{\rangle\!\rangle}}_{Y} $ converges to ${{\langle\!\langle}}\alpha, \widetilde{\gamma} {{\rangle\!\rangle}}_{Y} = {{\langle\!\langle}}\alpha, \gamma {{\rangle\!\rangle}}_{A_{\delta}} $, where $\widetilde{\gamma}$ is the zero extension of $\gamma$ to $Y$. Since $u_{{\varepsilon}} |_{A_{\delta}}$ converges to $\alpha |_{A_{\delta}}$, we obtain $$\|\alpha |_{A_{\delta}} \|_{h_{{\varepsilon}_{0}}, \omega} \leq \liminf_{{\varepsilon}\to 0}\|u_{{\varepsilon}} |_{A_{\delta}} \|_{h_{{\varepsilon}_{0}}, \omega}=0.$$ This is because the norm of the weak limit can be estimated by the limit inferior of norms of a sequence. Therefore we know that $\alpha |_{A_{\delta}} = 0$ for any $\delta>0$. By the definition of $A_{\delta}$, the union of $\{A_{\delta} \}_{\delta >0}$ is equal to $Y=X \setminus Z$. Hence the weak limit $\alpha $ is zero on $Y$. By using Claim \[zero\], we complete the proof of Theorem \[main\]. By the definition of $u_{{\varepsilon}}$, we have $$u = u_{{\varepsilon}} + {\overline{\partial}}v_{{\varepsilon}}.$$ Claim \[zero\] implies that ${\overline{\partial}}v_{{\varepsilon}}$ converges to $u$ with respect to the weak $L^{2}$-topology. Then we can easily see that $u$ is a ${\overline{\partial}}$-exact form (that is, $u \in {\rm{Im}} {\overline{\partial}}$). This is because the subspace ${\rm{Im}} {\overline{\partial}}$ is closed in $L_{(2)}^{n, i}(Y, F)_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}$ with respect to the weak $L^{2}$-topology. Indeed, for every $\gamma_{1} + D^{''*}\gamma_{2} \in \mathcal{H}^{n, i}(F)_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}} \oplus {\rm{Im}} D^{''*}_{h_{{\varepsilon}_{0}}}$, we have $${{\langle\!\langle}}u, \gamma_{1} + D^{''*}\gamma_{2} {{\rangle\!\rangle}}= \lim_{{\varepsilon}\to 0}{{\langle\!\langle}}{\overline{\partial}}v_{{\varepsilon}}, \gamma_{1} + D^{''*}\gamma_{2} {{\rangle\!\rangle}}=0.$$ Therefore we know $u \in {\rm{Im}} {\overline{\partial}}$. To summarize, we proved that $u$ is a ${\overline{\partial}}$-exact form in $L_{(2)}^{n, i}(Y, F)_{h_{{\varepsilon}_{0}}, {\widetilde{\omega}}}$, which implies that the cohomology class $\{u \}$ of $u$ is zero in $H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(h_{{\varepsilon}_{0}})})$. By property (c), we obtain the conclusion of Theorem \[main\]. Proof of Corollary ------------------ In this subsection, we prove that $h_{\min}$ and $h_{\rm{Siu}}$ satisfy the assumptions of Theorem \[main\]. First we recall the definition of Siu’s metrics $h_{\rm{Siu}}$ (which was first introduced and plays an central role in [@Siu98]). Let $F$ be a line bundle whose Kodaira dimension is non-negative (that is, some positive multiple of $F$ admits sections). For every positive integer $m>0$, let $h_{m}$ be the singular metric on $F$ induced by a basis of $H^{0}(X, F^m)$, namely $h_{m}$ is (locally) defined by $$- \log h_{m}:= \frac{1}{2m} \log \Big( \sum_{j=1}^{N_{m}} |s_{j}^{(m)}|^{2} \Big),$$ where $\{s_{j} \}_{j=1}^{N_{m}}$ is a basis of $H^{0}(X, F^m)$. Then with a choice of positive numbers $\{ \varepsilon_{m} \}_{m\geq 1}$, Siu’s metric $h_{\rm{Siu}}$ is defined by $$-\log h_{\rm{Siu}} = \log \sum_{m \geq 1} \varepsilon_{m} \frac{1}{h_{m}}.$$ This metric $h_{\rm{Siu}}$ and the multiplier ideal sheaf ${\mathcal{I}(h_{\rm{Siu}})}$ depend on the choice of $\{{\varepsilon}_{k}\}_{k=1}^{\infty}$ (see [@Kim]), but $h_{\rm{Siu}}$ always admits an analytic Zariski decomposition. Thus we can expect a Nadel vanishing theorem for ${\mathcal{I}(h_{\rm{Siu}})}$. \[main-co\] Let $F$ be a big line bundle on a smooth projective variety $X$ and let $h$ be either a metric with minimal singularities $h_{\min}$ or Siu’s metric $h_{\rm{Siu}}$ on $F$. Then we have $$H^{i}(X, K_{X} \otimes F \otimes {\mathcal{I}(h)}) = 0 \hspace{0.4cm} {\text{for}}\ {\text{any}}\ i >0.$$ It is sufficient to prove that the metrics $h_{\min}$ and $h_{\rm{Siu}}$ satisfy the assumptions in Theorem \[main\]. Since $F$ is big, some positive multiple $F^{m}$ of $F$ can be written as $F^{m}= A \otimes E$, where $A$ is a very ample line bundle and $E$ is an effective line bundle. Take a (non-zero) section $s_{A}$ (resp. $s_{E}$) of $A$ (resp. E). Let $h_{A}$ be the metric induced by a basis of $H^{0}(X, A)$. Then $h_{A}$ is smooth and has strictly positive curvature since $A$ is very ample. Further let $h_{E}$ be the singular metric defined by the section $s_{E}$. Now we consider the section $s$ of $F^m$ and the singular metric $h_{{{\rm{big}}}}$ on $F$ defined by $$s := s_{A} \otimes s_{E}\hspace{0.2cm} {\rm{and}} \hspace{0.2cm} h_{{{\rm{big}}}}: = (h_{A} h_{E})^{1/m}.$$ Since the curvature of $h_{{{\rm{big}}}}$ is equal to $(\sqrt{-1}\Theta_{h_{A}}(A) + [{\rm{div}}s_{E}])/m $, the metric $h_{{{\rm{big}}}}$ has strictly positive curvature. Further the point-wise norm $|s|_{h^{m}_{{{\rm{big}}}}}$ is bounded. This is because, the norm $|s|_{h^{m}_{{{\rm{big}}}}}$ is equal to $|s_{A}|_{h_{A}}$ and $h_{A}$ is a smooth metric. It remains to confirm assumption (2) in Theorem \[main\]. When $h$ is a metric with minimal singularities, assumption (2) is obvious by the definition of $h_{\min}$. We consider the case where $h$ is Siu’s metric $h_{\rm{Siu}}$. By the construction of $h_{\rm{Siu}}$, the metric $h_{\rm{Siu}}$ is less singular than $h_{m}$. When $\{ t_{j} \}_{j=1}^{N}$ is a basis of $H^{0}(X, A)$, sections $\{ s_{E}\otimes t_{j} \}_{j=1}^{N}$ become a part of a basis of $H^{0}(X, F^{m})$. Therefore, by the construction of $h_{m}$ and $h_{{{\rm{big}}}}$, the metric $h_{m}$ is less singular than $h_{{{\rm{big}}}}$. Appendix ======== In this section, we show that metrics with minimal singularities do not always have analytic singularities even if line bundles are big. The content of this section is known to specialists, but we give the references and proof for the reader’s convenience. Throughout this section, let $X$ be a compact Kähler manifold and $D$ be a divisor on $X$. We denote by $\mathcal{O}_{X}(D)$, the line bundle (the invertible sheaf) defined by $D$. \[birat\] Let $h_{\min}$ be a metric with minimal singularities on $\mathcal{O}_{X}(D)$. Assume that $h_{\min}$ has analytic singularities. Then $\mathcal{O}_{X}(D)$ admits a birational Zariski decomposition. That is, there exist a modification $\pi: \widetilde{X} \to X$, a nef $\mathbb{R}$-divisor $P$ and an $\mathbb{R}$-effective divisor $N$ on $\widetilde{X}$ with the following properties$:$\ $\bullet$ $\pi^{*}D = P + N$\ $\bullet$ For any positive integer $k > 0$, the map $$H^{0}(X, \mathcal{O}_{\widetilde{X}}(\lfloor kP \rfloor)) \to H^{0}(X, \mathcal{O}_{\widetilde{X}}( kD ))$$ induced by the section $e_{k}$ is an isomorphism, where $e_{k}$ is the natural section of $\lceil kN \rceil$. Here $\lfloor G \rfloor$ $($resp. $\lceil G \rceil$$)$ denotes the divisor defined by the round-downs $($resp. the round-ups$)$ of the coefficients of an $\mathbb{R}$-divisor $G$. Since $h_{\min}$ has analytic singularities, we can take an ideal sheaf $\mathcal{I} \subset \mathcal{O}_{X}$ such that a local weight $ \varphi _{\min}$ can be written as $$\varphi_{\min} = \frac{c}{2} \log \big( |f_{1}|^{2} + |f_{2}|^{2} + \cdots + |f_{k}|^{2}\big) +v,$$ where $f_{i}$ are local generators of $\mathcal{I}$ (see Definition \[s-met\]). By taking a resolution of $\mathcal{I}$, we obtain a modification $\pi : \widetilde{X} \to X$ with $$\sqrt{-1}\Theta_{\pi^{*}h_{\min}} (\pi^{*}D) = \beta + [N],$$ where $\beta $ is a smooth semi-positive $(1,1)$-form and $N$ is an effective $\mathbb{R}$-divisor. On the other hand, the pull-back $\pi^{*}h_{\min}$ of $h_{\min}$ is a metric with minimal singularities on $\mathcal{O}_{\widetilde{X}}(\pi^{*}D)$. Indeed, we fix a smooth metric $h_{0}$ on $\mathcal{O}_{X}(D)$ and take an arbitrary singular metric $g$ on $\mathcal{O}_{\widetilde{X}}(\pi^{*}D)$ with positive curvature. Then there exists an $L^{1}$-function $\varphi_{g}$ on $\widetilde{X}$ such that $g= \pi^{*}h_{0} e^{-\varphi_{g}}$ since $\widetilde{X}$ is also a compact Kähler manifold. Now we define the function $\psi_{g}$ on $X$ by $$\psi_{g}(x):= \sup_{y \in \pi^{-1}(x)} \varphi_{g}(y) \quad \text{for any}\ x\ \in X.$$ Then the curvature of $h_{0} e^{-\psi_{g}}$ is also positive. Hence $h_{\min} \leq C h_{0} e^{-\psi_{g}}$ for some positive constant $C>0$ by the definition of $h_{\min}$. It yields $\pi^{*} h_{\min} \leq C \pi^{*}h_{0} e^{-\varphi_{g}} =C g$. Therefore $\pi^{*} h_{\min}$ is a metric with minimal singularities. Put $P:= \pi^{*}D- N$. Then the first Chern class of $P$ contains the semi-positive form $\beta$ (in particular $P$ is nef). Finally we show that the map in Proposition \[birat\] is isomorphic. For an arbitrary section $s \in H^{0}(X, \mathcal{O}_{X}( kD ))$, the metric $\pi^{*}h_{\min}^{k}$ is less singular than the metric $h_{s}$ induced by the section $s$, since $\pi^{*}h_{\min}^{k}$ is a metric with minimal singularities $\mathcal{O}_{\widetilde{X}}(k \pi^{*}D)$. In particular, the Lelong number of a weight of $k \pi^{*}h_{\min}$ is less than or equal to that of $h_{s}$, thus we obtain $kN \leq {\rm{div}} s $. Since ${\rm{div}} s$ is a $\mathbb{Z}$-divisor, we have $\lceil kN \rceil \leq {\rm{div}} s $. It implies that $s/e_{k}$ is a (holomorphic) section. This completes the proof. In general, it is difficult to compute metrics with minimal singularities. Proposition \[birat\] says that metrics with minimal singularities on line bundles admitting no birational Zariski decomposition never have analytic singularities. 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--- abstract: | In these lectures, superconductivity in impure thin films close to the absolute zero of temperature is discussed. The behavior as function of the applied magnetic field and the amount of impurities suggests the presence of a superconductor-insulator transition at zero temperature. The theory of superconductivity in the limit where all the electrons become tightly bound in pairs is used to explain the main characteristics of the transition. In that limit, where the theory becomes equivalent to a phase-only theory, electron pairs exist on either side of the transition. The presentation is pedagogical in nature and includes exercises as a learning aid for those new to the field.\ author: - | Adriaan M. J. Schakel\ [*Low Temperature Laboratory, Helsinki University of Technology*]{},\ [*P.O. Box 2200, FIN-02015 HUT, Finland*]{}\ and\ [*Department of Electrophysics, National Chiao Tung University,* ]{}\ [*Hsinchu, 30050, Taiwan, R.O.C.*]{} title: 'Superconductor-Insulator Quantum Phase Transitions [^1]' --- PACS: 74.40.+k, 71.30.+h, 64.60.Fr Introduction ============ The topic of these lectures is superconductivity in impure thin films close to the absolute zero of temperature. Such superconducting films may by increasing either the amount of impurities, or the applied magnetic field go over to an insulating state. The transition is believed to signal the presence of a quantum critical point at zero temperature [@MPAFisher; @HPsu1]. Quantum critical phenomena differ from conventional critical phenomena taking place at finite temperature, in that quantum rather than thermal fluctuations are important in the critical region. Since quantum phase transitions occur at zero temperature, they cannot be induced by changing the temperature as can thermal phase transitions, and another parameter is to be varied to trigger the transition. It also means that one can never tune right through the transition as experiments are necessarily done at finite temperature. For low enough temperature though, the presence of a quantum critical point can nevertheless be detected by using finite-size scaling. A superconducting and an insulating state seems an unlikely combination to be present in the same system. Whereas superconductivity needs an [*attractive*]{} interaction for the pairing between electrons, the insulating state needs a [*repulsive*]{} interaction. It is a priori not clear how these two requirements can be fulfilled in a single system. Adding to the bewilderment is the striking similarity in the current-voltage characteristics in both phases close to the transition. By interchanging the current and voltage axes in one phase, a characteristic obtained at some value of the applied magnetic field say, can be mapped onto a characteristic of the other phase obtained at a different value of the field. This means that, although the physical properties of the superconducting and the insulating state are completely different, there is a close connection between the conduction mechanisms in the two phases. A last defining aspect of the superconductor-insulator transition is the presence of a $1/r$ Coulomb potential, which at low charge-carrier densities becomes very strong. In these lectures, we will argue that the main aspects of the superconductor-insulator transition just mentioned can be accounted for by a single theory. Namely, the theory of superconductivity in the limit where all the electrons become tightly bound in pairs—the so-called [*composite boson limit*]{} [@Eagles; @Leggett]. This is the opposite limit of the conventional weak-coupling BCS limit, where only electrons in a thin shell around the Fermi surface become loosely bound in Cooper pairs. In the composite boson limit, the superconducting state displays a quantum phase transition to an insulating state characterized not by an unbinding of electron pairs, but rather by a quenching of the condensate of composite bosons. In other words, electron pairs exist on both sides of the transition. In the superconducting state they are Bose-Einstein condensed, while in the insulating state they are localized. The attractive interaction responsible for the binding of the electrons in pairs translates into a repulsive interaction between the composite bosons. The theory describing these bosons is the Bogoliubov theory of superfluidity, which is equivalent to a phase-only theory. This implies that the superconductor-insulator transition can be described by a phase-only effective theory without ignoring any degrees of freedom [@Ramakrishnan; @CFGWY]. To be fair, we should mention at this point that no consensus exists to what extend the phase-only theory can be applied to the superconductor-insulator transition. Further experimental and theoretical studies are required to settle this point. General Scaling --------------- Since quantum rather than thermal fluctuations are relevant in a quantum phase transition, one has to work in spacetime rather than just in space as is appropriate for thermal phase transitions in equilibrium. As a result, in addition to a diverging correlation length $\xi$, quantum phase transitions have also a diverging correlation time $\xi_t$. They indicate, respectively, the distance and time period over which the system fluctuates coherently. The two are related, with the diverging correlation time scaling with the diverging correlation length as $$\label{zcrit} \xi_t \sim \xi^z,$$ where $z$ is the so-called dynamic exponent. It is a measure for the asymmetry between the time and space directions close to the critical point. The dynamic exponent is to be added to the set of critical exponents used to characterize a thermal phase transition. Since such transitions have only two independent exponents, a quantum phase transition is specified by three independent exponents. The traditional scaling theory of thermal, continuous phase transitions in equilibrium, first put forward by Widom [@Widom], is easily extended to include the time dimension [@Ma] because the relation (\[zcrit\]) implies the presence of only one independent diverging scale. Let $\delta = K - K_{\rm c}$, with $K$ the parameter triggering the phase transition, measure the distance from the critical value $K_{\rm c}$. A physical observable $O$ at the absolute zero of temperature depends on $K$ as well as on other variable, such as an external field, energy, or momentum. Let us denote these other variables collectively by $\Gamma$. In the critical region close to the critical point, $O$ can be written as $$\label{scaling0} O(\Gamma,K) = \xi^{d_O} {\cal O}(\hat{\Gamma}), \;\;\;\;\;\;\;\; (T=0),$$ where $d_O$ is the scaling dimension of the observable $O$, $\xi \sim |\delta|^{- \nu}$, with $\nu$ the correlation length exponent, and $\hat{\Gamma}$ is obtained from $\Gamma$ by rescaling it with factors of the correlation length, so that $\hat{\Gamma}$ is independent of that scale. To be specific, if an external field scales as $\Gamma \sim \xi^{d_\Gamma}$, then the rescaled field is defined as $\hat{\Gamma} = \xi^{-d_\Gamma} \Gamma $. The right side of Eq. (\[scaling0\]) does not depend explicitly on $K$, but only implicitly through $\xi$. The data of the observable $O$ as function of an external field $\Gamma$ obtained at different values of the parameter $K$ triggering the transition can be collapsed on a single curve when instead of $O(\Gamma,K)$, the rescaled quantity $|\delta|^{\nu d_O} O(\Gamma,K)$ is plotted as function not of $\Gamma$, but of $\hat{\Gamma}$. Indeed, because of Eq. (\[scaling0\]), the combination $|\delta|^{\nu d_O} O(\Gamma,K)$ depends only on $\hat{\Gamma}$ and is thus independent of the distance from the critical point. This is used to determine critical exponents experimentally. By rescaling the vertical and horizontal axis of the plot with $|\delta|^x$ and $|\delta|^y$, respectively, the best collapse obtained at some value $x_0$ and $y_0$ give the combination of critical exponents $\nu d_O = y_0$ and $\nu d_\Gamma =x_0$. Since a physical system is always at some finite temperature, we have to investigate how the scaling law (\[scaling0\]) changes when the temperature becomes nonzero. The easiest way to include the temperature in a quantum theory is to go over to imaginary time $\tau = it$, with $\tau$ restricted to the interval $0 \leq \tau \leq 1/T$. The temporal dimension thus becomes of finite extend. The behavior at a finite temperature is still controlled by the quantum critical point, provided the correlation time satisfies $\xi_t < 1/T$. If this condition is fulfilled, the system will not see the finite extent of the time dimension. This is what makes quantum phase transitions experimentally accessible. Instead of the zero-temperature scaling (\[scaling0\]), we now have the finite-size scaling $$\label{scalingT} O(\Gamma,K,T) = T^{-d_O/z} {\cal O}(\hat{\Gamma}_T, \xi_t T), \;\;\;\;\;\;\;\; (T \neq 0),$$ where instead of using the correlation length to convert dimensionfull quantities in dimensionless ones, the temperature is used: $\hat{\Gamma}_T = T^{d_\Gamma/z} \Gamma$. Notation {#chap:not .unnumbered} -------- We adopt Feynman’s notation and denote a spacetime point by $x=x_\mu =(t,{\bf x})$, $\mu = 0,1, \cdots,d$, with $d$ the number of space dimensions, while the energy $k_0$ and momentum ${\bf k}$ of a particle will be denoted by $k=k_\mu = (k_0,{\bf k})$. The time derivative $\partial_0 = \partial/\partial t$ and the gradient $\nabla$ are sometimes combined in a single vector $\tilde{\partial}_\mu = (\partial_0, -\nabla)$. The tilde on $\partial_\mu$ is to alert the reader for the minus sign appearing in the spatial components of this vector. We define the scalar product $k \cdot x = k_\mu x_\mu = k_0 t - {\bf k} \cdot {\bf x}$ and use Einstein’s summation convention. Because of the minus sign in the definition of the vector $\tilde{\partial}_\mu$ it follows that $\tilde{\partial}_\mu a_\mu = \partial_0 a_0 + \nabla \cdot {\bf a}$, with $a_\mu$ an arbitrary vector. Integrals over spacetime are denoted by $$\int_x = \int_{t,{\bf x}} = \int {\mbox{d}}t \, {\mbox{d}}^d x,$$ while those over energy and momentum by $$\int_k = \int_{k_0,{\bf k}} = \int \frac{{\mbox{d}}k_0}{2 \pi} \frac{{\mbox{d}}^d k}{(2 \pi)^d}.$$ When no integration limits are indicated, the integrals are assumed to run over all possible values of the integration variables. Similarly, for functional integrals we use the notation $$\int \mbox{D} \phi = \int_\phi.$$ We will work in natural units with the speed of light, Boltzmann’s constant $k_{\rm B}$, and Planck’s constant $\hbar$ set to unity. These lectures include exercises, which are clearly marked. Most of the solutions can be found in Ref. [@Pompo]. Superconductivity ================= In this section we study the theory of superconductivity in the limit where all the electrons become tightly bound in pairs [@Eagles; @Leggett]. The composite boson limit is to be distinguished from the usual weak-coupling BCS limit, where only electrons (of opposite momentum) in a thin shell around the Fermi surface become loosely bound in Cooper pairs. BCS Theory ---------- As starting point we take the microscopic BCS model specified by the Lagrangian [@BCS] $$\begin{aligned} \label{bcs:BCS} {\cal L} &=& \psi^{\ast}_{\uparrow} [i\partial_0 - \xi(-i \nabla)] \psi_{\uparrow} + \psi_{\downarrow}^{\ast} [i \partial_0 - \xi(-i \nabla)]\psi_{\downarrow} - \lambda \psi_{\uparrow}^{\ast}\,\psi_{\downarrow} ^{\ast}\,\psi_{\downarrow}\,\psi_{\uparrow} \nonumber \\ &:=& {\cal L}_0 + {\cal L}_{\rm int},\end{aligned}$$ where ${\cal L}_0$ is the free theory, and ${\cal L}_{\rm int} = - \lambda \psi_{\uparrow}^{\ast} \, \psi_{\downarrow}^{\ast} \, \psi_{\downarrow}\,\psi_{\uparrow}$ is a local electron-electron interaction term, representing the effective, phonon mediated, attraction between electrons with coupling constant $\lambda < 0$. The field $\psi_{\uparrow (\downarrow )}$ is an anticommuting field describing the electrons with mass $m$ and spin up (down), while $\xi(-i \nabla) = \epsilon(-i \nabla) - \mu$, with $\epsilon(-i \nabla) = - \nabla^2/2m$, is the kinetic energy operator with the chemical potential $\mu$ characterizing the ensemble of fermions subtracted. The theory is invariant under global U(1) transformations under which $$\label{bcs:3g} \psi_{\sigma} \rightarrow \mbox{e}^{i \alpha } \psi_{\sigma},$$ with $\sigma = \uparrow, \downarrow$ and $\alpha$ a constant transformation parameter. The superconducting state is characterized by a spontaneous breakdown of this symmetry. To investigate the superconducting state, we integrate out the fermionic degrees of freedom at the expense of a newly introduced auxiliary field, which is better equipped to describe this condensed matter state. It is convenient to first introduce the two-component field $$\label{bcs:32} \psi = \left( \begin{array}{c} \psi_{\uparrow} \\ \psi_{\downarrow}^{\ast} \end{array} \right) \:\:\:\:\:\: \psi^{\dagger} = (\psi_{\uparrow}^{\ast},\psi_{\downarrow}).$$ In this so-called Nambu representation, ${\cal L}_0$ becomes $$\label{bcs:33} {\cal L}_0 = \psi^{\dagger}\, \left(\begin{array}{cc} i \partial_0 - \xi(-i \nabla) & 0 \\ 0 & i \partial_0 + \xi(-i \nabla) \end{array}\right) \, \psi,$$ where the anticommuting character of the electron fields is used and total derivatives are neglected. We next replace the electron-electron interaction with an expression involving the auxiliary field $\Delta$ $$\label{bcs:35} \lambda \psi_{\uparrow}^{\ast} \, \psi_{\downarrow}^{\ast} \, \psi_{\downarrow}\, \psi_{\uparrow} \to \Delta^* \, \psi_{\downarrow}\,\psi_{\uparrow} + \psi_{\uparrow}^{\ast} \, \psi_{\downarrow}^{\ast} \, \Delta - \frac{1}{\lambda} |\Delta|^2.$$ The original interaction is regained when the auxiliary field is integrated out. Physically, $\Delta$, representing a product of two electron fields, describes electron pairs. We shall therefore refer to it as pair field. With this replacement, the partition function $$\label{bcs:34} Z = \int_{\psi^{\dagger},\psi} \exp \left( i \int_x \,{\cal L} \right),$$ becomes $$\begin{aligned} \label{bcs:36} Z = \int_{\psi^{\dagger},\psi, \Delta^*, \Delta} && \!\!\!\! \exp\left(\frac{i}{\lambda} \int_x |\Delta|^2 \right) \\ && \!\!\!\! \times \exp \left[ i \int_x \, \psi^{\dagger} \left( \begin{array}{cc} i \partial_0 - \xi(-i \nabla) & -\Delta \\ -\Delta^* & i \partial_0 + \xi(-i \nabla) \end{array} \right) \psi \right] \nonumber . \end{aligned}$$ Since the fermion fields appear quadratically now, they may be integrated out to yield an effective action $S_{\rm eff}$ for $\Delta$ and $\Delta^*$ $$\label{bcs:312} S_{\rm eff}[\Delta^*, \Delta] = -i \, {\rm Tr} \ln \left( \begin{array}{cc} p_{0} - \xi ({\bf p}) & -\Delta \\ -\Delta^* & p_{0} + \xi ({\bf p}) \end{array}\right),$$ where $p_0 = i \partial_0$ and $\xi({\bf p}) = \epsilon({\bf p}) - \mu$, with $\epsilon({\bf p}) = {\bf p}^2/2m$ and ${\bf p} = -i \nabla$, so that the partition function becomes $$\label{bcs:37} Z = \int _{\Delta^*, \Delta} \exp \left(i S_{\rm eff} [ \Delta^*, \Delta] + \frac{i}{\lambda} \int_x |\Delta|^2 \right).$$ The trace Tr appearing in Eq. (\[bcs:312\]) denotes the trace over discrete indices as well as the integration over spacetime and over energy $k_0$ and momentum ${\bf k}$. In the mean-field approximation, the functional integral over the pair field in Eq. (\[bcs:37\]) is approximated by the saddle point, where only the extremal value, satisfying the equation $$\label{bcs:gap} \frac{\delta S_{\rm eff} }{\delta \Delta_0^* (x) } = - \frac{1}{\lambda} \Delta_0(x),$$ is included. For a system homogeneous in spacetime, the pair field is a constant $\bar{\Delta}$, and Eq. (\[bcs:gap\]) reduces, after passing to the Fourier representation, to the BCS gap equation [@BCS]: $$\begin{aligned} \label{bcs:gape} \frac{1}{\lambda} &=& - i \int_k \frac{1}{k_0^{2} - E^{2}(k) + i \eta} \nonumber \\ &=& - \frac{1}{2} \int_{\bf k} \frac{1}{E({\bf k})}.\end{aligned}$$ Here, $\eta$ is an infinitesimal positive constant which is to be set to zero at the end of the calculation, and $$\label{bcs:spec} E({\bf k}) = \sqrt{\xi^2({\bf k}) + |\bar{\Delta}_0|^2}$$ is the spectrum[^2] of the fermionic excitations. A nontrivial solution to the gap equation signals the spontaneous symmetry breakdown of the global U(1) symmetry (\[bcs:3g\]). Composite Boson Limit --------------------- For studying the composite boson limit, it proves prudent to swap the coupling constant $\lambda$ in the gap equation (\[bcs:gape\]) for a more convenient parameter, namely the binding energy $\epsilon_a$ of an electron pair in vacuum [@RDS]. Both parameters characterize the strength of the electron-electron interaction. To establish the connection between the two, let us consider the Schrödinger equation for the problem at hand. In reduced coordinates, it reads $$\left[- \frac{\nabla^2}{m} + \lambda \, \delta({\bf x}) \right] \psi({\bf x}) = - \epsilon_a,$$ where the reduced mass is $m/2$ and the delta-function potential, with $\lambda < 0$, represents the attractive local electron-electron interaction ${\cal L}_{\rm int}$ in Eq. (\[bcs:BCS\]). We stress that this is a two-particle problem in vacuum and not the famous Cooper problem of two interacting fermions on top of a filled Fermi sea. The equation is most easily solved in the Fourier representation, yielding $$\psi({\bf k}) = - \frac{\lambda}{{\bf k}^2/m + \epsilon_a} \psi(0),$$ or $$\label{bound} - \frac{1}{\lambda} = \int_{\bf k} \frac{1}{{\bf k}^2/m + \epsilon_a} .$$ This bound-state equation allows us to replace the coupling constant $\lambda$ with the binding energy $\epsilon_a$. When substituted in the gap equation (\[bcs:gape\]), the latter becomes $$\label{bcs:reggap} \int_{\bf k} \frac{1}{{\bf k}^2/m + \epsilon_a} = \frac{1}{2} \int_{\bf k} \frac{1}{E({\bf k})}.$$ By inspection, it follows that this equation has a solution given by [@Eagles; @Leggett] $$\label{comp:self} \bar{\Delta}_0 \rightarrow 0, \;\;\;\;\; \mu \rightarrow - {{\textstyle\frac{1}{2}}} \epsilon_a,$$ where it should be noted that, in contrast to the weak-coupling limit, the chemical potential characterizing the ensemble of fermions is [*negative*]{} here. This is the composite boson limit. To appreciate the physical significance of the specific value found for the chemical potential in this limit and also its name, observe that the spectrum $E_{\rm B}({\bf q})$ of the two-fermion bound state measured relative to the pair chemical potential $2\mu$ reads $$E_{\rm B}({\bf q}) = \frac{{\bf q}^2}{4m} - \mu_{\rm B},$$ where $\mu_{\rm B}$ is defined as $\mu_{\rm B} = \epsilon_a + 2 \mu$ and may be understood as the chemical potential characterizing the ensemble of composite bosons. The negative value for $\mu$ found in Eq. (\[comp:self\]) is precisely the condition for a Bose-Einstein condensation of an [*ideal*]{} gas of composite bosons in the ${\bf q} = 0$ state. Including quadratic terms in $\bar{\Delta}_0$, we obtain as solution to Eq. (\[bcs:reggap\]) $$\mu = - \frac{1}{2} \epsilon_a + (1-d/4) \frac{|\bar{\Delta}_0|^2}{\epsilon_a}.$$ This leads to the chemical potential $$\label{chem} \mu_{\rm B} = (2-d/2) \frac{|\bar{\Delta}_0|^2}{\epsilon_a},$$ characterizing the now interacting Bose gas. Renormalization {#sec:ren} --------------- For a system homogeneous in spacetime, so that the field $\Delta_0(x)$ is constant, the effective action (\[bcs:312\]) is readily evaluated. Disassembling the argument of the logarithm as $$\left( \begin{array}{cc} p_{0} - \xi ({\bf p}) & -\bar{\Delta}_0 \\ -\bar{\Delta}_0^* & p_{0} + \xi ({\bf p}) \end{array}\right) = \left( \begin{array}{cc} p_{0} - \xi ({\bf p}) & 0 \\ 0 & p_{0} + \xi ({\bf p}) \end{array}\right) - \left( \begin{array}{cc} 0 & \bar{\Delta}_0 \\ \bar{\Delta}_0^* & 0 \end{array}\right),$$ and expanding the second logarithm in a Taylor series, we recognize the form $$\begin{aligned} \label{resum} S_{\rm eff}[\bar{\Delta}_0^*, \bar{\Delta}_0] = && -i \, {\rm Tr} \ln \left( \begin{array}{cc} p_{0} - \xi ({\bf p}) & 0 \\ 0 & p_{0} + \xi ({\bf p}) \end{array}\right) \nonumber \\ && - i \, {\rm Tr} \ln \left(1 - \frac{|\bar{\Delta}_0|^2}{p_0^2 - \xi^2({\bf p})} \right),\end{aligned}$$ apart from an irrelevant constant. Again passing to the Fourier representation, and carrying out the integral over the loop energy $k_0$, we obtain the effective Lagrangian $$\label{Exi} {\cal L}_{\rm eff} = \int_{\bf k} \left[ E({\bf k}) - \xi({\bf k}) \right].$$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ To the one-loop result (\[Exi\]), the tree term $|\bar{\Delta}_0|^2/\lambda$ is to be added. Expanding $E({\bf k})$ in Eq. (\[Exi\]) in a Taylor series, we see that the effective Lagrangian also contains a term quadratic in $\bar{\Delta}_0$. This term amounts to a renormalization of the coupling constant. Specifically, the renormalized coupling constant $\lambda_{\rm r}$ is to this order given by $$\begin{aligned} \label{bcs:reng} \frac{1}{\lambda_{\rm r}} &=& \frac{1}{\lambda} + \frac{1}{2} \int_{\bf k} \frac{1}{\xi({\bf k})} \nonumber \\ &=& \frac{1}{\lambda} + \frac{\Gamma(1-d/2)}{(4 \pi)^{d/2}} \frac{m^{d/2}}{\epsilon_a^{1-d/2}},\end{aligned}$$ with, as is appropriate in the composite boson limit, $$\label{approx} \xi({\bf k}) = \epsilon({\bf k}) + {{\textstyle\frac{1}{2}}} \epsilon_a,$$ and dimensional regularization is used in evaluating the momentum integral. Because of Eq. (\[bound\]), which may be viewed as the defining equation of the parameter $\epsilon_a$, the right side of Eq.(\[bcs:reng\]) is zero. This implies that in the composite boson limit, $\lambda_{\rm r} \to - \infty$ so that we indeed have tightly bound pairs here. Bogoliubov theory ================= In this section we show that in the composite boson limit, the effective theory obtained after integrating out the fermionic degrees of freedom, is the Bogoliubov theory of superfluidity in an interacting Bose gas. The gas consists of composite bosons with a mass twice the electron mass. The system is known to undergo a quantum phase transition from the superfluid to a (Mott) insulating state [@FWGF]. When translated back to the fermionic theory, this transition corresponds to one where the condensate is drained of composite bosons, without breaking them up. We include impurities in the Bogoliubov theory to show that it leads to localization without destroying the superfluid state completely. The insulating state, which is now a result not only of repulsive interactions, as in a Mott insulator, but also of (Anderson) localization, is called an Anderson-Mott insulator. Derivative Expansion -------------------- We next wish to relax the assumption of homogeneity in spacetime and consider a spacetime-dependent pair field. To this end, we study the effective action (\[bcs:312\]) and expand $\Delta(x)$ around the constant value $\bar{\Delta}_0$ satisfying the gap equation (\[bcs:gape\]), $$\Delta(x) = \bar{\Delta}_0 + \tilde{\Delta}(x).$$ We obtain in this way, $$\label{effact} S_{\rm eff} = i \, {\rm Tr} \sum_{\ell =1}^\infty \frac{1}{\ell} \left[ G_0(p) \left( \begin{array}{cc} 0 & \tilde{\Delta} \\ \tilde{\Delta}^* & 0 \end{array} \right) \right]^\ell,$$ where $G_0$ is the correlation function, $$\begin{aligned} \label{bcs:prop} G_0(k) &=& \left( \begin{array}{cc} k_0 - \xi ({\bf k}) & -\bar{\Delta}_0 \\ -\bar{\Delta}_0^* & k_0 + \xi ({\bf k}) \end{array} \right)^{-1} \\ &=& \frac{1}{k_0^2 - E^2({\bf k}) + i \eta } \left( \begin{array}{cc} k_0 \, {\rm e}^{i k_0 \eta } + \xi ({\bf k}) & \bar{\Delta}_0 \\ \bar{\Delta}_0^* & k_{0} \, {\rm e}^{-i k_0 \eta}- \xi ({\bf k}) \end{array} \right). \nonumber \end{aligned}$$ The exponential functions in the diagonal elements of the correlation function are additional convergence factors needed in nonrelativistic theories [@Mattuck]. When evaluating the effective action (\[effact\]), the precise meaning of the trace Tr appearing there should be kept in mind. Explicitly, it is defined as $$\label{bcs:explicit} S_{\rm eff} = -i {\rm Tr} \, \ln \left[K(p,x) \right] = -i {\rm tr} \ln\left[ K(p,x) \delta (x - y)\bigr|_{y = x} \right],$$ where the trace tr is the usual one over discrete indices. We abbreviated the matrix appearing in Eq. (\[bcs:312\]) by $K(p,x)$ so as to cover the entire class of actions of the form $$S = \int_x \psi^\dagger(x) K(p,x) \psi(x).$$ The delta function in Eq. (\[bcs:explicit\]) arises because $K(p,x)$ is obtained as a second functional derivative of the action $$\frac{\delta^{2} S}{\delta \psi^\dagger(x) \, \delta \psi(x)} = K(p,x) \, \delta (x - y) \bigr|_{y = x},$$ each of which gives a delta function. Since the action contains only one integral over spacetime, one delta function remains. Because it is diagonal, the delta function may be taken out of the logarithm and Eq. (\[bcs:explicit\]) can be written as $$\begin{aligned} \label{bcs:Trexplicit} S_{\rm eff} &=& -i {\rm tr} \, \int_x \ln \left[ K(p,x) \right] \delta (x - y) \bigr|_{y = x} \nonumber \\ &=& -i {\rm tr} \, \int_x \int_k {\rm e}^{i k \cdot x} \, \ln \left[ K(p,x) \right] {\rm e}^{-i k \cdot x}.\end{aligned}$$ In the last step, we used the integral representation of the delta function: $$\delta (x) = \int_k {\rm e}^{-i k \cdot x},$$ shifted the exponential function $\exp (i k \cdot y)$ to the left, which is justified because the derivative $p_\mu$ does not operate on it, and, finally, set $y_\mu$ equal to $x_\mu$. We thus see that the trace Tr in Eq. (\[bcs:explicit\]) stands for the trace over discrete indices as well as the integration over spacetime and over energy and momentum. The integral $\int_k$ arises because the effective action calculated here is a one-loop result with $k_\mu$ the loop energy and momentum. The integrals in Eq. (\[bcs:Trexplicit\]) cannot in general be evaluated in closed form because the logarithm contains energy-momentum operators and spacetime-dependent functions in a mixed order. To disentangle the integrals resort has to be taken to a derivative expansion [@FAF] in which the logarithm is expanded in a Taylor series. Each term contains powers of the energy-momentum operator $p_\mu$ which acts on every spacetime-dependent function to its right. All these operators are shifted to the left by repeatedly applying the identity $$f(x) p_\mu g(x) = (p_\mu - i \tilde{\partial}_\mu) f(x) g(x),$$ where $f(x)$ and $g(x)$ are arbitrary functions of spacetime and the derivative $\tilde{\partial}_\mu = (\partial_0,-\nabla)$ acts [*only*]{} on the next object to the right. One then integrates by parts, so that all the $p_\mu$’s act to the left where only a factor $\exp(i k \cdot x)$ stands. Ignoring total derivatives and taking into account the minus signs that arise when integrating by parts, one sees that all occurrences of $p_\mu$ (an operator) are replaced with $k_\mu$ (an integration variable). The exponential function $\exp(-i k \cdot x)$ can at this stage be moved to the left where it is annihilated by the function $\exp(i k \cdot x)$. The energy-momentum integration can now in principle be carried out and the effective action be cast in the form of an integral over a local density ${\cal L}_{\rm eff}$: $$S_{\rm eff} = \int_x {\cal L}_{\rm eff}.$$ This is in a nutshell how the derivative expansion works. ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ Map onto Bogoliubov Theory -------------------------- We are interested in terms in the effective action (\[effact\]) quadratic in the field $\tilde{\Delta}$. Using the derivative expansion, we find $$\begin{aligned} \label{comp:Seff} S_{\rm eff}^{(2)}(q) &=& \frac{i}{2} \, {\rm Tr} \, \frac{1}{p_0^2 - E^2({\bf p})} \frac{1}{(p_0 + q_0)^2 - E^2({\bf p} - {\bf q})} \\ && \;\;\;\; \times \Bigr\{ \bar{\Delta}_0^2 \, \tilde{\Delta}^* \tilde{\Delta}^* + [p_0 + \xi({\bf p})] [p_0 + q_0 - \xi({\bf p} - {\bf q})] \tilde{\Delta} \tilde{\Delta}^* \nonumber \\ && \;\;\;\;\;\;\;\;\; + \bar{\Delta}_0^{*^{\scriptstyle{2}}} \tilde{\Delta} \tilde{\Delta} + [p_0 - \xi({\bf p})] [p_0 + q_0 + \xi({\bf p} - {\bf q})] \tilde{\Delta}^* \tilde{\Delta} \Bigl\}, \nonumber \end{aligned}$$ where $q_\mu = i\tilde{\partial}_\mu$. Let us first ignore the derivatives in this expression. After carrying out the integral over the loop energy $k_0$ and using the gap equation (\[bcs:gape\]), we obtain $$\label{comp:Lag1} {\cal L}^{(2)}(0) = -\frac{1}{8} \int_{\bf k} \frac{1}{E^3({\bf k})} \left(\bar{\Delta}_0^2 \, \tilde{\Delta}^*{}^2 + \bar{\Delta}_0^{*^{\scriptstyle{2}}} \tilde{\Delta}^2 + 2 |\bar{\Delta}_0|^2 |\tilde{\Delta}|^2 \right).$$ In the composite boson limit, where the spectrum of the fermionic excitations is given by Eq. (\[approx\]), the integral over the loop momentum becomes elementary, yielding $$\int_{\bf k} \frac{1}{E^3({\bf k})} = \frac{4 \Gamma(3-d/2)}{(4 \pi)^{d/2}} m^{d/2} \epsilon_a^{d/2-3}.$$ We next consider the terms in Eq. (\[comp:Seff\]) involving derivatives. Following Ref. [@Haussmann] we set $\bar{\Delta}_0$ to zero here. The integral over the loop energy is easily carried out, with the result $$\begin{aligned} {\cal L}^{(2)}(q) &=& - \frac{1}{2} \int_{\bf k} \frac{1}{q_0 - {\bf k}^2/m + 2 \mu - {\bf q}^2/4m} \tilde{\Delta} \tilde{\Delta}^* \nonumber \\ && - \frac{1}{2} \int_{\bf k} \frac{1}{-q_0 - {\bf k}^2/m + 2 \mu - {\bf q}^2/4m} \tilde{\Delta}^* \tilde{\Delta}.\end{aligned}$$ In the composite boson limit, the remaining momentum integrals become elementary again and after expanding in derivatives we find $$\begin{aligned} \label{derivatives} \lefteqn{\int_{\bf k} \frac{1}{q_0 - {\bf k}^2/m - \epsilon_a - {\bf q}^2/4m} =} \\ && - \frac{ \Gamma(1-d/2)}{(4 \pi)^{d/2}} m^{d/2} \epsilon_a^{d/2-1} - \frac{ \Gamma(2-d/2)}{(4 \pi)^{d/2}} m^{d/2} \epsilon_a^{d/2-2} \left(q_0 - \frac{{\bf q}^2}{4m} \right). \nonumber \end{aligned}$$ When combined with the tree term $|\tilde{\Delta}|^2/\lambda$, the first term at the right side of this equation yields the renormalization (\[bcs:reng\]) of the coupling constant. The second term at the right side of Eq. (\[derivatives\]) gives, when combined with the contribution (\[comp:Lag1\]), the result [@Haussmann], $$\label{fin} {\cal L}^{(2)} = \frac{1}{2} \frac{\Gamma(2-d/2)}{(4 \pi)^{d/2}} m^{d/2} \epsilon_a^{d/2-2}\, \tilde{\Psi}^\dagger \, M_0(q) \, \tilde{\Psi}, \;\;\;\;\; \tilde{\Psi} = \left(\begin{array}{l} \tilde{\Delta} \\ \tilde{\Delta}^* \end{array} \right),$$ where $M_0(q)$ stands for the $2 \times 2$ matrix, $$\begin{aligned} \label{comp:M} \lefteqn{M_0(q) =} \nonumber \\ && \left( \begin{array}{cc} q_0 - {\bf q}^2/4m - (2-d/2) |\bar{\Delta}_0|^2/ \epsilon_a & - (2-d/2) \bar{\Delta}_0^2/ \epsilon_a \\ - (2-d/2) \bar{\Delta}_0^{*^{\scriptstyle{2}}} / \epsilon_a & -q_0 - {\bf q}^2/4m - (2-d/2) |\bar{\Delta}_0|^2/ \epsilon_a \end{array} \right). \nonumber \\ && \end{aligned}$$ As we shall show now, this is the Bogoliubov theory of superfluidity in an interacting Bose gas. That is to say, after integrating out the fermionic degrees of freedom from the theory of superconductivity, we obtain in the composite boson limit a theory describing a gas of repulsively interacting (composite) bosons. Quantum Phase Transition ------------------------ The Bogoliubov theory is specified by the Lagrangian [@Bogoliubov] $$\label{bec:Lagr} {\cal L} = \phi^* \bigl[i \partial_0 - \epsilon(-i \nabla) + \mu_{\rm B} \bigr] \phi - \lambda_{\rm B} |\phi|^4,$$ where $\mu_{\rm B}$ is the chemical potential characterizing the Bose gas. The self-coupling is taken to be positive, $\lambda_{\rm B} > 0$, so that the local interaction is repulsive. At the mean-field, or classical level, where quantum fluctuations are ignored, the theory (\[bec:Lagr\]) undergoes a phase transition when the chemical potential changes sign. For $\mu_{\rm B}>0$, the global U(1) symmetry is spontaneously broken by a nontrivial ground state, while for $\mu_{\rm B}<0$ the symmetry is unbroken. The change in $\mu_{\rm B}$ can be induced by varying the temperature, as in a thermal phase transition, but it can also be induced at zero temperature by varying, for example, the number of charge carriers, or the amount of impurities. The zero-temperature quantum phase transition describes a transition between a superfluid and an insulating state [@FWGF]. The ground state of a system homogeneous in spacetime is obtained by considering the shape of the potential energy $$\label{bec:V} {\cal V} = - \mu_{\rm B} |\bar{\phi}|^2 + \lambda_{\rm B} |{\bar \phi}|^4.$$ For $\mu_{\rm B} > 0$ it indeed has a minimum away from the origin $\phi = 0$ given by $$\label{bec:min} |\bar{\phi}_0|^2 = \frac{1}{2} \frac{\mu_{\rm B}}{\lambda_{\rm B} },$$ and the potential becomes $$\label{V0} {\cal V}_0 = - \frac{\mu_{\rm B}^2}{4 \lambda_{\rm B}}.$$ Since the total particle number density $n_{\rm B}$ is represented by $$n_{\rm B}(x) = |\phi(x)|^2,$$ the quantity $n_0 := |\phi_0|^2$ physically denotes the number density of particles residing in the ground state. A nonzero value for $n_0$ thus signals Bose-Einstein condensation. For a homogeneous system in its ground state, we see that at the mean-field level $\bar{n}_0 = \bar{n}$ so that all the particles reside in the condensate. This will change when quantum fluctuations are included as a result of which particles are knocked out of the condensate (see below). To account for the nontrivial ground state, we introduce the shifted field[^3] $\tilde{\phi}(x)$: $$\label{bec:newfields} \phi(x) = \bar{\phi}_0 + \tilde{\phi}(x).$$ The terms in the Lagrangian (\[bec:Lagr\]) quadratic in this shifted field may be cast in the matrix form $$\label{bec:L0} {\cal L}_0 = {{\textstyle\frac{1}{2}}} \tilde{\Phi}^{\dagger} M_0(p) \tilde{\Phi}, \;\;\;\;\;\; \tilde{\Phi} = \left(\begin{array}{l} \tilde{\phi} \\ \tilde{\phi}^* \end{array} \right),$$ with $$\label{bec:M} M_0(p) = \left( \begin{array}{cc} p_0 - \epsilon({\bf p}) + \mu_{\rm B} - 4 \lambda_{\rm B} |\bar{\phi}_0|^2 & - 2 \lambda_{\rm B} \bar{\phi}_0^2 \\ - 2 \lambda_{\rm B} \bar{\phi}_0^{*^{\scriptstyle{2}}} & -p_0 - \epsilon ({\bf p}) + \mu_{\rm B} - 4 \lambda_{\rm B} |\bar{\phi}_0|^2 \end{array} \right).$$ Taking into account only the quadratic terms in the field and neglecting higher-order terms, as we just did, is known as the Bogoliubov approximation. Comparing this expression with Eq. (\[comp:M\]) obtained in the composite boson limit after integrating out the fermionic degrees of freedom from the theory of superconductivity, we conclude that the composite bosons have—as expected—a mass $m_{\rm B}=2m$ twice the fermion mass $m$, and a small chemical potential given by Eq.(\[chem\]), which we there derived from the gap equation. It also follows that the number density of composite bosons condensed in the ground state reads $$\bar{n}_0 = \frac{\Gamma(2-d/2)}{(4 \pi)^{d/2}} m^{d/2} \epsilon_a^{d/2-2} |\bar{\Delta}_0|^2,$$ and that the interaction $\lambda_{\rm B}$ between the composite bosons is $$\label{comp:lambda} \lambda_{\rm B} = (4 \pi)^{d/2} \frac{1-d/4}{\Gamma(2-d/2)} \frac{\epsilon_a^{1-d/2}}{m^{d/2}},$$ or, using Eq. (\[bound\]), $$\lambda_{\rm B} = - \frac{1-d/4}{1-d/2} \lambda .$$ Note that the parameter $\lambda (<0)$ characterizing the attractive electron-electron interaction appears below $d=2$ with a minus sign here, leading to a repulsive interaction between the composite bosons. (In the next subsection, we will see that $d=2$ is the upper critical dimension of the $T=0$ Bogoliubov theory.) This brings us to the important conclusion that for $d<2$ the same interaction responsible for the formation of electron pairs, is also responsible for the stability of the superfluid state, and when this state ceases to exist, for that of the insulating state, which both need a repulsive interaction. The quantum phase transition encoded in the Bogoliubov theory corresponds, when translated back to the fermionic theory, to one where the condensate is drained of composite bosons, without breaking them up. In other words, composite bosons exist on both sides of the transition, either condensed (superfluid state) or localized (insulating state) [@Ramakrishnan; @CFGWY]. Beyond mean-field theory {#sec:beyond} ------------------------ We can continue now and improve on the usual mean-field approximation of the theory of superconductivity, where the functional integral over the pair field in the partition function (\[bcs:37\]) is approximated by the saddle point, by integrating out the field $\tilde{\Psi}$ in Eq.(\[fin\]), or to simplify notation, the field $\tilde{\Phi}$ in Eq.(\[bec:L0\]). This leads to the effective potential $$\label{eff:Veff} {\cal V}_{\rm eff} = -\frac{i}{2} {\rm tr} \int_k \ln[M_0(k)] = \frac{1}{2} \int_{\bf k} E({\bf k}).$$ Here, $E({\bf k})$ is the famous single-particle Bogoliubov spectrum [@Bogoliubov], $$\begin{aligned} \label{eff:bogo} E({\bf k}) &=& \sqrt{ \epsilon ^2({\bf k}) + 2 \mu_{\rm B} \epsilon({\bf k}) } \nonumber \\ &=& \sqrt{ \epsilon ^2({\bf k}) + 4 \lambda_{\rm B} |\bar{\phi}_0|^2 \epsilon({\bf k}) }.\end{aligned}$$ In the limit of large momentum, the spectrum behaves in a way $$\label{eff:med} E({\bf k}) \sim \epsilon({\bf k}) + 2 \lambda_{\rm B} |\bar{\phi}_0|^2$$ typical for a nonrelativistic particle of mass $m$ moving in a background medium, provided by the condensate in this case. The most notable feature of the Bogoliubov spectrum is that it is gapless, behaving for small momentum as $$\label{eff:micror} E({\bf k}) \sim c \, |{\bf k}|,$$ with $c = \sqrt{\mu_{\rm B}/m}$. ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ The integral over the loop momentum in Eq. (\[eff:Veff\]) can be carried out using the integral representation of the Gamma function $$\label{gamma} \frac{1}{a^z} = \frac{1}{\Gamma(z)} \int_0^\infty \frac{{\mbox{d}}\tau}{\tau} \tau^z {\rm e}^{-a \tau}.$$ In arbitrary space dimension $d$ this yields, using dimensional regularization: $$\label{regularized} {\cal V}_{\rm eff} = - L_d m^{d/2} \mu_{\rm B}^{d/2 + 1}, \;\;\; L_d = \frac{\Gamma(1-d/2) \Gamma(d/2 + 1/2)}{2 \pi^{d/2 + 1/2} \Gamma(d/2+2)}.$$ For $d=2$, the effective potential diverges. To investigate this, we set $d=2- \varepsilon$, with $\epsilon$ small and positive, and expand ${\cal V}_{\rm eff}$ around $d=2$, giving $$\label{Vep} {\cal V}_{\rm eff} = - \frac{m}{4 \pi \varepsilon} \frac{\mu_{\rm B}^2}{\kappa^{\varepsilon/2}} ,$$ with $\kappa$ an arbitrary renormalization group scale parameter which enters for dimensional reasons. If the Bogoliubov spectrum had not been gapless, but had an energy gap instead, this parameter would have appeared in Eq. (\[Vep\]) in the place of $\kappa$. As always in dimensional regularization, the divergence shows up as a pole in $\epsilon$. Comparing the one-loop contribution with the classical contribution (\[V0\]), we conclude that Eq. (\[Vep\]) leads to a renormalization of the coupling constant $\lambda$, yielding the renormalized coupling $\lambda_{\rm r}$ [@Uzunov] $$\frac{1}{\hat{\lambda}_{\rm r}} = \frac{1}{\hat{\lambda}} + \frac{m}{\pi} \frac{1}{\varepsilon},$$ where $\hat{\lambda} = \lambda /\kappa^{\varepsilon/2}$ and a similar definition for $\hat{\lambda}_{\rm r}$. The quantum critical point is approached by letting the renormalized group scale parameter $\kappa \to 0$. For fixed coupling $\lambda$, it then follows that upon approaching the critical point, the renormalized coupling tends to $\hat{\lambda}_{\rm r} \to \pi \varepsilon /m$. For $d<2$, or equivalently $\epsilon >0$, the fixed point is nontrivial. In the limit $d\to 2$, $\hat{\lambda}_{\rm r}\to 0$ and the theory becomes Gaussian, identifying $d=2$ as the upper critical dimension. Due to quantum fluctuations not all the particles are known to reside in the condensate [@FW]. Specifically, in $d$ space dimensions, the (constant) particle number density $\bar{n}$ at the one-loop level is given by [@TN] $$\label{depletion} \bar{n} = |\bar{\phi}_0|^2 - 2^{d/2-2} \frac{d^2-4}{d-1} L_d m^{d/2} \lambda_{\rm B}^{d/2} |\bar{\phi}_0|^d.$$ Since the quantum-induced term is positive for $1<d<4$, the number of particles residing in the condensate given $\bar{n}$ is reduced compared to the classical result $\bar{n} = |\bar{\phi}_0|^2$. This shows that due to quantum fluctuations, particles are knocked out of the condensate. ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ Despite that due to quantum fluctuations not all the particles reside in the condensate, all the particles do in the absence of impurities and at zero temperature participate in the superflow, and move on the average with the superfluid velocity. Put differently, the superfluid mass density $\rho_{\rm s}$ is given by the total particle number density $n$: $\rho_{\rm s} = m n$. ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ Impurities ---------- One of the ways to trigger a superconductor-insulator transition is to change the amount of impurities. This means that, [*e.g.*]{}, the correlation length $\xi$ diverges as $|\hat{\alpha}^* - \hat{\alpha}|^{-\nu}$ when the parameter $\hat{\alpha}$ characterizing the impurities approaches the critical value $\hat{\alpha}^*$. To account for impurities, we include a term [@Ma] $$\label{Dirt:dis} {\cal L}_{\alpha} = \psi({\bf x}) \, |\phi(x)|^2$$ in the bosonic theory (\[bec:Lagr\]), where $\psi({\bf x})$ is a space-dependent random field with a Gaussian distribution $$\label{random} P[\psi] = \exp \left[-\frac{1}{\alpha} \int_{\bf x} \, \psi^2({\bf x}) \right],$$ characterized by the impurity strength $\alpha$ ($\hat{\alpha}$ alluded to above is a rescaled version of $\alpha$). Notice that the random field does not depend on time. This is because it is introduced to mimic impurities, which are randomly distributed in space, not in time. We shall treat the impurities in the so-called quenched approximation [@Ma], where the average of an observable $O(\phi^*,\phi)$ is obtained as follows $$\langle O(\phi^*,\phi) \rangle = \int_\psi P[\psi] \, \langle O(\phi^*,\phi) \rangle_\psi,$$ with $\langle O(\phi^*,\phi) \rangle_\psi$ indicating the grand-canonical average for a given impurity configuration. That is to say, first the ensemble average is taken for fixed $\psi$, and only after that the averaging over the random field is carried out. In terms of the shifted field (\[bec:newfields\]), the random term (\[Dirt:dis\]) becomes $${\cal L}_{\alpha} = \psi({\bf x}) (|\bar{\phi}_0|^2 + |\tilde{\phi}|^2 + \bar{\phi}_0 \tilde{\phi}^* + \bar{\phi}_0^* \tilde{\phi} ).$$ The first two terms lead to an irrelevant change in the chemical potential, so that only the last two terms need to be considered, which can be cast in the matrix form $${\cal L}_{\alpha} = \psi({\bf x}) \, \bar{\Phi}_0^\dagger \tilde{\Phi}, \;\;\;\;\;\;\; \bar{\Phi}_0 = \left(\begin{array}{l} \bar{\phi}_0 \\ \bar{\phi}_0^* \end{array} \right).$$ The integral over $\tilde{\Phi}$ is Gaussian in the Bogoliubov approximation and therefore easily performed to yield an additional term to the effective action $$S_{\alpha} = -\frac{1}{2} \int_{x,y} \psi({\bf x}) \bar{\Phi}_0^\dagger \, G_0(x-y) \bar{\Phi}_0 \psi({\bf y}),$$ where the correlation function $G_0$ is the inverse of the matrix $M_0$ introduced in Eq. (\[bec:M\]). To proceed, we pass to the Fourier representation: $$\begin{aligned} G_0(x-y) &=& \int_k {\rm e}^{-i k \cdot (x-y)} \, G_0(k) \\ \psi({\bf x}) &=& \int_{\bf k} {\rm e}^{i {\bf k} \cdot {\bf x}} \psi({\bf k}).\end{aligned}$$ The contribution to the effective action then appears in the form $$\label{S_d} S_{\alpha} = -\frac{1}{2} \int_{\bf k} |\psi({\bf k})|^2 \bar{\Phi}_0^\dagger G(0,{\bf k}) \bar{\Phi}_0.$$ Since the random field is Gaussian distributed, the average over this field representing quenched impurities yields: $$\langle |\psi({\bf k})|^2 \rangle = {{\textstyle\frac{1}{2}}} \Omega \alpha,$$ with $\Omega$ the volume of the system. The remaining integral over the loop momentum in Eq. (\[S\_d\]) is readily carried out to yield in arbitrary space dimensions the contribution to the Lagrangian $$\label{L_D} \langle {\cal L}_\alpha \rangle = \frac{1}{2} \Gamma(1-d/2) \left(\frac{m}{2 \pi} \right)^{d/2} |\bar{\phi}_0|^2 (6 \lambda_{\rm B} |\bar{\phi}_0|^2 - \mu_{\rm B})^{d/2-1} \alpha.$$ The divergence in the limit $d \to 2$ shows that also in the presence of impurities, the two-dimensional case is special. This expression can be used to obtain the additional depletion due to impurities. To this end, we differentiate it with respect to the chemical potential, giving [@GPS; @pla] $$\bar{n}_\alpha = \frac{\partial \langle {\cal L}_\alpha \rangle}{\partial \mu_{\rm B}} = \frac{2^{d/2-5}\Gamma(2-d/2)}{\pi^{d/2}} m^{d/2} \lambda_{\rm B}^{d/2-2} \bar{n}_0^{d/2-1} \alpha,$$ where we recall that $\bar{n}_0= |\bar{\phi}_0|^2$ denotes the (constant) number density of particles residing in the condensate. Because this contribution is positive, it amounts to an additional depletion of the condensate. The divergence in the limit $\lambda_{\rm B} \rightarrow 0$ for $d <4$ signals the collapse of the system with impurities when the interparticle repulsion is removed. To determine the superfluid mass density $\bar{\rho}_{\rm s}$ in the presence of impurities, we replace, as in the last exercise of Sec.\[sec:beyond\], $\mu_{\rm B}$ with $\mu_{\rm eff}$ defined in Eq.(\[bec:mureplacement\]) and $i\partial_0$ with $i\partial_0 - ({\bf u} - {\bf v}_{\rm s}) \cdot (-i \nabla)$ in the contribution (\[S\_d\]) to the effective action. Differentiating it with respect to the externally imposed velocity, $-{\bf u}$, we find to linear order in the difference ${\bf u}- {\bf v}_{\rm s}$: $${\bf g} = \bar{\rho}_{\rm s} {\bf v}_{\rm s} + \bar{\rho}_{\rm n} {\bf u},$$ with the superfluid and normal mass density [@pla] $$\bar{\rho}_{\rm s} = m\left(\bar{n} - \frac{4}{d} \bar{n}_\alpha \right), \;\;\;\; \bar{\rho}_{\rm n} = \frac{4}{d} m \bar{n}_\alpha.$$ As expected, $\bar{\rho}_{\rm s} \neq m\bar{n}$ in the presence of impurities. Moreover, the normal mass density is a factor $4/d$ larger than the mass density $m\bar{n}_\alpha$ knocked out of the condensate by the impurities. For $d=3$ this gives the factor ${{\textstyle\frac{4}{3}}}$ first found in Ref. [@HuMe]. As argued there, this indicates that part of the zero-momentum state belongs not to the condensate, but to the normal fluid. Being trapped by the impurities, the fraction $(4-d)/d \times \bar{n}_\alpha$ of the zero-momentum state is localized. This is an important conclusion as it shows that the phenomenon of localization can be accounted for in the Bogoliubov theory of superfluidity by including a random field, without necessarily destroying that state. Phase-only theory ================= In this section we show that the Bogoliubov theory, which we obtained in the composite boson limit after integrating out the fermionic degrees of freedom from the theory of superconductivity, contains only one degree of freedom, viz. the phase of the order parameter. Physically, it describes the Goldstone mode of the spontaneously broken global U(1) symmetry. In the context of superconductivity, this mode is called Anderson-Bogoliubov mode. The Bogoliubov theory may therefore, at least in the superfluid state, be represented by a phase-only effective theory. We continue to account for the $1/r$ Coulomb potential in the effective theory and give general scaling arguments for the physical quantities represented by that theory. Derivation ---------- It was first shown by Beliaev [@Beliaev] that the gaplessness of the single-particle spectrum first found by Bogoliubov at the classical level persists at the one-loop order and later proven by Hugenholtz and Pines [@HP] to hold to all orders in perturbation theory. In fact, as was proven by Gavoret and Nozières [@GN], the Bogoliubov spectrum is identical to that of the Goldstone mode accompanying the spontaneous breakdown of the global U(1) symmetry, thus explaining its gaplessness. Also from the perspective of degrees of freedom, this conclusion makes sense. Although the normal phase is described by a complex $\phi$-field, having two components, it contains only one degree of freedom [@Leutwyler]. This is because the energy $E({\bf k}) \sim {\bf k}^2$ is always positive. As a result, only positive energies appear in the Fourier decomposition of the field, and one needs—as is well known from standard quantum mechanics—a complex field to describe a single spinless particle. In the superfluid phase, on the other hand, where $E^2({\bf k}) \sim {\bf k}^2$, the Fourier decomposition contains positive as well as negative energies so that a single real field suffices to describe this mode. In other words, although the number of fields is different, the number of degrees of freedom is the same in both phases. This implies that the superfluid state can be described by a phase-only theory as it captures all the degrees of freedom, ignoring vortices for the moment which are easily incorporated in the theory as will be discussed in the next section. To obtain the phase-only theory, we set, [*cf.*]{} Eq.(\[bec:newfields\]) $$\phi(x) = {\rm e}^{i \varphi(x)} \, (\bar{\phi}_0 + \tilde{\phi}),$$ with $\varphi(x)$ a background field representing the Goldstone mode accompanying the spontaneous symmetry breakdown of the global U(1) symmetry. Inserting this in the Lagrangian (\[bec:Lagr\]) and expanding it, we obtain $$\label{quick} {\cal L}^{(2)} = - {\cal V}_0 - |\bar{\phi}_0|^2 U - U (\bar{\phi}_0 \tilde{\phi}^* + \bar{\phi}_0^* \tilde{\phi} ) - \lambda_{\rm B} |\bar{\phi}_0|^2 (\bar{\phi}_0 \tilde{\phi}^* + \bar{\phi}_0^* \tilde{\phi} )^2,$$ where the field $U(x)$ stands for the combination $$\label{eff:U} U(x) = \partial_0 \varphi(x) + \frac{1}{2m} [\nabla \varphi(x)]^2.$$ In deriving Eq. (\[quick\]), we used the mean-field equation $\mu_{\rm B} = 2 \lambda_{\rm B} |\bar{\phi}_0|^2$. We continue to integrate out the tilde field (which is tantamount to substituting its field equation back into the Lagrangian) to obtain the phase-only theory $$\label{eff:quick} {\cal L}_{\rm eff} = - \bar{n} U(x) + \frac{1}{4} U(x) \frac{1}{\lambda_{\rm B}} U(x),$$ where we ignored the irrelevant constant term ${\cal V}_0$ and substituted $|\bar{\phi}_0|^2 (= \bar{n}_0) = \bar{n}$ to this order. Using the mean-field equation again, we can write the coefficient of the last term as: $$\frac{1}{4} \frac{1}{\lambda_{\rm B}} = \frac{1}{2} \frac{\bar{n}}{m c^2} = \frac{1}{2} \bar{n}^2 \kappa,$$ with $c$ the speed of sound introduced in Eq. (\[eff:micror\]). Standard thermodynamics relates $c$ to the compressibility $\kappa$ via $$\kappa = \frac{1}{m \bar{n} c^2}.$$ The phase-only theory (\[eff:quick\]) can thus be cast in the equivalent form $$\label{eff:Leff} {\cal L}_{{\rm eff}} = -\bar{n}\left[\partial_{0}\varphi + \frac{1}{2m}( {\bf \nabla} \varphi)^{2} \right] + \frac{1}{2} \bar{n}^2 \kappa \left[\partial_{0}\varphi + \frac{1}{2m}( {\bf \nabla}\varphi)^{2}\right]^{2},$$ which turns out to be exact [@effbos]. The theory describes a sound wave, with the dimensionless phase field $\varphi$ representing the Goldstone mode of the spontaneously broken global U(1) symmetry. It has the gapless spectrum $E^2({\bf k}) = c^2 {\bf k}^2$. The effective theory gives, ignoring vortices for the moment, a complete description of the superfluid at low energies and small momenta. When one goes to higher energies and momenta, additional terms with higher-order derivatives should be included in the effective theory, but it remains a phase-only theory. Coulomb potential ----------------- It is straightforward to generalize the result (\[eff:quick\]) to include long-ranged interactions. A case of particular interest to us is the 3-dimensional Coulomb potential $$V({\bf x}) = \frac{q^2}{|{\bf x}|},$$ whose Fourier transform in $d$ space dimensions reads $$V({\bf k}) = 2^{d-1} \pi^{(d-1)/2} \Gamma[{{\textstyle\frac{1}{2}}}(d-1)] \frac{q^2}{|{\bf k}|^{d-1}}.$$ Here, $q$ stands for the electric charge, which in the case of Cooper pairs is twice the electron charge. The simple contact interaction $L_{\rm int} = - \lambda_{\rm B}\int_{\bf x} |\phi(x)|^4$ in Eq.(\[bec:Lagr\]) is now replaced by $$L_{\rm int} = - \frac{1}{2} \int_{{\bf x}, {\bf y}} |\phi(t,{\bf x})|^2 V({\bf x} - {\bf y}) |\phi(t,{\bf y})|^2.$$ The rationale for using the 3-dimensional Coulomb potential, even when considering charges confined to move in a lower dimensional space, is that the electromagnetic interaction remains 3-dimensional. The effective theory then becomes after passing over to the Fourier representation $$\label{effCoul} {\cal L}_{\rm eff} = - \bar{n} U(k) + \frac{1}{2} U(k_0,{\bf k}) \frac{1}{V({\bf k})} U(k_0,-{\bf k}),$$ and leads to the spectrum $$E^2({\bf k}) = 2^{d-1} \pi^{(d-1)/2} \Gamma [{{\textstyle\frac{1}{2}}}(d-1)] \frac{\bar{n} q^2}{m} |{\bf k}|^{3-d}.$$ For $d=3$, this yields the famous plasma mode, with an energy gap given by the plasma frequency $\omega_{\rm p}^2 = 4 \pi \bar{n} q^2/m$. For $d=2$ on the other hand, the spectrum behaves as $E({\bf k}) \propto \sqrt{|{\bf k}|}$, implying that the mode it describes is much harder that the sound wave with the spectrum $E({\bf k}) \propto |{\bf k}|$ obtained for the system without the $1/r$ Coulomb interaction included. To appreciate under which circumstances the Coulomb interaction becomes important, we note that for electronic systems $1/|{\bf x}| \sim k_{\rm F}$ for dimensional reasons and the fermion number density $\bar{n} \sim k_{\rm F}^d$, where $k_{\rm F}$ is the Fermi momentum. The ratio of the Coulomb interaction energy to the Fermi energy $\epsilon_{\rm F} = k_{\rm F}^2/2m$ is therefore proportional to $\bar{n}^{-1/d}$. This means that the lower the electron number density is, the more important the Coulomb interaction becomes. Hyperscaling ------------ Let us consider the two terms in the effective theory (\[eff:Leff\]) quadratic in the Goldstone field $\varphi$ and write them in the most general form [@FF] $$\label{general} {\cal L}_{\rm eff}^{(2)} = - \frac{1}{2} \frac{\rho_{\rm s}}{m^2} (\nabla \varphi)^2 + \frac{1}{2} \bar{n}^2 \kappa (\partial_0 \varphi)^2.$$ The coefficient $\rho_{\rm s}$ is the superfluid mass density, which is, as we saw in the previous section, a response function and in general does not equal $m \bar{n}$. The other coefficient, $$\bar{n}^2 \kappa = \frac{\partial \bar{n}}{\partial \mu_{\rm B}} ,$$ can be related to the (0,0)-component of the polarization tensor $\Pi_{0 0}$. This can be understood by noting that an electromagnetic field is included via the minimal substitution $\tilde{\partial}_\mu \to \tilde{\partial}_\mu + q A_\mu$, with $A_\mu$ the electromagnetic vector potential. Since the polarization tensor (times $q^2$) is obtained by differentiating the effective theory twice with respect to the vector potential, we obtain $$\label{Pi} \lim_{|{\bf k}| \to 0} \Pi_{0 0} (0,{\bf k}) = \bar{n}^2 \kappa,$$ where, as is typical for response functions, the energy transfer is put to zero before the momentum transfer ${\bf k}$ is. Equation (\[general\]) leads to the general expression for the speed of sound $$\label{speed} c^2 = \frac{\rho_{\rm s}}{m^2 \bar{n}^2 \kappa}.$$ The singular behavior of the system close to the critical point is encoded in the phase-only theory. Simple dimensional analysis shows that near the phase transition it scales as $${\cal L}_{\rm eff} \sim \xi^{-(d+z)},$$ while $$\label{scaling} (\nabla \varphi)^2 \sim \xi^{-2}, \;\;\;\; (\partial_0 \varphi)^2 \sim \xi_t^{-2} \sim \xi^{-2z},$$ with $\xi_t$ the correlation time and $z$ the dynamic exponent. Combining these hyperscaling arguments, and remembering that the mass parameter is inessential with regards to the critical behavior, one arrives [@FF] at the scaling laws for the two coefficients appearing in the effective theory (\[general\]): $$\label{hyperrho} \rho_{\rm s} \sim \xi^{-(d+z-2)}, \;\;\;\; \kappa \sim \xi^{-(d-z)} .$$ The first conclusion is consistent with the universal jump predicted by Nelson and Kosterlitz [@NeKo] which corresponds to taking $z=0$ and $d=2$. In the presence of impurities it is believed that the compressibility stays finite at the critical point, implying $z=d$ [@FF]. This remarkably simple argument thus predicts an exact and nontrivial value for the dynamic exponent. Without impurities, the dynamic exponent is $z=2$ [@Uzunov]. This agrees with what one naively expects, given that in the nonrelativistic theory (\[bec:Lagr\]) we started with, one time derivative appears in combination with two space derivatives, $i \partial_0 + \nabla^2/2m$. This last argument should, however, be treated with care when applied to the phase-only theory (\[eff:Leff\]). In that theory, the time and space derivatives appear in a symmetrical form, yet $z$ is in general not unity, as we just saw. The difference is that in the effective theory, the relative coefficient $c^2$ scales according to Eq.(\[speed\]) with the scaling laws (\[hyperrho\]) as $$c^2 \sim \xi^{2(1-z)},$$ while the relative coefficient $m$ in the microscopic theory does not scale. Incidentally, the (quantum) XY model has a dynamic exponent $z=1$, so that $c$ in that case does not scale. In experiments on charged systems, instead of the superfluid mass density, usually the conductivity $\sigma$ is measured. To see the relation between the two, we introduce a vector potential in the effective theory by replacing $\nabla \varphi$ with $\nabla \varphi - q {\bf A}$ in Eq. (\[general\]), and allow the superfluid mass density to vary in space and time. The term in the action quadratic in ${\bf A}$ then becomes after passing to the Fourier representation $$S_\sigma = - \frac{1}{2} \frac{q^2}{m^2} \int_k {\bf A}(-k) \rho_{\rm s} (k) {\bf A}(k).$$ The electromagnetic current, $${\bf j}(k) = \frac{\delta S_\sigma}{\delta {\bf A}(-k)}$$ obtained from this action can be written as $${\bf j}(k) = \sigma(k) {\bf E}(k),$$ with the conductivity $$\label{conductivity} \sigma(k) = i \frac{q^2}{m^2} \frac{\rho_{\rm s}(k)}{k_0}$$ essentially given by the superfluid mass density. So if we know the scaling of the electric charge, we can determine the scaling of the conductivity. With the $1/r$ Coulomb potential included, the quadratic terms in the effective theory (\[effCoul\]) may, after passing to the Fourier representation, be cast in the general form $$\label{L2} {\cal L}_{\rm eff}^{(2)} = \frac{1}{2} \left(\frac{\rho_{\rm s}}{m^2} {\bf k}^2 - \frac{|{\bf k}|^{d-1}}{q'^{\scriptstyle{2}}} k_0^2\right) |\varphi(k)|^2,$$ where $q'$ is the redefined charge parameter $$q'^{\scriptstyle{2}} = 2^{d-1} \pi^{(d-1)/2} \Gamma\left[{{\textstyle\frac{1}{2}}}(d-1)\right] q^2.$$ The charge is connected to the (0, 0)-component of the polarization tensor via $$q'^{\scriptstyle{2}} = \lim_{|{\bf k}| \rightarrow 0} \frac{|{\bf k}|^{d-1}}{\Pi_{0 0} (0,{\bf k})} .$$ A simple hyperscaling argument like the one given above for the case without Coulomb interaction shows that near the transition, the charge scales as [@FGG] $$\label{escaling} q'^{\scriptstyle{2}} \sim \xi^{1-z},$$ independent of the number of space dimensions $d$. It then follows from Eq. (\[conductivity\]) that the conductivity scales as $$\label{sgeneral} \sigma \sim \xi^{3 - (d+z)}.$$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ ------------------------------------------------------------------------ In the presence of random impurities, the charge is expected to be finite at the transition, so that $z=1$ [@FGG]. This is again an exact result, which replaces the value $z=d$ of an impure system without Coulomb interaction. The prediction was first confirmed for impure superconducting films [@HPsu1], and has subsequently also been observed in other 2-dimensional systems such 2-dimensional Josephson-junction arrays [@Delft], quantum Hall systems [@WET], and 2-dimensional electron systems [@KSSMF]. We will refer to a quantum critical point with a $1/r$ Coulomb interaction as CQCP. In the vicinity of such a critical point, the conductivity scales as [@Wegner] $$\sigma \sim \xi^{2 - d},$$ implying that in two space dimensions, the conductivity is a marginal operator which remains finite at the CQCP. Scaling of magnetic vector potential {#sec:magscale} ------------------------------------ Let us finish this section by determining the scaling of the magnetic vector potential. We start with the observation that close to a CQCP, the electric field $E$ scales as $E \sim \xi_t^{-1} \xi^{-1} \sim \xi^{-(z+1)}$ (for a review, see Ref. [@SGCS]). Thus conductivity measurements [@YaKa; @KSSMF] close to a CQCP collapse onto a single curve when plotted as function of the dimensionless combination $|\delta|^{\nu (z+1)}/E$, where as before $\delta=K-K_{\rm c})$ measures the distance from the critical point $K_{\rm c}$, and $\nu$ is the correlation length exponent, $\xi \sim |\delta|^{- \nu}$. (For a field-controlled transition, $K$ stands for the applied magnetic field, while for a density-controlled transition it stands for the charge-carrier density.) The scaling of the electric field with the correlation length expresses the more fundamental result that the anomalous scaling dimension $d_{\bf A}$ of the magnetic vector potential ${\bf A}$ is unity, $d_{\bf A} = 1$. Because the magnetic vector potential always appears in the gauge-invariant combination $\nabla - q {\bf A}$, the anomalous scaling dimension of the electric charge $q$ of the charge carriers times the vector potential is unity too, $d_{q {\bf A}} = 1$. Writing the anomalous scaling dimension of the vector potential as a sum $d_{\bf A} = d^0_{\bf A} + {{\textstyle\frac{1}{2}}} \eta_{\bf A}$ of its canonical scaling dimension $d^0_{\bf A} = {{\textstyle\frac{1}{2}}} (d + z -2)$, obtained by simple power counting, and (half) the critical exponent $\eta_{\bf A}$, describing how the correlation function decays at the critical point, we conclude that $d_q = d^0_q - \frac{1}{2}\eta_{\bf A}$. Here, $d^0_q = 1 - d^0_{\bf A}$ stands for the canonical scaling dimension of the electric charge. Now, for a $1/r$ Coulomb potential, the charge scales according to Eq. (\[escaling\]) as $q^2 \sim \xi^{1-z}$ independent of the number $d$ of space dimensions [@FGG]. Combined with the previous result, this fixes the value of the exponent $\eta_{\bf A}$ in terms of the number of space dimensions and the dynamic exponent: $$\label{eta} \eta_{\bf A}=5 -d - 2z.$$ Since in the presence of impurities, the electric charge is finite at a CQCP, leading to $z=1$, it follows that $\eta_{\bf A}=1$ in two space dimensions. As we shall see in the next section, this exponent becomes important when considering the interaction between vortices close to the CQCP. Experimental status ------------------- For a critical discussion of the experimental status of the phase-only theory, see Ref. [@LG]. A more recent discussion can be found in Ref. [@MG]. According to the phase-only theory discussed here, no electronic excitations exist in the critical region. However, electron tunneling measurements on superconducting films of varying thickness apparently probed the energy gap of these excitations [@VDG]. Moreover, the gap was found to approach zero as the transition to the insulating state is approached. Similar experiments [@HCV] for the field-tuned transition showed the presence of a large number of electronic excitations near the Fermi energy, thus raising doubts about the applicability of the phase-only theory. Experimental support for the presence of electron pairs in the insulating state comes from Hall effect studies on superconducting films, which show two critical fields [@PHR]. The lower critical field is seen in the longitudinal resistance and is believed to mark the superconductor-insulator transition. The higher critical field is seen in the transverse or Hall resistance and is believed to signal the crossover from a bosonic to a fermionic insulator without pairing. At the higher critical field, the longitudinal resistance has its maximum. The critical exponents determined in earlier experiments on the superconductor-insulator transition [@HPsu1; @YaKa] had the value $z =1$ for the dynamic exponent, in accord with the prediction in Ref.[@MPAFisher], and $\nu =1.3$ for the correlation length exponent. More recent studies [@MCMHG], however, find agreement with these results only for the transition tuned by changing the film thickness. For the field-tuned transition the value $z \nu = 0.7$ was found instead, which is about half the value one expects. The cause for this discrepancy is not clear. It implies that, contrary to common believe, the critical exponents depend on how the phase transition is crossed, by tuning the field or the film thickness. Clearly, more experimental and theoretical studies are required to fully understand the superconductor-insulator transition, and to establish to what extend the phase-only theory is applicable. Duality ======= One of the most intriguing results found in experiments on quantum phase transitions in superconducting films, as well as in 2-dimensional Josephson-junction arrays [@Delft], quantum Hall systems [@IVQH], and 2-dimensional electron systems [@KSSMF] is the striking similarity in the current-voltage ($I$-$V$) characteristics on both sides of the transition. By interchanging the $I$ and $V$ axes in one phase, an $I$-$V$ characteristic of that phase at a given value of the applied magnetic field (in superconducting films, 2-dimensional Josephson-junction arrays, and quantum Hall systems) or charge carrier density (in 2-dimensional electron systems) can be mapped onto an $I$-$V$ characteristic of the other phase at a different value of the magnetic field or charge-carrier density. This reflection symmetry hints at a deep connection between the conduction mechanisms in the two phases that can be understood by invoking a duality transformation [@MPAFisher; @WeZe]. Whereas the conducting phase is most succinctly described in terms of charge carriers of the system, the insulating phase is best formulated in terms of vortices. At zero temperature, these topological defects should, just like the charge carriers, be thought of as quantum point particles. The duality transformation links the two descriptions, which turn out to be very similar. Vortices {#sec:Vortices} -------- Let us now include vortices in the phase-only theory. This is achieved by introducing the so-called plastic field $\varphi_\mu^{\rm P}$ via the minimal substitution $\tilde{\partial}_\mu \varphi \to \tilde{\partial}_\mu \varphi + \varphi_\mu^{\rm P}$ [@GFCM]. The plastic field is defined such that its curl gives a delta function at the location of the vortices. Specifically, in two space dimensions, where vortices are point objects, located at the positions ${\bf x}^\alpha$ say: $$\label{th2d} \nabla \times {\mbox{\boldmath $\varphi$}}^{\rm P} = -2 \pi \sum_\alpha \delta({\bf x} - {\bf x}^\alpha),$$ while in three dimensions, where vortices are line objects, located along the curves $C_\alpha$ say: $$\nabla \times {\mbox{\boldmath $\varphi$}}^{\rm P} = -2 \pi \sum_\alpha \int_{C_\alpha} \mbox{d} {\bf x}^\alpha \, \delta({\bf x} - {\bf x}^\alpha).$$ Let us concentrate on static phenomena so that we can ignore the time derivatives in the effective theory (\[general\]). When besides vortices also the magnetic vector potential is included, the effective theory becomes in three dimensions $$\label{starting} {\cal L}_{\rm eff}^{(2)} = - \frac{1}{2} \frac{\rho_{\rm s}}{m^2} (\nabla \varphi - {\mbox{\boldmath $\varphi$}}^{\rm P} - q {\bf A})^2 - \frac{1}{2} (\nabla \times {\bf A})^2,$$ or after the canonical transformation $q {\bf A} \to q {\bf A} - {\mbox{\boldmath $\varphi$}}^{\rm P}$: $$\label{dressed} {\cal L}_{\rm eff}^{(2)} = - \frac{1}{2} \frac{\rho_{\rm s}}{m^2} (\nabla \varphi - q {\bf A})^2 - \frac{1}{2} (\nabla \times {\bf A}- {\bf B}^{\rm P})^2 ,$$ where the plastic field ${\bf B}^{\rm P}$ stands for $${\bf B}^{\rm P} = - \Phi_0 \sum_{\alpha} \int_{C_\alpha} \mbox{d} {\bf x}^\alpha \, \delta({\bf x} - {\bf x}^\alpha),$$ with $\Phi_0 = 2 \pi/q$ the magnetic flux quantum in units where the speed of light and Planck’s constant $\hbar$ are set to unity. \[In two dimensions, this plastic field is a scalar and stands for $$\label{BP} B^{\rm P} = - \Phi_0 \sum_{\alpha} \delta({\bf x} - {\bf x}^\alpha),$$ as follows from Eq. (\[th2d\]).\] After integrating out the phase field $\varphi$ in Eq. (\[dressed\]), we obtain the magnetic part of the effective action $S_{\rm mag}$. Written as a functional integral over the magnetic vector potential, it reads in the Coulomb gauge $\nabla \cdot {\bf A}=0$ $$\label{mag} {\rm e}^{i S_{\rm mag}} = \int_{\bf A} \, \exp \left\{i \int_x \left[-\frac{1}{2}(\nabla \times {\bf A} - {\bf B}^{\rm P})^2 - \frac{1}{2} \frac{1}{\lambda^2}{\bf A}^2 \right]\right\},$$ with $\lambda$ the magnetic penetration depth. The mass term, with $\lambda^{-2} = q^2\rho_{\rm s}/m^2$, is generated through the Anderson-Higgs mechanism in the process of integrating out the phase mode $\varphi$. With this construction, we can now calculate the interaction between two vortices. To facilitate the calculation in the case of a superconducting film below, we linearize the first term in Eq.(\[mag\]) by introducing an auxiliary field $\tilde{\bf h}$ via a Hubbard-Stratonovich transformation: $$- {{\textstyle\frac{1}{2}}} (\nabla \times {\bf A} - {\bf B}^{\rm P})^2 \to i (\nabla \times {\bf A} - {\bf B}^{\rm P}) \cdot \tilde{\bf h} - {{\textstyle\frac{1}{2}}} \tilde{\bf h}^2.$$ The original form is regained after integrating out the auxiliary field again. After integrating out the magnetic vector potential, we arrive at a form appropriate for a dual description in terms of magnetic vortices rather than electric charges [@KKS] $$\label{3d} {\rm e}^{i S_{\rm mag}} = \int_{\tilde{\bf h}} \, \exp\left\{i \int_x \left[-{{\textstyle\frac{1}{2}}} \lambda^2 (\nabla \times \tilde{\bf h})^2 - {{\textstyle\frac{1}{2}}}\tilde{\bf h}^2 - i \tilde{\bf h} \cdot {\bf B}^{\rm P} \right] \right\}.$$ Physically, $\tilde{\bf h}$, which satisfies the condition $\nabla \cdot \tilde{\bf h}=0$, represents ($i$ times) the fluctuating local induction. The vortices described by ${\bf B}^{\rm P}$ couple to $\tilde{\bf h}$ with a coupling constant $g=\Phi_0/\lambda$ independent of the electric charge. Observe the close similarity between the original (\[mag\]) and the dual form (\[3d\]). This becomes even more so when an external electric current ${\bf j}^{\rm P}$ is coupled to the ${\bf A}$ field by including a term $- {\bf A} \cdot {\bf j}^{\rm P}$ in Eq. (\[mag\]), and ${\bf B}^{\rm P}$ describing the vortices is set to zero there. Finally, also integrating out the local induction, one obtains the well-known Biot-Savart law for the interaction potential $S_{\rm mag} = - \int_t V$ between two static vortices in a bulk superconductor [@deGennes], $$\begin{aligned} \label{V3d} V(r) &=& \frac{1}{2\lambda^2} \int_{{\bf x},{\bf y}} B_i^{\rm P}({\bf x}) G({\bf x} - {\bf y}) B_i^{\rm P}({\bf y}) \nonumber \\ &=& \frac{g^2}{4 \pi} \int_{C_1} \int_{C_2} \mbox{d} {\bf l}^1 \cdot \mbox{d} {\bf l}^2 \; \frac{{\rm e}^{-R/\lambda}}{R} \nonumber \\ &=& - \frac{g^2}{2 \pi} L \left[\ln(r/2\lambda) + \gamma \right] + {\cal O}(r/\lambda)^2 ,\end{aligned}$$ where we ignored the self-interaction. In Eq. (\[V3d\]), $G({\bf x})$ is the vortex-vortex correlation function with Fourier transform $G({\bf k})=1/({\bf k}^2 + \lambda^{-2} )$, $R$ denotes the distance between the differential lengths $\mbox{d} {\bf l}^1$ and $\mbox{d} {\bf l}^2$, $L$ is the length of each of the two vortices, and $\gamma$ is Euler’s constant. For distances smaller than the magnetic penetration depth, which is the length scale for variations in the electric current and the magnetic field, the interaction is logarithmic as in a superfluid. If the system size is smaller than $\lambda$, it will replace the penetration depth as infra-red cutoff in the logarithm, and there will be no reference to the electric charge anymore. To describe magnetic vortices in a film of thickness $w$ [@Pearl], the bulk result (\[3d\]) has to be adjusted in two ways to account for the fact that both the vortices and the screening currents, which produce the second term in (\[3d\]), are confined to the plane. This is achieved by including a Dirac delta function $w \delta(x_3)$ in the second and third term. Instead of Eq. (\[3d\]), we then arrive at the interaction potential [@Pearl; @deGennes] $$\begin{aligned} \label{V2d} V_\perp(r) &=& \frac{1}{2 \lambda_\perp} \int_{{\bf x}_\perp, {\bf y}_\perp} B_\perp^{\rm P}({\bf x}_\perp) G_\perp({\bf x}_\perp - {\bf y}_\perp) B_\perp^{\rm P}({\bf y}_\perp) \nonumber \\ &=& -\frac{g_\perp^2}{2 \pi} \left[ \ln(r/4 \lambda_\perp) + \gamma\right] + {\cal O}(r/\lambda_\perp)^2,\end{aligned}$$ where $B_\perp^{\rm P} = - \Phi_0 \sum_{\alpha} \delta({\bf x}_\perp - {\bf x}_\perp^\alpha)$ describes the vortices in the film with coordinates ${\bf x}_\perp$, $\lambda_\perp = \lambda^2/w$ is the transverse magnetic penetration depth, $g_\perp^2 = \Phi_0^2/\lambda_\perp$ the coupling constant squared, and $$\begin{aligned} G_\perp({\bf x}_\perp) &=& \int_{x_3} \, G_\perp({\bf x}_\perp, x_3) \nonumber \\ &=& \int_{{\bf k}_\perp} \, {\rm e}^{- i {\bf k}_\perp \cdot {\bf x}_\perp} G_\perp({\bf k}_\perp,0),\end{aligned}$$ with $G_\perp({\bf k}_\perp,0) = 2/ k_\perp (2 k_\perp +\lambda_\perp^{-1})$. For small distances, the interaction is seen to be identical to that in a bulk superconductor [@Pearl], and also to that in a superfluid film. As in the bulk, the vortex coupling constant $g_\perp$ in the film is independent of the electric charge. The logarithmic interaction between vortices we found in Eq.(\[V2d\]) appears to pose a severe problem to the duality picture we alluded to in the introduction of this section as the charges interact via a $1/r$ Coulomb potential. The difference should spoil the experimentally observed reflection symmetry in the $I$-$V$ characteristics. However, it should be realized that the results derived in this subsection are valid only in the mean-field region, where $\eta_{\bf A}=0$. In the critical region governed by a CQCP, the value of this exponent was found in Sec. \[sec:magscale\] to be unity. As we will now demonstrate, this leads to a qualitative change in the interaction potential between two vortices from logarithmic in the mean-field region to $1/r$ in the vicinity of the CQCP. Changing vortex interaction --------------------------- Close to a CQCP we have to include the field renormalization factor $Z_{\bf A}$ in the vortex-vortex correlation function $G_\perp$ appearing in the expression (\[V2d\]) for the vortex interaction. It then becomes $$\label{Gr} G_\perp({\bf k}_\perp,0) = \frac{2}{k_\perp } \frac{Z_{\bf A}}{ 2 k_\perp +\lambda_\perp^{-1}},$$ with $Z_{\bf A} \sim k_\perp^{\eta_{\bf A}}$. Because the magnetic vector potential and the local induction renormalize in the same way, their renormalization factor is identical. Due to this extra factor, the interaction between two vortices in the film takes the form of a $1/r$ Coulomb potential [@PRL2k] $$\label{renV} V_\perp(r) = \frac{g_\perp^2}{2 \pi} \frac{a}{r},$$ where $a$ is some microscopic length scale which accompanies the renormalization factor $Z_{\bf A}$ for dimensional reasons [@Goldenfeld]. The absence of any reference to the electric charge in the renormalized and bare interaction (at least for small enough systems) implies that the same results should be derivable from our starting theory (\[starting\]) with $q$ set to zero. By directly integrating out the Anderson-Bogoliubov mode, and ignoring the momentum dependence of $\rho_{\rm s}$, which is valid outside the critical region, one easily reproduces the bare interaction potential (\[V2d\]). The renormalized interaction (\[renV\]) is obtained by realizing that according to Eq.(\[hyperrho\]), $\rho_{\rm s} \sim k_\perp$ for $d=2$ and $z=1$. In other words, the extra factor of $k_\perp$ that came in via the renormalization factor $Z_{\bf A}$ in our first calculation to produce the $1/r$ potential, comes in via $\rho_{\rm s}$ here. One might wonder if perhaps also the Coulomb interaction between electric charges changes in the vicinity of a CQCP. We do not expect this to happen. Since the $1/r$ Coulomb interaction is genuine 3-dimensional, this interaction cannot be affected too much by what happens in the film, which constitutes a mere slice of 3-dimensional space. The reason that the interaction between vortices is susceptible to the presence of a CQCP, is that this interaction is a result of currents around the vortex cores which are confined to the plane. A similar change in the $r$-dependence of the interaction between two vortices upon entering a critical region has been observed numerically in the 3-dimensional Ginzburg-Landau model [@OlTe]. Near the charged fixed point of that theory, $\eta_{\bf a}=1$ [@HeTe], as in our case. This is a very pleasing coincidence as the (2+1)-dimensional Ginzburg-Landau model constitutes the dual formulation of the system. Dual Theory ----------- The [*dynamics*]{} of the charged degrees of freedom is described by the effective Lagrangian (\[general\]) with the speed of sound given by Eq. (\[speed\]). In accord with the above findings, we have ignored the coupling to the magnetic vector potential, so that the effective theory essentially describes a superfluid. In the dual formulation, the roles of charges and vortices are interchanged. And the Anderson-Bogoliubov mode mediating the interaction between two vortices is represented as a photon associated with a fictitious gauge field $a_\mu$, i.e., $\tilde{\partial}_\mu \varphi \sim \epsilon_{\mu \nu \lambda} \tilde{\partial}_\nu a_\lambda$. In 2+1 dimensions, this identification makes sense as a photon has only one transverse direction and thus only one degree of freedom. It therefore represents the same number of degrees of freedom as does the Anderson-Bogoliubov mode. The elementary excitations of the dual theory are the vortices, described by a complex scalar field $\psi$. Specifically, the dual theory of Eq. (\[general\]) turns out to be the Ginzburg-Landau model [@dualGL; @GFCM; @WeZe; @KKS] $${\cal L}_{\rm dual} = -{{\textstyle\frac{1}{4}}} f_{\mu \nu}^2 + |(\partial_\mu -i g a_\mu) \psi|^2 - m_\psi^2 |\psi|^2 - u|\psi|^4,$$ with $f_{\mu \nu} = \tilde{\partial}_\mu a_\nu - \tilde{\partial}_\nu a_\mu$, $m_\psi$ a mass parameter, and $u$ the strength of the self-coupling. Both the gauge part as well as the matter part of the dual theory are of a relativistic form. The gauge part is because the effective theory (\[general\]), obtained after ignoring nonlinear terms, is Lorentz invariant. The matter part is because vortices of positive and negative circulation can annihilate, and can also be created. In this sense they behave as relativistic particles. As was pointed out in Ref. [@WeZe], the speed of “light” in the gauge and matter part need not to be identical and will in general differ. The interaction potential (\[V2d\]) between two external vortices is now being interpreted as the 2-dimensional Coulomb potential between charges. The observation concerning the critical behavior of the Ginzburg-Landau model implies that the qualitative change in $V(r)$ upon entering the critical region is properly represented in the dual formulation. Whereas in the conducting phase, the charges are condensed, in the insulating phase, the vortices are condensed [@MPAFisher]. In the dual theory, the vortex condensate is represented by a nonzero expectation value of the $\psi$ field, which in turn leads via the Anderson-Higgs mechanism to a mass term for the gauge field $a_\mu$. Because $(\epsilon_{\mu \nu \lambda} \tilde{\partial}_\nu a_\lambda)^2 \sim (\tilde{\partial}_\mu \varphi)^2$, the mass term $a_\mu^2$ with two derivatives less implies that the Anderson-Bogoliubov mode has acquired an energy gap. That is to say, the phase where the vortices are condensed is incompressible and indeed an insulator. Since electric charges are seen by the dual theory as flux quanta, they are expelled from the system as long as the dual theory is in the Meissner state. Above the critical field $h = \nabla_\perp \times {\bf a} = h_{c_1}$ they start penetrating the system and form an Abrikosov lattice. In the original formulation, this corresponds to a Wigner crystal of the charges. Finally, when more charges are added and the dual field reaches the critical value $h_{c_2}$, the lattice melts and the charges condense in the superfluid phase described by the effective theory (\[general\]). Acknowledgments {#acknowledgments .unnumbered} =============== It is a pleasure to thank J. Spa[ł]{}ek and the other organizers of the XL Cracow School of Theoretical Physics for inviting me to lecture at the renowned summer school at Zakopane, Poland. I am indebted to M. Krusius for the hospitality at the Low Temperature Laboratory, Helsinki University of Technology, Finland where the lectures were prepared, and to B. Rosenstein for the hospitality at the Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan where these notes were written. I would also like to acknowledge helpful discussions with G. Honig and M. Paalanen. This work was funded in part by the EU sponsored programme Transfer and Mobility of Researchers under contract No. ERBFMGECT980122, and by the National Science Council (NCS) of Taiwan. [99]{} M. P. A. Fisher, Phys. Rev. Lett. [**65**]{}, 923 (1990). A. F. Hebard and M. A. Paalanen, Phys. Rev. Lett. [**65**]{}, 927 (1990). D. M. Eagles, Phys. 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[^2]: To avoid confusion, let us repeat that the bar in $\bar{\Delta}_0$ indicates that the pair field is a constant, while the subscript $0$ indicates that it satisfies the extremal condition (\[bcs:gap\]). [^3]: Similar as before, the bar in $\bar{\phi}_0$ denotes a constant value of the field, while the subscript $0$ indicates that it satisfies the mean-field equation (\[bec:min\]).

--- abstract: | We point out that the $B\to\rho\rho$ data have seriously constrained the possibility of resolving the $B\to\pi\pi$ puzzle from the large observed $B^0\to\pi^0\pi^0$ branching ratio in the available theoretical approaches. The next-to-leading-order (NLO) contributions from the vertex corrections, the quark loops, and the magnetic penguin evaluated in the perturbative QCD (PQCD) approach have saturated the experimental upper bound of the $B^0\to\rho^0\rho^0$ branching ratio, and do not help. The NLO PQCD predictions for the $B^0\to\rho^\mp\rho^\pm$ and $B^\pm\to\rho^\pm\rho^0$ branching ratios are consistent with the data. The inclusion of the NLO jet function from the soft-collinear effective theory into the QCD-improved factorization approach, though enhancing the $B^0\to\pi^0\pi^0$ branching ratio sufficiently, overshoots the bound of the $B^0\to\rho^0\rho^0$ branching ratio, and deteriorates the predictions for the $B^\pm\to \pi^0 K^\pm$ and $B^0\to \pi^\mp K^\pm$ direct CP asymmetries. author: - 'Hsiang-nan Li$^{1}$' - Satoshi Mishima$^2$ title: 'Implication of the $B\to\rho\rho$ data on the $B\to \pi \pi$ puzzle' --- =10000 INTRODUCTION ============ The observed direct CP asymmetries and branching ratios of the $B\to \pi K$, $\pi\pi$ decays [@HFAG], $$\begin{aligned} A_{CP}(B^0\to \pi^\mp K^\pm)&=&(-10.8\pm 1.7)\%\;,\nonumber\\ A_{CP}(B^\pm\to \pi^0 K^\pm)&=&(4\pm 4)\%\;,\nonumber\\ B(B^0\to\pi^\mp\pi^\pm)&=&(4.9\pm 0.4)\times 10^{-6}\;,\nonumber\\ B(B^0\to\pi^0\pi^0)&=&(1.45\pm 0.29)\times 10^{-6}\;,\label{data}\end{aligned}$$ were regarded as puzzles, since they obviously contradict to the expected relations $A_{CP}(B^0\to \pi^\mp K^\pm)\approx A_{CP}(B^\pm\to \pi^0 K^\pm)$ and $B(B^0\to\pi^\mp\pi^\pm)\gg B(B^0\to\pi^0\pi^0)$. These puzzles have been analyzed in the perturbative QCD (PQCD) approach [@KLS; @LUY] up to next-to-leading-order (NLO) accuracy recently [@LMS05], where the contributions from the vertex corrections, the quark loops, and the magnetic penguin were taken into account. It was found that the vertex corrections modify the color-suppressed tree contribution, such that the relative strong phase between the tree and penguin amplitudes involved in the $B\to\pi K$ decays decreases. The predicted magnitude of the $B^\pm \to\pi^0 K^\pm$ direct CP asymmetry then becomes smaller, and matches the data in Eq. (\[data\]). Though the $B\to\pi K$ puzzle has been resolved, the $B\to\pi\pi$ puzzle remains, because the NLO color-suppressed tree amplitude does not increase the predicted $B^0\to\pi^0\pi^0$ branching ratio sufficiently. A resolution to a puzzle usually demands an introduction of new mechanism. It is thus essential to investigate whether the proposed new mechanism deteriorates the consistency of theoretical results with other data. To make sure the above NLO effects are reasonable, we apply the same PQCD formalism to more two-body nonleptonic $B$ meson decays, concentrating on the $B\to\rho\rho$ branching ratios, which are also sensitive to the color-suppressed tree contribution. It will be shown that the NLO PQCD predictions are in agreement with the data of the $B^0\to\rho^\mp\rho^\pm$ and $B^\pm\to\rho^\pm\rho^0$ branching ratios, and saturate the experimental upper bound of the $B^0\to\rho^0\rho^0$ branching ratio, $B(B^0\to\rho^0\rho^0)<1.1\times 10^{-6}$ [@HFAG]. Therefore, our resolution to the $B\to\pi K$ puzzle makes sense, and the $B\to\pi\pi$ puzzle is confirmed. The dramatic difference between the $B\to\pi\pi$ and $\rho\rho$ data has been also noticed in [@BLS0602], which stimulates the proposal of a new isospin amplitude with $I = 5/2$. The possible new physics signals from the $B\to\pi\pi$ decays have been discussed in [@BBLS; @YWL2; @CGHW]. It has been claimed that the $B\to\pi\pi$ puzzle is resolved in the QCD-improved factorization (QCDF) approach [@BBNS] with an input from soft-collinear effective theory (SCET) [@BY05]: the inclusion of the NLO jet function, one of the hard coefficients of SCET$_{\rm II}$, into the QCDF formula for the color-suppressed tree amplitude leads to enough enhancement of the $B^0\to\pi^0\pi^0$ branching ratio. Following the argument made above, we apply the same formalism [@BY05] to the $B\to\pi K$, $\rho\rho$ decays as a check. It turns out that the effect of the NLO jet function deteriorates the QCDF results for the direct CP asymmetries in the $B^\pm\to \pi^0 K^\pm$ and $B^0\to \pi^\mp K^\pm$ decays: the magnitude of the former increases, while that of the latter decreases, contrary to the tendency indicated by the data. This NLO effect also overshoots the upper bound of the $B^0\to\rho^0\rho^0$ branching ratio very much. This observation is expected: the $B^0\to\rho^0\rho^0$ and $B^0\to\pi^0\pi^0$ decays have the similar factorization formulas, so the branching ratio $B(B^0\to\rho^0\rho^0)$ ought to be larger than $B(B^0\to\pi^0\pi^0)$ due to the meson decay constants $f_\rho > f_\pi$. Therefore, the $B\to\rho\rho$ data have seriously constrained the possibility of resolving the $B\to\pi\pi$ puzzle in the available theoretical approaches. There exists an alternative phenomenological application of SCET [@BPRS; @BPS05], where the jet function, characterized by the scale of $O(\sqrt{m_b\Lambda})$, $m_b$ being the $b$ quark mass and $\Lambda$ a hadronic scale, is regarded as being incalculable. Its contribution, together with other nonperturbative parameters, such as the charming penguin, were then determined by the $B\to\pi\pi$ data. That is, the color-suppressed tree amplitude can not be explained, but the data are used to fit for the phenomenological parameters in the theory. Predictions for the $B\to\pi K$, $KK$ decays were then made based on the obtained parameters and partial SU(3) flavor symmetry [@BPS05]. Final-state interaction (FSI) is certainly a plausible resolution to the $B\to\pi\pi$ puzzle, but the estimate of its effect is quite model-dependent. Even opposite conclusions were drawn sometimes. When including FSI either into naive factorization [@CHY] or into QCDF [@CCS], the $B^0\to\pi^0\pi^0$ branching ratio was treated as an input in order to fix the involved free parameters. Hence, no resolution was really proposed. It has been found that FSI, evaluated in the Regge model, is insufficient to account for the observed $B^0\to\pi^0\pi^0$ branching ratio [@DLLN]. We conclude that there is no satisfactory resolution in the literature: the available proposals are either data fitting, or can not survive the constraints from the $B\to\pi K$, $\rho\rho$ data under the current theoretical development. In Sec. II we compute the branching ratios, the direct CP asymmetries, and the polarization fractions of the $B\to\rho\rho$ decays using the NLO PQCD formalism. The branching ratios and the direct CP asymmetries of the $B\to\pi K$, $\pi\pi$, $\rho\rho$ decays are calculated in Sec. III by including the NLO jet function from SCET$_{\rm II}$ into the QCDF formulas. Section IV is the discussion, where we comment on and compare the various analyses of the FSI effects in the $B\to\pi K$, $\pi\pi$ decays. $B\to\rho\rho$ IN NLO PQCD ========================== The NLO contributions from the vertex corrections, the quark loops, and the magnetic penguin to the $B\to\pi K$ and $\pi\pi$ decays have been calculated in the naive dimensional regularization (NDR) scheme in the PQCD approach [@LMS05], and the results for the branching ratios and the direct CP asymmetries are quoted in Tables \[br1\] and \[cp1\], respectively. We have taken this chance to correct a minor numerical mistake in the vertex corrections for the $B\to\pi K$ decays, whose branching ratios become smaller by $2\sim 4\%$. Note that a minus sign is missing for the $q=t$ term in the expression for the quark-loop contributions in Eq. (27) of [@LMS05]. Nevertheless, this typo has nothing to do with the numerical outcomes. Our observations are summarized below. The corrections from the quark loops and from the magnetic penguin come with opposite signs, and sum to about $-10\%$ of the leading-order (LO) penguin amplitudes. They mainly reduce the penguin-dominated $B\to\pi K$ branching ratios, but have a minor influence on the tree-dominated $B\to\pi\pi$ branching ratios, and on the direct CP asymmetries. On the contrary, the vertex corrections do not change the branching ratios, except the $B^0\to\pi^0\pi^0$ one. They modify only the direct CP asymmetries of the $B^\pm\to\pi^0 K^\pm$, $B^0\to\pi^0K^0$, and $B^0\to\pi^0\pi^0$ modes by increasing the color-suppressed tree amplitude $C'$ few times. The larger $C'$, leading to the nearly vanishing direct CP asymmetry $A_{CP}(B^\pm\to \pi^0K^\pm)$, resolves the $B\to\pi K$ puzzle within the standard model. Mode Data [@HFAG] LO LO$_{\rm NLOWC}$ +VC +QL +MP +NLO --------------------------- ------------------------------ ------------------- ------------------- ------------------- ------------------- ------------------- -------------------------------------------------------- $B^\pm \to \pi^\pm K^0$ $ 24.1 \pm 1.3 $ $17.0$ $32.3$ $30.1$ $34.2$ $24.1$ $23.6^{+14.5\,(+13.8)}_{-\ 8.4\,(-\ 8.2)}$ $B^\pm \to \pi^0 K^\pm$ $ 12.1 \pm 0.8 $ $10.2$ $18.4$ $17.1$ $19.4$ $14.0$ $13.6^{+10.3\,(+\ 7.3)}_{-\ 5.7\,(-\ 4.3)}$ $B^0 \to \pi^\mp K^\pm$ $ 18.9 \pm 0.7 $ $14.2$ $27.7$ $26.1$ $29.4$ $20.5$ $20.4^{+16.1\,(+11.5)}_{-\ 8.4\,(-\ 6.7)}$ $B^0 \to \pi^0 K^0 $ $ 11.5 \pm 1.0 $ $\phantom{0}5.7$ $12.1$ $11.4$ $12.8$ $\phantom{0}8.7$ $\phantom{0}8.7^{+\ 6.0\,(+\ 5.5)}_{-\ 3.4\,(-\ 3.1)}$ $B^0 \to \pi^\mp \pi^\pm$ $ \phantom{0}4.9 \pm 0.4 $ $\phantom{0}7.0$ $\phantom{0}6.8$ $\phantom{0}6.6$ $\phantom{0}6.9$ $\phantom{0}6.7$ $\phantom{0}6.5^{+\ 6.7\,(+\ 2.7)}_{-\ 3.8\,(-\ 1.8)}$ $B^\pm \to \pi^\pm \pi^0$ $ \phantom{0}5.5 \pm 0.6 $ $\phantom{0}3.5$ $\phantom{0}4.1$ $\phantom{0}4.0$ $\phantom{0}4.1$ $\phantom{0}4.1$ $\phantom{0}4.0^{+\ 3.4\,(+\ 1.7)}_{-\ 1.9\,(-\ 1.2)}$ $B^0 \to \pi^0 \pi^0$ $ \phantom{0}1.45 \pm 0.29 $ $\phantom{0}0.12$ $\phantom{0}0.27$ $\phantom{0}0.37$ $\phantom{0}0.29$ $\phantom{0}0.21$ $\phantom{0}0.29^{+0.50\,(+0.13)}_{-0.20\,(-0.08)}$ : Branching ratios from PQCD in the NDR scheme in units of $10^{-6}$. The label LO$_{\rm NLOWC}$ means the LO results with the NLO Wilson coefficients, and +VC, +QL, +MP, and +NLO mean the inclusions of the vertex corrections, of the quark loops, of the magnetic penguin, and of all the above NLO corrections, respectively. The errors in the parentheses represent only the hadronic uncertainty [@LMS05].[]{data-label="br1"} Mode Data [@HFAG] LO LO$_{\rm NLOWC}$ +VC +QL +MP +NLO --------------------------- ------------------------------ --------------------- --------------------- --------------------- --------------------- --------------------- --------------------------------------------- $B^\pm \to \pi^\pm K^0$ $ -2 \pm 4$ $\ \, -1$ $-1$ $-1$ $\phantom{-}0$ $\ \, -1$ $\ \ \ \ \, \phantom{-}0\pm 0\,(\pm 0)$ $B^\pm \to \pi^0 K^\pm$ $ \phantom{-}4 \pm 4 $ $\ \, -8$ $-6$ $-2$ $-5$ $\ \, -8$ $\ \, -1^{+3\,(+3)}_{-6\,(-5)}$ $B^0 \to \pi^\mp K^\pm$ $ -10.8 \pm 1.7 $ $-12$ $-8$ $-9$ $-6$ $-10$ $-10^{+7\,(+5)}_{-8\,(-6)}$ $B^0 \to \pi^0 K^0 $ $ \phantom{-}2 \pm 13 $ $\ \, -2$ $\phantom{-}0$ $-7$ $\phantom{-}0$ $\ \, \phantom{-}0$ $\ \, -7^{+3\,(+1)}_{-4\,(-2)}$ $B^0 \to \pi^\mp \pi^\pm$ $ \phantom{-}37 \pm 10$ $\phantom{-}14$ $\phantom{-}19$ $\phantom{-}21$ $\phantom{-}16$ $\phantom{-}20$ $\phantom{-}18^{+20\,(+\ 7)}_{-12\,(-\ 6)}$ $B^\pm \to \pi^\pm \pi^0$ $ \phantom{-}1 \pm 6 $ $\ \, \phantom{-}0$ $\ \, \phantom{-}0$ $\ \, \phantom{-}0$ $\ \, \phantom{-}0$ $\ \, \phantom{-}0$ $\ \ \ \ 0\pm 0\,(\pm 0)$ $B^0 \to \pi^0 \pi^0$ $ \phantom{-}28^{+40}_{-39}$ $\ \, -4$ $-34$ $\phantom{-}65$ $-41$ $-43$ $\phantom{-}63^{+35\,(+\ 9)}_{-34\,(-15)}$ : Direct CP asymmetries from PQCD in the NDR scheme in percentage.[]{data-label="cp1"} The above observations can be easily understood as follows. The $B^0\to \pi^\mp K^\pm$ decays involve the color-allowed tree $T'$ and the QCD penguin $P'$ in the topological amplitude parametrization. The data of $A_{CP}(B^0\to \pi^\mp K^\pm)\approx -11.5\%$ imply a sizable relative strong phase between $T'$ and $P'$. The $B^\pm\to \pi^0 K^\pm$ decays involve $C'$ and the electroweak penguin amplitude $P'_{ew}$, in addition to $T'$ and $P'$. If $C'$ is large enough, and more or less orthogonal to $T'$, it may orient the sum $T'+C'$ roughly along with $P'+P'_{ew}$. The smaller relative strong phase between $T'+C'$ and $P'+P'_{ew}$ then gives $A_{CP}(B^\pm\to \pi^0 K^\pm)\approx 0$. We found in PQCD that the vertex corrections indeed modify $C'$ in this way. Because our analysis shows the sensitivity of $C'$ to the NLO corrections, it is worthwhile to investigate the direct CP asymmetries of other charged $B$ meson decays. The results will be published elsewhere. The color-suppressed tree amplitude $C$ involved in the $B\to\pi\pi$ decays, despite of being increased few times too by the vertex corrections, remains subleading with the ratio $|C/T|\approx 0.2$, where $T$ represents the color-allowed tree amplitude. This ratio is not enough to explain the observed $B^0\to\pi^0\pi^0$ branching ratio as shown in Table \[br1\] [@LMS05]. A much larger $|C/T|\approx 0.8$ must be achieved in order to resolve the $B\to\pi\pi$ puzzle [@Charng2]. We mention that a different source for the large relative strong phase between $C$ and $T$ has been proposed in [@GHZP], which arises from charm- and top-mediated penguins. Helicity Amplitudes ------------------- We examine whether the observations made in [@LMS05] are solid by applying the same NLO PQCD formalism to the $B\to\rho\rho$ decays, which are also sensitive to the color-suppressed tree contribution. The $B\to\rho\rho$ decays have been analyzed at LO in [@LILU; @rho]. The numerical results in the two references differ a bit due to the different choices of the characteristic hard scales, which can be considered as one of the sources of theoretical uncertainties (from higher-order corrections). The $B\to V_2(\epsilon_2,P_2) V_3(\epsilon_3,P_3)$ decay rate is written as $$\Gamma =\frac{G_{F}^{2}P_c}{64\pi m^{2}_{B} } \sum_{\sigma}{\cal M}^{(\sigma)\dagger }{\cal M^{(\sigma)}}\;, \label{dr1}$$ where $P_c=|{\bf P}_2|=|{\bf P}_3|=m_B/2$ is the momentum of either of the vector mesons $V_2$ and $V_3$, $m_B$ being the $B$ meson mass. $\epsilon_{2}$ $(\epsilon_{3})$ are the polarization vectors of the meson $V_2$ $(V_3)$. The amplitudes $\cal M^{(\sigma)}$ corresponding to the polarization configurations $\sigma$ with both $V_2$ and $V_3$ being longitudinally polarized, and being transversely polarized in the parallel and perpendicular directions are written as $$\begin{aligned} {\cal M}^{\sigma} \ =\ \left( m_{B}^{2}{\cal M}_{L}\;,\ \ m_{B}^{2}{\cal M}_{N} \epsilon^{*}_{2}(T)\cdot\epsilon^{*}_{3}(T)\;,\ \ - i{\cal M}_{T}\epsilon^{\alpha \beta\gamma \rho} \epsilon^{*}_{2\alpha}(T)\epsilon^{*}_{3\beta}(T) P_{2\gamma }P_{3\rho } \right) \;,\end{aligned}$$ respectively. In the above expressions $\epsilon(T)$ denote the transverse polarization vectors, and we have adopted the convention $\epsilon_{0123} = 1$. Define the velocity $v_2=P_2/m_{V_2}$ $(v_3=P_3/m_{V_3})$ in terms of the $V_2$ $(V_3)$ meson mass $m_{V_2}$ $(m_{V_3})$. The helicity amplitudes, $$\begin{aligned} A_{L}&=&-G m^{2}_{B}{\cal M}_{L}, \nonumber\\ A_{\parallel}&=&G \sqrt{2}m^{2}_{B}{\cal M}_{N}, \nonumber \\ A_{\perp} &=& G m_{V_2} m_{V_3} \sqrt{2[(v_2\cdot v_3)^{2}-1]} {\cal M }_{T} \;, \label{ase3}\end{aligned}$$ with the normalization factor $G=\sqrt{G_F^2P_c/(64\pi m^2_{B}\Gamma)}$, satisfy the relation, $$\begin{aligned} |A_{L}|^2+|A_{\parallel}|^2+|A_{\perp}|^2=1\;.\end{aligned}$$ We also need to employ another equivalent set of helicity amplitudes, $$\begin{aligned} H_{0}\ =\ m^{2}_{B} {\cal M}_L\;, \ \ \ \ \ \ \ H_{\pm}\ =\ m^{2}_{B} \left( {\cal M}_{N} \mp \frac{{\cal M}_{T}}{2} \right) \;,\label{ase2}\end{aligned}$$ with the helicity summation, $$\begin{aligned} \sum_{\sigma}{\cal M}^{(\sigma)\dagger }{\cal M^{(\sigma)}} = |H_{0}| ^{2}+|H_{+}|^{2} + | H_{-}|^{2}\;.\end{aligned}$$ The definitions in Eq. (\[ase3\]) are related to those in Eq. (\[ase2\]) via $$\begin{aligned} A_{L}\ =\ -G H_0 \;,\ \ \ \ \ A_{\parallel}\ =\ \frac{G}{\sqrt{2}}(H_++H_-) \;,\ \ \ \ \ A_{\perp}\ =\ -\frac{G}{\sqrt{2}}(H_+-H_-) \;. \label{ase}\end{aligned}$$ The explicit expressions of the distribution amplitudes $\phi_\rho$, $\phi_{\rho}^t$, and $\phi_{\rho}^s$ for a longitudinally polarized $\rho$ meson, and $\phi_\rho^T$, $\phi_{\rho}^v$, and $\phi_{\rho}^a$ for a transversely polarized $\rho$ meson are referred to [@TLS; @PB1]. However, for the twist-3 distribution amplitudes $\phi_{\rho}^t$, $\phi_{\rho}^s$, $\phi_{\rho}^v$, and $\phi_{\rho}^a$, we adopt their asymptotic models as shown below: $$\begin{aligned} \phi_\rho(x)&=&\frac{3f_\rho}{\sqrt{2N_c}} x(1-x)\left[1+ 0.18\, C_2^{3/2}(2x-1)\right]\;, \label{pwr}\\ \phi_{\rho}^t(x)&=&\frac{f^T_{\rho}}{2\sqrt{2N_c}} 3(2x-1)^2 \;, \label{pwt}\\ \phi_{\rho}^s(x) &=&\frac{3f_\rho^T}{2\sqrt{2N_{c}}} (1-2x) \;, \label{pws}\\ \phi_\rho^T(x)&=&\frac{3f_\rho^T}{\sqrt{2N_c}} x(1-x)\left[1+ 0.2\, C_2^{3/2}(2x-1)\right]\;, \label{pwft}\\ \phi_{\rho}^v(x)&=&\frac{f_{\rho}}{2\sqrt{2N_c}} \frac{3}{4}[1+(2x-1)^2] \;, \label{pwv}\\ \phi_{\rho}^a(x) &=&\frac{3f_\rho}{4\sqrt{2N_{c}}} (1-2x) \;, \label{pwa}\end{aligned}$$ with the decay constants $f_\rho=200$ MeV and $f_\rho^T=160$ MeV, and the Gegenbauer polynomial $C_2^{3/2}(t)=3(5 t^2-1)/2$. On one hand, the sum-rule derivation of light-cone meson distribution amplitudes suffer sizable theoretical uncertainty, so that the asymptotic models are acceptable. On the other hand, the asymptotic models for twist-3 distribution amplitudes were also adopted in QCDF [@BBNS], and the comparison of our results with theirs will be more consistent. ------------------------------------------------------------------------------------------ $\rho^+\rho^-$ $H_{h}^{(u)} $ ---------------------- ------------------------------------------------------------------- $F_{e}^h$ $F^h_{e4} \left( a_1 \right)$ ${\cal M}_{e}^h$ ${\cal M}^h_{e4} \left( a_1' \right)$ $F_{a}^h$ $\eta_T F^h_{a4} \left( a_2 \right)$ ${\cal M}_{a}^h$ ${\cal M}^h_{a4} \left( a_2' \right)$ $\rho^+\rho^-$ $H_{h}^{(t)} $ $F_e^{P,h}$ $F^h_{e4} \left( a_4^{(u)} \right) $ ${\cal M}_{e}^{P,h}$ ${\cal M}^h_{e4}\left( a_4^{\prime (u)} \right) + {\cal M}^h_{e6}\left( a_6^{\prime (u)} \right)$ $F_a^{P,h}$ $ \eta_T F^h_{a4} \left( a_4^{(d)} \right) + F^h_{a6} \left(a_6^{(d)} \right)$ ${\cal M}_{a}^{P,h}$ ${\cal M}^h_{a4} \left( a_3^{\prime (u)} + a_3^{\prime (d)} + a_4^{\prime (d)} + a_5^{\prime (u)} + a_5^{\prime (d)}\right) + \eta_T {\cal M}^h_{a6} \left( a_6^{\prime (d)} \right)$ $\rho^+\rho^0$ $\sqrt{2}H_{h}^{(u)} $ $F^h_e$ $F^h_{e4} \left( a_1 + a_2 \right)$ ${\cal M}^h_{e}$ ${\cal M}^h_{e4} \left( a_1' + a_2' \right)$ $F^h_a$ 0 ${\cal M}^h_{a}$ 0 $\rho^+\rho^0$ $\sqrt{2}H_{h}^{(t)} $ $F_e^{P,h}$ $F^h_{e4} \left( a_3^{(u)} - a_3^{(d)} + a_4^{(u)} - a_4^{(d)} + a_5^{(u)} - a_5^{(d)} \right)$ ${\cal M}_{e}^{P,h}$ ${\cal M}^h_{e4}\left( a_3^{\prime (u)} - a_3^{\prime (d)} + a_4^{\prime (u)} - a_4^{\prime (d)} - a_5^{\prime (u)} + a_5^{\prime (d)} \right) + {\cal M}^h_{e6}\left( a_6^{\prime (u)} - a_6^{\prime (d)}\right) $ $F_a^{P,h}$ 0 ${\cal M}_{a}^{P,h}$ 0 $\rho^0\rho^0$ $\sqrt{2}H_{h}^{(u)} $ $F^h_e$ $F^h_{e4} \left(- a_2 \right)$ ${\cal M}^h_{e}$ ${\cal M}^h_{e4} \left( -a_2' \right)$ $F^h_a$ $\eta_T F^h_{a4} \left( a_2 \right)$ ${\cal M}^h_{a}$ ${\cal M}^h_{a4} \left( a_2' \right)$ $\rho^0\rho^0$ $\sqrt{2}H_{h}^{(t)} $ $F_e^{P,h}$ $ F^h_{e4} \left( -a_3^{(u)} + a_3^{(d)} + a_4^{(d)} - a_5^{(u)} + a_5^{(d)} \right)$ ${\cal M}_{e}^{P,h}$ ${\cal M}^h_{e4}\left( - a_3^{\prime (u)} + a_3^{\prime (d)} + a_4^{\prime (d)} + a_5^{\prime (u)} - a_5^{\prime (d)} \right) + {\cal M}^h_{e6}\left( a_6^{\prime (d)} \right)$ $F_a^{P,h}$ $ \eta_T F^h_{a4} \left( a_4^{(d)} \right) + F^h_{a6} \left(a_6^{(d)} \right)$ ${\cal M}_{a}^{P,h}$ ${\cal M}^h_{a4} \left( a_3^{\prime (u)} + a_3^{\prime (d)} + a_4^{\prime (d)} + a_5^{\prime (u)} + a_5^{\prime (d)}\right) + \eta_T {\cal M}^h_{a6} \left( a_6^{\prime (d)} \right)$ ------------------------------------------------------------------------------------------ : LO $B\to \rho\rho$ decay amplitudes with $\eta_{T}=0\ (1)$ for the longitudinal (transverse) components.[]{data-label="rhorho"} For the $\bar b \to \bar d$ transition, the helicity amplitudes have the general expression, $$\begin{aligned} H_{h} &=& V_{ub}^*V_{ud}\, H_{h}^{(u)} +V_{cb}^*V_{cd}\, H_{h}^{(c)} +V_{tb}^*V_{td}\, H_{h}^{(t)} \;, \label{eq:amp}\end{aligned}$$ with $h=0$ or $\pm$, and $V$’s being the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The amplitudes $H^{(u)}_{h}$, $H^{(c)}_{h}$, and $H^{(t)}_{h}$ are decomposed at LO into $$\begin{aligned} H^{(u)}_{h} &=& m_B^2 \left( f_\rho F^h_e + {\cal M}^h_e + f_BF^h_a + {\cal M}^h_a \right) \;, \nonumber\\ H^{(c)}_{h} &=& 0 \;, \nonumber\\ H^{(t)}_{h} &=& - m_B^2 \left( f_\rho F^{P,h}_e + {\cal M}^{P,h}_e + f_BF^{P,h}_a + {\cal M}^{P,h}_a \right) \;.\end{aligned}$$ The LO PQCD factorization formulas for the $B\to\rho\rho$ helicity amplitudes associated with the final states $\rho^+\rho^-$, $\rho^+\rho^0$, and $\rho^0\rho^0$ are summarized in Table \[rhorho\]. The Wilson coefficients $a^{(q)}$ for the factorizable contributions, and $a^{\prime(q)}$ for the nonfactorizable contributions can be found in [@LMS05], where $q=u$ or $d$ denotes the quark pair produced in the electroweak penguin. The explicit expressions of the LO factorizable amplitudes $F^0_{e4,a4,a6}$ and of the LO nonfactorizable amplitudes ${\cal M}^0_{e4,e6,a4,a6}$ are similar to those for the $B\to PP$ decays [@LMS05] but with the replacements of the distribution amplitudes and the masses, $$\begin{aligned} & &\phi^A(x)\to\phi(x)\;,\;\;\;\;\phi^P(x)\to\phi^s(x)\;,\;\;\;\; \phi^T(x)\to\phi^t(x)\;,\nonumber\\ & &m_{02}\to -m_\rho\;,\;\;\;\;m_{03}\to m_\rho\;. \label{REPL}\end{aligned}$$ In the above replacement $m_{02}$ ($m_{03}$) is the chiral enhancement scale associated with the pseudo-scalar meson involved in the $B\to P$ transition (emitted from the weak vertex), and $m_\rho=0.77$ GeV the $\rho$ meson mass. Note that the amplitude $F^0_{e6}$ from the operators $O_{5-8}$ vanishes at LO. The LO factorization formulas for the transverse components are collected in Appendix A, whose relations to $F^\pm$ and to ${\cal M}^\pm$ in Table \[rhorho\] follow Eq. (\[ase2\]). For example, the amplitude $F^\pm_{e4}$ is given by $$\begin{aligned} F^\pm_{e4}\ =\ F_{Ne4} \mp \frac{F_{Te4}}{2} \;.\end{aligned}$$ NLO Corrections --------------- The vertex corrections to the $B\to \rho\rho$ decays modify the Wilson coefficients for the emission amplitudes in the standard definitions [@LMS05] into $$\begin{aligned} a_1(\mu) &\to& a_1(\mu) +\frac{\alpha_s(\mu)}{4\pi}C_F\frac{C_{1}(\mu)}{N_c} V_1(\rho) \;, \nonumber\\ a_2(\mu) &\to& a_2(\mu) +\frac{\alpha_s(\mu)}{4\pi}C_F\frac{C_{2}(\mu)}{N_c} V_2(\rho) \;, \nonumber\\ a_i(\mu) &\to& a_i(\mu) +\frac{\alpha_s(\mu)}{4\pi}C_F\frac{C_{i\pm 1}(\mu)}{N_c} V_i(\rho) \;,\;\;\;\;i=3 - 10\;,\label{wnlo}\end{aligned}$$ where $V_i(\rho)$ in the NDR scheme are in agreement with those in [@BN] for the longitudinal component, $$\begin{aligned} V_i(\rho) &=& \left\{ {\renewcommand\arraystretch{2.5} \begin{array}{ll} 12\ln\displaystyle{\frac{m_b}{\mu}}-18 +\frac{2\sqrt{2N_c}}{f_\rho}\int_0^1 dx\, \phi_\rho(x)\, g(x)\;, & \mbox{\rm for }i=1-4,9,10\;, \\ -12\ln\displaystyle{\frac{m_b}{\mu}}+6 -\frac{2\sqrt{2N_c}}{f_\rho}\int_0^1dx\, \phi_\rho(x)\, g(1-x)\;, & \mbox{\rm for }i=5,7\;, \\ \displaystyle{ -\frac{2\sqrt{2N_c}}{f_\rho^T}\int_0^1 dx\, \phi_\rho^{s}(x)\, [-6+h(x)] }\;, & \mbox{\rm for }i=6,8\;, \end{array} } \right.\label{vim}\end{aligned}$$ and with those in [@YWL] for the transverse components, $$\begin{aligned} V_i^\pm(\rho) &=& \left\{ {\renewcommand\arraystretch{2.5} \begin{array}{ll} 12\ln\displaystyle{\frac{m_b}{\mu}}-18 +\frac{2\sqrt{2N_c}}{f_\rho}\int_0^1 dx\, [\phi_\rho^v(x)\pm\phi_\rho^a(x)]\, g(x)\;, & \mbox{\rm for }i=1-4,9,10\;, \\ -12\ln\displaystyle{\frac{m_b}{\mu}}+6 -\frac{2\sqrt{2N_c}}{f_\rho}\int_0^1dx\, [\phi_\rho^v(x)\pm\phi_\rho^a(x)]\, g(1-x)\;, & \mbox{\rm for }i=5,7\;. \end{array} } \right.\label{vit}\end{aligned}$$ We do not show $V_{6,8}^\pm$, because of the associated factorizable emission amplitudes $F^{\pm}_{e6}=0$. Moreover, the vertex corrections introduce the additional contributions resulting from the penguin operators $O_{5-8}$, $$\begin{aligned} &\rho^+\rho^-: &f_\rho F^{P,h}_e\to f_\rho F^{P,h}_e+f_\rho^T F^h_{e6} \left( a_{6\rm VC}^{(u)} \right)\;,\nonumber\\ &\rho^+\rho^0: &f_\rho F^{P,h}_e\to f_\rho F^{P,h}_e+f_\rho^T F^h_{e6} \left( a_{6\rm VC}^{(u)}-a_{6\rm VC}^{(d)} \right)\;,\nonumber\\ &\rho^0\rho^0: &f_\rho F^{P,h}_e\to f_\rho F^{P,h}_e+f_\rho^T F^h_{e6} \left( a_{6\rm VC}^{(d)} \right)\;,\end{aligned}$$ where the arguments $a_{6\rm VC}$ represent only the vertex-correction piece in Eq. (\[wnlo\]). Taking into account the NLO contributions from the quark loops and from the magnetic penguin, the helicity amplitudes are modified into $$\begin{aligned} {\renewcommand\arraystretch{2.0} \begin{array}{lll} \displaystyle \rho^+\rho^-: &H^{(u,c)}_{h} \, \to\, H^{(u,c)}_{h} +m_B^2{\cal M}^{(u,c)}_{h}\;, & \displaystyle H^{(t)}_{h} \, \to\, H^{(t)}_{h} - m_B^2 {\cal M}^{(t)}_{h} - m_B^2{\cal M}^{(g)}_{h}\;, \\ \displaystyle \rho^+\rho^0:& H^{(u,c,t)}_{h} \, \to\, H^{(u,c,t)}_{h}\;, & \\ \displaystyle \rho^0\rho^0:& \displaystyle H^{(u,c)}_{h} \, \to\, H^{(u,c)}_{h} + \frac{m_B^2}{\sqrt{2}}{\cal M}^{(u,c)}_{h}\;, &\displaystyle H^{(t)}_{h} \, \to\, H^{(t)}_{h} -\frac{m_B^2}{\sqrt{2}}{\cal M}^{(t)}_{h} -\frac{m_B^2}{\sqrt{2}}{\cal M}^{(g)}_{h} \;, \end{array} }\label{qlmp}\end{aligned}$$ where ${\cal M}_{h}^{(u)}$, ${\cal M}_{h}^{(c)}$, ${\cal M}_{h}^{(t)}$, and ${\cal M}_{h}^{(g)}$ denote the up-loop, charm-loop, QCD-penguin-loop, and magnetic-penguin corrections, respectively. The magnetic-penguin contribution to the $B\to PV$ modes was computed in [@MISHIMA03]. ${\cal M}_{h}^{(u,c,t)}$ and ${\cal M}_{h}^{(g)}$ for $h=0$ are similar to those for the $B\to PP$ decays [@LMS05] with the replacements in Eq. (\[REPL\]). Those for the transverse components are presented in Appendix A. The choices of the $B$ meson wave function, of the $B$ meson lifetimes, and of the CKM matrix elements, including the allowed ranges of their variations, are the same as in [@LMS05]. We vary the Gegenbauer coefficients in $\phi_\rho$ and in $\phi_\rho^T$ by 100% as analyzing the theoretical uncertainty. The resultant $B\to\rho$ form factors at maximal recoil, $$\begin{aligned} A_0\ =\ 0.31^{+0.07}_{-0.06} \;,\;\;\;\; A_1\ =\ 0.21^{+0.05}_{-0.04} \;,\;\;\;\; V\ =\ 0.26^{+0.07}_{-0.05} \;,\label{form}\end{aligned}$$ associated with the longitudinal, parallel, and perpendicular components of the $B\to\rho\rho$ decays, respectively, are similar to those derived from QCD sum rules [@sumrho; @BZ0412], and almost the same as adopted in the QCDF analysis [@AK]. Compared to [@sumrho], one-loop radiative corrections to the two-parton twist-3 contributions have been considered in [@BZ0412]. The central value of the form factor $V$ in Eq. (\[form\]) is a bit smaller than those in [@sumrho; @BZ0412]. We emphasize that this difference is not essential, since the perpendicular component corresponding to $V$ contributes roughly less than 10% of the total $B\to\rho\rho$ branching ratios as shown below. The PQCD results for the $B\to\rho\rho$ branching ratios, together with the BABAR and Belle data, are listed in Table \[br2\]. It is obvious that the NLO PQCD values are consistent with the data of the $B^0 \to \rho^\mp \rho^\pm$ and $B^\pm \to \rho^\pm \rho^0$ branching ratios. The color-suppressed tree amplitude is also enhanced by the vertex corrections here, but the ratio $|C/T|\approx 0.2$ for the longitudinal component, similar to that in the $B\to\pi\pi$ decays, is still small. However, the central value of the predicted $B^0 \to \rho^0 \rho^0$ branching ratio has almost saturated the experimental upper bound. We conclude that it is unlikely to accommodate the measured $B^0\to\pi^0\pi^0$, $\rho^0\rho^0$ branching ratios simultaneously in PQCD. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Mode BABAR [@HFAG] Belle [@HFAG] LO LO$_{\rm NLOWC}$ +VC +QL +MP +NLO ----------------------------- ---------------------- -------------------- -------- ------------------ -------- -------- -------- --------------------------------------------- -- $B^0 \to \rho^\mp \rho^\pm$ $ 30 \pm 4\pm 5 $ $22.8\pm $27.8$ $26.1$ $25.2$ $26.6$ $25.9$ $25.3^{+25.3\,(+12.1)}_{-13.8\,(-\ 7.9)}$ 3.8^{+2.3}_{-2.6}$ $B^\pm \to \rho^\pm \rho^0$ $17.2\pm 2.5\pm 2.8$ $31.7\pm $13.7$ $16.2$ $16.0$ $16.2$ $16.2$ $16.0^{+15.0\,(+\ 7.8)}_{-\ 8.1\,(-\ 5.3)}$ 7.1^{+3.8}_{-6.7}$ $B^0 \to \rho^0 \rho^0$ $ <1.1 $ — $0.33$ $0.56$ $1.02$ $0.62$ $0.45$ $0.92^{+1.10\,(+0.64)}_{-0.56\,(-0.40)}$ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : $B\to\rho\rho$ branching ratios from PQCD in the NDR scheme in units of $10^{-6}$.[]{data-label="br2"} We obtain the direct CP asymmetries $A_{CP}(B^0 \to \rho^\mp \rho^\pm)=-0.02\, (-0.07)$, $A_{CP}(B^\pm \to \rho^\pm\rho^0)=0.00\, (0.00)$, and $A_{CP}(B^0 \to \rho^0 \rho^0)=0.56\, (0.80)$, where the values (in the parentheses) are from LO (NLO) PQCD. We have also computed the polarization fractions. The NLO corrections have a minor impact on the $B^0 \to \rho^\mp \rho^\pm$ and $B^\pm \to \rho^\pm \rho^0$ decays: their longitudinal polarization contributions remain dominant, reaching 93% and 97%, respectively. However, the ${\bar B}^0 \to \rho^0 \rho^0$ polarization fractions are sensitive to the NLO corrections as indicated in Table \[pof\], where the average longitudinal, parallel, and perpendicular polarization fractions, $f_L$, $f_\parallel$, and $f_\perp$, respectively, are defined by $$\begin{aligned} f_{L,\parallel,\perp}=\frac{B(B^0 \to \rho^0 \rho^0)_{L,\parallel,\perp}+B({\bar B}^0 \to \rho^0 \rho^0)_{L,\parallel,\perp}}{B(B^0 \to \rho^0 \rho^0)+B({\bar B}^0 \to \rho^0 \rho^0)}\;.\end{aligned}$$ The average longitudinal polarization fraction of the $B^0 \to \rho^0 \rho^0$ decays was also found to be smaller in LO PQCD [@LILU; @rho]. It is easy to understand the changes due to the NLO effects. As stated before, the color-suppressed tree amplitude, being the main tree contribution in the $B^0 \to \rho^0 \rho^0$ decay, is enhanced by the vertex corrections. The $B^0 \to \rho^0 \rho^0$ polarization fractions should then approach the naive counting rules [@CKL2; @AK; @LM04]: $f_L\sim 1$ and $f_\parallel\sim f_\perp \sim \lambda^2$ obeyed by a tree-dominated decay, where $\lambda\approx 0.22$ is the Wolfenstein parameter. Mode $f_L$ $f_\parallel$ $f_\perp$ -------------------------------- ------------- --------------- ------------- $B^0 \to \rho^0 \rho^0$ 0.71 (0.67) 0.14 (0.15) 0.15 (0.18) ${\bar B}^0 \to \rho^0 \rho^0$ 0.09 (0.79) 0.45 (0.10) 0.46 (0.11) Average 0.23 (0.78) 0.38 (0.11) 0.39 (0.11) : LO and NLO (in the parentheses) polarization fractions of the $B^0\to\rho^0\rho^0$ decays from PQCD.[]{data-label="pof"} JET FUNCTION IN SCET ==================== In this section we investigate the resolution to the $B\to\pi\pi$ puzzle claimed in QCDF with the input of the NLO jet function from SCET [@BY05]. The leading-power SCET formalism has been derived for two-body nonleptonic $B$ meson decays [@BPRS]. However, there exist different opinions on the calculability of the hard coefficients in SCET$_{\rm II}$, one of which is the jet function characterized by a scale of $O(\sqrt{m_b\Lambda})$. In [@BPS05] the jet function is regarded as being incalculable, and treated as a free parameter. Together with other hadronic parameters, it is determined by fitting the SCET formalism to the $B\to\pi\pi$ data. Therefore, the large ratio $|C/T|$ obtained in [@BPS05] is an indication of the data, instead of coming from an explicit evaluation of the amplitudes. In this analysis the QCD penguin amplitude, receiving a significant contribution from the long-distance charming penguin [@CNPR], was also found to be important. Similarly, the large charming penguin, as one of the fitting parameters in SCET, also arises from the data fitting. A global analysis of the $B\to\pi\pi$, $\pi K$ decays based on the leading-power SCET parametrization has been performed recently in [@WZ0610], where a smaller branching ratio $B(B^0\to\pi^0\pi^0)\approx 0.84\times 10^{-6}$ was obtained. A plausible mechanism in SCET for enhancing the ratio $|C/T|$ was provided in [@BY05]: the jet function could increase the nonfactorizable spectator contribution to the color-suppressed tree amplitude $C$ at NLO. This significant effect was implemented into QCDF [@BY05]. Because of the end-point singularities present in twist-3 spectator amplitudes and in annihilation amplitudes, these contributions have to be parameterized in QCDF [@BBNS]. Different scenarios for choosing the free parameters, labelled by “default", “S1", “S2", $\cdots$, “S4", were proposed in [@BN]. As shown in Table \[tab1\], the large measured $B^0\to\pi^0\pi^0$ branching ratio can be accommodated, when the parameter scenario S4 is adopted. It has been emphasized in the Introduction that the same formalism should be applied to other decay modes for a check, among which we focus on the quantities sensitive to $C$: the $B\to \pi K$ direct CP asymmetries and the $B^0\to\rho^0\rho^0$ branching ratio. The QCDF formulas for the $B\to VV$ decays with the NLO contributions from the vertex corrections, the quark loops, and the magnetic penguin can be found in [@AK; @CY01; @YWL], which appear as the $O(\alpha_s)$ terms of the Wilson coefficients $a_i$, $i=1,\cdots,10$. The vertex corrections are the same as in Eqs. (\[vim\]) and (\[vit\]). Note that the expressions of the Wilson coefficients $a_{6,8}$ differ between [@AK] and [@CY01; @YWL]: $a_{6,8}$ for both the longitudinal and transverse components in [@CY01; @YWL] do not receive any $O(\alpha_s)$ correction. We disagree on this result as shown in Eqs. (\[vim\]) and (\[qlmp\]). Hence, we adopt the expressions in [@AK] for the contributions from the quark loops, the magnetic penguin, and the annihilation. We also employ the $B\to\rho$ form factor values in [@AK]. Since the spectator amplitudes were not shown explicitly in [@AK], we use those from [@BN]. The parameter sets default and S4 have been defined for the $B\to PP$ decays [@BN], but have not for the $B\to VV$ ones. Therefore, we assume that the parameters for the latter are the same as for the former in the following analysis. Fortunately, the predicted $B^0\to\rho^0\rho^0$ branching ratio is insensitive to the variation of the annihilation phase $\phi_A$, which is one of the most essential parameters in QCDF: varying $\phi_A$ between 0 and $2\pi$, the $B^0\to\rho^0\rho^0$ branching ratio changes by less than 10%. The jet function $j_\parallel$ derived in [@BY05] is relevant to the $B\to PP$ decays and to the $B\to VV$ decays with longitudinally polarized final states. The jet function $j_\perp$ is relevant to the $B\to VV$ decays with transversely polarized final states. These jet functions apply not only to the color-suppressed tree amplitudes, but to the color-allowed tree and penguin amplitudes, which are free of the end-point singularities. We mention that the NLO corrections to the hard coefficients of SCET$_{I}$ have been derived in [@BJ05; @BJ052]. This new piece modifies the QCDF outcomes slightly, comparing the color-allowed and color-suppressed tree contributions obtained in [@BY05] and in [@BJ05]. Hence, we consider the NLO correction only from the jet function for simplicity. Furthermore, since the jet function enhances the color-suppressed tree amplitude, the $B^0\to \rho^0\rho^0$ polarization fractions are expected to approach the naive counting rules. That is, the longitudinal component dominates. This tendency has been confirmed in PQCD as indicated by Table \[pof\]. To serve our purpose, it is enough to evaluate only the $B\to \rho_L\rho_L$ branching ratios here. Mode Data [@HFAG] default, LO jet default, NLO jet S4, LO jet S4, NLO jet ------------------------------ ------------------------------ ----------------- ------------------ -------------- ------------- $B^\pm\to\pi^\pm\pi^0$ $ \phantom{0}5.5 \pm 0.6 $ 6.02 (6.03) 6.24 (6.28) 5.07 (5.07) 5.77 (5.87) $B^0\to\pi^\mp\pi^\pm$ $ \phantom{0}4.9 \pm 0.4 $ 8.90 (8.86) 8.69 (8.62) 5.22 (5.17) 4.68 (4.58) $B^0\to\pi^0\pi^0$ $ \phantom{0}1.45 \pm 0.29 $ 0.36 (0.35) 0.40 (0.40) 0.72 (0.70) 1.07 (1.13) $B^\pm \to \pi^\pm K^0$ $ 24.1 \pm 1.3 $ 20.50 (19.3) 20.13 21.60 (20.3) 20.50 $B^\pm \to \pi^0 K^\pm$ $ 12.1 \pm 0.8 $ 11.79 (11.1) 11.64 12.48 (11.7) 12.02 $B^0 \to \pi^\mp K^\pm$ $ 18.9 \pm 0.7 $ 17.33 (16.3) 17.21 19.60 (18.4) 19.23 $B^0\to\pi^0 K^0$ $ 11.5 \pm 1.0 $ 7.49 (7.0) 7.41 8.56 (8.0) 8.36 $B^\pm\to\rho^\pm_L\rho^0_L$ $19.1\pm 3.5$ 18.51 19.48 16.61 18.64 $B^0\to\rho^\mp_L\rho^\pm_L$ $25.2^{+3.6}_{-3.7}$ 25.36 24.42 18.48 16.76 $B^0\to\rho^0_L\rho^0_L$ $<1.1$ 0.43 0.66 0.92 1.73 : Branching ratios from QCDF with the input of the SCET jet function in units of $10^{-6}$. The values in the parentheses are quoted from [@BY05; @BN] for comparison. The data for the $B\to\rho\rho$ decays include all polarizations.[]{data-label="tab1"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Mode Data [@HFAG] default, LO jet default, NLO jet S4, LO jet S4, NLO jet ------------------------ ------------------------------ -------------------------------------- ------------------- ---------------------------------------- ----------------------- $B^\pm\to\pi^\pm\pi^0$ $ \phantom{-}1 \pm 6 $ $\ \, -0.02$ ($-0.02$) $-0.02$ $\ \, -0.02$ ($-0.02$) $\ \, -0.02$ $B^0\to\pi^\mp\pi^\pm$ $ \phantom{-}37 \pm 10$ $-6.57$ ($-6.5$) $-6.65$ $\phantom{-}10.60$ ($\phantom{-}10.3$) $\phantom{-}10.91$ $B^0\to\pi^0\pi^0$ $ \phantom{-}28^{+40}_{-39}$ $\ 44.67$ ($\ 45.1$) $\ 41.95$ $-19.58$ ($-19.0$) $-18.48$ $B^\pm\to \pi^\pm K^0$ $ -2 \pm 4$ $\phantom{-}0.84$ ($\phantom{-}0.9$) $\phantom{-}0.85$ $\phantom{-}0.29$ ($\phantom{-}0.3$) $\ \,\phantom{-}0.29$ $B^\pm\to \pi^0 K^\pm$ $ \phantom{-}4 \pm 4 $ $\phantom{-}6.88$ ($\phantom{-}7.1$) $\phantom{-}7.04$ $-3.53$ ($-3.6$) $\ \,-4.08$ $B^0\to \pi^\mp K^\pm$ $ -10.8 \pm 1.7 $ $\phantom{-}4.28$ ($\phantom{-}4.5$) $\phantom{-}4.24$ $-4.06$ ($-4.1$) $\ \,-3.89$ $B^0\to \pi^0 K^0$ $ \phantom{-}2 \pm 13 $ $-3.15$ ($-3.3$) $-3.37$ $\phantom{-}0.78$ ($\phantom{-}0.8$) $\ \,\phantom{-}1.60$ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Direct CP asymmetries from QCDF with the input of the SCET jet function in percentage. The values in the parentheses are quoted from [@BN] for comparison.[]{data-label="tab2"} The predictions for the $B\to\pi\pi$, $\pi K$, $\rho_L\rho_L$ decays from QCDF with the input of the SCET jet function are summarized in Tables \[tab1\] and \[tab2\]. The values in the parentheses are quoted from [@BY05] for the $B\to\pi\pi$ branching ratios, and from [@BN] for the $B\to\pi\pi$ direct CP asymmetries and for the $B\to\pi K$ decays. The small differences between our results and those from [@BY05; @BN] are attributed to the different choices of the CKM matrix elements, meson masses, etc. All the calculations performed in this work, except those of the $B\to\pi\pi$ branching ratios, are new. It is found that the scenario S4 plus the NLO jet function lead to the ratio $C/T\approx 0.8$, and accommodate at least the BABAR data of the $B^0\to\pi^0\pi^0$ branching ratio. Nevertheless, the same configuration overshoots the experimental upper bound of the $B^0\to\rho^0\rho^0$ branching ratio apparently, implying that the color-suppressed tree amplitude is enhanced overmuch by the NLO jet function. Adopting the default scenario, QCDF satisfies the $B^0\to\rho^0\rho^0$ bound, but the predicted $B^0\to\pi^0\pi^0$ branching ratio becomes too small. We have surveyed the other scenarios, and found the results from S1 and S3 (S2) similar to those from the default (S4). That is, it is also unlikely to accommodate the $B\to\pi\pi$, $\rho\rho$ data simultaneously in QCDF. The $B\to\pi K$ branching ratios are not affected by the NLO jet function, because the color-suppressed tree amplitude is still subleading in the penguin-dominated modes. The $B^\pm\to\rho^\pm_L\rho^0_L$ and $B^0\to\rho^\mp_L\rho^\pm_L$ branching ratios are not either, since they involve the larger color-allowed tree amplitude. Another indication against the resolution in [@BY05] is given by the direct CP asymmetries of the $B\to\pi K$ decays shown in Table \[tab2\]: both $A_{CP}(B^\pm\to \pi^0 K^\pm)$ and $A_{CP}(B^0\to \pi^\mp K^\pm)$ deviate more from the data, a consequence expected from the discussion in [@LMS05]. As explained in Sec. II, the color-suppressed tree amplitude $C'$ needs to be roughly orthogonal to $T'$ in order to have a vanishing $A_{CP}(B^\pm\to \pi^0 K^\pm)$. The NLO jet function, though increasing $C'$, does not introduce a large strong phase relative to $T'$. That is, $C'/T'$ is large, but remains almost real as in the $B\to\pi\pi$ case [@BY05]. This is exactly the same reason the $B\to\pi K$ puzzle can not be resolved in SCET [@BPS05; @WZ0610]: the leading-power SCET formalism demands a real ratio $C'/T'$, such that a large $C'/T'$ just pushes the SCET prediction for $A_{CP}(B^\pm\to \pi^0 K^\pm)$, about $-18\%$ [@BPS05], further away from the data. The direct CP asymmetry of the $B^0\to\pi^0 K^0$ decays, whose tree contribution comes only from $C'$, is sensitive to the NLO jet function as indicated in Table \[tab2\]. The direct CP asymmetries of the $B\to\pi\pi$ decays, which are tree-dominated, are relatively insensitive to the NLO jet function. DISCUSSION ========== Before concluding this work, we comment on and compare the various analyses of the FSI effects in the $B\to\pi K$, $\pi\pi$ decays. The tiny $B^0\to\pi^0\pi^0$ branching ratio obtained in perturbative calculations naturally leads to the conjecture that FSI may play an essential role. Though the estimate of FSI effects is very model-dependent, the simultaneous applications to different decay modes can still impose a constraint. The FSI effects from both the elastic and inelastic channels have been computed in the Regge model for the $B\to\pi\pi$ decays [@DLLN] and for the $B\to VV$ decays [@LLNS]. The conclusion is that FSI improves the agreement between the theoretical predictions and the experimental data, but does not suffice to resolve the $B\to\pi\pi$ puzzle: the $B^0\to\pi^0\pi^0$ branching ratio is increased by FSI only up to 0.1–0.65 [@DLLN]. Moreover, the inelastic FSI through the long-distance charming penguin was found to be negligible in the $B\to\pi\pi$ decays, though it might be important in the $B\to\pi K$ ones. The reason is that the contribution from the intermediate $D\bar D$ states is CKM suppressed in the former compared to the $D_s\bar D$ states in the latter. This observation differs from that in [@BPRS; @BPS05], where a significant charming-penguin contribution was claimed. We have pointed out in Sec. III that the large charming penguin in [@BPRS; @BPS05] is a consequence of fitting the SCET parametrization to the data. ![Contributions from inelastic FSI. []{data-label="fig1"}](nlofsi.eps) The inelastic FSI has been also evaluated as the absorptive part of charmed meson loops shown in Figs. \[fig1\](a) and \[fig1\](b) [@CCS]. The two unknown cutoff parameters, appearing in the form factors associated with the three-meson vertices, were fixed by the measured $B\to\pi K$ branching ratios. Note that these parameters should be the same for $B \to \pi K$ and $B \to \pi \pi$ in the SU(3) limit. Applying the same formalism to the latter, FSI can not resolve the $B\to\pi \pi$ puzzle, even allowing reasonable SU(3) breaking effects for the cutoff parameters. This result is understandable: the absorptive amplitudes from Figs. \[fig1\](a) and \[fig1\](b) are more or less orthogonal to the short-distance QCD penguin amplitudes in the $B\to\pi \pi$ decays, so that their effect is minor. Hence, the conclusion in [@CCS] is the same as in [@DLLN]. That is, the charming penguin is not enough to explain the observed $B\to\pi\pi$ branching ratios. Then additional dispersive amplitudes must be taken into account in [@CCS]. Those from Figs. \[fig1\](a) and \[fig1\](b), though calculable in the framework of [@CCS], were not considered. If considered, they, also contributing to the $B\to\pi K$ decays, would change the earlier predictions. Therefore, a brand new mechanism, the dispersive amplitude from the meson annihilation $D\bar D\to \pi\pi$ shown in Fig. \[fig1\](c), was introduced. There is no corresponding diagram for the $B\to\pi K$ decays. However, this amplitude is beyond the theoretical framework, i.e., it can not be expressed in terms of the Feynman rules derived in [@CCS]. The four free parameters, namely, the two cutoff parameters involved in Figs. \[fig1\](a) and \[fig1\](b), and the real and imaginary contributions from Fig. \[fig1\](c), were then determined by the four pieces of the $B\to\pi\pi$ data: the three branching ratios and the direct CP asymmetry $A_{CP}(B^0\to\pi^\mp\pi^\pm)$. That is, the $B^0\to\pi^0 \pi^0$ branching ratio has been treated as an input. The point of [@CCS] is to predict the direct CP asymmetries of the $B^0\to\pi^0 \pi^0$ and $B^\pm\to \pi^\pm\pi^0$ decays, using the parameters fixed above. The rescattering among the final states of the $B\to PP$ decays with $P=\pi$, $K$, and $\eta$ has been studied in [@CHY]. These elastic FSI effects were parameterized in terms of two strong phases, which, together with the $B\to\pi$ and $B\to K$ form factors, and the chiral enhancement scale, were then determined by a global fit to the data, including the measured $B^0\to\pi^0 \pi^0$ branching ratio. Nevertheless, the feature of the elastic FSI effects, i.e., the correlated decrease and increase of the $B^0\to\pi^\mp\pi^\pm$ and $B^0\to\pi^0 \pi^0$ branching ratios, respectively, was noticed [@CHY]. A FSI phase difference between the two $B\to\pi\pi$ isospin amplitudes with $I=0$, $1$ has been introduced in [@KP0601], which was then varied to fit the $B\to\pi\pi$ data. Therefore, no explanation for the large $B^0\to\pi^0 \pi^0$ branching ratio was provided from the viewpoint of FSI. There exist other global fits based on different parametrizations for the charmless $B$ meson decays. For example, the large ratio $C/T$ was extracted by fitting the quark-amplitude parametrization to the $B\to\pi\pi$ data [@BFRPR; @Y03; @Charng; @HM04; @CGRS; @Ligeti04; @WZ; @BHLD]. No responsible mechanism was addressed, though the largeness of $C$ was translated into the largeness of the QCD penguin with an internal $t$ quark and/or of the exchange amplitudes in [@BFRPR]. The QCDF formalism, in which the twist-3 spectator and annihilation amplitudes with the end-point singularities were parameterized as mentioned in Sec. III, has been implemented into a global fit to the data [@Du; @Alek; @CWW]. To reach a better fit, the free parameters involved in QCDF must take different values for the $B\to PP$, $PV$, $VP$ modes. These parameters have been tuned to account for the $B\to\pi\pi$ data in [@KP0601]. As emphasized before, the analysis must be also applied to other modes in order to obtain a consistent picture: the parameters preferred in [@KP0601] lead to a large real $C/T$, which is not favored by the data of the $B\to\pi K$ direct CP asymmetries as stated in Sec. III. After carefully investigating the proposals available in the literature, we have found that none of them can really resolve the $B\to\pi\pi$ puzzle. The NLO PQCD analysis has confirmed that it is unlikely to accommodate the $B\to\pi\pi$, $\rho\rho$ data simultaneously (the NLO PQCD predictions are consistent with the $B\to\rho\rho$ data). The $B\to\pi\pi$ decays have been studied in the framework of light-cone sum rules (LCSR) [@KMMM], where a small $B^0\to\pi^0\pi^0$ branching ratio was also observed. Since there is only little difference between the sum rules for the $B\to\pi\pi$ and $B\to\rho\rho$ modes, we expect that the conclusion from LCSR will be the same as from PQCD. The resolution with the input of the NLO SCET jet function into QCDF [@BY05] does not survive the constraint from the $B\to\rho\rho$ data, and renders the $B^0\to\pi^\mp K^\pm$ and $B^\pm\to\pi^0 K^\pm$ direct CP asymmetries deviate more away from the measured values. We conclude that the $B\to\rho\rho$ data have seriously constrained the possibility of resolving the $B\to\pi\pi$ puzzle in the available theoretical approaches. 1.0cm We thank H.Y. Cheng, C.K. Chua, D. Pirjol, D. Yang, and R. Zwicky for useful discussions. This work was supported by the National Science Council of R.O.C. under Grant No. NSC-94-2112-M-001-001, by the Taipei branch of the National Center for Theoretical Sciences, and by the U.S. Department of Energy under Grant No. DE-FG02-90ER40542. Transverse Helicity Amplitudes ============================== In this Appendix we present the factorization formulas for the transverse helicity amplitudes: $$\begin{aligned} F_{Ne4}(a) &=& 16 \pi C_F m_B^2 \int_0^1 dx_1 dx_2 \int_0^{\infty} b_1db_1\, b_2db_2\, \phi_B(x_1,b_1) \nonumber \\ & &\times r_3\, \bigg\{ \left[ \phi_{2}^T(\overline{x_2}) + 2 r_{2} \phi_{2}^v(\overline{x_2}) + r_{2} x_2 \left(\phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \right] E_{e}(t) h_{e}(A,B,b_1,b_2,x_2) \nonumber\\ & &\;\;\;\;\;\;\;\;\;\;\;\; + r_{2} \left(\phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) E_{e}(t') h_{e}(A',B',b_2,b_1,x_1) \bigg\}\;, \\ F_{Te4}(a) &=& 32 \pi C_F m_B^2 \int_0^1 dx_1 dx_2 \int_0^{\infty} b_1db_1\, b_2db_2\, \phi_B(x_1,b_1) \nonumber \\ & &\times r_3 \, \bigg\{ \left[ \phi_{2}^T(\overline{x_2}) - 2 r_{2} \phi_{2}^a(\overline{x_2}) - r_{2} x_2 \left(\phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \right] E_{e}(t) h_{e}(A,B,b_1,b_2,x_2) \nonumber\\ & &\;\;\;\;\;\;\;\;\;\;\;\; + r_{2} \left(\phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) E_{e}(t') h_{e}(A',B',b_2,b_1,x_1) \bigg\}\;, \\ F_{Ne6}(a) &=& F_{Te6}(a)\ =\ 0 \;, \\ F_{Na4}(a) &=& 16 \pi C_F m_B^2\, r_{2}r_{3} \int_0^1 dx_2 dx_3 \int_0^{\infty} b_2db_2\, b_3db_3\, \nonumber \\ & &\times \bigg\{ \left[ (1-x_3) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^a(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^v(\overline{x_3}) \right) + (1+x_3) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^v(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^a(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t) h_{e}(A,B,b_2,b_3,x_3) \nonumber\\ & &\;\;\;\;\;\; - \left[ (2-x_2) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^v(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^a(\overline{x_3}) \right) - x_2 \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^a(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^v(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t') h_{e}(A',B',b_3,b_2,x_2) \bigg\}\;, \nonumber\\ &=& 16 \pi C_F m_B^2\, r_{2}r_{3} \int_0^1 dx_2 dx_3 \int_0^{\infty} b_2db_2\, b_3db_3\, \nonumber \\ & &\times \bigg\{ \left[ x_3 \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) + \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t) h_{e}(A,B,b_2,b_3,x_3) \nonumber\\ & &\;\;\;\;\;\; - \left[ (1-x_2) \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) + \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t') h_{e}(A',B',b_3,b_2,x_2) \bigg\}\;, \\ F_{Ta4}(a) &=& -\, 32 \pi C_F m_B^2\, r_{2}r_{3} \int_0^1 dx_2 dx_3 \int_0^{\infty} b_2db_2\, b_3db_3\, \nonumber \\ & &\times \bigg\{ \left[ (1-x_3) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^v(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^a(\overline{x_3}) \right) + (1+x_3) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^a(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^v(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t) h_{e}(A,B,b_2,b_3,x_3) \nonumber\\ & &\;\;\;\;\;\; - \left[ (2-x_2) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^a(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^v(\overline{x_3}) \right) - x_2 \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^v(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^a(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t') h_{e}(A',B',b_3,b_2,x_2) \bigg\}\;, \nonumber\\ &=& 32 \pi C_F m_B^2\, r_{2}r_{3} \int_0^1 dx_2 dx_3 \int_0^{\infty} b_2db_2\, b_3db_3\, \nonumber \\ & &\times \bigg\{ \left[ x_3 \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) - \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t) h_{e}(A,B,b_2,b_3,x_3) \nonumber\\ & &\;\;\;\;\;\; + \left[ (1-x_2) \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) - \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E_{a}(t') h_{e}(A',B',b_3,b_2,x_2) \bigg\}\;, \\ F_{Na6}(a) &=& 32 \pi C_F m_B^2 \int_0^1 dx_2 dx_3 \int_0^{\infty} b_2db_2\, b_3db_3\, \nonumber \\ & &\times \bigg\{ r_{2} \left(\phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \phi_{3}^T(\overline{x_3}) E_{a}(t) h_{e}(A,B,b_2,b_3,x_3) \nonumber\\ & &\;\;\;\;\;\; + r_{3} \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) E_{a}(t') h_{e}(A',B',b_3,b_2,x_2) \bigg\}\;, \\ F_{Ta6}(a) &=& 2\,F_{Na6}(a)\;, \\ {\cal M}_{Ne4}(a') &=& 32 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_3 \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\, \phi_B(x_1,b_1) \nonumber \\ & &\times \bigg\{ (1-x_3) \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) E'_{e}(t) h_n(A,B,b_1,b_3) \nonumber \\ & &\;\;\;\;\;\; + \left[ x_3 \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) - 2 r_2 (x_2+x_3) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^v(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^a(\overline{x_3}) \right) \right] \nonumber\\ & &\;\;\;\;\;\;\times E'_{e}(t') h_n(A',B',b_1,b_3) \bigg\}\;, \nonumber\\ &=& 32 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_3 \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\, \phi_B(x_1,b_1) \nonumber \\ & &\times \bigg\{ (1-x_3) \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) E'_{e}(t) h_n(A,B,b_1,b_3) \nonumber \\ & &\;\;\;\;\;\; + \left[ x_3 \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) - r_2 (x_2+x_3) \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \right.\nonumber\\ & &\left.\;\;\;\;\;\;\;\;\;\;\; - r_2 (x_2+x_3) \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right] E'_{e}(t') h_n(A',B',b_1,b_3) \bigg\}\;, \\ {\cal M}_{Te4}(a') &=& 64 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_3 \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\, \phi_B(x_1,b_1) \nonumber \\ & &\times \bigg\{ (1-x_3) \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) E'_{e}(t) h_n(A,B,b_1,b_3) \nonumber \\ & &\;\;\;\;\;\; + \left[ x_3 \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) + 2 r_2 (x_2+x_3) \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^a(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^v(\overline{x_3}) \right) \right] \nonumber \\ & &\;\;\;\;\;\;\times E'_{e}(t') h_n(A',B',b_1,b_3) \bigg\}\;, \nonumber\\ &=& 64 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_3 \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\, \phi_B(x_1,b_1) \nonumber \\ & &\times \bigg\{ (1-x_3) \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) E'_{e}(t) h_n(A,B,b_1,b_3) \nonumber \\ & &\;\;\;\;\;\; + \left[ x_3 \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) + r_2 (x_2+x_3) \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \right.\nonumber\\ & &\left.\;\;\;\;\;\;\;\;\;\;\; - r_2 (x_2+x_3) \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right] E'_{e}(t') h_n(A',B',b_1,b_3) \bigg\}\;, \\ {\cal M}_{Ne5}(a') &=& - {\cal M}_{Ne4}(a') \;, \\ {\cal M}_{Te5}(a') &=& - {\cal M}_{Te4}(a') \;, \\ {\cal M}_{Ne6}(a') &=& -\, 32 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_{2} \int_0^1 dx_1dx_2dx_3\int_0^{\infty} b_1 db_1\, b_3 db_3\,\phi_B(x_1,b_1) \nonumber \\ & &\times x_2 \left(\phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \phi_{3}^T(\overline{x_3}) \left[ E'_{e}(t) h_n(A,B,b_1,b_3) + E'_{e}(t') h_n(A',B',b_1,b_3) \right]\;, \\ {\cal M}_{Te6}(a') &=& 2\, {\cal M}_{Ne6}(a')\;, \\ {\cal M}_{Na4}(a') &=& -\, 64 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_{2} r_{3} \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\,\phi_B(x_1,b_1) \nonumber \\ & &\times \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^v(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^a(\overline{x_3}) \right) E'_{a}(t') h_n(A',B',b_3,b_1)\;, \nonumber\\ &=& -\, 32 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_{2} r_{3} \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\,\phi_B(x_1,b_1) \nonumber \\ & &\times \bigg\{ \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) + \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \bigg\} \nonumber \\ & &\;\;\;\;\;\;\times E'_{a}(t') h_n(A',B',b_3,b_1)\;, \\ {\cal M}_{Ta4}(a') &=& 128 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_{2} r_{3} \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\,\phi_B(x_1,b_1) \nonumber \\ & &\times \left( \phi_{2}^v(\overline{x_2}) \phi_{3}^a(\overline{x_3}) + \phi_{2}^a(\overline{x_2}) \phi_{3}^v(\overline{x_3}) \right) E'_{a}(t') h_n(A',B',b_3,b_1)\;, \nonumber\\ &=& 64 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2\, r_{2} r_{3} \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\,\phi_B(x_1,b_1) \nonumber \\ & &\times \bigg\{ \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) - \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \bigg\} \nonumber \\ & &\;\;\;\;\;\;\times E'_{a}(t') h_n(A',B',b_3,b_1)\;, \\ {\cal M}_{Na5}(a') &=& {\cal M}_{Na4}(a')\;, \\ {\cal M}_{Ta5}(a') &=& {\cal M}_{Ta4}(a')\;, \\ {\cal M}_{Na6}(a') &=& 32 \pi C_F \frac{\sqrt{2N_c}}{N_c}m_B^2 \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1 db_1\, b_3 db_3\,\phi_B(x_1,b_1) \nonumber \\ & &\times \bigg\{ \left[ r_3 x_3 \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) - r_2 (1-x_2) \left(\phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \phi_{3}^T(\overline{x_3}) \right] \nonumber \\ & &\;\;\;\;\;\;\times E'_{a}(t) h_n(A,B,b_3,b_1) \nonumber \\ & &\;\;\;\;\;\; + \left[ r_3 (2-x_3) \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) - r_2 (1+x_2) \left(\phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \phi_{3}^T(\overline{x_3}) \right] \nonumber \\ & &\;\;\;\;\;\;\times E'_{a}(t') h_n(A',B',b_3,b_1) \bigg\}\;, \\ {\cal M}_{Ta6}(a') &=& 2\, {\cal M}_{Na6}(a')\;. $$ The quark-loop corrections ${\cal M}^{(q)}_{N,T}$ for $q=u$, $c$, and $t$, and the magnetic-penguin corrections ${\cal M}^{(g)}_{N,T}$ to the transverse components are written as $$\begin{aligned} {\cal M}^{(q)}_{N} &=& - 16 m_B^2 \frac{C_F^2}{\sqrt{2N_c}}\, r_3 \int_0^1 dx_1 dx_2 dx_3\int_0^{\infty} b_1db_1 b_2db_2 \phi_B(x_1,b_1) \nonumber \\ & & \times \big\{\left[ \phi_2^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \right. \nonumber \\ & & \;\;\;\;\;\;\;\;\;\;\; \left. + r_2 (2+x_2) \phi_{2}^v(\overline{x_2})\phi_{3}^v(\overline{x_3}) + r_2 x_2 \phi_{2}^a(\overline{x_2})\phi_{3}^v(\overline{x_3}) + 4 r_2 \phi_{2}^a(\overline{x_2})\phi_{3}^a(\overline{x_3}) \right]\nonumber \\ & & \;\;\;\;\;\; \times E^{(q)}(t_q,l^2) h_{e}(A,B,b_1,b_2,x_2) \nonumber\\ & & \;\;\;\;\;\; +\, r_2 \phi_{2}^v(\overline{x_2}) \phi_{3}^v(\overline{x_3}) E^{(q)}(t_q^{\prime},l^{\prime 2}) h_{e}(A',B',b_2,b_1,x_1) \big\} \;, \\ {\cal M}^{(q)}_{T} &=& 0 \;, \\ {\cal M}^{(g)}_{N} &=& 16 m_B^4 \frac{C_F^2}{\sqrt{2N_c}} \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1db_1\, b_2db_2\, b_3db_3\, \phi_B(x_1,b_1) \nonumber \\ & &\times\, \Big\{ \left[ - r_2 (1-x_2^2) \left(\phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \phi_{3}^T(\overline{x_3}) - r_3 (1+x_2) x_3 \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right.\nonumber\\ & &\left.\;\;\;\;\;\;\;\;\;\;\; - r_2 r_3 (1-x_2) \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \right.\nonumber\\ & &\left.\;\;\;\;\;\;\;\;\;\;\; - r_2 r_3 x_3(1-2x_2) \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right] E_{g}(t_q) h_g(A,B,C,b_1,b_2,b_3,x_2) \nonumber\\ & & \;\;\;\;\;\; - r_2 r_3 x_3 \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) E_{g}(t_q^{\prime}) h_g(A',B',C',b_2,b_1,b_3,x_1) \Big\} \;, \\ {\cal M}^{(g)}_{T} &=& 32 m_B^4 \frac{C_F^2}{\sqrt{2N_c}} \int_0^1 dx_1dx_2dx_3 \int_0^{\infty} b_1db_1\, b_2db_2\, b_3db_3\, \phi_B(x_1,b_1) \nonumber \\ & &\times\, \Big\{ \left[ - r_2 (1-x_2^2) \left(\phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \phi_{3}^T(\overline{x_3}) - r_3 (1+x_2) x_3 \phi_{2}^T(\overline{x_2}) \left(\phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right.\nonumber\\ & &\left.\;\;\;\;\;\;\;\;\;\;\; + r_2 r_3 (1-x_2) \left( \phi_{2}^v(\overline{x_2}) + \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) + \phi_{3}^a(\overline{x_3}) \right) \right.\nonumber\\ & &\left.\;\;\;\;\;\;\;\;\;\;\; - r_2 r_3 x_3(1-2x_2) \left( \phi_{2}^v(\overline{x_2}) - \phi_{2}^a(\overline{x_2}) \right) \left( \phi_{3}^v(\overline{x_3}) - \phi_{3}^a(\overline{x_3}) \right) \right] E_{g}(t_q) h_g(A,B,C,b_1,b_2,b_3,x_2) \nonumber\\ & & \;\;\;\;\;\; - r_2 \left[ (1-x_2) \phi_{2}^v(\overline{x_2})\phi_{3}^T(\overline{x_3}) - r_3 (1-2x_2) \phi_{2}^v(\overline{x_2})\phi_{3}^v(\overline{x_3}) - r_3 \phi_{2}^v(\overline{x_2})\phi_{3}^a(\overline{x_3}) \right.\nonumber\\ & &\left.\;\;\;\;\;\;\;\;\;\;\; - r_3 \phi_{2}^a(\overline{x_2})\phi_{3}^v(\overline{x_3}) + r_3 \phi_{2}^a(\overline{x_2})\phi_{3}^a(\overline{x_3}) \right] E_{g}(t_q^{\prime}) h_g(A',B',C',b_2,b_1,b_3,x_1) \Big\} \;.\end{aligned}$$ The definitions of all the variables and the convolution factors in the above expressions are referred to [@LMS05]. 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--- abstract: 'In this paper we study entire radial solutions for the quasilinear $p$-Laplace equation $\Delta_p u + k(x) f(u) = 0$ where $k$ is a radial positive weight and the nonlinearity behaves e.g. as $f(u)=u|u|^{q-2}-u|u|^{Q-2}$ with $q<Q$. In particular we focus our attention on solutions (positive and sign changing) which are infinitesimal at infinity, thus providing an extension of a previous result by Tang (2001).' author: - 'Andrea Sfecci [^1]' title: On the structure of radial solutions for some quasilinear elliptic equations --- **Key Words:** supercritical equations, radial solution, ground states, Fowler transformation, invariant manifold.\ **MR Subject Classification**: 35J70, 35J10, 37J10. Introduction ============ In this paper we are going to discuss the structure of radial solutions of the following quasilinear elliptic equation $$\label{plap} \Delta_p u(\x) + k(|\x|) f(u(\x)) = 0\,,$$ where $\Delta_p u = \operatorname{div}( |\nabla u|^{p-2} \nabla u)$ is the so-called $p$-Laplace operator, $\x\in\RR^n$ with $n>p>1$, $k:(0,+\infty) \to \RR$ and $f:\RR\to\RR$ are $C^1$-functions. Therefore, we will consider the following ordinary differential equation $$\label{plap.rad} (r^{n-1} u'|u'|^{p-2})' + r^{n-1} k(r) f(u) = 0\,, $$ where, with a little abuse of notation, we have set $u(r)=u(\x)$, with $r=|\x|$, and $'$ denotes the derivative with respect to $r$. We assume the following hypotheses on the function $f$: $$\label{howisf} \tag{\bf F} f(u)=u|u|^{q-2} b(u)\,, \text{ with } \begin{cases} q>2\,, b\in C^1(\RR)\,, \\ b(u)>0 \text{ in } (-d^-,d^+)\,, & d^\pm>0\,, \\ b(-d^-)=b(d^+)=0\,. \end{cases}$$ Notice that if $f(u)=u|u|^{q-2}-u|u|^{Q-2}$, with $Q>q>2$, then is fulfilled. The following conditions on the weight $k$ are required: $$\label{howisk} \tag{\bf K} k(r)= h(r) r^\delta>0 \,, \text{ with } \begin{cases} h>0\,, \delta>-p\,, \\ h_0 := \lim_{r\to 0} h(r) \in (0,+\infty)\,, \\ h_\infty := \lim_{r\to \infty} h(r) \in (0,+\infty) \,, \\ \limsup_{r\to 0} h'(r)r <+\infty\,, \\ \lim_{r\to \infty} h'(r) r^{1+\varpi} = 0\,, \quad \varpi>0\,. \end{cases}$$ The structure of radial solutions of in the case of power-type nonlinearities $f(u)=u|u|^{q-2}$ and $k\equiv 1$ is strictly related to the following constants $$p_*=p \frac{n-1}{n-p}\,,\qquad p^*=\frac{np}{n-p}\,,$$ which are respectively known as the Serrin and the Sobolev critical exponents. Such values change when we consider a non-constant weight of the type $k(r)=r^\delta$, i.e. in the case of Hénon equation (see e.g. [@BSS; @BW; @NiHenon]). In this paper we will discuss the existence of solutions of vanishing at infinity. In particular we classify solutions in two classes depending on their behaviour at infinity: [*fast decay solutions*]{} which satisfy $\lim_{r\to\infty} u'(r)r^{\frac{n-1}{p-1}}=\widetilde L\in \RR$, and [*slow decay solutions*]{} satisfying $\lim_{r\to\infty} u'(r) r^{\frac{n-1}{p-1}} =\infty$. Notice that, setting $L=-\widetilde L \frac{n-p}{p-1}$, the formers satisfy also $\lim_{r\to\infty} u(r) r^{\frac{n-p}{p-1}} =L\in \RR$. We will denote fast decay solutions by $v(r,L)$. Moreover, we call [*regular solutions*]{} of , the ones satisfying $u(0)=d\in\RR$, and we denote them by $u(r,d)$. The problem of existence of radial solutions for equation presents a wide literature and different approaches. We address the interested reader to the following papers and the references therein. In [@AlaQua; @DolFlo; @FQT; @KYY; @Tang00; @Tang01], different situations with $f(u)=u|u|^{q-2}$ are considered, while the case of a [*sign-changing*]{} nonlinearity as $f(u)=u|u|^{Q-2}-u|u|^{q-2}$, with $Q>q$, is treated in [@AccPuc; @CDGHM; @CGHH2015; @CGHY2013; @SerTang]. In [@DaDu; @KMPT; @Tang00; @Tang01] nonlinearities of the type $f(u)=u|u|^{q-2}-u|u|^{Q-2}$, with $Q>q$, are considered. The present paper will focus on this kind of nonlinearities providing a generalizations of [@Tang01]. We will enter in such details below. If we introduce the following change of variable, which reminds a Fowler-type transformation borrowed from [@BidVer] (see also [@BaFloPino2000POIN; @Fow31; @F2010JDE]), $$\label{FowT} \begin{cases} x_l(t)=u(r)r^{\alpha_l}\\ y_l(t)=u'(r)|u'(r)|^{p-2}r^{\beta_l}\\ \end{cases} \qquad r=e^t$$ with $l>p_*$, where $$\alpha_l=\frac{p}{l-p}>0, \qquad \beta_l = (\alpha_l+1)(p-1), \qquad \gamma_l =\beta_l - (n-1)<0,$$ then equation can be written in the form of a dynamical system which is not anymore singular: $$\label{sist} \tag{${\rm S}_l$} \left( \begin{array}{c} \dot{x_l} \\\dot{y_l} \end{array}\right) = \left( \begin{array}{cc} \alpha_l & 0 \\ 0 & \gamma_l \end{array} \right) \left( \begin{array}{c} x_l \\ y_l \end{array}\right) +\left( \begin{array}{c} y_l |y_l|^{\frac{2-p}{p-1}} \\ - g_l(x_l,t) \end{array}\right)$$ where $g_l(x_l,t)=k(\eu^t) \, f(x_l\eu^{-\alpha_l t}) \, \eu^{\alpha_l(l-1)t}$. In order to ensure uniqueness of the solutions, we will assume in the whole paper $p\in(1,2]$. Such a restriction is just technical, we assume it to avoid cumbersome technicalities (cf. [@F2009AMPA; @F2010JDE]). We will see how the research of regular fast decay solutions of corresponds to the research of homoclinic trajectories of system , see Remark \[corrisp\] below. Here is the main result of this paper. In the statement, we present the result for regular solutions which are positive near zero (+) and the symmetric situation for solutions which are negative near zero (–). \[main\] Consider the differential equation . Assume , . If $l:=p\frac{q+\delta}{p+\delta}>p^*$, then (+) : there exists an increasing sequence $(A_k)_{k\geq0}$ of positive numbers, such that $u(r,A_k)$ is a regular fast decay solution with $k$ non degenerate zeros. In particular $u(r,A_0)$ is a regular positive fast decay solution. Moreover, $u(r,d)$ is a regular positive slow decay solution for any $0<d<A_0$, and there is $A^*_k \in [A_{k-1},A_k)$ such that $u(r,d)$ is a regular slow decay solution with $k$ nondegenerate zeros whenever $A^*_k<d<A_{k}$, for any $k \ge 1$. (–) : there exists an increasing sequence $(B_k)_{k\geq0}$ of positive numbers, such that $u(r,-B_k)$ is a regular fast decay solution with $k$ non degenerate zeros. In particular $u(r,-B_0)$ is a regular negative fast decay solution. Moreover, $u(r,-d)$ is a regular negative slow decay solution for any $0<d<B_0$, and there is $B^*_k \in [B_{k-1},B_k)$ such that $u(r,-d)$ is a regular slow decay solution with $k$ nondegenerate zeros whenever $B^*_k<d<B_{k}$, for any $k \ge 1$. Our main theorem partially extends a result obtained by Tang in [@Tang01 Theorem 1], where the author considers equation $\Delta_p u + f(u) = 0$, i.e. with $k(r)\equiv 1$. There, existence of radial ground states is obtained assuming $f(0)=0$, $f(u)>0$ in an interval $(0,d^+)$ and $\Phi(u)=u f'(u)/f(u)$ non-increasing in the interval of positivity of $f$. The solutions provided by Tang in [@Tang01] correspond to regular solutions $u(r,d)$ with $d\in(0,A_0]$ in the statement of our main theorem above. In this paper we extend the discussion to nodal solutions, also introducing the weight $k$. Notice that we do not require the monotonicity assumption on the function $\Phi$, so that we can consider, e.g., the case $f(u)= u^{q_1}+u^{q_2}-u^{q_3}$ with $q_1<q_2<q_3$, but we can have $A_1^*>A_0$ in the statement of Theorem \[main\], which means that we can loose the uniqueness of the positive fast decay solution ensured by the assumption on $\Phi$, cf. [@Tang01 Theorem 2]. We underline that, in order to prove our result, the adopted techniques are completely different. As a final remark, we recall that the case of equation , where and hold with $\delta=0$, has been investigated in [@DaDu] in the case $l=q<p^*$. Notice that, in Theorem \[main\] we can consider also an [*apparently subcritical*]{} situation with $q<p^*$ and $\delta<0$ such that $l=p\frac{q+\delta}{p+\delta}>p^*$. The paper is organized as follows. In the next section we are going to introduce the main tools needed in order to prove our main theorem. The proof is based on the study of the invariant manifolds associated to the saddle-type equilibrium $(x,y)=(0,0)$ in . In particular, in Section \[sub21\] we draw the phase portrait of some systems which are autonomous, then in Section \[sub22\], using invariant manifold theory, we provide the needed background in the non-autonomous case. Section \[sec3\] contains the proof of the main theorem. The proof is divided in five steps: in Section \[step1\] we introduce a truncated problem, then in Sections \[step2\] and \[step3\] we prove respectively the existence of fast and slow decay solutions for this problem; in Section \[step4\] we provide the estimates on the number of zeroes of such solutions, finally in Section \[step5\] we prove that the solutions we have found are indeed solutions of the original problem. Introduction of invariant manifolds =================================== In the following subsections we will introduce the main tools we need in order to prove our main result. In this paper, we will denote by $\bs{x_l}(t,\tau,\Q)=(x_l(t,\tau,\Q),y_l(t,\tau,\Q))$ the trajectory of which is in $\Q$ for $t=\tau$, i.e. such that $\bs{x_l}(\tau,\tau,\Q)=\Q$. Moreover, it is easy to verify that applying with different values $l\neq L$ we get the following identities $$\begin{aligned} \label{xyll} &&\bs{x_L}(t)= \bs{x_l}(t) \eu^{(\alpha_{L}-\alpha_l)t}\,, \label{gll} \\ &&g_{L}(x,t) = g_l(x \eu^{-(\alpha_{L}-\alpha_l)t},t) \eu^{[\alpha_{L}(L-1)-\alpha_l(l-1)]t}\,, \\ \label{Gll} &&G_L(x,t)=G_l(x \eu^{-(\alpha_{L}-\alpha_l)t},t) \eu^{p(\alpha_L-\alpha_l)t}\,.\end{aligned}$$ where $G_{l}(x,t) = \int_0^x g_{l}(\xi,t)\, d\xi$. The autonomous superlinear case {#sub21} ------------------------------- In this section we focus our attention on systems which are autonomous, i.e. such that $g_l(x,t)=g_l(x)$. In particular we recall some of the results contained in [@F2010JDE]. We refer also to [@DalFra; @FraSfe1] for the picture in the classical case $p=2$. We assume the following hypotheses on the nonlinearity $g_l$. : There is $l>p_*$ such that $g_l(x,t) \equiv g_l(x)$ is $t$-independent, satisfying $$\lim_{x \to 0} \frac{g_l(x)}{x} = 0 \qquad \text{and}\qquad \lim_{|x| \to +\infty} \frac{g_l(x)}{x}=+\infty\,.$$ Moreover $g_l(x)/|x|^{p-1}$ is an increasing function, positive for $x>0$. It can be verified, with a short computation, that if $k(r)= k_0 r^\delta$, with $k_0\in\RR$ and $f(u)=u|u|^{q-2}$ then, setting $l=p \, \frac{q+\delta}{p+\delta}$, we obtain $g_l(x,t)=x|x|^{q-2}$ which satisfies . In the case of a constant function $k(r)=k_0$ we get $l=q$. Assume and fix the corresponding $l>p_*$ in . The origin $\O=(0,0)$ is a saddle and admits a $1$-dimensional unstable manifold $M^u$ and a $1$-dimensional stable manifold $M^s$. Moreover, by , which implicitly gives $$\lim_{x \to 0} \frac{g_l(x)}{x|x|^{p-2}} = 0 \qquad \text{and}\qquad \lim_{|x| \to +\infty} \frac{g_l(x)}{x|x|^{p-2}}=+\infty\,,$$ we have two non-trivial critical points $\boldsymbol{P^+} =(P_x^+,P_y^+)$ with $P_x^+>0>P_y^+$ and $\boldsymbol{P^-} =(P_x^-,P_y^-)$ with $P_x^-<0<P_y^-$. In particular $P_x^\pm$ solves $g_l(x)=x|x|^{p-2} \alpha_l^{p-1} |\gamma_l|$ and $P_y^\pm = \mp \left(\alpha_l |P_x^\pm|\right)^{p-1} $. They are stable for $l>p^*$, centers for $l=p^*$ and unstable for $p_*<l<p^*$. In particular, $$\label{alfagamma} \alpha_l + \gamma_l \quad \begin{cases} > 0 & \text{ if } p_*<l<p^*\,, \\ = 0 & \text{ if } l=p^*\,, \\ < 0 & \text{ if } l>p^*\,. \end{cases}$$ We define the following energy function $$H_{l}(x,y) = \frac{n-p}{p}\, xy + \frac{p-1}{p}\, |y|^{\frac{p}{p-1}} + G_{l}(t,x)\,,$$ which is strictly related to the Pohozaev function $$\mathcal{P}(u,u',r) = r^n \left[ \frac{n-p}{p} \, \frac{u\, u' \, |u'|^{p-2}}{r} + \frac{p-1}{p} \, |u'|^p + F(u,r)\right] ,$$ where $F(u,r)= \int_0^u f(\upsilon,r) \,d\upsilon$. In fact we have $$\label{Po-rel} \mathcal{P}(u(r),u'(r),r) = H_{p^*} ( x_{p^*}(t),y_{p^*}(t),t)= H_l(x_l(t), y_l(t), t) \eu^{-(\alpha_l+\gamma_l)t} \,.$$ A computation gives $$\frac{d}{dt}H_{p^*}(x_{p^*}(t),y_{p^*}(t)) = \frac{\partial}{\partial t} G_{p^*}(t,x_{p^*}(t))\,,$$ If holds with $l=p^*$, then $G_{p^*}$ is independent of $t$, so that (${\rm S}_{p^*}$) is a Hamiltonian system and the unstable and the stable manifolds coincide: we have the existence of two homoclinc trajectories (see Figure \[figHham\] for the phase portrait in this case). Moreover we can compute $H_{p^*}(\O)=0$ and $H_{p^*}(\bs{P^\pm})<0$. Now, let us assume with $l\neq p^*$. By , we obtain $$\begin{gathered} \label{Hl-est} \frac{d}{dt} H_l(x_l(t), y_l(t)) = \frac{d}{dt} \left[ \eu^{(\alpha_l+\gamma_l)t} \, H_{p^*}(x_{p^*}(t),y_{p^*}(t)) \right]\\ = (\alpha_l+\gamma_l) H_l(x_l(t), y_l(t), t) + \eu^{(\alpha_l+\gamma_l)t} \frac{\partial}{\partial t} \, G_{p^*}(t,x_{p^*}(t)) \,.\end{gathered}$$ We stress that, in this case, (${\rm S}_{p^*}$) is non-autonomous. We introduce the function $\mathcal G_l(x)= G_l(x)/x|x|^{p-1}$. Since $g_l(x)/|x|^{p-1}$ is increasing, then $\mathcal G_l$ is increasing too: indeed, one has $\mathcal G_l'(x)=\frac{1}{|x|^{p+1}}\left[ x g_l(x) - p\, G_l(x) \right]$ which is non-negative, since $$G_l(x) = \int_0^x \frac{g_l(s)}{|s|^{p-1}} |s|^{p-1} \,ds \leq \frac{g_l(x)}{|x|^{p-1}} \int_0^x |s|^{p-1}\,ds = \frac{x \,g_l(x)}{p}\,.$$ By , we obtain $$\begin{aligned} \frac{\partial}{\partial t} G_{p^*}(t,x) &=& \frac{\partial}{\partial t} \left( G_l(x \eu^{-(\alpha_{p^*}-\alpha_l)t}) \eu^{p(\alpha_{p^*}-\alpha_l)t} \right) \\ &=& \frac{\partial}{\partial t} \left(\mathcal G_l(x \eu^{-(\alpha_{p^*}-\alpha_l)t}) x|x|^{p-1} \right) \,,\end{aligned}$$ which has the sign of $\alpha_l -\alpha_{p^*}$: $$\alpha_l -\alpha_{p^*} \ \begin{cases} > 0 & \text{ if } p_*<l<p^*\,, \\ < 0 & \text{ if } l>p^*\,. \end{cases}$$ Hence, by and , $$\frac{d}{dt} H_l(\xl(t)) \ \begin{cases} > 0 & \text{ if } p_*<l<p^*\, \\ < 0 & \text{ if } l>p^* \end{cases} \qquad \text{ when } H_l(\xl(t))\geq 0 \,.$$ Let us consider the case $l>p^*$ and fix $\Q\in\RR^2\setminus\{{\bs O}\}$. Suppose that $\lim_{t\to-\infty} \xl(t,0,\Q)=\O$ (in particular $\lim_{t\to-\infty} H_{l}(\xl(t,0,\Q))=0$) then, by the previous computation, $H_{l}(\xl(t,0,\Q))<0$ for every $t\in\RR$ so that $\lim_{t\to+\infty} \xl(t,0,\Q)=\bs{P^\pm}$. Conversely, if $\lim_{t\to+\infty} \xl(t,0,\Q)=\O$, since $\alpha_l+\gamma_l\neq 0$, thanks to the Poincaré-Bendixson criterion, $\xl$ is not a homoclinic and there are not heteroclinic cycles. Therefore, $\lim_{t\to-\infty} |\xl(t,0,\Q)|=+\infty$. Arguing similarly, in the case $p_*<l<p^*$, if $\lim_{t\to+\infty} \xl(t,0,\Q)=\O$ then $\lim_{t\to-\infty} \xl(t,0,\Q)=\bs{P^\pm}$, while if we assume $\lim_{t\to-\infty} \xl(t,0,\Q)=\O$ then $\lim_{t\to+\infty} |\xl(t,0,\Q)|=+\infty$. Finally, introducing polar coordinates, it is immediately verified that the angular velocity of the solutions is unbounded as $|\xl|\to+\infty$, so that, if $|\xl(t,0,\Q)| \to +\infty$ then the trajectory draws an infinite number of rotations around the origin having the shape of a spiral. Hence we can draw the stable and unstable manifolds for the autonomous system , when is satisfied with $l\neq p^*$, as in Figure \[fignoncrit\]. In particular, when $l>p^*$, the stable manifold has the shape of an unbounded double spiral. The next remark underlines the correspondence between solutions of  converging to the origin and solutions of . \[corrisp\] Assume . Consider the trajectory $\xl(t,\tau,\Q)$ of  and let $u(r)$ be the corresponding solution of ; then $u(r)$ is a regular solution if and only if $\Q \in M^u$, while it has fast decay if and only if $\Q \in M^s$. Such a result can be proved by standard arguments of invariant manifold theory, see e.g. [@DalFra; @FraSfe1; @FraSfe2]. In fact $x_l(t) \propto \eu^{\alpha_l t}$, for $t\sim -\infty$, implies $u(r) \propto 1$ for $r\sim 0$ and $y_l(t) \propto \eu^{\gamma_l t}$, for $t\sim +\infty$, implies $u'(r) \propto r^{-\frac{n-1}{p-1}}$ for $r\sim \infty$. Moreover, the next remark gives the corresponding result for the non-trivial critical points $\bs{P^\pm}$. \[corrP\] Assume , with $l\neq p^*$. Consider the trajectory $\xl(t,\tau,\Q)$ of  and let $u(r)$ be the corresponding solution of . Then, $u(r)$ is a slow decay solution if and only if $\lim_{t\to+\infty} \xl(t,\tau,\Q) = \bs{P^\pm}$. Analogously, $u(r)$ is a singular solution if and only if $\lim_{t\to-\infty} \xl(t,\tau,\Q) = \bs{P^\pm}$. The non-autonomous case {#sub22} ----------------------- In this section we provide the construction of invariant manifolds in the non-autonomous case. The contents of this section collect only the hypotheses we need in the present paper, we refer to [@F2013DIE; @FraSfe2] for an overview on this topic. In order to simplify the exposition we will treat the case $p\neq 2$ without further mentioning. The approach for the classical case $p=2$ gives a slightly different picture of the phase portrait since, in system , the term $y|y|^{\frac{2-p}{p-1}}=y$ is linear when $p=2$ (cf. [@DalFra; @F2013DIE; @FraSfe1]). The following hypotheses on the function $g_l$ permit us to introduce stable manifolds for non-autonomous systems . : Assume that there is $l>p_*$ such that $$\begin{aligned} && g_{l}(0,t)=\partial_x g_{l}(0,t) = 0\,, \qquad \text{ for any $t \in \RR$,}\\ && \ds \lim_{t\to +\infty} g_{l}(x,t) = g^{+\infty}_{l}(x) \quad\text{and}\quad \lim_{t\to +\infty} \eu^{\varpi t} \, {\partial_t} \, g_{l}(x,t)=0 \,,\end{aligned}$$ uniformly on compact sets, where the function $g^{+\infty}_{l}$ is a non-trivial $C^1$ function satisfying and $\varpi$ is a suitable positive constant. Introducing a [*time-type*]{} variable $z= \eu^{-\varpi t}$, we obtain a $3$-dimensional autonomous system: $$\label{sist.na} \left( \begin{array}{c} \dot{x}_{l} \\ \dot{y}_{l} \\ \dot{z} \end{array}\right) = \left( \begin{array}{ccc} \alpha_{l} & 0 &0 \\ 0 & \gamma_{l} & 0 \\ 0 & 0 & -\varpi \end{array} \right) \left( \begin{array}{c} x_{l} \\ y_{l} \\ z \end{array}\right) +\left( \begin{array}{c} y_l|y_l|^{\frac{2-p}{p-1}} \\- g_{l}(x_{l},-\ln(z)/\varpi)\\ 0\end{array}\right) .$$ We have thus obtained an autonomous system in $\RR^3$ such that all its trajectories converge to the $z=0$ plane as $t \to +\infty$. Hence,  is useful to investigate the asymptotic behavior of the solutions of in the future. Assume . The origin admits a $2$-dimensional stable manifold: we denote it by $\boldsymbol{W^s_l}$. From standard arguments of dynamical system theory, we see that the set $W^s_{l}(\tau)=\boldsymbol{W^s_l} \cap \{ z= \eu^{-\varpi \tau} \}$ is a curve, for any $\tau \in \RR$, see e.g. [@BaFloPino2000POIN; @F2013DIE; @FraSfe2; @JPY]. Let us denote by $W^s_l(+\infty)$ the stable manifold $M^s$ of the autonomous system where $g_l(x,t) \equiv g_l^{+\infty}(x)$. Then we have the following, cf. [@F2013DIE; @JPY]. \[allinfinitoWS\] Assume ; then $W^s_{l}(\tau)$ approaches $W^s_{l}(+\infty)$ as $\tau \to +\infty$. More precisely, if $W^s_{l}(\tau_0)$ intersects transversally a certain line $L$ in a point $\Q(\tau_0)$ for $\tau_0 \in(-\infty,+\infty]$, then there is a neighborhood $I$ of $\tau_0$ such that $W^s_{l}(\tau)$ intersects $L$ in a point $\Q(\tau)$ for any $\tau \in I$, and $\Q(\tau)$ is continuous (in particular it is as smooth as $g_l$). The proof is a consequence of standard facts in dynamical system theory (see e.g. [@CodLev §13] or [@JSell]). As a consequence we have the following characterization. $$\label{Ws} W^s_{l}(\tau):= \left\{ \Q \in \RR^2 \mid \lim_{t\to +\infty} \boldsymbol{x_{l}}(t,\tau, \Q)=(0,0) \right\} \,.$$ Arguing similarly it is possible to introduce unstable manifolds. For our purposes, we need to require a different behaviour for $g_l$ as $t\to-\infty$. : Assume that there is $l>p_*$ such that $$\begin{aligned} && g_{l}(0,t)=\partial_x g_{l}(0,t) = 0\,, \qquad \text{ for any $t \in \RR$,}\\ && \ds \lim_{t\to -\infty} g_{l}(x,t) = 0 \quad\text{and}\quad \lim_{t\to-\infty} \eu^{-\varpi t} \, {\partial_t} \, g_{l}(x,t)=0 \,,\end{aligned}$$ uniformly on compact sets, where $\varpi$ is a suitable positive constant. Arguing as above, denoting by $W^u_l(-\infty)$ the unstable manifold $M^u$ of the autonomous system where $g_l(x,t) \equiv 0$, we have the corresponding properties for the unstable manifold $\boldsymbol{W^u_l}$ and the curves $W^u_l(\tau) =\boldsymbol{W^u_l}\cap \{z=\eu^{\varpi\tau}\}$. Notice that, in this case, $M^u$ consists of the $x$ axis. \[allinfinitoWU\] Assume ; then $W^u_{l}(\tau)$ approaches $W^u_{l}(-\infty)$, i.e. the $x$ axis, as $\tau \to -\infty$. More precisely, if $W^u_{l}(\tau_0)$ intersects transversally a certain line $L$ in a point $\Q(\tau_0)$ for $\tau_0 \in[-\infty,+\infty)$, then there is a neighborhood $I$ of $\tau_0$ such that $W^u_{l}(\tau)$ intersects $L$ in a point $\Q(\tau)$ for any $\tau \in I$, and $\Q(\tau)$ is continuous (in particular it is as smooth as $g_l$). As above, we can characterize the curves $W^u_l(\tau)$ as follows: $$\label{Wu} W^u_{l}(\tau):= \left\{ \Q \in \RR^2 \mid \lim_{t\to-\infty} \boldsymbol{x_{l}}(t,\tau, \Q)=(0,0) \right\} \,.\\ $$ Let us list some properties of the manifolds we have introduced. See [@F2010JDE; @F2013DIE; @FraSfe2] for more details. As in the autonomous case, we have the following correspondence between solutions of and . \[corr-non-aut\] Assume $l>p_*$. If $u(r,d)$ is a regular solution of , then the corresponding trajectory $\xl(t)$ of satisfies $\xl(t)\in W^u_l(t)$ for every $t\in\RR$. Correspondingly, if $v(r,L)$ is a fast decay solution, then the corresponding trajectory satisfies $\xl(t)\in W^s_l(t)$ for every $t\in\RR$. Hence $\xl(t,\tau,\Q)$ is a regular fast decay solution of if and only if $\Q\in W^u_l(\tau)\cap W^s_l(\tau)$. Moreover, if a trajectory $\xl(t)$ of satisfies $\lim_{t\to+\infty} \xl(t) = \bs{P^\pm}$, then the corresponding solution $u(r)$ of is a slow decay solution. The set $W^u_l(\tau)$ is tangent to the $x$-axis at the origin, while $W^s_l(\tau)$ is tangent to the $y$-axis at the origin (notice that, in the classical case $p=2$, the latter is tangent to the line $y+(n-2)x=0$). The set $W^u_l(\tau)$ is split by the origin into two connected components, we will denote by $W^{u,+}_l(\tau)$ the one which leaves the origin and enters the $x>0$ semi-plane (corresponding to regular solutions which are positive for $r$ small), and by $W^{u,-}_l(\tau)$ the other which enters the $x<0$ semi-plane (corresponding to regular solutions which are negative for $r$ small). Similarly $W^{s}_{l}(\tau)$ is split by the origin into $W^{s,+}_{l}(\tau)$ and $W^{s,-}_{l}(\tau)$, which leave the origin and enter respectively in $x> 0$ and in $x< 0$ (corresponding to fast decay solutions which are definitively positive and definitively negative respectively). Following e.g. [@FraSfe2 Lemma 4.2], we introduce now some parametrizations of the manifolds. Fix $\tau\in\RR$ and consider the branch $W^{u,+}_l(\tau)$. For every $d>0$, there exists a point $\Q(d,\tau)\in W^{u,+}_l(\tau)$ corresponding to the regular solution $u(r,d)$ at $r=\eu^\tau$, i.e. $\Q(d,\tau)=(u(r,d)r^{\alpha_l},u'(r,d)|u'(r,d)|^{p-2} r^{\beta_l})$. Hence, we can find a parametrization $\Sigma_l^{u,+}(\cdot, \tau): (0,+\infty) \to \RR^2$ such that $\Sigma_l^{u,+}(d, \tau)=\Q(d,\tau)\in W^{u,+}_l(\tau)$. In particular, we have by construction that $\Sigma_l^{u,+}: (0,+\infty)\times \RR \to \RR^2$ is continuous. Similarly, we can introduce a continuous parametrization of $W^{s,+}_l(\tau)$, through the parameter $L$ associated to every fast decay solution $v(r,L)$, thus obtaining $\Sigma_l^{s,+}: (0,+\infty)\times \RR \to \RR^2$ such that $\Sigma_l^{s,+}(L,\tau)\in W^{s,+}_l(\tau)$ for every $L>0$ and $\tau\in\RR$. Again, $\Sigma_l^{u,-}$ and $\Sigma_l^{s,-}$ parametrize $W^{u,-}_l(\cdot)$ and $W^{s,-}_l(\cdot)$ respectively, considering the regular solutions $u(r,-d)$ for every $d>0$ and the fast decay solutions $v(r,-L)$ for every $L>0$. Moreover, we will consider the polar coordinates, associated to the sets $W^{u,\pm}_l(\tau)$ and $W^{s,\pm}_l(\tau)$ as follows: $$\label{sigmap} \begin{array}{lll} \Sigma^{u,\pm}_{l}(d, \tau)& \!\!\!\!= \rho^{u,\pm}_l(d, \tau) \big( \cos( \theta^{u,\pm}_l(d, \tau)), \sin( \theta^{u,\pm}_l(d, \tau))\big)\,, & d>0,\, \tau\in \RR,\\ \Sigma^{s,\pm}_{l}(L, \tau)& \!\!\!\!= \rho^{s,\pm}_l(L, \tau) \big( \cos( \theta^{s,\pm}_l(L, \tau)), \sin( \theta^{s,\pm}_l(L, \tau))\big) \,, & L>0,\, \tau\in \RR. \end{array}$$ In particular we set for definiteness $$\label{starting} \begin{array}{ll} \theta^{u,+}_l(0, \tau)=0 \,, & \theta^{u,-}_l(0, \tau)=-\pi \,, \\ \theta^{s,+}_l(0, \tau)=-\pi/2 \,, & \theta^{s,-}_l(0, \tau)=-3\pi/2 \,. \end{array}$$ Similarly we introduce polar coordinates associated to the solutions of : $$\label{xlpolar} \xl(t,\tau,\Q) = \rho_l(t, \tau,\Q) \big( \cos( \phi_l(t, \tau,\Q)), \sin( \phi_l (t, \tau,\Q))\big)\,.$$ Proof of the main result {#sec3} ======================== The proof is divided in five parts. At first we introduce a truncation of the nonlinearity $f$, and we prove the theorem for the truncated problem, introducing invariant manifolds and studying their shape. We show the existence of regular fast decay solutions looking for intersections between the unstable manifold and the stable one. The existence of regular slow decay solutions follows by topological arguments. Then, we discuss their nodal properties. Finally, we prove that all the solutions of the truncated equation are solutions of the original one, too. The truncated problem {#step1} --------------------- In order to ensure the continuability of the solutions of we introduce the following truncation of the nonlinearity $f$: $$\label{trunc} \bar f(u) = \begin{cases} f(u) & u\in (-d^-,d^+)\,, \\ 0 & u\in (-\infty,-d^--1)\cup(d^++1,+\infty) \,,\\ & \text{smooth and non-positive elsewhere}\,. \end{cases}$$ We will prove our main theorem for the truncated nonlinearity $\bar f$. Then, providing some a priori estimates, we will show that such solutions solve the original equation , too. Without loss of generality we assume $$\label{maxf} \max_{\RR}|\bar f(u)| = f_\infty := \max_{u\in[-d^-,d^+]}|f(u)|\,,$$ When we consider the truncation of $f$ introduced in , applying the Fowler transformation with $l=p\frac{q+\delta}{p+\delta}>p^*$, we obtain system with $$\label{truncg} g_l(x,t)= \begin{cases} 0 & \text{ if } x\eu^{-\alpha_l t}\leq -d^- \!-\!1 \,, \\ k(\eu^t) \, \bar f(x\eu^{-\alpha_l t}) \, \eu^{\alpha_l(l-1)t} & \text{ if } -d^- \!-\!1<x\eu^{-\alpha_l t}<d^+ \!+\!1 \,,\\ 0 & \text{ if } x\eu^{-\alpha_l t}\geq d^+ +1\,, \end{cases}$$ Assume the hypotheses of Theorem \[main\]. The function $g_l$ in satisfies both and with $l=p\frac{q+\delta}{p+\delta}$. Clearly, $g_l(0,t)=\partial_x g_l(0,t)=0$ for every $t\in\RR$. Let us first prove that holds. Concerning the behavior of $g_l$ as $t\to -\infty$, we can find a constant $C>0$ such that $$g_l(x,t) \leq C \eu^{\alpha_l(q-1)t} \quad \text{and} \quad |\partial_t g_l(x,t)| \leq C \eu^{\alpha_l(q-2)t} \,, $$for every $t\in\RR$ and $x\in\RR$. Indeed $$|g_l(x,t)| = h(\eu^t) \eu^{\delta t} \bar f(x \eu^{-\alpha_l t}) \eu^{\alpha_l(l-1)t} = h(\eu^t) \bar f(x \eu^{-\alpha_l t}) \eu^{\alpha_l(q-1)t}$$ and $$\begin{gathered} \partial_t g_l(x,t) = [h'(\eu^t)\eu^t +\alpha_l(q-1) h(\eu^t)] \bar f(x \eu^{-\alpha_l t}) \eu^{\alpha_l(q-1)t} \\ + h(\eu^t) x\bar f'(x \eu^{-\alpha_l t})\eu^{\alpha_l(q-2)t}\end{gathered}$$ hold when $g_l(x,t) \neq 0$. Hence, using , we see that holds and we get the existence of the unstable manifold $W^u_l(\tau)$ for every $\tau\in\RR$. In order to prove the validity of , let us now consider the limit as $t\to +\infty$. We have $$g_l(x,t) = h(\eu^t) x|x|^{q-2} \bar b(x \eu^{-\alpha_l t})\,, \qquad \text{when } g_l(x,t)\neq 0\,,$$ where $\bar f(u) = u|u|^{q-2} \bar b(u)$. One has $\lim_{t\to+\infty} g_l(x,t) = h_\infty x|x|^{q-2} b(0)$, uniformly on compact set. Moreover, $$\begin{gathered} \eu^{\varpi t} \partial_t g_l(x,t) = h'(\eu^t)\eu^{(1+\varpi)t} x|x|^{q-2} b(x \eu^{-\alpha_l t}) \\ - \alpha_l h(\eu^t) x|x|^{q-2} b'(x \eu^{-\alpha_l t}) x \eu^{(\varpi-\alpha_l)t}\,, \qquad \text{when } g_l(x,t)\neq 0\end{gathered}$$ where $\varpi$ is given by , suitably reduced in order to guarantee that $\varpi<\alpha_l$. Then follows. Being $f(-d^-)=f(d^+)=0$, we have the existence of the constant solutions $u\equiv d^+$ and $u\equiv -d^-$ which correspond respectively to the trajectories $\Eqp(t)=\xl(t,0,\bs{S_0^+})=(d^+ \, \eu^{\alpha_l t},0)$ and $\Eqm(t)=\xl(t,0,\bs{S_0^-})=(-d^- \, \eu^{\alpha_l t},0)$. In particular $\Eqp(t)\in W^{u,+}_l(t)$ and $\Eqm(t)\in W^{u,-}_l(t)$ for every $t\in \RR$. By , we have $$\label{unpogiu} \begin{array}{l} u'(r,d)<0 \text{ for $r$ small if } d\in(0,d^+)\,,\\ u'(r,-d)>0 \text{ for $r$ small if } d\in(0, d^-)\,. \end{array}$$ We consider the switched polar coordinates $\Omega^{u,\pm}_{l}(d, \tau)=( \theta^{u,\pm}_l(d, \tau) , \rho^{u,\pm}_l(d, \tau))$ of $W^{u,+}_l(\tau)$, introduced as in , defined in the stripe $\stripe = \RR \times [0,+\infty)$. We introduce the sets $$\begin{aligned} F^+(\tau)= \left\{ \Omega^{u,+}_{l}(d, \tau) \,:\, d\geq 0 \right\} \,, \qquad F^-(\tau)= \left\{ \Omega^{u,-}_{l}(d, \tau) \,:\, d\geq 0 \right\}\,,\end{aligned}$$ which are paths in the stripe $\stripe$. Moreover $$\label{flow.x.axis} x_l(t) \dot y_l(t) < 0 \text{ whenever } x_l(t)\in (-d^- \eu^{\alpha_l t}, d^+ \eu^{\alpha_l t})\setminus\{0\} \text{ and } y_l(t)= 0\,,$$ so that by we necessarily have $$\label{thu} \theta^{u,+}_l(d, \tau) \leq 0 \text{ and } \theta^{u,-}_l(d, \tau) \leq -\pi \text{ for every } d>0 \text{ and } \tau\in\RR\,.$$ Using , we can draw the sets corresponding to $W^u_l(\tau)$ for every $\tau\in\RR$ as follows. \[locateWu\] For every $\tau\in\RR$, $F^+(\tau)$ connects the point $(0,0)$ to $\SEqp(\tau)=(0,d^+ \,\eu^{\alpha_l t})$, respectively $F^-(\tau)$ connects $(-\pi,0)$ to $\SEqm(\tau)=(-\pi,d^-\,\eu^{\alpha_l t})$ (see Figure \[figint\]). Arguing as above we can consider some curves associated to the sets $W^{s,+}_l(\tau)$ and $W^{s,-}_l(\tau)$. So, we denote by $$\begin{aligned} \Omega^{s}_{l,2j}(L, \tau)=\big( \theta^{s,+}_l(L, \tau) - 2\pi j , \rho^{s,+}_l(L, \tau)\big)\,,\\ \Omega^{s}_{l,2j+1}(L, \tau)=\big( \theta^{s,-}_l(L, \tau) - 2\pi j , \rho^{s,-}_l(L, \tau)\big)\,,\end{aligned}$$ where $\Omega^{s}_{l,0}(\cdot, \tau)$ and $\Omega^{s}_{l,1}(\cdot, \tau)$ are respectively the natural representation of $W^{s,+}_l(\tau)$ and $W^{s,-}_l(\tau)$ on the stripe $\stripe$, by the choice . The others are simply their traslations of an angle $\Delta\theta=-2j\pi$. By with $l>p^*$, $W^s_l(+\infty)$ has the shape of a double spiral. Therefore, for every $N\in\N$, there exists a compact set $\mathcal E \subset W^s_l(+\infty)$, containing the origin, such that both the branches $\mathcal E \cap W^{s,+}_l(+\infty)$ and $\mathcal E \cap W^{s,-}_l(+\infty)$ perform more than $N+1$ complete rotations in the plane. By Remark \[allinfinitoWS\], the unstable manifold $W^s_l(\tau)$ exists for any $\tau\in\RR$ and converges to $W^s_l(+\infty)$, as $\tau\to+\infty$. Therefore, $W^s_l(\tau)$ perform at least $N+1$ rotations for $\tau$ sufficiently large. As a consequence we have the following remark. For every integer $N$ we can find a time $\bar\tau_N$ with the following property: $$\label{Lpm} \begin{array}{l} \text{for every $\tau>\bar\tau_N$ there exist $L^+(N,\tau)>0$ and $L^-(N,\tau)>0$}\\ \qquad \text{such that $\theta^{s,\pm}_l(L^\pm(N,\tau), \tau)= 2\pi N + \pi/2$} \end{array}$$ (we assume that $L^\pm(N,\tau)$ is the minimum value with such a property). Moreover there exists $\chi_N>0$ such that $$\rho^{s,\pm}_l(L, \tau)< \chi_N\,, \quad \text{for every $L<L^\pm(N,\tau)$ and $\tau>\bar\tau_N$.}$$ Further, since $$\label{flow.y.axis} \dot x_l(t) y_l(t) > 0 \text{ whenever } x_l(t)=0 \text{ and } y_l(t)\neq 0\,,$$ we have $$\label{ths} \theta^{s,+}_l(L, \tau)>-{\pi}/{2} \text{ and } \theta^{s,-}_l(L, \tau)>-3\pi/2 \text{ for every } L>0 \text{ and }\tau\in\RR \,.$$ Let us choose $\tau_N\geq \bar\tau_N$ large enough to have $$\label{tauNlarge} d^\pm \eu^{\alpha_l \tau_N} > \chi_N \,.$$ We fix an integer $N$ and define $$\begin{array}{rl} E_{2j}(\tau) &= \left\{ \Omega^{s}_{l,2j}(L, \tau) \,:\, 0\leq L\leq L^+(N,\tau) \right\}\,,\\[2mm] E_{2j+1}(\tau) &= \left\{ \Omega^{s}_{l,2j+1}(L, \tau) \,:\, 0\leq L\leq L^-(N,\tau) \right\}\,, \end{array}$$ for every $\tau>\tau_N$, corresponding to subsets of $W_l^s(\tau)$. The previous reasoning provide the following conclusion. \[locateWs\] For every integer $0\leq j \leq N$, the paths $E_{2j}(\tau)$ and $E_{2j+1}(\tau)$ intersect the line $\theta=\pi/2$ for every $\tau>\tau_N$. Moreover, $E_k(\tau)\subset \RR \times [0,\chi_N]$ for every integer $k\in[0,2N+1]$ (see Figure \[figint\]). From , we have $\SEqpm(\tau)\notin \RR \times [0,\chi_N]$, for every $\tau>\tau_N$. Hence, from Propositions \[locateWu\] and \[locateWs\], we expect to find intersections between $F^\pm(\tau)$ and $E_k(\tau)$, cf. Figure \[figint\]: the next section enters in such details. The existence of regular fast decay solutions {#step2} --------------------------------------------- Let us fix a positive integer $N$ and consider a time $\tau>\tau_N$, where $\tau_N$ is given by . We are going to prove, the existence of intersections between $F^\pm(\tau)$ and $E_k(\tau)$ for $0\leq k \leq 2N+1$. For every integer $k\in[0,2N+1]$, denote by $\Gamma_k(\tau)$ the region enclosed between $E_k(\tau)$, $\theta=\pi/2$ and $\rho=0$ (see Figure \[figint\]). Being $W^{s,+}_l(\tau) \cap W^{s,-}_l(\tau)=\{\O\}$, the paths $E_k(\tau)$ do not intersect each other. In particular, we have $$\label{Gamma-boxed} \Gamma_0(\tau) \subset \Gamma_1(\tau) \subset \cdots \subset \Gamma_{k-1}(\tau) \subset \Gamma_{k}(\tau) \subset \cdots \,.$$ Notice that the first part of the path $F^+(\tau)$ is contained in $\Gamma_k(\tau)$. Since $\SEqp(\tau)\notin \Gamma_k(\tau)$, $F^+(\tau)$ must leave $\Gamma_k(\tau)$ at a certain point. A similar reasoning can be done for $F^-(\tau)$: indeed, it starts inside $\Gamma_k(\tau)$ (except the case $F^-(\tau)\cap \Gamma_0(\tau)=\varnothing$) and $\SEqm(\tau)\in F^-(\tau)$ is such that $\SEqm(\tau)\notin \Gamma_k(\tau)$ (see Figure \[figint\]). We denote by $\bs{R_k^\pm}(\tau)$ the first intersection (in the sense of the parameter $d$) between the paths $F^\pm(\tau)$ and $E_k(\tau)$. More precisely, we get the following lemma. \[lemint\] For every integer $N$, we can find constants $A_0,A_1,\ldots,A_{2N+1}$ and $B_0,B_1,\ldots,B_{2N}$ such that, for every $\tau> \tau_N$, $$\begin{array}{l} \bs{R_{k}^+}(\tau) = \Omega^{u,+}_{l}(A_{k}, \tau) \in F^+(\tau) \cap E_k(\tau)\,, \\ \bs{R_{k}^-}(\tau) = \Omega^{u,-}_{l}(B_{k}, \tau) \in F^-(\tau) \cap E_{k+1}(\tau)\,, \end{array}$$ (see Figure \[figint\]). We assume without loss of generality that they are the smallest positive constants with such a property (i.e. $F^\pm(\tau)$ exits from $\Gamma_k(\tau)$ for the first time at these points). Correspondingly, $u(r,A_k)$ and $u(r,-B_k)$ are regular fast decay solutions. The last assertion follows as an immediate consequence of Remark \[corr-non-aut\]. Indeed, we can define also the corresponding points in the plane $(x_l,y_l)$: $\bs{Q_{k}^+}(\tau) = \Sigma^{u,+}_{l}(A_{k}, \tau)$ and $\bs{Q_{k}^-}(\tau) = \Sigma^{u,-}_{l}(B_{k}, \tau)$. In particular $\bs{Q_{k}^+}(\tau) \in W^{u,+}_l(\tau)\cap W^s_l(\tau)$ and $\bs{Q_{k}^-}(\tau) \in W^{u,-}_l(\tau)\cap W^s_l(\tau)$ for every $\tau>\tau_N$, so that they correspond to homoclinic orbit for system . We have proved, for every positive integer $N$, the existence of intersections between $F^\pm(\tau)$ and $E_k(\tau)$, for $0\leq k \leq 2N+1$, when $\tau>\tau_N$. We stress that the integer $N$ can be chosen arbitrarily large (thus enlarging $\tau_N$ correspondingly), so that the previous constants $A_k$ and $B_k$ can be found for every choice of the integer $k$ as required by Theorem \[main\]. The correct estimate on the number of nondegenerate zeroes will be provided in Section \[step4\]. The existence of regular slow decay solutions {#step3} --------------------------------------------- We focus now our attention on the existence of slow decay solutions following the arguments presented in [@DalFra; @FraSfe2]. In the previous section, we have proved that $F^\pm(\tau)$ leaves $\Gamma_k(\tau)$ in some points $\bs{R_{k}^\pm}(\tau)$. In Figure \[figint\], a simple situation is pictured: the intersections $F^\pm(\tau) \cap E_k(\tau)$ consist of a unique point. Unfortunately, we cannot prove in general such a uniqueness property. Indeed, a more complex situation can arise: the intersections $F^\pm(\tau)\cap \Gamma_k(\tau)$ can consist of many disconnected paths, as in Figure \[figintdiff\]. However, from , there exists necessarily at least one sub-path of $F^\pm(\tau)$ linking $E_{k-1}(\tau)$ to $E_k(\tau)$. Hence, once fixed an integer $N$ and $\tau>\tau_N$ as above, for every integer $k\in(0,2N+1]$, we can find $A_{k}^*\in [A_{k-1},A_{k})$ such that $$A_k^* := \max \{d\in[0,A_k) \,:\, \text{exists } L >0 \text{ such that } \Omega^{u,+}_l(d,\tau)= \Omega^{s}_{l,k-1}(L,\tau) \} \,.$$ Set $\bs{R^{+,*}_{k}} := \Omega^{u,+}_l(A_k^*,\tau)$ and $a_k^+(\tau):=\{ \Omega^{u,+}_{l}(d, \tau) \,:\, d\in(A_{k}^*,A_{k}) \}$. Then $a_k^+(\tau) \subset\Gamma_{k}(\tau) \setminus \Gamma_{k-1}(\tau)$ (see Figure \[figintdiff\]). Similarly, for every integer $k\in(0,2N]$, we can find $B_{k}^*\in [B_{k-1},B_{k})$ such that $$B_k^* := \max \{d\in[0,B_k) \,:\, \text{exists } L >0 \text{ such that } \Omega^{u,-}_l(d,\tau)= \Omega^{s}_{l,k}(L,\tau) \} \,.$$ Correspondingly, define $\bs{R^{-,*}_{k}} := \Omega^{u,-}_l(B_k^*,\tau)$ and $a_k^-(\tau):=\{ \Omega^{u,-}_{l}(d, \tau) \,:\, d\in(B_{k}^*,B_{k}) \} \subset\Gamma_{k+1}(\tau) \setminus \Gamma_{k}(\tau)$. Possibly we can have $A_k^*=A_{k-1}$ or $B_k^*=B_{k-1}$ for some $k$’s (in the simple situation presented in Figure \[figint\], they hold for every $k$). \[lemslow\] For every $d\in(A_k^*,A_k)$, $u(r,d)$ is a regular slow decay solution. Similarly, for every $d\in(B_k^*,B_k)$, $u(r,-d)$ is a regular slow decay solution. By Lemma \[corr-non-aut\], we need to prove that $\xl(t,\tau,\Q)\to \bs{P^+}$ or $\xl(t,\tau,\Q)\to \bs{P^-}$ as $t\to+\infty$, where $\Q=\Sigma^{u,+}_l(d,\tau)$. For every $\Q\in\RR^2$, let us denote by $\Omega(t,\tau,\Omega_\Q)$ the trajectory of $\xl(t,\tau,\Q)$ in the stripe $\stripe$, where $\Omega_\Q=(\phi_\Q,\rho_\Q)\in\stripe$ corresponds to the switched polar coordinates of $\Q$, i.e. $\Q= \rho_\Q (\cos \phi_\Q,\sin \phi_\Q)$. Notice that $$\label{FEinvariant} \begin{array}{l} \Omega_\Q \in E_k(\tau) \Rightarrow \Omega(t,\tau,\Omega_\Q) \in E_k(t) \ \forall t\in\RR \,,\\ \Omega_\Q \notin E_k(\tau) \Rightarrow \Omega(t,\tau,\Omega_\Q) \notin E_k(t) \ \forall t\in\RR \,, \\ \Omega_\Q \in F^\pm(\tau) \Rightarrow \Omega(t,\tau,\Omega_\Q) \in F^\pm(t) \ \forall t\in\RR \,,\\ \Omega_\Q \notin F^\pm(\tau) \Rightarrow \Omega(t,\tau,\Omega_\Q) \notin F^\pm(t) \ \forall t\in\RR \,. \\ \end{array}$$ Indeed, they correspond to the sets $W^{s,\pm}_l(\tau)$ and $W^{u,\pm}_l(\tau)$, which satisfy the same property, cf. and . By , the flow on $\{\theta=\pi/2\}\subset\stripe$ points towards $\{\theta<\pi/2\}$. Hence, using also , we see that if $\Omega_\Q\in \Gamma_k(\tau) \setminus \Gamma_{k-1}(\tau)$ then $\Omega(t,\tau,\Omega_\Q)\in \Gamma_k(t) \setminus \Gamma_{k-1}(t)$ for every $t>\tau$. Moreover, since $W_l^{s}(t) \to W_l^s(+\infty)$, we have that $E_k(t) \to E_k(+\infty)$, where $E_k(+\infty)$ corresponds to a polar representation of a subset of $W^s(+\infty)$. In particular $\Gamma_k(t) \setminus \Gamma_{k-1}(t)$ remains bounded and $\Gamma_k(t) \setminus \Gamma_{k-1}(t) \to \Gamma_k(+\infty) \setminus \Gamma_{k-1}(+\infty)$, where $\Gamma_j(+\infty)$ is the region enclosed by $E_j(+\infty)$, $\theta=\pi/2$ and $\rho=0$. We introduce the polar coordinates of the non-trivial critical points $\bs{P^\pm}= \rho_\pm (\cos\phi_\pm, \sin\phi_\pm)$ and the corresponding points $\Omega_{\bs{P},2j}=(\phi_+ -2\pi j,\rho_+)$ and $\Omega_{\bs{P},2j+1}=(\phi_- -2\pi j,\rho_-)$ on the stripe $\stripe$. The unique attractor of $\Gamma_k(+\infty) \setminus \Gamma_{k-1}(+\infty)$ is $\Omega_{\bs{P},k}$, so we have $\Omega(t,\tau,\Omega_\Q)\to \Omega_{\bs{P},k}$ as $t\to+\infty$ for every $\Omega_\Q\in \Gamma_k(\tau) \setminus \Gamma_{k-1}(\tau)$. The previous limits corresponds to $\xl(t,\tau,\Q) \to \bs{P^+}$ for $k$ even and to $\xl(t,\tau,\Q) \to \bs{P^-}$ for $k$ odd. Hence, the corresponding solution has a slow decay, cf. Remark \[corr-non-aut\]. Choosing $\Omega_\Q=\Omega^{u,\pm}_l(d,\tau) \in a_k^\pm(\tau)$ we find a regular slow decay solution $u(r,\pm d)$. As a consequence of the previous argument, we also have the following lemma. \[pri\] For every $d\in(A_k^*,A_k]$, $u(r,d)$ satisfies $-d^- < u(r,d) <d^+ $ for every $r > \eu^{\tau_N}$. Similarly, for every $d\in(B_k^*,B_k]$, $u(r,-d)$ satisfies $-d^- < u(r,-d) <d^+ $ for every $r > \eu^{\tau_N}$. For every $\tau>\tau_N$ we have $F^\pm(\tau) \subset \RR \times [0,\chi_N]$. So, by , we have $\rho_l^{u,\pm}(d,\tau)< d^\pm \, \eu^{\alpha_l \tau}$. Consequently, $-d^-<x_l(t,\tau,\Sigma_l^{u,\pm}(d,\tau)) \eu^{-\alpha_l t}<d^+$ for every $\tau>\tau_N$ and the assertion follows. The number of non-degenerate zeros {#step4} ---------------------------------- The correct estimates on the number of nondegenerate zeros of the fast decay solutions is given by the following lemma, see e.g. [@DalFra Lemma 3.3] (see also [@BaFloPino2000POIN; @F2013DIE; @FraSfe2] for a full fledged proof). \[anglecontrol\] Let us consider system  and assume . Consider the trajectory $\xl(\cdot ,\tau,\Q)$ with $\Q = \Sigma^{u,+}_{l}(d, \tau)$ and its polar coordinates . Then, the angle $\vartheta:=\theta^{u,+}_l(d,\tau)-\theta^{u,+}_l(0,\tau)=\theta^{u,+}_l(d,\tau)$ performed by the unstable manifold $W_{l}^{u,+}(\tau)$ equals the angle $\phi:=\phi_l(t,\tau,\Q)-\phi_l(-\infty,\tau,\Q)$ performed by the trajectory $\xl(t,\tau,\Q)$ in the interval $(-\infty,\tau]$. Similarly, assume and consider $\xl(\cdot ,\tau,\Q)$ with $\Q = \Sigma^{s,+}_{l}(L, \tau)$. Then, the angle $\vartheta:=\theta^{s,+}_l(L,\tau)-\theta^{s,+}_l(0,\tau)=\theta^{s,+}_l(L,\tau)+\pi/2$ performed by the stable manifold $W_{l}^{s,+}(\tau)$ equals, but with reversed sign, the angle $\varphi:=\phi_l(+\infty,\tau,\Q)-\phi_l(\tau,\tau,\Q)$ performed by the trajectory $\xl(t,\tau,\Q)$ in the interval $[\tau,+\infty)$. A similar reasoning holds for $\Q = \Sigma^{u,-}_{l}(d, \tau)$ and $\Q = \Sigma^{s,-}_{l}(L, \tau)$. The previous lemma is the key point in order to prove the nodal properties of the regular solutions we have found. Let us start with regular fast decay ones. We consider $u(r,A_k)$ and the associated points $\bs{Q^+_k}=\Sigma^{u,+}_l(A_k,\tau)\in\RR^2$ and $\bs{R^+_k}=\Omega^{u,+}_l(A_k,\tau)\in\stripe$. By the previous lemma, in order to obtain the angle performed by $\xl(\cdot,\tau,\bs{Q^+_k})$ in the whole time interval $(-\infty,+\infty)$ we have to consider the angle variation along the path $F^+(\tau)$ between $(0,0)\in \stripe$ and $\bs{R^+_k}$ and then the one along $E_k(\tau)$ between $\bs{R^+_k}$ and $(-\pi(k+1/2),0)\in\stripe$. We easily obtain a tolal angle of $-\pi(k+1/2)$. So, by the flow condition , the trajectory $\Omega(\cdot,\tau,\bs{R^+_k})$ intersects the vertical lines $\theta=-\pi/2-j\pi$ once for every integer $j\in\{1,\ldots,k\}$, thus finding exactly $k$ zeros of $\xl(\cdot,\tau,\bs{Q^+_k})$ which are non-degenerate. A similar reasoning holds for $u(r,-B_k)$. We turn now to consider $u(r,d)$ with $d\in(A^*_k,A_k)$ and, correspondingly, $\Q=\Sigma^{u,+}_l(d,\tau)$ and $\R=\Omega^{u,+}_l(d,\tau)$. The total angle performed by the solution $\xl(\cdot,\tau,\Q)$ in the time interval $(-\infty,\tau)$ is given by the angle variation along the path $F^+(\tau)$. Then, for every $t>\tau$, $\Omega(t,\tau,\R)$ is forced to remain between the paths $E_{k-1}(t)$ and $E_k(t)$ and to converge to $\Omega_{\bs{P},k}$ as $t\to+\infty$: thus the total angle variation is $-2j\pi+\phi_+\in(-(2j+1)\pi,-2j\pi)$ if $k=2j$, resp $-2j\pi+\phi_-\in(-(2j+2)\pi,-(2j+1)\pi)$ if $k=2j+1$. By , the trajectory $\Omega(\cdot,\tau,\bs{R})$ intersects all the lines $\theta= -\pi/2 - j\pi$ only once and correspondingly all the zeros are non-degenerate. A similar reasoning holds for $u(r,-d)$ with $d\in(B^*_k,B_k)$. Back to the orginal problem {#step5} --------------------------- We have proved Theorem \[main\] for the differential equation with $f$ replaced by its truncation $\bar f$ introduced in . We are going now to prove that such solutions solve also the original equation . To this aim we need Lemma \[apriori\] below. We underline that it is well-known in literature that, under hypotheses and , a positive solution of with $u(0)<d^+$ necessarily satisfies $u(r)<d^+$ for every $r>0$ (and correspondingly negative ones with $u(0)>-d^-$ satisfy $u(r)>-d^-$). The situation is more complicated if we treat nodal solutions. In the case of a decreasing weight $k$, we can easily provide the same [*a priori*]{} estimate by introducing an energy function. For a more general weight $k$, such a situation is not ensured and we argue as follows. \[apriori\] Consider $\Q=(Q_x,Q_y)\in\RR^2$ with $-d^- < Q_x \eu^{-\alpha_l \sigma} < d^+$ for a certain $\sigma$. Suppose that there exists $\sigma'>\sigma$ such that $\xl(\sigma',\sigma,\Q)$ satisfies $-d^-<x_l(\sigma',\sigma,\Q) \eu^{-\alpha_l \sigma'} < d^+$. Then, the solution $\xl(t,\sigma,\Q)$ of , with $g_l$ as in , satisfies $-d^-<x_l(t,\sigma,\Q) \eu^{-\alpha_l t} < d^+$, for every $t\in(\sigma,\sigma')$. Defining $\xi_l(t) = x_l(t) - d^+ \eu^{\alpha_l t}$ we obtain the system $$\label{sist.xi} \begin{cases} \dot \xi_l = \alpha_l \xi_l + y_l|y_l|^{\frac{2-p}{p-1}}\\ \dot y_l = \gamma_l y_l - h_l(t,\xi_l) \end{cases}$$ where $h_l(t,\xi_l)=g_l(t,\xi_l+d^+ \eu^{\alpha_l t})$. Notice that, by , $h_l(t,\xi_l)\leq 0$ if $\xi_l>0$. Moreover, as in , $$\label{flow.xi} \dot \xi_l(t) y(t) > 0 \text{ whenever } \xi_l(t)=0 \text{ and } y_l(t)\neq 0\,,$$ In particular $\mathcal Q_1 = \{ (\xi,y) \,:\, \xi \geq 0\,, y\geq 0 \}$ and $\mathcal Q_2 = \{ (\xi,y) \,:\, \xi \geq 0\,, y\leq 0 \}$ are respectively positively and negatively invariant sets. In this setting we have to prove that if a solution $(\xi_l(t),y_l(t))$ of satisfies $\xi_l(\sigma)<0$ and $\xi_l(\sigma')<0$ then $\xi_l(t)<0$ for every $t\in(\sigma,\sigma')$. Arguing by contradiction, let $t_0\in(\sigma,\sigma')$ be such that $\xi_l(t_0)\geq 0$. If $y_l(t_0)\geq 0$ then $(\xi_l(t_0),y_l(t_0))\in \mathcal Q_1$ which is invariant in the future and we get a contradiction with $\xi_l(\sigma')<0$. Conversely, if $y_l(t_0)< 0$ then $(\xi_l(t_0),y_l(t_0))\in \mathcal Q_2$ which is invariant in the past and we get a contradiction with $\xi_l(\sigma)<0$. The estimates with respect $d^-$ is analogous. We consider a regular solution $u(r,d)$ of with $f=\bar f$ as in and the corresponding solution $\xl$ of system (notice that $g_l$ is as in ). If $d\in(-d^-,d^+)$, by , we have $u(r,d)\in(-d^-,d^+)$ for every $r\in[0,\eu^{t_1}]$, with $\eu^{t_1}$ sufficiently small. Hence, we have $-d^- <x_l(t) \eu^{-\alpha_l t}<d^+ $ for every $t \leq t_1$. Now, given a regular solution $u(r,d)$ with $d\in(A_k^*,A_k]$, setting $N>k/2$ and $\Q=\Sigma_l^{u,+}(d,\tau)$, by Lemma \[pri\], we have $-d^-<x_l(t,\tau,\Q) \eu^{-\alpha_l t}<d^+$ for every $t \geq \tau > \tau_N$. 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Tang, [*Uniqueness and global structure of positive radial solutions for quasilinear elliptic equations*]{}, Comm. Partial Differential Equations **26** (2001), 909–938. [^1]: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona - Italy. Partially supported by G.N.A.M.P.A.

[**An extinction-survival-type phase transition in\ the probabilistic cellular automaton $p182$–$q200$**]{} [[^1][^2] [**and**]{} [**Mário J. de Oliveira**]{}[^3] ]{} [*[Instituto de Física, Universidade de São Paulo]{}\ *]{} [**Abstract\ **]{} Introduction {#intro} ============ Cellular automata (CA, also for cellular automaton) have been a major theme of investigation since their introduction in the late 40’s and early 50’s of the last century as model systems for parallel processing machines and algorithms, memory storage devices, and self-reproducing simple organisms [@jvn]. Meanwhile, CA have become deeply intertwined with the subject matter of modern equilibrium and nonequilibrium statistical mechanics, such as with the theory of phase transitions, irreversibility, ergodicity, chaos, percolation, and physical kinetics, to name a few [@discrete; @wolfram; @dkpca; @grinstein; @rujan; @lebowitz; @chen; @helbing; @droz]. Probabilistic versions of CA (PCA) with absorbing states, [*i.e.*]{}, with states that once reached trap the dynamics definitely, are a whole chapter in CA theory. The single most representative PCA in this class is the Domany-Kinzel (DK) PCA, because its two-dimensional parameter space encompasses both the directed bond and the directed site percolation processes [@dkpca]. More general PCA with more than one absorbing state have also been investigated, with larger neighbourhoods and more than two parameters, at the expense of some symmetries present in the DK PCA. In particular, a class of two and three parameter PCA with two absorbing states was investigated in [@boccara], where many general results on the phase diagrams of the models were obtained. In usual PCA modeling, the freedom to set the transition probabilities directly in the rule table allows one to embody competing interactions in the PCA from the outset, such that a resulting complex dynamic behaviour becomes an expected treat. However, elementary one-dimensional CA are more charming in their simplicity, since most of them can be implemented with a few deterministic binary operations that also make them of greater technological relevance.[^4] Moreover, it has been found that by composing simple CA in space and time, probabilistically or not, we may obtain complex behaviour out of simple components. A striking example of this possibility is given by the composite deterministic CA of H. Fukś that solves the density classification problem, an impossible problem for locally interacting single CA [@fuks; @noisy]. These observations triggered our interest in probabilistic mixtures of simple CA. One PCA that we considered preliminarly combines CA rules $150$ and $200$ in Wolfram’s enumeration scheme [@wolfram] and is briefly mentioned in Section \[mcarlo\]. Another possible combination is given by CA rules $23$ and $254$, that when combined as a PCA in the guise of a model for an organism ($1$’s) consuming finite renewable resources ($0$’s) and dying from overcrowding may display an active-inactive type phase transition. As far as we are aware, this kind of composite PCA has not yet been fully explored in the literature. In this paper we investigate in detail a probabilistic mixture of CA $182$ and CA $200$ that is left-right symmetric, has a single scalar order paramenter, no conserved quantities, and two absorbing states, both reachable dynamically as long as the mixed PCA contains finite portions of both CA $182$ and CA $200$ dynamics. After some exploratory work, we found that this PCA displays some interesting features, such as its behaviour in a certain small parameter region besides an extinction-survival-type phase transition as we probabilistically switch off one CA and switch on the other. We characterize this phase transition by mean-field analysis and Monte Carlo simulations and determine that it belongs to the directed percolation universality class of critical behaviour. The article is organized as follows. In Section \[model\], we define the PCA and in Section \[mean\] we analyse it at the mean-field level of approximation. Section \[mcarlo\] presents the results of direct Monte Carlo simulations of the PCA for its critical point and critical exponents, and in Section \[summary\] we make a few final remarks, summarize our results, and identify some perspectives for further investigation. Model description {#model} ================= Let $\eta_{\ell}(t) \in \{0,1\}$ denote the state of the site $\ell \in \Lambda\subset {\mathbb Z}$ at instant $t \in {\mathbb N}$, with $\Lambda$ a finite lattice of $L$ sites with periodic boundary conditions $\ell+L \equiv \ell$. The state of the system at instant $t$ is given by $\bm{\eta}(t) = (\eta_{1}(t), \eta_{2}(t), \ldots, \eta_{L}(t)) \in \{0,1\}^{\Lambda}$. The model we are interested in is the probabilistic cellular automaton defined by the rules in Table \[TAB-W\]. We dub this system PCA $p182$–$q200$, with $p+q=1$. When $p=1$, the mixed PCA reduces to the deterministic CA $182$, which has a stationary density of 1’s given by $\rho_{182}^{\rm exact} = 3/4$ [@wolfram]. CA $200$ ($p=0$ in Table \[TAB-W\]), otherwise, is quite an uninteresting CA, since most initial configurations die out quickly or do not evolve at all under its rules. The role of CA $200$ in our mixture of CAs is to provide a route to the absorbing state devoid of 1’s, a state that cannot be reached by CA $182$ except from the initial empty configuration itself. So, as $p$ varies between $p=0$ and $p=1$ we expect a competition to set up between the two rules to dominate the dynamics, with an extinction-survival-type phase transition somewhere in between. We indeed found such a phase transition, as long as the initial condition is not one of the absorbing configurations of the model. PCA $p182$–$q200$ can be related with the DK PCA only at $p = 1/2$, a point that is off the mixed site-bond percolation parameter subspace of the DK PCA, since this subspace requires that a certain parameter $x$ of the DK PCA be zero, which is impossible in our setting, because the match between the two PCAs requires that $p = 1/2$ and $x=1-p$ simultaneously. The point $p=1/2$ in PCA $p182$–$q200$, however, corresponds to a completely uncorrelated dynamics, since bits will flip with probability $1/2$ irrespective of their neighborhood, with the exception of the bits in the bulk of blocks $11{\cdots}1$, [*cf.*]{} Table \[TAB-W\]. These blocks, however, are not stable at $p \neq 0$, since they will eventually be eroded from the boundaries. We thus expect that at $p=1/2$ the stationary density of active sites fluctuates around $\rho = 1/2$. 111 110 101 100 011 010 001 000 ----- ----- ----- ----- ----- ----- ----- ----- ----- $p$ 1 0 1 1 0 1 1 0 $q$ 1 1 0 0 1 0 0 0 : \[TAB-W\]Rule table for PCA $p182$–$q200$, $p+q=1$. The first row gives the initial neighborhood, the other two rows give the final state reached by the central bit of the initial neighborhood with the probability given at the leftmost column. Clearly, the configurations $00{\cdots}0$ and $11{\cdots}1$ are absorbing configurations of the PCA. Mean-field analysis {#mean} =================== We begin by analysing PCA $p182$–$q200$ in the mean-field approximation to obtain some first information on its critical behaviour. For a nice brief exposition and application of the technique see, [*e.g.*]{}, [@tania]. The dynamics of the probability distribution $P_{t}(\bm{\eta})$ of the states $\bm{\eta}$ of the PCA is ruled by the equation $$\label{PPP} P_{t+1}(\bm{\eta}') = \sum_{\bm{\eta}}W(\bm{\eta}'|\bm{\eta})P_{t}(\bm{\eta}),$$ where the summation runs over all $\bm{\eta} \in \{0,1\}^{\Lambda}$ and $W(\bm{\eta}'|\bm{\eta}) \geq 0$ is the conditional probability for a transition $\bm{\eta}$ $\to$ $\bm{\eta}'$ to occur in one time step. Since in a CA or PCA all sites are updated simultaneously and independently, we can write $$\label{WWW} W(\bm{\eta}'|\bm{\eta}) = \prod_{\ell=1}^{L}W_{\ell}({\eta_{\ell}'}|\bm{\eta}), \quad {\rm with} \quad \sum_{\eta_{\ell}'}W_{\ell}({\eta_{\ell}'}|\bm{\eta}) = 1.$$ For PCA $p182$–$q200$ we have $W_{\ell}({\eta_{\ell}'}|\bm{\eta})=$ $W({\eta_{\ell}'}|\eta_{\ell-1}, \eta_{\ell}, \eta_{\ell+1})$, independent of $\ell$. The time evolution of the marginal probability distribution $P_{t}(\eta_{\ell},$ $\eta_{\ell+1},$ …, $\eta_{\ell+n-1})$ of observing $n$ consecutive sites in state $(\eta_{\ell}, \eta_{\ell+1}, \ldots, \eta_{\ell+n-1})$ is, from Eqs. (\[PPP\]) and (\[WWW\]), given by $$\begin{aligned} P_{t+1}(\eta_{\ell}', \eta_{\ell+1}', \ldots, \eta_{\ell+n-1}') = \nonumber \\ = \sum_{\eta_{\ell-1}} \sum_{\eta_{\ell}} \cdots \sum_{\eta_{\ell+n}} W(\eta_{\ell}'|\eta_{\ell-1},\eta_{\ell},\eta_{\ell+1})\, W(\eta_{\ell+1}'|\eta_{\ell},\eta_{\ell+1},\eta_{\ell+2}) \ldots \nonumber \\ \ldots W(\eta_{\ell+n-1}'|\eta_{\ell+n-2},\eta_{\ell+n-1},\eta_{\ell+n})\, P_{t}(\eta_{\ell-1}, \eta_{\ell}, \ldots, \eta_{\ell+n}). \label{MARG}\end{aligned}$$ We see from equation (\[MARG\]) that to determine the probability of observing $n$ consecutive sites in a given state at instant $t+1$ we need to know the probabilities of observing the state of $n+2$ sites at instant $t$. To proceed with the calculations in an approximate fashion, we truncate this hierarchy at some point to split the correlations and get a closed set of equations. The simplest approximation ($n=1$) is obtained by taking $$\label{APPROX} P_{t}(\eta_{\ell-1},\eta_{\ell},\eta_{\ell+1}) \approx P_{t}(\eta_{\ell-1})\, P_{t}(\eta_{\ell})\, P_{t}(\eta_{\ell+1}).$$ Higher order approximations ($n \geq 2$) are obtained (assuming spatial homogeneity) by the generalized splitting scheme $$\label{MFN} P_t(\eta_{\ell-1}, \ldots, \eta_{\ell+n}) \approx \frac{P_t(\eta_{\ell-1}, \ldots, \eta_{\ell+n-2})\, P_t(\eta_{\ell}, \ldots, \eta_{\ell+n-1})\, P_t(\eta_{\ell+1}, \ldots, \eta_{\ell+n})} {P_t(\eta_{\ell}, \ldots, \eta_{\ell+n-2})\,P_t(\eta_{\ell+1}, \ldots, \eta_{\ell+n-1})}.$$ From the rules in Table \[TAB-W\] and the above equations, the single-site ($n=1$) approximation for $\rho^{(1)}_{t} = P_{t}(\eta_{\ell}=1)$ (where the superscript refers to the order of the approximation) reads $$\rho^{(1)}_{t+1} = (\rho^{(1)}_{t})^{3} + (2-p)(\rho^{(1)}_{t})^{2}(1-\rho^{(1)}_{t}) + 3p\rho^{(1)}_{t}(1-\rho^{(1)}_{t})^{2}.$$ In the stationary state, $\rho^{(1)}_{t+1} = \rho^{(1)}_{t} = \rho^{(1)}$, and the above equation becomes $$\label{STAT} \rho^{(1)} = (\rho^{(1)})^{3} + (2-p)(\rho^{(1)})^{2}(1-\rho^{(1)}) + 3p\rho^{(1)}(1-\rho^{(1)})^{2},$$ with solutions $\rho^{(1)}=0$, $\rho^{(1)}=1$ and $\rho^{(1)}=(3p-1)/(4p-1)$. The first two solutions correspond to the two absorbing states of the dynamics, whereas the last, nontrivial solution corresponds to the active state and is valid as long as $p \geq 1/3$. The single-site mean-field approximation for PCA $p182$–$q200$ thus predicts an extinction-survival-type phase transition at $p_{c}^{(1)} = 1/3$. Notice that at $p=1$, $\rho^{(1)}=2/3$, not far from the exact stationary value $\rho_{182}^{\rm exact} = 3/4$ for CA $182$. We have also considered higher-order approximations with $n=2$, $3$, and $4$. In these cases, however, the equations are too cumbersome to be written down here. The $n=2$ approximation was solved analytically and gives the same results as the $n=1$ case—somehow, the two-sites marginal probability $P(\eta_{\ell},\eta_{\ell+1})$ factors into $P(\eta_{\ell})P(\eta_{\ell+1})$. The cases $n=3$ and $4$ were solved numerically. The densities $\rho^{(n)}(p)$ of active sites for these approximation are shown in Figure \[fig:pc\]. As expected, at $p = 1/2$ all approximations (as well as the Monte Carlo simulation results, [*cf.*]{} Section \[mcarlo\]) give $\rho(1/2) = 1/2$. In Table \[TAB-PC\] we show the values of $p_c^{(n)}$ and $\rho^{(n)}$ at $p=1$ for these approximations together with the corresponding values obtained by Monte Carlo simulations. As one can see from Table \[TAB-PC\], the $n$-th order mean-field approximation to the exact PCA converges slowly with $n$. $n$ $p_{c}^{(n)}$ $\rho^{(n)}(p=1)$ ------- --------------- ------------------- -- 1 1/3 2/3 2 1/3 2/3 3 0.4015 0.7135 4 0.4203 0.7166 MC 0.48810(5) 0.7500(5) Exact NA 3/4 : \[TAB-PC\]Critical parameter $p_{c}^{(n)}$ and value of the density of active sites $\rho^{(n)}(p)$ at $p=1$ obtained by mean-field approximations or order $n$ and Monte Carlo simulation ([*cf.*]{} Section \[mcarlo\]). The numbers between parentheses (in this table and elsewhere in this article) indicate the uncertainty in the last digit(s) of the data. For $p=1$, PCA $p182$–$q200$ reduces to CA $182$, for which the exact stationary density $\rho_{182}^{\rm exact} = 3/4$. Direct Monte Carlo simulation {#mcarlo} ============================= The density profile ------------------- Our Monte Carlo simulations of PCA $p182$–$q200$ ran as follows. For each $p$, the PCA is initialized according to a Binomial($L$,$s$) distribution (mostly with $s=\frac{1}{2}$), [*i.e.*]{}, each site is initialized as $\eta_{\ell}=1$ with probability $s$, and relaxed through $L^{2}/10$ Monte Carlo steps, with one Monte Carlo step equal to a synchronous update of the states of all $L$ sites of the lattice. We then sample $\rho_{L} = L^{-1}\sum_{\ell}\eta_{\ell}$ every other Monte Carlo step. Our data for $\rho_{L}(p)$ in a lattice of $L=8000$ sites appear in Figure \[fig:rho\]. We can identify two regions in this figure: an “active region I” that goes from $p=0$ up to $p \simeq 0.03$ and an “active region II” that goes from $p \simeq 0.48$ up to $p=1$. Region I is easily understood as an artifact of our initialization of the PCA with a Binomial($L$,$\frac{1}{2}$) distribution. At $p=0$, PCA $p182$–$q200$ becomes CA $200$, that washes out all local configurations but those of neighbouring 1’s ([*cf.*]{} Table \[TAB-W\]). The only neighborhood that evolves under CA $200$ dynamics is 010, that becomes 000 with probability $q=1-p=1$. For a Binomial($L$,$s$) initial distribution, the expected initial density of 1’s is $\mathbb{E}[\eta_{\ell}=1] = s$ and of triplets 010 is $\mathbb{E}[\eta_{\ell}\eta_{\ell+1}\eta_{\ell+2}=010] = (1-s)s(1-s)$. After one single time step, all 010 go into 000 and the stationary density of 1’s becomes $\mathbb{E}[\eta_{\ell}=1] = s-(1-s)s(1-s)$. For $s=1/2$, $\rho(p=0) = 3/8$, in accordance with the value in Figure \[fig:rho\]. As $p$ increases from zero, the noise brough up by the CA $182$ rules disturbs the dynamics and the PCA eventually converges to the absorbing state devoid of active sites. For very small $p > 0$, after the triplets $010$ quickly become $000$ with probability $q=1-p$, only the boundaries of the remaining clusters $11{\cdots}1$ move, performing random walks $100 \rightleftharpoons 110$ at the right edge and $001 \rightleftharpoons 011$ at the left edge with probability $p$. Under this dynamics, the number of active sites just fluctuates about a certain value. When these wandering boundaries meet, however, either two clusters coalesce with a low probability $p$ through the $101 \to 111$ channel or one cluster vanishes with a high probability $1-p$ through the $010 \to 000$ channel. This behaviour is reminiscent of the long time behaviour of the one-species lattice gas where $k$ particles coalesce into $\ell$ particles, $k{A} \to \ell{A}$ with $k > \ell$, which can be mapped into the problem of the reunion of $k$ random walkers bounded to move in a limited region [@mario]. The average density of active sites for very small $p$ is then expected to decay very slowly towards zero as $\rho(t) \sim 1/\sqrt{pt}$ or, equivalently, the extinction time $\tau_{\rm ext}(L,p) \sim L^{2}/p$, with prefactors depending on the sizes and spatial distribution of clusters $11{\cdots}1$ on the initial configuration. Since we sample only a very tiny fraction of all possible initial configurations (for $L=400$ and $\rho(0)=0.3$, the number of possible configurations under periodic boundary conditions is $\sim 1.4 \times 10^{102}$), these prefactors vary widely from one simulation to the other, even if we average over several thousands of realizations. This became evident as the data for $\tau_{\rm ext}(L,p)$ turned out to be overdispersed, [*i.e.*]{}, with sample variance exceeding the mean. Numerical evidence for the scaling of $\tau_{\rm ext}$ is given in Figure \[fig:tau\]. We believe that the deviation from the expected exponents is due both to finite-size effects and to the prefactor issue. While it is a little bothering to obtain such non-expected figures, at least we got the correct functional form $\tau_{\rm ext}(L,p) \sim L^{a}/p^{b}$ with $a >1$ and $b \approx 1$, corroborating our qualitative analysis. Notice that $a \simeq z_{\rm DP} \simeq 1.581$ ([*cf.*]{} Section \[indices\]), but since this region of small $p$ is very far from the critical region there is no reason to confuse the two values—the figures are similar by accident and by the numerical limitations and issues mentioned before. -- -- -- -- The unusual bump in $\rho_{L}(p)$ visible in Figure \[fig:rho\] near $p \simeq 0.57$ does not have a simple explanation. We thought at first that it could be signaling a first order phase transition, but further simulation data convinced us that the phase transition is continuous. Notice that the non-monotonic behaviour of $\rho(p)$ for $p \gtrsim 0.57$ is also observed in the mean-field approximations of orders $n=3$ and $4$. We found, however, that the closely related PCA $p150$–$q200$ may provide a clue to this bump. PCA $p150$–$q200$ differs from PCA $p182$–$q200$ in the transition $101 \to 111$, that lacks in the former. It means that in PCA $p150$–$q200$, clusters $11{\cdots}1$ cannot coalesce upon encountering. Preliminary numerical simulations indicate that this PCA suffers an extinction-survival phase transition by $p \simeq 0.572$, in the vicinity of the bump seen in Figure \[fig:rho\]. The density of active cells in PCA $p150$–$q200$ is smooth at $p > p_{c}$, without features. A possible scenario for the dynamics around $p \simeq 0.572$ in PCA $p182$–$q200$ is then that the effect of sites surviving and spreading together with that of clusters $11{\cdots}1$ coalescing increase the density of active sites by producing larger clusters that both diminish the erosion from the boundaries (by diminishing the total number of boundaries) and create more stable segments $111$. As $p$ increases from $p \simeq 0.572$, boundaries fluctuate more widely, eventually leading to the annihilation of previously existing clusters that cannot recover easily, leading to a decrease in the density of active sites and to the non-monotonic behaviour seen in Figure \[fig:rho\]. Increasing $p$ makes coalescence to become more probable and single active cells in triplets $010$ more stable, leading to a resume of the increase in the density of active sites. We shall henceforth focus on the transition around $p \simeq 0.48$ only. Critical indices {#indices} ---------------- We determined the critical point $p_{c}$ through an examination of the stationary density of active sites $\rho_{L}(p)$ near $p \simeq 0.48$ for different lattice sizes and also by perusal of the scaling relation $$\rho_{L}(t) \sim t^{-\beta/\nu_{\|}}\Phi(\varepsilon t^{1/\nu_{\|}},\, t^{\nu_{\perp}/\nu_{\|}}/L),$$ with $\varepsilon = p-p_{c} \geq 0$ [@haye]. For a very large system, this scaling relation assumes the form $\rho(t) \sim t^{-\beta/\nu_{\|}}\Phi(\varepsilon t^{1/\nu_{\|}})$, with $\Phi(x \ll 1) \sim {\rm constant}$ and $\Phi(x \gg 1) \sim x^{\beta}$. The first approach gives $p_{c}$ and $\beta$ more directly but is harder computationally, due to the very long simulation runtimes to attain stationarity close to the critical point. The second approach is more computer-friendly and usually more precise but requires first the simultaneous determination of $p_{c}$ and $\delta = \beta/\nu_{\|}$ and then the determination of $\nu_{\|}$. The third exponent $\nu_{\perp}$ (or, equivalently, $z = \nu_{\|}/\nu_{\perp}$) is obtained by finite-size scaling. Figure \[fig:pxl\] shows the estimated critical points $p_{c}(L)$ obtained directly from the stationary density profiles $\rho_{L}(p)$. A linear regression (LR) fit to the data gives $p_{c}(\infty) = 0.4880(1)$ (with a puny $R^{2}=0.776$, though). The inset in the figure exhibits the data from the mean-field approximation of orders $n=2$, 3, and 4 (remember that the $n=1$ and $n=2$ approximations give the same results). The extrapolated $p_{c}^{(\infty)} = 0.5138$ from an LR fit to these data ($R^{2}=0.984$) overshoots the more reliable value obtained from the Monte Carlo simulations and is given here just to illustrate the slow convergence of the mean-field approximation with $n$. As we have already mentioned, the determination of $\beta$ through a log-log plot of $\rho(\varepsilon) \sim \varepsilon^{\beta}$ close to the critical point is a poor choice of method. A better option is to plot the time-dependent profile $\rho_{L}(t)$ close to $p_{c}$ for some large $L$. On the critical point, $\rho(t) \sim t^{-\delta}$ and we can estimate $p_{c}$ and $\delta$ simultaneously by plotting $\log_{b}[\rho_{L}(t/b)/\rho_{L}(t)]$ against $1/t$ for some small $b$. Our data for $\rho_{L}(t)$ appear in Figure 5a and represent averages over $1000$ realizations of the dynamics, all with initial density $\rho_{L}(0)=0.2$ in a lattice of $L=16000$ sites. From these data we estimated $p_{c}=0.48810(5)$. The data for the instantaneous values of $\delta$ using $b=10$ appear in Figure 5b. Our best estimate for this exponent comes from the curve with $p=0.4881$ and is given by $\delta = 0.17(1)$. -- -- -- -- The exponent $\nu_{\|}$ can be obtained by plotting $t^{\delta}\rho_{L}(t)$ versus $t \varepsilon^{\nu_{\|}}$ and tuning $\nu_{\|}$ to achieve data collapse with different $\varepsilon$. Similarly, by plotting $t^{\delta}\rho_{L}(t)$ versus $t/L^{z}$ at $p_{c}$ for different $L$ and tuning $z$ until data collapse we can obtain $z = \nu_{\|}/\nu_{\perp}$. The collapsed curves for our data appear in Figure \[fig:zet\]. Figure 6a was obtained with $p=0.48812$, $\delta=0.16$, and $\nu_{\|}=1.7$. We can comfortably estimate $\nu_{\|}$ only within $\pm 0.05$, so that our best estimate for this exponent is $\nu_{\|} = 1.70(5)$. Combining $\delta$ and $\nu_{\|}$ furnishes $\beta = \delta \nu_{\|} = 0.29(2)$. Data collapse in Figure 6b was obtained for data taken at $p = 0.4881$ with $\delta = 0.165$ and $z = 1.55$. As before, we could determine $z$ only within $\pm 0.05$, so that our best estimate for it reads $z = 1.55(5)$. The exponents $\delta$, $\nu_{\|}$, and $z$ together determine the universality class of critical behaviour of the model, other exponents following from well known hyperscaling relations [@haye]. The best values available for the critical exponents of the DP process on the $(1+1)D$ square lattice are $\delta_{\rm DP} = 0.159\,464(6)$, $\nu_{\|{\rm DP}} = 1.733\,847(6)$, $\beta_{\rm DP} = 0.276\,486(8)$, and $z_{\rm DP} = 1.580\,745(10)$ [@jensen]. Thus, within the error bars our data put the critical behaviour of PCA $p182$–$q200$ in the DP universality class. Our estimation of $\delta$ is somewhat high and limitrophe to the accepted $\delta_{\rm DP}$. We probably underestimated the uncertainty in this quantity. This higher value of $\delta$ impacted the estimation of $\nu_{\|}$ and $z$, which turned out to be smaller than the accepted values, but in these cases still within the error bars. Anyway, we can safely establish the universality class of critical behaviour of PCA $p182$–$q200$ as that of the DP process, as no other universality class has exponents close to the values found for it. -- -- -- -- Summary and perspectives {#summary} ======================== We showed that the mixture of CA rules $182$ and $200$ into a PCA suffers an extinction-survival-type phase transition at $p_{c}=0.48810(5)$ in the $(1+1)D$ directed percolation universality class of critical behaviour. Although PCA $p182$–$q200$ has two absorbing configurations, $00{\cdots}0$ and $11{\cdots}1$, this second absorbing configuration is hardly achieved, since neither CA $182$ nor CA $200$ have a stationary density of active sites close to $1$ except if the initial configuration is $11{\cdots}1$ itself. We also found that PCA $p182$–$q200$ has some unusual features, notably its slow dynamics for very small $p$, when it approaches the stationary state diffusively with a large relaxation time proportional to $1/p$, and the bump in its density profile. We would like to delve further into the small $p$ behaviour of PCA $p182$–$q200$. In this regard, it would not be entirely without interest to consider a stochastic reaction-diffusion version of the PCA consisting of the reactions $100 \rightleftharpoons 110$, $001 \rightleftharpoons 011$, and $010 \to 000$, making up a sort of constrained, partially reversible contact process. In this model, clusters $11{\cdots}1$ never coalesce and can only be eroded from the boundaries. It is also related with an RSOS interface model introduced some time ago [@aehm]. We guess that, depending on the rate of the process $010 \to 000$, it displays an extinction-survival phase transition. 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Ódor, “Universality classes in nonequilibrium lattice systems,” [[*Rev. Mod. Phys.*]{} [**76**]{}, 663–724 (2004)](http://dx.doi.org/10.1103/RevModPhys.76.663). [^1]: Corresponding author. [^2]: Email: [[](mailto:[email protected])]{} [^3]: Email: [[](mailto:[email protected])]{} [^4]: An “atlas” of elementary CA can be consulted at [http://atlas.wolfram.com/01/01/]{}.

--- abstract: 'The spin weighted spherical harmonic (SWSH) description of angular functions typically is associated with the Newman-Penrose (NP) null tetrad formalism. Recently the SWSH description, but not the NP formalism, has been used in the study of the polarization anisotropy of the cosmic microwave background. Here we relate this application of SWSHs to a description of electromagnetic radiation and polarization in the NP formalism. In particular we introduce NP Stokes fields that are the NP equivalent of the Stokes parameters. In addition to giving a more coherent foundation for the recent cosmological SWSH application, the NP formalism aids in the computation of the Lorentz transformation properties of polarization.' author: - 'Ezra T. Newman' - 'Richard H. Price' title: NP Stokes fields for radio astronomy --- Introduction {#sec:intro} ============ To describe relativistic fields, the null tetrad, or Newman-Penrose(NP) formalism[@NP], uses a set of four null spacetime vectors ${\vec\ell},\vec{n}, \vec{m}, \vec{m^*}$ defined at every point in some region of spacetime. In the NP formalism, rather than deal with the components of a tensor, say the Maxwell tensor $F_{\mu\nu}$ for electromagnetism, one uses the projections of tensors on this tetrad. Because the tetrad field is chosen to satisfy certain properties, it turns out that the NP fields, the projected quantities, are convenient for the mathematics of radiation fields. Since the tetrad legs themselves have angular properties, projection with them adds some angular dependence not inherent in the physical field themselves. This additional angular dependence means that in general NP fields should not be expanded in ordinary spherical harmonics. Rather, all the simplicity of spherical harmonic expansions is regained if NP fields are expanded in sets of angular functions called spin weighted spherical harmonics (SWSHs)[@Goldbergetal67]. SWSHs have recently played a significant role in research on cosmological anisotropies. The directional nature of linear polarization complicated comparisons of linear polarization in different sky directions, and limited polarization studies to small angular comparisons. In 1996 Seljak[@Seljak96] found that certain combinations of the Stokes parameters $Q$ and $U$ were particularly well suited to dealing with the angular properties of linear polarization. This insight was still limited, as had been earlier work, to comparisons over small angular regions. That restriction was removed soon after in the breakthrough work by Seljak and Zaldarriaga[@SeljaZaldarriaga97PRL; @ZaldSeljak97] in which SWSHs were introduced. There soon followed papers[@HuWhite97; @NgLiu] with further mathematical details of the use of SWSHs for cosmological anisotropies. This application of SWSHs to cosmology exploited their connection to the rotation group [@GelfanMinlosShapiro58], rather than the role that the SWSHs typically play in physics. To our knowledge, none of the papers on the application of SWSHs to cosmological anisotropies makes reference to the NP formalism itself. One of our purposes here is to show the very natural connection that exists. In this paper we introduce a NP formalism for linear polarization and for Stokes parameters, and we show in this formalism how SWSHs very naturally arise. We argue, furthermore, that the NP formalism has advantages beyond that of giving an explanatory view of a technique already in use. Since we will only be considering a description of the fields at an observer location, not the propagation of the field, it is appropriate, for simplicity, to limit ourselves to the notation of special, not general relativity. But we make this simplification with some regrets since the NP formalism is extremely useful for dealing with the propagation of fields in curved spacetime. For some problems in curved spacetime, in fact, it is almost indispensable. We will use the standard special relativistic metric with sign convention +--- and with the speed of light $c$ set to unity, so that in Minkowski coordinates $$\label{eq:srtmetric} ds^2=dt^2-dx^2-dy^2-dz^2\,.$$ Greek letters $\{\mu,\nu...\}$ will indicate indices on 4-vectors, and Latin letters $\{i,j,...\}$ will indicate indices on 3-vectors, also called “spatial vectors.” Arrows over symbols indicate 4-vectors; boldface symbols will be used for 3-vectors. For 3-vectors, such as the electric and magnetic vectors, subscript and superscript indices are equivalent since the bases used are orthonormal and the 3-metric is $\delta_{ij}$. The 3-vector components are equivalent to the spatial contravariant (superscript) components of a corresponding 4-vector. We use the same root symbol, $\ell$, for the multipole index and for one of the null legs of the NP tetrad. The difference will be clear from context, since in the latter usage it will either appear as a 4-vector $\vec{\ell}$ or will be represented by its components $\ell^\mu$. ![Direction of wave propagation and basis vectors.[]{data-label="fig:pvector"}](pvector){width=".3\textwidth"} The NP formalism {#sec:NPform} ================ The situation we consider is that of an observer and an antenna at our coordinate origin. Radiation from all directions arrives at the origin, although not necessarily isotropically. This radiation can be considered to be a distribution of plane waves, and we restrict attention to waves propagating in a narrow range of directions around some particular direction $\widehat{{\boldsymbol{r}}}$. As shown in Fig. \[fig:pvector\], this direction can also be specified with the angles $\theta$ and $\phi$. The Cartesian components of the vector $\widehat{\bf r}$ can be written as $$\{\widehat{r}_x,\widehat{r}_y,\widehat{r}_z\}=\{\sin\theta\cos\phi,\sin\theta\sin\phi, \cos\theta\}\,.$$ It will be very useful to have, in addition to the Cartesian basis, a basis consisting of $\widehat{{\boldsymbol{r}}}$ and the two orthogonal unit vectors tangent to the celestial sphere, $\widehat{{\boldsymbol{\theta}}}$ and $\widehat{{\boldsymbol{\phi}}}$, as shown in Fig. \[fig:pvector\]. The direction in which an antenna must be pointed to receive this radiation is, of course, $-\widehat{{\boldsymbol{r}}}$, so that the sky position the observer would assign to the source of this radiation is the antenna direction $\theta_{(A)},\phi_{(A)}$ given by $\theta_{(A)}\equiv\pi-\theta$ and $\phi_{(A)}\equiv\phi+\pi$. We now use the null tetrad to help in the discussion of radio waves propagating in the $\widehat{{\boldsymbol{r}}}$ direction. The NP null tetrad consists of two real null 4-vectors $\vec{\ell}$ and $\vec{n}$, and the complex null vector $\vec{m}$, which can usefully be considered to be equivalent to two null 4-vectors $\vec{m}$, and its complex conjugate $\vec{m}^*$. The four null vectors must have all dot products to be zero except for the dot products in the following normalization conditions $$\label{eq:normcond} \vec{\ell}\cdot\vec{n}=1\quad\quad \vec{m}\cdot\vec{m}^*=-1\ .$$ In terms of the notation just defined, we choose our NP tetrad to be the null vectors with the following contravariant components $$\begin{aligned} \ell^{\mu}=\{\ell^0,\ell^j\}&=&2^{-1/2} \{1,\widehat{r}_j\}\label{tetraddef1}\\ n^{\mu}&=&2^{-1/2} \{1,-\widehat{r}_j\}\label{tetraddef2}\\ m^{\mu}&=&2^{-1/2} \{0,-\widehat{\theta}_j-i\widehat{\phi}_j\}\,.\label{tetraddef3}\end{aligned}$$ The Minkowski components of these tetrad legs are $$\begin{aligned} \ell^{\mu}&=&\{\ell^0,\ell^x,\ell^y,\ell^z\}=2^{-1/2}\label{minkcomp1} \{1,\; \sin{\theta}\cos{\phi},\;\sin{\theta}\sin{\phi},\;\cos{\theta}\}\label{minkcomp2}\\ n^{\mu}&=&2^{-1/2}\{1,\; -\sin{\theta}\cos{\phi},\;-\sin{\theta}\sin{\phi},\;-\cos{\theta} \}\\ m^{\mu}&=&2^{-1/2}\{0,\; -\cos{\theta}\cos{\phi}+i\sin{\phi},\; -\cos{\theta}\sin{\phi}-i\cos{\phi},\;\sin{\theta}\}\,.\label{minkcomp3}\end{aligned}$$ It should be noted that all physical fields in the beam can be functions of spacetime location $x^\mu$ only through the combination $k_\mu x^\mu\propto t-r_jx^j$, where $\vec{k}$ is the propagation 4-vector of the beam. It follows that for the beam propagating in the $\widehat{{\boldsymbol{r}}}$ spatial direction $k_\mu\propto\{1,-r_j\}$, and $k^\mu\propto\{1,r_j\}$. Thus our tetrad leg $\vec{\ell}$ is proportional to the propagation 4-vector $\vec{k}$. In the NP formalism the six independent pieces of information about the electromagnetic field at a spacetime point are represented by the following complex projections[@Fmunuconvention] of the Maxwell tensor $F_{\mu\nu}$ (see the Appendix of Ref. [@NP]): $$\begin{aligned} \Phi_0&\equiv& F_{\mu\nu}\ell^\mu m^\nu= \textstyle{\frac{1}{2}}\left[ -E^{{{\hat{\theta}}}}+B^{{{\hat{\phi}}}}-i\left(E^{{{\hat{\phi}}}}+B^{{{\hat{\theta}}}}\right) \right]\nonumber\\ \Phi_1&\equiv &\textstyle{\frac{1}{2}}F_{\mu\nu}\left( \ell^\mu n^\nu-m^\mu m^{*\nu}\right)= -\textstyle{\frac{1}{2}} \left[E^{\hat{r}}+iB^{\hat{r}} \right] \label{eq:Phidefs}\\ \Phi_2&\equiv& F_{\mu\nu}m^{*\mu} n^\nu=\textstyle{\frac{1}{2}} \left[ E^{{{\hat{\theta}}}}+B^{{{\hat{\phi}}}} +i\left(-E^{{{\hat{\phi}}}}+B^{{{\hat{\theta}}}} \right)\right]\,.\nonumber\end{aligned}$$ Here the components $E^{\hat{r}}, B^{\hat{\theta}}$ etc., represent the ordinary components of the electric and magnetic fields with respect to the orthonormal basis system shown in Fig. \[fig:pvector\]. Any of the NP projections is characterized by two important properties. The first is its [*spin weight*]{}, the number of $\vec{m}$ vectors used in the projection, minus the number of $\vec{m}^*$s used. This spin weight is connected to the spin weight of the SWSHs. In fact, SWSHs consist of sets of spherical harmonics of specific spin weights. A spin weight $s$ quantity must be expanded in spin weight $s$ spherical harmonics to realize the simplicity mentioned in Sec. \[sec:intro\]. In view of the close connection of projected fields of specific spin weight, and SWSHs, it is understandable that the use of SWSHs in physics is traditionally associated with the NP formalism. The properties of SWSHs have been described in detail both in connection with the NP formalism[@Goldbergetal67; @NewmanPenrose66] and for use with cosmological anisotropies[@ZaldSeljak97; @HuWhite97; @NgLiu]. We will not need those details here; we need only the motivation that the SWSHs have turned out to be important in the description of the linear polarization of the CMB. The second important property of NP fields is that each has a definite [*boost weight*]{} (also called [*conformal weight*]{}). This is the number of $\vec{\ell}$s minus the number of $\vec{n}$s that are used in projecting the NP field. Just as spin weight determines rotation properties, the boost weight determines properties under Lorentz transformations. By using the NP description of linear polarization, as we show in Sec. \[sec:Lorentz\], it is straightforward to calculate the changes measured under a general Lorentz boost. Stokes parameters in the NP formalism {#sec:NPrad} ===================================== We note that the plane waves traveling in the direction shown in Fig. \[fig:pvector\] must have their Poynting vector directed in the $\widehat{{\boldsymbol{r}}}$ direction, and hence must have $E^{\hat{r}}=B^{\hat{r}}=0$, $E^{\hat{\theta}}=B^{\hat{\phi}}$ and $E^{\hat{\phi}}=-B^{\hat{\theta}}$. In this case, $\Phi_0$ and $\Phi_1$ vanish, and all the electromagnetic information is carried in $\Phi_2$ which reduces to $$\label{eq:Phi0rad} \Phi_2= E^{\hat{\theta}}-iE^{\hat{\phi}}\,.$$ In the description of Stokes parameters a complex representation of the electric field is typically used. Because the NP quantities have their own complex nature, we avoid the more typical approach and describe a frequency component of the electric field as $$\label{calEjdef} E^{\hat{\theta}}={\cal E}^\theta \cos{(\omega t+\delta_\theta(\omega))}\quad\quad\quad E^{\hat{\phi}}={\cal E}^\phi \cos{(\omega t+\delta_\phi(\omega))}\,.$$ We next define a field that is phase shifted by $\pi/2$: $$\begin{gathered} \label{phaseshift} \widetilde{\Phi_2}= {\cal E}^\theta \cos{(\omega t+\delta_\theta(\omega)+\pi/2)} -i {\cal E}^\phi \cos{(\omega t+\delta_\phi(\omega)+\pi/2)} \\ = - {\cal E}^\theta \sin{(\omega t+\delta_\theta(\omega))} +i {\cal E}^\phi \sin{(\omega t+\delta_\phi(\omega))}\,.\end{gathered}$$ To construct the NP quantities for radio astronomy we average over the range of frequencies of interest to form the following “NP Stokes fields” $$\label{allcals} {\cal I}=2\Bigl\langle\Phi_2\Phi_2^* \Bigr\rangle \quad\quad\quad {\cal S}=2\Bigl\langle\Phi_2\Phi_2 \Bigr\rangle \quad\quad\quad {\cal V}=2i\Bigl\langle{\Phi_2}\widetilde{\Phi_2}^* \Bigr\rangle\,.$$ The angle brackets $\langle\rangle$ denote the average over some frequency interval. Unless the radiation of interest is totally unpolarized we assume that $\delta_\theta\equiv\langle\delta_\theta(\omega)\rangle$ and $\delta_\phi\equiv\langle\delta_\phi(\omega)\rangle$ are not both zero. The first of these quantities is straightforward to evaluate with Eq.  to show that $${\cal I}=2\left\langle(E^{\hat{\theta}})^2 +(E^{\hat{\phi}})^2 \right\rangle =({\cal E}^\theta)^2+({\cal E}^\phi)^2\equiv I.$$ The last symbol in this equation is the usual Stokes parameter $I$. (See, e.g., Sec. 7.2 of Ref. [@Jackson3rd]). Up to multiplicative factors this quantity is clearly just the intensity of radiative energy, i.e., the magnitude of the Poynting vector. The NP quantity ${\cal I}$ has spin weight zero, in complete accord with what one would expect on physical grounds, since the intensity, or temperature, is a scalar under rotations. In terms of the electric field, the NP quantity ${\cal S}$ has the form $${\cal S}=2\left\langle(E^{\hat{\theta}})^2 -(E^{\hat{\phi}})^2 -2i E^{\hat{\theta}}E^{\hat{\phi}} \right\rangle = ({\cal E}^\theta)^2-({\cal E}^\phi)^2- 2i {\cal E}^\theta {\cal E}^\phi \cos{(\delta_\theta - \delta_\phi )} =Q-iU\,,$$ where $Q$ and $U$ are the usual Stokes parameters of the radiation. (See, e.g., Sec. 7.2 of Ref. [@Jackson3rd]). Unlike ${\cal I}$, the ${\cal S}$ field has spin weight -2, which means that an expansion of ${\cal S}$ in functions on the sphere is best carried out with spin weight -2 harmonics[@orplus2]. The fact that this spin weight -2 NP Stokes field (or its spin weight +2 complex conjugate) carries the information about linear polarization explains the applicability of spin weight $\pm2$ SWSHs to the studies of the linear polarization of the CMB. In terms of the electric field the third of our NP Stokes fields is $$\label{calVdef} {\cal V}=2{\cal E}^\theta {\cal E}^\phi \sin{(\delta_\phi - \delta_\theta )}= V\,,$$ where $V$ is the last of the usual Stokes parameters. Like ${\cal I}$, and unlike ${\cal S}$, the field ${\cal V}$ has spin weight 0. The mathematical reason for this is that the Stokes parameter $V$ is invariant with respect to rotations about the direction of propagation. Like ${\cal I}$, the circular polarization measure ${\cal V}$ should be expanded in ordinary spherical harmonics. (We mention, only as an aside, that at present cosmological models do not include processes that produce circular polarization.) All three NP Stokes fields have boost weight -2. We will see below that the boost weight of an NP quantity indicates the quantity’s transformation property under a Lorentz boost. In particular a boost weight $w$ quantity is multiplied by the Doppler shift (as defined in Sec. \[sec:Lorentz\]) raised to the power $w$. Thus, all the Stokes fields must be multiplied by the square of the Doppler shift under a boost. This is in accord with the fact that all the Stokes fields are quadratic in the transverse electric fields, which themselves are multiplied by a Doppler shift factor under a boost. (An alternate viewpoint is that every photon is Doppler shifted, and the rate of photon arrival is Doppler shifted, hence all radiative intensities gets doubly Doppler shifted.) Lorentz tranformations {#sec:Lorentz} ====================== Here we consider the behavior of the NP radiative quantities under a Lorentz transformation, the transformation of measured quantities associated with a change of observer rest frame, in particular for a “boost,” a transformation from the reference frame of an observer to the frame of a relatively moving observer. Transformations of Stokes parameters have been considered by Challinor and van Leeuwen [@ChallinorvanLeeuwen], who did not use SWSHs or the NP formalism; rather they confined their study to small angular scales. We show here that the NP formalism removes the restriction of small angular scales for Lorentz transformations, just as it does for angular correlations. The complete set of Lorentz transformations of reference frame can be considered to be equivalent to the transformations of the null tetrad that maintain the conditions in Eq. . The group of transformations of the tetrad is usefully divided into several separate subclasses[@Kinnersley1969; @Price1972II], as follows.\ Class I: $$\begin{aligned} \vec{n}^{\,\prime}&= & \vec{n} \label{eq:I1}\\ \vec{m}^{\,\prime}&=&\vec{m}+b\,\vec{n}\label{eq:I2}\\ \vec{\ell}^{\,\prime}&= & \vec{\ell}+b\,\vec{m}^*+b^*\vec{m}+bb^*\vec{n}\,,\label{eq:I3}\end{aligned}$$ Class II: $$\begin{aligned} \vec{\ell}^{\,\prime}&=&\vec{\ell}\label{firstclass1}\\ \vec{m}^{\,\prime}&=&\vec{m}+a\vec{\ell}\label{firstclass2}\\ \vec{n}^{\,\prime}&=&\vec{n}+a\,\vec{m}^*+{a}^*\vec{m} +a{a}^*\vec{\ell} \label{firstclass4}\,,\end{aligned}$$ Class III: $$\label{eq:III} \vec{m}^\prime= e^{i\lambda}\vec{m}\,,\quad\quad\vec{\ell}^{\,\prime}=\vec{\ell}\,, \quad\quad\vec{n}^{\,\prime}=\vec{n}\,,$$ Class IV: $$\vec{m}^\prime= \vec{m}\,,\quad\quad\vec{\ell}^{\,\prime}=K\vec{\ell}\,, \quad\quad\vec{n}^{\,\prime}=K^{-1}\vec{n}\,.\label{eq:IV}$$ The two complex parameters $a,b$ and the two real parameters $\lambda$ and $K$ contain the six degrees of freedom of the Lorentz group. They can be related to the usual boost and rotation parameters of the coordinate basis vectors of observers’ reference frames if a specific association is made between the null tetrad $\{\vec{\ell}, \vec{n},\vec{m},\vec{m}^*\}$ and the reference frame of an observer. Since the radiative quantities defined in Eqs.  are constructed from the tetrad legs, the transformations in Eqs. – induce transformations to new radiative quantities $\Phi_0^{\prime}, \Phi_1^{\prime},\Phi_2^{\prime}$, as follows.\ Class I: $$\begin{aligned} \Phi_0^\prime&=&\Phi_0+2b\Phi_1+b^2\Phi_2\label{ClassIPhifirst}\\ \Phi_1^\prime&=&\Phi_1+b\Phi_2\\ \Phi_2^\prime&=&\Phi_2\end{aligned}$$ Class II: $$\begin{aligned} \Phi_0^\prime&=& \Phi_0\\ \Phi_1^\prime&=&\Phi_1+a^*\Phi_0\\ \Phi_2^\prime&=&\Phi_2+2a^*\Phi_1+a^{*2}\Phi_0\end{aligned}$$ Class III: $$\Phi_0^\prime=e^{i\lambda}\Phi_0\,, \quad\quad \Phi_1^\prime=\Phi_1\,, \quad\quad \Phi_2^\prime=e^{-i\lambda}\Phi_2$$ Class IV: $$\Phi_0^\prime=K\Phi_0\,, \quad\quad \Phi_1^\prime=\Phi_1\,, \quad\quad \Phi_2^\prime=K^{-1}\Phi_2 \label{ClassIPhilast}\,.$$ We are now in a position to apply this mathematical infrastructure to the radiation problem. We consider two relatively moving observers ${\cal O}$ and ${\cal O}^{\,\prime}$, with 4-velocities $\vec{u}$ and $\vec{u}^{\,\prime}$, and we focus our attention on a particular beam of radiation. The interpretation of the NP Stokes fields given in Secs. \[sec:NPrad\] requires that $\vec{\ell}$ be in the null direction of propagation of the radiation and that $\vec{\ell}\cdot\vec{u}=1/\sqrt{2\;}$. Since we want both observers to be making the same physical measurements, only in different frames, we must have that $\vec{\ell}^{\,\prime}$ is also in the null direction of propagation (the same spacetime direction as $\vec{\ell}$) and that $\vec{\ell}^{\,\prime}\cdot\vec{u}^{\,\prime}=1/\sqrt{2\;}$. From the requirement that $\vec{\ell}$ and $\vec{\ell}^{\,\prime}$ be in the same null direction, we infer that the tetrads of the two observers must be related only by transformations of Classes II, III, and IV, since the transformations of Class I in Eqs. –, change the direction of $\vec{\ell}$. Equivalently, $b$ must vanish and the relationship of the measurements of the two observers must depend only on the real parameters $K,\lambda$ and on the complex parameter $a$. An interesting conclusion now follows. As pointed out at the start of Sec. \[sec:NPrad\], $\Phi_0=\Phi_1=0$. The transformations in Eqs. –, with $b=0$, then tell us that $\Phi_0^\prime=\Phi_1^\prime=0$. This, of course, must be the case on physical grounds. Furthermore, these relations tell us that the most general nontrivial transformation of $\Phi_2$ is $\Phi_2^\prime=e^{-i\lambda}K^{-1}\Phi_2$. For the NP Stokes fields this means that $$\label{NPStokesxform} {\cal I}^\prime=K^{-2}{\cal I}\,, \quad\quad {\cal S}^\prime=e^{-2i\lambda}K^{-2}{\cal S}\,, \quad\quad {\cal V}^\prime=K^{-2}{\cal V}\,.$$ In a Lorentz transformation, then, the polarization properties in ${\cal S}$ are affected by both $K$ and by the $\lambda$ parameter of a Class III transformation, while the other two NP Stokes fields, as well as the magnitude of ${\cal S}$, are affected only by $K$. The detailed relationship of the parameters $K$ and $\lambda$, and the frames of observers, are given in Appendix \[genLXform\]. Here we only give the result for $K$. If ${{\boldsymbol{\beta}}}$ is the 3-velocity of a frame ${\cal O}^\prime$, as observed by ${\cal O}$, then $$K =\frac{\sqrt{1-\beta^2\;}}{1-{\bf\hat{r}}\cdot{{\boldsymbol{\beta}}}}\,.$$ This is the well known Doppler factor that relates observations of radiation by the two observers, and is the ratio $E/E^\prime$ of the observed energies. Although the formal transformation properties in Eq.  are correct, they cannot directly be related to observations of the CMB. The Stokes parameters, and the NP Stokes fields, measure radiative power per unit area. Radio telescopes, however, measure radiative power per unit area [*per unit frequency interval $d\nu$, per unit solid angle $d\Omega$*]{}. In considering transformation properties we must take into account that $$\label{nuOmegaxform} \nu^\prime=K^{-1}\,\nu\quad\quad d\Omega^\prime=K^2\,d\Omega\,.$$ Thus, a radio telescope does not measure e.g., the Stokes parameter $I$, but rather the specific intensity $$I_\nu\equiv \frac{dI}{d\nu d\Omega}\,.$$ From the transformation properties in Eqs.  and it follows that the specific intensity $I_\nu$ transforms according to $$I_\nu^\prime=K^{-3}I_\nu\,.$$ We can apply this to the NP Stokes fields and can introduce measureable quantities ${\cal I}_\nu$, ${\cal S}_\nu$, ${\cal V}_\nu$, defined per unit frequency per unit solid angle. The transformation laws for these quantities are those of Eq. , with $K^{-2}$ replaced by $K^{-3}$. These specific intensities are directly related to what is measured by radiotelescopes, but it is simple, and practical, to construct other quantities directly related to telescope observables. The specific intensities can be combined with the frequency at which the specific intensity is measured to form quantities $I_\nu/\nu^N$, and the equivalent for NP Stokes fields. For these fields the transformation properties are $$\label{genNPStokesxform} \left({\cal I}_\nu/\nu^N\right)^\prime=K^{N-3}\left({\cal I}_\nu/\nu^N\right) \quad\quad \left({\cal S}_\nu/\nu^N\right)^\prime=e^{-2i\lambda}K^{N-3}\left({\cal S}_\nu/\nu^N\right) \quad\quad \left({\cal V}_\nu/\nu^N\right)^\prime=K^{N-3}\left({\cal V}_\nu/\nu^N\right)\,.$$ In cosmology, the Lorentz transformation properties of radiation quantities are important for transforming observations to a “rest frame” of the CMB. In principle this is the frame that must be used for comparisons of observations and theories of early universe processes. The general prescription for finding this frame is to identify a dipole in the radiation. A transformation can then be made to a frame in which this dipole vanishes. The process, however, is not unique. One can, for example, choose to elminate the dipole in circular polarization rather than intensity. The problem of finding a rest frame is essentially the problem of understanding how the radiation gives a representation of the Lorentz group. The general theory of such representations can be based on the work of Gel’fand, Graev and Vilenkin[@GGV] and is sketched in Appendix \[app:GGV\]. With this theory, Lorentzian vectors and tensors are constructed from the multipoles, the 2-sphere integrals of radiation quantities. This issue of a choice of rest frames is taken up in greater detail in Appendix \[app:GGV\]. Summary and Conclusions {#sec:Conc} ======================= The introduction[@ZaldSeljak97] of SWSHs to the analysis of polarization anisotropy followed from a consideration of the explicit rotation properties of the Stokes parameters $Q,U$ and to the subsequent realization that a combination of the two had convenient transformation properties under rotation. We have shown here that the NP formalism leads very naturally to an NP formulation of the Stokes parameters, and that in this formulation the use of spin weight $\pm2$ SWSHs is immediate. We have also shown the convenience of the NP formalism for “boosts,” the transformtion of radiation properties from an oberver’s frame to the frame of a relatively moving observer. More specifically, we have shown that the NP formalism allows for the computation of boosts of any radiation quantity without the limitation of small angular scale. The convenience of the NP formalism is not a coincidence. Its mathematical underpinnings are deeply rooted in the structure of spacetime, so the formalism is very naturally suited to the description of fields that propagate at the speed of light. This can be taken as the reason that Stokes parameters arise simply as fields quadratic in the NP field describing electromagnetic radiation. This even more certainly explains the relative simplicity of boost transformations. The brief discussion given of the nonuniqueness of the “rest frame” of the CMB is closely associated with the convenience of the NP formalism for boosts. Since the advantages, or conveniences, follow from the appropriateness of the NP formalism for fields propagating along null directions in spacetime, they apply also to non-electromagnetic fields. In particular, gravitational waves, whether the standard transverse-traceless waves of Einstein’s theory, or the full set of six possible polarization states, all have gravitational Stokes parameters that follow from fields quadratic in the NP projections of the Riemann tensor. Lastly, we mention that the NP formalism is very naturally suited to calculations of the propagation of radiation in curved spacetime. In the cosmological context this means that the NP formalism should greatly simplify, for example, the calculation of the effect on polarization of gravitational lensing or inhomogeneous cosmological expansion. Acknowledgment ============== One of us (RHP) gratefully acknowledges support for this work under NSF grants PHY-0554367 and by the Center for Gravitational Wave Astronomy. We thank Arthur Kosowsky for very helpful discussions about the cosmological background for this work, and Teviet Creighton and Fredrick Jenet for discussions of statistical issues in connection with anisotropies. The general Lorentz transformation {#genLXform} ================================== To relate the value of $K$ to the usual parameterization of a Lorentz transformation we consider, as in Sec. \[sec:Lorentz\], a frame ${\cal O}^\prime$ observed in frame ${\cal O}$ to be moving with 3-velocity ${{\boldsymbol{\beta}}}$. The propagation 4-vector $\vec{k}$ of a plane electromagnetic wave (equivalently, the 4-momentum of a photon) must be parallel to $\vec{\ell}$, so we can write $\vec{k}=\kappa\,\vec{\ell}$, where $\kappa$ is proportional to the energy $E$ of a “photon” in frame ${\cal O}$. With $\vec{\ell}^{\,\prime}=K\vec{\ell}$, we have from this $$\frac{E^{\,\prime}}{E}=\frac{\vec{k}\cdot\vec{u}^{\,\prime}}{\vec{k}\cdot\vec{u}} =\frac{\vec{\ell}\cdot\vec{u}^{\,\prime}}{\vec{\ell}\cdot\vec{u}} =K^{-1}\,\frac{\vec{\ell}^{\,\prime}\cdot\vec{u}^{\,\prime}}{\vec{\ell}\cdot\vec{u}} =K^{-1}\,.$$ To get the specific dependence of $K$ on the relative direction of motion, and the direction of observation, we can use a straightforward evaluation of $\vec{\ell}\cdot\vec{u}^\prime$ in the ${\cal O}$ frame, in which $\vec{\ell}$ has the contravariant components in Eq. , and $\vec{u}^{\,\prime}$ has the contravariant components $$u^{\prime\mu}=\gamma\{1,\beta^j\}\,,$$ where $\gamma$ is the usual Lorentz factor $1/\sqrt{1-\beta^2\;}$. From these components we get $$\label{eq:elldotu} \vec{\ell}\cdot\vec{u}^{\,\prime}=\gamma \left(1-{\bf \hat{r}}\cdot{{\boldsymbol{\beta}}}\right)/\sqrt{2\;}.$$ But we have $\vec{\ell}= K^{-1}\vec{\ell}^{\,\prime}$ so that $\vec{\ell}\cdot \vec{u}^{\,\prime} =K^{-1}\vec{\ell}^{\,\prime}\cdot\vec{u}^{\,\prime}=K^{-1}/\sqrt{2\;}$. From this we have $K^{-1}=\sqrt{2\;}\, \vec{\ell}\cdot \vec{u}^{\,\prime}$ and $$\label{Kgammabeta} K=\frac{1}{\sqrt{2\;}\,\vec{\ell}\cdot\vec{u}^\prime} =\frac{1}{\gamma\left(1-{\bf\hat{r}}\cdot{{\boldsymbol{\beta}}}\right)}\,.$$ It should be noticed that we have not completely specified the Lorentz transformation between the reference frames of observers ${\cal O}$ and ${\cal O}^\prime$; we have specified only the relative velocity, not the rotations that relate them. From the point of view of the transformations in Eqs. –, we have fixed the complex parameter $b$ to be zero (since the direction of $\vec{\ell}$ is fixed) and we have fixed the real parameter $K$ with Eq. . What remains is the three degrees of freedom in the parameters $\lambda$ and $a$. These three degrees of freedom can be fixed independently for each direction $\theta, \phi$. That is, $\lambda$ and $a$ can, like $K$ in Eq. , be functions of ${\bf\hat{r}}$. These three degrees of freedom are equivalent to the freedom to make spatial rotations connecting frames ${\cal O} $ and ${\cal O}^\prime$. The choice of the orientations of these frames is equivalent to a specification of how the null tetrad legs are chosen. We start by considering observer ${\cal O}$ with 4-velocity $\vec{u}$ and radiation in a particular direction. In Eq.  we chose $\vec{\ell}$ in the direction of propagation, and normalized $\vec{\ell}$ so that $\vec{u}\cdot\vec{\ell}=1/\sqrt{2\;}$, for reasons already discussed. We next, in Eq. , defined $\vec{n}$ by $$\label{ndef} \vec{n}\equiv\sqrt{2\;}\vec{u}-\vec{\ell}\,.$$ This choice guarantees that $\vec{n}$ is null and satisfies the normalization condition in Eq. . It also means that for any direction of incoming radiation we have a unit “outward” spatial vector $\widehat{{\boldsymbol{r}}}\equiv(\vec{n}-\vec{\ell})/\sqrt{2\;}$ that is orthogonal to $\vec{u}$, and that is parallel to the spatial direction of propagation. (The vector $\widehat{{\boldsymbol{r}}}$ is just $\sqrt{2\;}$ times the projection of $\vec{\ell}$ orthogonal to $\vec{u}$.) In addition, the form of $\vec{n}$ in Eq.  provides a unit timelike vector $\widehat{t}\equiv(\vec{n}+\vec{\ell})/\sqrt{2\;}=\vec{u}$ that is independent of the direction of propagation of the radiation. In short, we have a tetrad compatible with the $\vec{\ell}$ and $\vec{n}$ legs of Eqs.  and . We also make the choice $$\label{nprimedef} \vec{n}^{\,\prime} =\sqrt{2\;}\vec{u}^{\,\prime}-\vec{\ell}^{\,\prime}$$ so that the tetrad for ${\cal O}^\prime$ is also compatible with the conditions in Eqs.  and . From Eqs. –, the set[@orderofXform] of allowed transformations (with $b=0$) of the tetrad for ${\cal O}^\prime$ in terms of the tetrad for ${\cal O}$ is $$\begin{aligned} \vec{\ell}^{\,\prime}&=& K \vec{\ell}\label{eq:full1}\\ \vec{m}^{\,\prime}&=& e^{i\lambda} \left(\vec{m}+ a\vec{\ell}\right)\label{eq:full2}\\ \vec{n}^{\,\prime}&=& K^{-1} \left(\vec{n}+a\vec{m}+a^*\vec{m}^{\,*} +aa^*K\vec{\ell}\right)\label{eq:full3}\,.\end{aligned}$$ For simplicity, we first make the restriction that the $z$ axis is chosen in the direction of ${\boldsymbol{\beta}}$. (The ${\boldsymbol{\beta}}$ direction can always be put in the $z$ direction by using pure rotations.) We then set equal the right-hand sides of the expressions for $\vec{n}^{\,\prime}$ given in Eqs.  and $$\sqrt{2\;}\gamma\{1,{{\boldsymbol{\beta}}}\}-K\{1,\widehat{r}\} =K^{-1}\left[ \{1,-\widehat{r}\} +a\{0,-\widehat{\theta}-i\widehat{\phi}\} +a^*\{0,-\widehat{\theta}+i\widehat{\phi}\} +aa^*K\{1,\widehat{r}\} \right]\,.$$ Taking the dot product of this equation, first with $\widehat{\theta}$ and then with $\widehat{\phi}$, for ${{\boldsymbol{\beta}}}$ in the $z$ direction, gives us $$\begin{aligned} \sqrt{2\;}\gamma\beta\sin\alpha&=&K^{-1}\left[\left(a/\sqrt{2\;}\right) +\left(a^*/\sqrt{2\;}\right) \right] \\ 0&=&K^{-1}\left[\left(a/\sqrt{2\;}\right) \left(i \right) +\left(a^*/\sqrt{2\;}\right)\left(-i\right) \right] \,,\end{aligned}$$ where $\alpha$ is the angle between the directions of ${\boldsymbol{\beta}}$ and $\widehat{r}$. (This is also $\theta$ for ${\boldsymbol{\beta}}$ in the $z$ direction.) From this we conclude $$\label{aforzxform} a=\beta\gamma K\sin\alpha=\frac{\beta \sin\alpha}{1-\beta\cos\alpha}\,.$$ Finally, we note that the direction of $\widehat{\phi}$ is invariant under a Lorentz boost, since $\widehat{\phi}$ is orthogonal to ${\boldsymbol{\beta}}$. Since $\vec{m}\cdot\widehat{\phi}=\vec{m}^{\prime}\cdot \widehat{\phi}^\prime$, this means that $\vec{m}\cdot\widehat{\phi}=\vec{m}^{\prime}\cdot \widehat{\phi}$. In addition, $\vec{m}\cdot\vec{\ell} =0$, so that the dot product of $\widehat{\phi}$ with Eq.  gives us $\lambda=0$. In summary, for a pure boost in the $z$ direction the tetrad transformation parameters are $$\label{zxformresults} \lambda=b=0\quad\quad\quad K=\frac{1}{\gamma(1-\beta\cos\alpha)} \quad\quad a=\frac{\beta\sin\alpha}{(1-\beta\cos\alpha)}\,.$$ It is interesting that $\lambda=0$ for this pure boost in the $z$ direction. According to Eq.  this means that ${\cal S}/{\cal I}$ is invariant for such a transformation, and therefore that both observers will agree on the linear polarization of the radiation. We now turn to the general transformation, with ${{\boldsymbol{\beta}}}$ not necessarily in the $z$ direction. We have already constructed $\vec{\ell}^{\,\prime}$ and $\vec{n}^{\,\prime}$ from the propagation vector and the 4-velocity $\vec{u}^{\,\prime}$. What remains is to find $\vec{m}^{\,\prime}$. To do this we note that from the forms of $\vec{m}$ and $\vec{\ell}$ given in Sec. \[sec:NPform\] we have $$\label{mfromell} \vec{m}\left(\theta,\phi\right) =-\,\left(\partial_\theta+(i/\sin\theta)\partial_\phi\right) \vec{\ell}\left(\theta,\phi\right)\,.$$ An analogous relationship must obtain in the primed frame, so we can write $$\vec{m}^{\prime}\left(\theta^{\prime},\phi^{\prime}\right) =-\,\left(\partial_{\theta^{\prime}}+(i/\sin\theta^{\prime})\partial_{\phi^{\prime}}\right) \vec{\ell}^{\,\prime}\left(\theta^{\prime},\phi^{\prime}\right) =-\,\left(\partial_{\theta^{\prime}}+(i/\sin\theta^{\prime})\partial_{\phi^{\prime}}\right) K(\theta,\phi)\vec{\ell}\left(\theta,\phi\right)\,.$$ In order to compute the right hand side of this equation, we must know $\theta,\phi$ as functions of $\theta^{\prime},\phi^{\prime}$ for an arbitrary Lorentz transformation. The general Lorentz transformation, for an arbitrary boost combined with an arbitrary rotation, requires the specification of six parameters. It is convenient here to utilize the description and parameterization related to the “aberration transform,” as described e.g., in Ref. [@NPpedapaper] with a slightly different notation[@caps]. In this description the direction of photon propagation $\theta^\prime,\phi^\prime$ in frame ${\cal O^\prime}$ is related to the direction $\theta,\phi$ for the same photon in frame ${\cal O}$ by $$\label{aberration} e^{i\phi^\prime}\cot{(\theta^\prime/2)} =\frac{ A\, e^{i\phi}\cot(\theta/2)+B }{ C\,e^{i\phi}\cot(\theta/2)+D }\,.$$ Here $A,B,C,D$ are complex numbers. The eight degrees of freedom in these complex numbers are constrained by the two conditions in $AD-BC=1$, leaving six unconstrained degrees of freedom in the transformation . Any proper (non-time reversing, non-parity reversing) Lorentz transformation can be represented with these six degrees of freedom. Carrying out the differentiations indicated gives $$\label{genmxform} \vec{m}^{\prime}\left(\theta^\prime,\phi^\prime\right)=e^{i(\sigma-\phi+\phi^\prime)} \left(\vec{m}\left(\theta,\phi\right)+ J(\theta,\phi)\,\vec{\ell}\left(\theta,\phi\right)\right)\,,$$ where $$\label{edth3} J(\theta,\phi)=-\,\left(\partial_\theta+(i/\sin\theta)\partial_\phi\right) \left[ \log{K\left(\theta,\phi\right)} \right]\,,$$ and $$\label{sigma} e^{i\sigma}=\sqrt{ \frac{{C}^*e^{-i\phi}\cot(\theta/2)+{D}^*}{C\;e^{i\phi}\cot(\theta/2)\,+{D}} }\,.$$ The values of $a$ and $\lambda$ are inferred from a comparison of the result in Eq.  and Eq. . The special case of the pure boost in the $z$ direction is given by $B=C=0$ and $D=1/A$, with $A$ a real number. Equation  gives $\sigma=0$. Since $K$, from Eq. , in a slighly different notation, is $K=[\gamma(1-\beta\cos\theta)]^{-1}$ we have, from Eq. , $$J=\partial_\theta[\log{\gamma(1-\beta\cos\theta)}] =\frac{\beta\sin\theta}{1-\beta\cos\theta}\,,$$ which means, according to Eq.  that $$\vec{m}^\prime= \vec{m}+\frac{\beta\sin\theta}{1-\beta\cos\theta}\vec{\ell}$$ in agreement with Eqs. , for the case of a pure boost in the $z$ direction. Representations of the Lorentz group and the relevance to the CMB {#app:GGV} ================================================================= Following the beautiful theory of representations of the Lorentz group developed by Gel’fand, Graev and Velinkin (GGV)[@GGV], one finds that the different Stokes fields lying on the celestial sphere, lie in different vector spaces of infinite dimensional representations of the Lorentz group. In the notation of GGV, representations are given by homogeneous functions of two complex variables ($z_{1,}z_{2,}\overline{z}_{1,}\overline{z}_{2,}$), each representation being labeled by the homogeneity degree, ($n_{1}-1,n_{2}-1$) of each pair of the variables. We consider only the so-called integer representations, for which $n_{1}$ and $n_{2}$ are either both positive or both negative integers. These representations, via homogeneous functions, can be mapped into functions on the sphere with well defined spin and conformal weights, $s$ and $w$. The $s$ and $w$ are related to $n_{1}$ and $n_{2}$ by[@HeldNewmanPosadas70] $$(n_{1},n_{2})=(w-s+1,w+s+1). \label{A1}$$ A variety of results can be extracted from the GGV work. If $n_{1}$ and $n_{2} $ are either both positive or both negative, one can find (or construct), from the infinite dimensional representation spaces, specific finite dimensional vector spaces that are equivalent to the standard tensor representations. (A simple example is that the $\ell=0$ harmonic coefficient of a $w=-2,s=0$ function is a Lorentz scalar. See below.) We review the basic argument and results for negative integer representations, taking for simplicity the special case of $s=0$. First, we note that the area element on the unit sphere (or equivalently, the solid angle as seen on the celestial sphere)[@HeldNewmanPosadas70] $$d\Omega =\sin \theta d\theta d\varphi , \label{A2}$$transforms (under the aberration transformation of Eq. ) as $$d\Omega ^{\prime }=K^{2}d\Omega . \label{A3}$$Second, we consider functions $F_{(-n-2)}$ and $G_{(n)}$ on the sphere, respectively with boost weight $-n-2$ and $n$, that is with Lorentz transformation properties $$F_{(-n-2)}^{\,\prime } =K^{-n-2}F_{(-n-2)}\quad\quad G_{(n)}^{\,\prime } =K^{n}G_{(n)}\,.$$ The integral $$\int F_{(-n-2)}G_{(n))}\,d\Omega$$ over the 2-sphere is therefore a Lorentz invariant: $$\int F_{(-n-2)}G_{(n))}\,d\Omega= \int F^\prime_{(-n-2)}G_{(n))}^\prime\,d\Omega^\prime\,.$$ In a similar manner, we can create Lorentz vectors and tensors. The following are several simple specific examples: \(a) If we choose $G_{(0)}=1$ and choose $F_{(-2)}$ arbitrarily, we have that $${\rm Harmonic}_{(\ell=0)}=\int F_{(-2)}d\Omega =\int F_{(-2)}^{\,\prime }d\Omega^{\prime } \,, \label{A7}$$is a Lorentz invariant. That is, we have that that the monopole, the $\ell=0$ harmonic coefficient, of $F_{(-2)}$ is Lorentz invariant. \(b) If we choose $G_{(1)}=\ell^{a}$ and choose $F_{(-3)}$ arbitrarily we have that $$w^{a}\equiv \int F_{(-3)}l^{a}d\Omega\,, \label{A8}$$ is a Lorentzian 4-vector extracted from $F_{(-3)}$. \(c) If we choose $G_{(2)}=\ell^{a}\ell^{b}$ and choose $F_{(-4)}$ arbitrarily we have that $$w^{ab}\equiv \int F_{(-4)}\ell^{a}\ell^{b}d\Omega\,, \label{A9}$$a trace-free symmetric tensor extracted from $F_{(-4)}.$ A subtle issue arises in the above constructions. In the body of the paper we took the components $\ell^{a}$ in one particular Minkowski coordinate system to have the canonical form $$\ell^{a}=\frac{\sqrt{2}}{2}(1,\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )\,. \label{canonical}$$For $\vec{\ell}$ in the primed frame, we must have $\vec{\ell}^{\;\prime}=K{\vec\ell}$, and hence the components of $\vec{\ell}^{\,\prime}$ must be ($\theta,\phi$-dependent) linear combinations of the same harmonics, i.e., of $(1,\sin \theta\cos \varphi, \sin \theta\sin \varphi,\cos \theta)$. But the components of $\vec{\ell}^{\;\prime}$, must have the canonical form $2^{-1/2}(1,\sin \theta ^{\prime }\cos \varphi ^{\prime },\sin \theta ^{\prime }\sin \varphi ^{\prime },\cos \theta ^{\prime })$, in the primed Minkowski frame. This means that a Lorentz transformation of coordinates $$\ell^{\prime a^{\prime }}=L_{a}^{a^{\prime }}\ell^{\prime a}=\frac{\sqrt{2}}{2}(1,\sin \theta ^{\prime }\cos \varphi ^{\prime },\sin \theta ^{\prime }\sin \varphi ^{\prime },\cos \theta ^{\prime })$$ is necessary to give the canonical form for the components $\vec{\ell}^{\,\prime}$. This in turn implies, via $\ell_{a^{\prime }}^{\prime }w^{\prime a^{\prime }}=\ell_{a}w^{a},$ a (coordinate) Lorentz transformation on the $w^{\prime a},$ $$w^{\prime a^{\prime }}=L_{a}^{a^{\prime }}w^{\prime a}.$$ These relations are easily translated to the Stokes fields, $\mathcal{I}\mathcal{(\theta }$,$\mathcal{\varphi )}$, $\mathcal{S}\mathcal{(\theta }$, $\mathcal{\varphi )}$ and $\mathcal{V}\mathcal{(\theta }$,$\mathcal{\varphi )} $ discussed in the text and, as well, to the closely associated quantity, the intensity per unit solid angle per unit frequency interval $\mathcal{I}_{\nu }\mathcal{(\theta },\mathcal{\varphi )}.$ The $\mathcal{I}_{\nu}$ transforms as with $s=0$, amd $w=-3$. Since $\mathcal{I(\theta }$,$\mathcal{\varphi )}$ has boost weight $w=-2$ and spin weight $s=0,$ a formal set of relations arise from the GGV work, and a set of Lorentzian tensors can be directly extracted as follows from $\mathcal{I}$, $\mathcal{I}_{\nu }$, or more generally from $\mathcal{I}/\nu^N$. \(1) From Eq. , it follows that the coefficient of the $\ell=0$ harmonic, the monopole of $\mathcal{I}$, is a Lorentz invariant, $T_{0}$. \(2) From Eq. , in the spherical harmonic expansion of the $w=-3$ quantity $\mathcal{I}^{\frac{3}{2}},$ the coefficients of the ($\ell=0,1)$ harmonics form and transform as a Lorentzian four-vector, say $T^{a}$. In the same manner, the coefficients of the ($\ell=0,1)$ harmonics in $\mathcal{I}_{\nu}$ determine a four-vector, $v^{a}$ that can be considered as defining the rest-frame for the radiation bath. \(3) From Eq. , the spherical harmonic expansion of either of the $w=-4$ quantities $\mathcal{I}^{2}$, $\nu\,\mathcal{I}$, the coefficients of the ($\ell=0,1,2)$ harmonics form and transform as a Lorentzian trace-free, symmetric two-index tensor, $T^{ab}$ and $V^{ab}$ respectively. \(4) These results generalize to all higher powers of $\mathcal{I}$, i.e. to all cases of integer $w\,<-4,$ and $s=0.$ These results allow us to assign a physical meaning for the vector defined in (2). The quantity $\mathcal{I}_{\nu}$, *the intensity per unit solid angle per unit frequency interval,* is the physical quantity that is usually measured in CMB observations. Since it is a quantity of weight $w=-3\ $ it has, as was mentioned above, an invariantly defined Lorentzian four-vector, $v^{a}$, the $\ell=0,1$ harmonics. Since the monopole is far larger than the dipole term, this vector is time-like and can be identified with the observers velocity with respect to the CMB background rest-frame. To use the Lorentzian vector $v^{a}\ $as a ‘rest-frame’ for the background radiation field one finds a moving frame in which $\mathcal{I}_{\nu}$ has no dipole component. More explicitly, the velocity of the rest frame, thus defined, is $$v^{a}\equiv \int \mathcal{I}_{\nu}l^{a}d\Omega$$This in turn could be used to find the $K$ and then determine the rest-frame distribution$$\mathcal{I}_{\nu }^{\prime }=K^{-3}\mathcal{I}_{\nu}.$$ [20]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , , , , , ****, (). , ****, (), . , ****, (), . , ****, (), . , ****, (), . , ****, (), . . , ****, (). , ** (, ). , ****, (), . , Ph.D. thesis, (). , ****, (). , , , ** (, ), . , ****, (). , , , ****, ().

--- abstract: 'We discuss the thermodynamic properties of dark energy (DE) with Quintom matter in spinor scenario. (1).Using the Cardy-Verlinde formula, we investigate the conditions of validity of the Generalized Second Law of thermodynamics (GSL) in the four evolutionary phases of Spinor Quintom-B model. We also clarify its relation with three cosmological entropy bounds. (2). We take thermodynamic stability of the combination between Spinor Quintom DE and the generalized Chaplygin Gas (GCG) perfect fluid into account, and we find that in the case of $\beta>0$ and $0<T<T_0$, the system we consider is thermodynamically stable. (3) Making use of the Maxwell Relation and integrability condition, we derive all thermal quantities as functions of either entropy or volume, and present the relation with quantum perturbation stability.' author: - 'Jing Wang$^{1,2}$[^1],   Shu-wang Cui$^2$,  Cheng-min Zhang$^1$' title: Thermodynamics of Spinor Quintom --- Introduction ============ There are mounting data from type Ia supernovae, cosmic microwave background (CMB) radiation, and so on[@1998snia; @Spergel; @Riess; @Seljak], have provided strong evidences for the present spatially flat and accelerated expanding universe, corresponding to $\ddot{a}>0$, which is dominated by dark sectors. Combined analysis of the above cosmological observations support that the energy of our universe is occupied by dark energy(DE) about $73\%$, dark matter about $23\%$ and usual baryon matter only about $4\%$ which can be described by the well known particle theory. In the context of Friedmann-Robertson-Walker(FRW) cosmology, the evolution of scale factor is governed by the temporal part of Enistein equation $3\frac{\ddot{a}}{a}=-4\pi G(\rho+3p)$, this acceleration may be attributed to the exotic form of negative pressure satisfying $p<-3\rho$, the so-called DE. So far, the nature of DE remains a mystery. To describe the property of this component, a significant parameter $w=\frac{p}{\rho}$, called Equation of State (EoS), was introduced. And it is need to be $w<-\frac{1}{3}$ theoretically. Based on different evolution of the EoS we can obtain different candidate for DE. Currently, it is widely taken the candidate as a small cosmological constant $\Lambda$ (or vacuum energy) with EoS $w=-1$ as well as a dynamical component such as the Quintessence with $-1<w<1$[@Wetterich:1987fm; @Ratra:1987rm], Phantom with $w<-1$[@Caldwell:1999ew], K-essence with both $w\geq-1$ and $w<-1$ but never crossing $-1$[@ArmendarizPicon:2000ah; @Chiba:1999ka]. Although the recent fits to the data in combination of WMAP[@Spergel:2006hy; @Komatsu:2008hk], the recently released 182 SNIa Gold sample[@Riess:2006fw] and also other cosmological observational data show remarkably the consistence of the cosmological constant, it is worth of noting that a class of dynamical models with the EoS across $-1$ [*Quintom*]{} is mildly favored [@Feng:2004ad; @Zhao:2006qg; @Zhao:2006bt; @Wang:2006ts]. In the literature there have been a lot of theoretical studies of Quintom-like models. Especially, a No-Go theorem has been proved to constrain the model building of Quintom[@Xia:2007km], and according to this No-Go theorem there are models which involve higher derivative terms for a single scalar field [@Li:2005fm], models with vector field [@ArmendarizPicon:2004pm], making use of an extended theory of gravity [@Cai:2005ie], non-local string field theory [@Aref'eva:2005fu], and others (see e.g. [@Guo:2004fq; @Quintom_tf; @Cai:2006dm; @Quintom_1; @Cai:2007qw; @Cai:2007gs; @Cai:2007zv; @Quintom_others; @Xiong:2007cn]). The similar work applied in scalar-tensor theory is also studied in Ref. [@Elizalde:2004mq]. Except that many works have been done in pursuit of establishing concrete model to understanding the theoretical nature and origin of this special fluid, there also are a number of papers committing themselves to investigating the thermodynamic properties of DE fluid. The thermodynamics of de Sitter space-time was first investigated by Gibbons and Hawking[@Gibbons:1977mu] and [@Verlinde:2000wg; @Cai:2009rd; @Pollock:1989pn; @Frolov:2002va] extended the study to quasi-de Sitter space-time. Based on an assumption that DE is a thermallized ensemble at certain temperature with an associated thermodynamical entropy, Ref. [@Brevik:2004sd; @Lima:2004wf; @GonzalezDiaz:2004eu; @Izquierdo:2005ku; @MohseniSadjadi:2005ps; @Setare:2006vz; @Setare:2006rf; @Wang:2005pk; @Santos:2006ce; @Santos:2007jy; @Bilic:2008zk] made various aspects of the thermodynamic discussions. The papers[@Sheykhi:2008qr; @Sheykhi:2008qs] have studied the GSL of modified gravity. In the literature[@Husain:2008qx], the thermodynamics of Quantum Gravity has been investigated. Ref.[@Li:2008tc] considered the apparent horizon of the Friedmann-Robertson-Walker universe as a thermodynamical system and investigate the thermodynamics of LQC in the semiclassical region. Previously, it have been considered that a Quintom dark energy with non-regular spinor matter[@Cai:2008gk]. In succession, to understand the possible combinations among different types of Quintom model in spinor field we study the implications of cosmic duality with this class of models and realize additional Quintom models by the aid of this dual properties. In the meantime, we also perform the statefinder diagnostic for this Spinor Quintom model[@wang:2008cs]. In this paper, we will discuss the thermodynamics of the Spinor Quintom model. From the thermodynamical point of view, our universe can be considered as a thermodynamical system filled with DE perfect fluid, we will examine the GSL and thermodynamic stability in this system. This letter is organized as follows. In section 2, we investigate the validity of GSL in spinor field with Quintom DE model, we indicate that the conditions under which the GSL can be satisfied. In section 3, we explore the conditions for thermodynamic stability of the combination between Quintom model with spinor field and the GCG perfect fluid. Some thermodynamic parameters, as functions of entropy and volume, are given in section 4, and we also display the relation with the stability from the point of view of quantum perturbation stability. Section 5 contains our conclusions and prospects. GSL in a System Filled with Spinor Quintom Matter ================================================= One of the distinguishing features of the driver of current accelerating expansion, the alleged DE, lies in violating the strong energy condition, $\rho+3p>0$[@Riess; @Sper]. As a result of the dependence on theoretical models this strength of acceleration is a question in debating. While most model independent analysis suggest that it be below the De Sitter value[@Daly:2003iy], it is certainly true that the body of observational data allows of a wide parameter space compatible with an acceleration larger than de Sitter’s[@Caldwell:1999ew; @Hannestad:2004cb]. If eventually it is proven to be the case, this dark component would violate not only the strong energy condition $\rho+3p>0$ but also the dominated energy condition $\rho+p>0$. In the literature, component with the above specialities was dubbed Phantom[@Caldwell:1999ew; @Caldwell:2003vq], suffering from a long list of pathologies such as quantum instabilities[@Carroll:2003st; @Frampton:2003xg] which leads to supersonic and causes a super accelerating universe ending in a big rip or big crunch along the cosmic evolution. Attracting many attentions, the interesting fluid has been widely discussed recent years[@Dabrowski:2003jm; @Meng:2003tc], and Ref.[@GonzalezDiaz:2004eu; @Myung:2008km] investigated the thermodynamics on phantom dark energy dominant universe. The thermodynamics of DE with constant EoS in the range of $-1<w<-\frac{1}{3}$ was considered in [@Danielsson:2004xw], and that of K-essence also was studied in Ref.[@Bilic:2008zk]. Based on the relation between the event of horizon and the thermodynamics of a black hole assumed by Bekenstein in 1973 [@Bekenstein:1973ur], the event of horizon of a black hole is a measure of its entropy. This idea has been generalized to horizons of cosmological models, so that each horizon corresponds to an entropy. Correspondingly, the second law of thermodynamics was modified in the way that in generalized form, the sum of all time derivative of entropies related to horizons plus time derivative of normal entropy must be positive, i.e., the sum of entropies must be increasing with time. Ref. [@Davies:1987ti] investigated the validity of GSL for the cosmological models which departs slightly from de Sitter space. Ref.[@Izquierdo:2005ku] explored the thermodynamics of DE taking the existence of the observer’s event horizon in accelerated universes into account. The conditions of validity of generalized second law in phantom dominated era was studied in [@MohseniSadjadi:2005ps]. The validity of the GSL of thermodynamics for the Quintom DE model with two scalar fields without coupling potential term was considered by [@Setare:2006rf]. In this section, we will discuss the validity of the GSL of thermodynamics for a Quintom-dominated universe in spinor field and clarify its relation with three cosmological entropy bounds: the Bekenstein bound[@Bekenstein:1980jp], the holographic Bekenstein-Hawking bound, and the Hubble bound[@Verlinde:2000wg]. To begin with the discussion, we deal with the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) space-time, then the space-time metric reads, $$ds^{2}=dt^{2}-a^{2}(t)d\vec{x}^2.$$ Assuming that the dynamics of gravity is governed by the Einstein-Hilbert action, for a spinor minimally coupled to general relativity[@ArmendarizPicon:2003qk; @Vakili:2005ya; @Ribas:2005vr], we have, $$S=S_\psi+S_m-\frac{1}{6}\int d^4 x\sqrt{-g}R.$$ where $R$ is the scalar curvature, $S_\psi$ is given by the the Dirac action and $S_m$ describes additional matter fields, such as scalar fields and gauge fields.[^2] We consider the spinor component as the thermodynamical system we may discuss, which is filled with Quintom DE fluid. With the aid of the dynamics of a spinor field which is minimally coupled to Einstein’s gravity[@Weinberg; @BirrellDavies; @GSW], we can write down the following Dirac action in a curved space-time background $$\begin{aligned} S_\psi&=&\int d^4 x~e~[\frac{i}{2}(\bar\psi\Gamma^{\mu}D_{\mu} \psi-D_{\mu}\bar\psi\Gamma^{\mu}\psi)-\Phi]\nonumber\\ &=&\int d^4 x ~e~{\cal L}_{\psi},\end{aligned}$$ Here, $e$ is the determinant of the vierbein $e_{\mu}^{a}$ and $\Phi$ stands for any scalar function of $\psi$, $\bar\psi$ and possibly additional matter fields. We will assume that $\Phi$ only depends on the scalar bilinear $\bar\psi\psi$. From the expression of the Dirac action, we have the energy density and the pressure of the spinor field: $$\begin{aligned} \label{density}\rho_\psi&=&T_{0}^{0}=\Phi~,\\ \label{pressure}p_\psi&=&-T_{i}^{i}=\Phi'\bar\psi\psi-\Phi~,\end{aligned}$$ For a gauge-transformed homogeneous and a space-independent spinor field, the equation of motion of spinor reads[@Cai:2008gk] $$\begin{aligned} \dot{\psi}+\frac{3}{2}H\psi+i\gamma^{0} \Phi' \psi&=&0,\\ \dot{\bar\psi}+\frac{3}{2}H\bar\psi-i\gamma^{0}\Phi' \bar\psi&=&0,\end{aligned}$$ where a dot denotes a time derivative while a prime denotes a derivative with respect to $\bar\psi\psi$, and $H$ is Hubble parameter. In the framework of FRW cosmology, the Friedmann constraint equation will be[^3] $$H^2=\frac{1}{3}\rho_\psi~,$$ From the equation of motion of spinor and the Friedmann constraint equation, we can obtain the the derivative of Hubble parameter with respect to time, $$\dot{H}=\frac{\dot{\rho_\psi}}{6H}=\frac{\Phi'\bar{\psi}\psi}{2}.$$ So we have $$\rho_\psi+p_\psi=-2\dot{H}.$$ According to the Gibbons equation $$Tds=dE+p_\psi dV=(p_\psi+\rho_\psi)dV+Vdp_\psi,$$ combined with the above relations and the expression of volume $V=\frac{4}{3}\pi R_H^3$ ($R_H$ is the event of the horizon), we may rewriting the first law of thermodynamics as, $$\begin{aligned} Tds&=&-2\dot{H}d(\frac{4}{3}\pi R_H^3)+\frac{4}{3}\pi R_H^3 d\rho_\psi\nonumber\\ &=&-8\pi R_H^2\dot{H}dR_H+8\pi HR_H^3 dH,\end{aligned}$$ where $T$ is the temperature of the background of Spinor fluid. Therefore, the derivative of normal entropy is given as follows: $$\dot{s}=\frac{ds}{dt}=\frac{1}{T}8\pi\dot{H}R_H^2(HR_H-\dot{R_H}).$$ Now we turn to consider the entropy corresponding to the event horizon. The definition of event horizon in a de Sitter space-time is $$R_H=a(t)\int_t^\infty\frac{dt'}{a(t')}.$$ So the time derivative of event of horizon in a spinor field approaching to de Sitter space satisfies the following equation: $$\dot{R_H}=\dot{a(t)}\int_t^\infty\frac{dt'}{a(t')}+a(t)\dot{\int_t^\infty\frac{dt'}{a(t')}}=HR_H-1.$$ i. In the parameter range of $HR_H\leq1$, the Bekenstein bound, which is supposed to hold for systems with limited self-gravity, is appropriate. And the EoS of spinor larger than $-1$, corresponding to a Quintessence dominant universe[@Davies:1987ti]. ii. While for $HR_H\geq1$, corresponding to a strongly self-gravitating universe, the Bekenstein bound has to be replaced by holographic Bekenstein-Hawking bound in which one has $S_B\geq S_{BH}$. And one can get a Phantom phase[@MohseniSadjadi:2005ps]. iii. If $HR_H=1$, the Bekenstein bound $S_B$ is equal to holographic Bekenstein-Hawking bound $S_{BH}$. Then we can write the final form of the time derivative of normal entropy, $$\dot{s}=\frac{8\pi R_H^2\dot{H}}{T}.$$ As we well know, the entropy is proportional to the area of its event horizon. If the horizon entropy corresponding to $R_H$ is defined as $s_H=\pi R_H^2$, the GSL can be stated as: $$\dot{s}+\dot{s_H}=\frac{8\pi R_H^2\dot{H}}{T}+2\pi R_H\dot{R_H}\geq0.$$ In the following, we will take the Quintom-B model realized in Ref. [@Cai:2008gk] to discuss the validity of GSL in spinor field. The temperature of Spinor Quintom-B is assumed to be positive. (1). Phantom dominated evolution:\ In this phase $\dot{R_H}\leq0$, so $\dot{s_H}\leq0$. From $V'<0$ one can get $\dot{H}>0$. So the condition for validity of GSL can be expressed as: $$\dot{H}\geq\mid\frac{\dot{R_H}T}{4R_H}\mid.$$ (2). Quintessence dominated evolution:\ In this period of evolution $\dot{R_H}\geq0$, then we have a negative time derivative of Hubble parameter but that of horizon entropy is not a negative value. Thus the condition for validity of GSL is: $$\mid\dot{H}\mid\leq\frac{\dot{R_H}T}{4R_H}.$$ (3). Phase transition from Phantom to Quintessence:\ At the transition point, we have $w=-1$ and $V'=0$, that is to say $\dot{H}=0$, so $\dot{s}=0$. Assuming that the event horizon $R_H$ varies continuous, one may expect that $\dot{R_H}=0$ in transition time, so the horizon entropy is continuous and differentiable[@Setare:2006rf]. Therefore, to realize the transition, it need to be continuous and differentiable in transition time for the total entropy of the universe. (4). The final phase–an approximate de Sitter universe:\ In such a state, the temperature is [@Davies:1987ti], $$T=\frac{bH}{2\pi},$$ where $b$ is a parameter. During this period, the universe lies in the Quintessence phase, so $$b\geq\frac{8\pi\mid\dot{H}\mid R_H}{H\dot{R_H}},$$ in de Sitter space-time case $R_H=\frac{1}{H}$, one can get $b\geq8\pi$, which should be satisfied if GSL is valid. In conclusion, one can find that the conditions for the validity GSL of Spinor Quintom model are similar to that of the Quintom DE model constructed by two scalar fields without coupling potential term which was considered in [@Setare:2006rf]. Thermodynamic Stability of The Combination between Spinor Quintom and GCG Perfect Fluid ======================================================================================= Since the Chaplygin gas was generalized people have make many correlative studies[@Bento:2003dj; @Chimento:2003ta] to reconcile the standard model with observations. Ref.[@Santos:2006ce] discusses the behavior of temperature and the thermodynamic stability of a generalized Chaplygin gas considering only general thermodynamics — the corresponding thermal equation of state for the GCG and analyzed its temperature behavior as well as its thermodynamic stability considering both adiabatic and thermal equations of state. While in the literature [@Santos:2007jy], Chaplygin gas was modified again, and a scenario was set up to determine the corresponding thermal equation of state of the modified Chaplygin gas(MCG) and it reveals that the MCG only presents thermodynamic stability during any expansion process if its thermal equation of state depends on temperature only, $P=P(T)$. Moreover, the modified Chaplygin gas may cool down through any thermodynamic process without facing any critical point or phase transition. We have established a combination between Chaplygin gas and Spinor Quintom in Ref.[@Cai:2008gk], in this section we will investigate the thermodynamic stability in a universe filled with the fluid combined by both Quintom and GCG in spinor field. In Ref.[@Cai:2008gk],we took the form of potential as $\Phi=\sqrt[1+\beta]{\Phi_0(\bar{\psi}\psi)^{1+\beta}+c}$, and got the EoS of GCG model $$p=-\frac{c}{\rho^{\beta}}~,$$ where parameter $\beta$ is a constant and positive $\beta>0$ and $c$ is also positive and a universal constant[@Santos:2006ce]. Here we consider a closed thermodynamic system full of dark energy fluid, in which the combination of Spinor Quintom with GCG play important role. Assuming the internal energy $U$ and pressure $p$ as only the functions of their natural viables entropy $s$ and volume $V$: $U=U(s,V), p=p(s,V)$, and the energy density of DE fluid is $$\rho=\frac{U}{V}~.$$ From general thermodynamics[@Kubo; @Landau], we know that $$(\frac{\partial U}{\partial V})_s=-p~.$$ Combined the above three equations, we can get the following form, $$(\frac{\partial U}{\partial V})_s=c\frac{V^\beta}{U^\beta},$$ and the expression of the internal energy of this system is also given by its solution, $$U=\sqrt[1+\beta]{cV^{1+\beta}+b},$$ where $b=b(s)$ is an integration parameter. It can be proven that even $c=c(s)$ is not a universal constant, the above expression remains valid. The Eq. (25) also can be written as[@Santos:2006ce]: $$U=V\sqrt[1+\beta]{c[1+(\frac{\sigma}{V})^{1+\beta}]},$$ where parameter $\sigma^{1+\beta}=\frac{b}{c}$. Then we may deduce the expressions of energy density and pressure with respect to this parameter, $$\begin{aligned} \label{density}\rho&=&\sqrt[1+\beta]{c[1+(\frac{\sigma}{V})^{1+\beta}]}~,\\ \label{pressure}p&=&-\sqrt[1+\beta]{\frac{c}{[1+(\frac{\sigma}{V})^{1+\beta}]^{\beta}}}~.\end{aligned}$$ By these two equations, we could understand the behavior of both past and future of our universe. In the early time with small scale factor and volume, the energy density and pressure behave as the below form: $$\begin{aligned} \rho&\approx&c^{\frac{1}{1+\beta}}\frac{\sigma}{V},\\ p&\approx&c^{\frac{1}{1+\beta}}(\frac{V}{\sigma})^{\beta}\sim0,\end{aligned}$$ corresponding to a high energy density and approximative pressureless matter dominant phase. During this period the energy density reduces as its entropy and volume adiabatically. Along with the cosmological expansion through to some late times, these two parameters are approximate respectively to $$\begin{aligned} \rho&\approx&c^{\frac{1}{1+\beta}}+\frac{c^{\frac{1}{1+\beta}}}{1+\beta}(\frac{\sigma}{V})^{1+\beta},\\ p&\approx&c^{\frac{1}{1+\beta}}.\end{aligned}$$ During this period of the evolution, the total system can be seen as constituted by two components: one with constant energy density and the other with an alterable energy density with respect to volume. While for a large value of scale factor, the energy density may rather lower and EoS is $p=-\rho=c^{\frac{1}{1+\beta}}$ which is a de Sitter Space-time. Consequently, we realize a transformation from dust-like matter- dominated universe to a de Sitter phase in the point of view of thermodynamics. In what follows, we will extensively examine the conditions for the thermodynamic stability of this combined system. (1). We determine how the pressure change with volume through the adiabatic expansion.\ Using Eq. (28), one can get $$(\frac{\partial p}{\partial V})_s=\beta\frac{p}{V}[1-\frac{1}{1+(\frac{\sigma}{V})^{1+\beta}}],$$ it is obvious that we exclude the case of $\beta=0$ due to a constant pressure and the disappearing derivative. While in the case of $\beta>0$ the above derivative is always negative value. (2). To make a system stable, it is necessary for the thermal capacity at constant volume to be positive $c_V>0$, the pressure reduces as volume at constant temperature, as well.\ For this purpose, we calculate the formula of temperature $T$ and entropy $s$ to determine how the temperature depends on its entropy and volume. In the thermodynamics and statistical physics, the temperature of a system is defined as: $$T=(\frac{\partial U}{\partial s})_V,$$ combined with the expression of internal energy, the formula of temperature can be written as follows[@Santos:2006ce]: $$T=\frac{1}{1+\beta}(cV^{1+\beta}+\varepsilon)^{-\frac{\beta}{1+\beta}}(V^{1+\beta}\frac{dc}{ds}+\frac{d\varepsilon}{ds}).$$ Clearly, if we take parameter as both $c$ and $\varepsilon$ are universal constant, the temperature equals to $0$ for any value of pressure and volume. As a result, the isotherm $T=0$ is simultaneously an isentropic curve at $s=const$, which violates the third law of thermodynamics[@Santos:2006ce]. Taking this factor into account, we choose $c$ as a universal constant and $\frac{d\varepsilon}{ds}>0$. From dimensional analysis it can be understood that $\varepsilon$ has a dimension of energy, $[\varepsilon]^{1+\beta}=[U]$. In this case, we take it as[@Santos:2006ce] $$b=(T_0s)^{1+\beta},$$ so, $$\frac{d\varepsilon}{ds}=(1+\beta)(T_0s)^{\beta}T_0.$$ Then the formulae of temperature and entropy of this system can be written as: $$\begin{aligned} T&=&T_0^{1+\beta}s^{\beta}[cV^{1+\beta}+(T_0s)^{1+\beta}]^{-\frac{\beta}{1+\beta}},\\ s&=&\frac{c^{\frac{1}{1+\beta}}}{T_0}\frac{T^{\frac{1}{\beta}}}{(T_0^{\frac{1+\beta}{\beta}}-T^{\frac{1+\beta}{\beta}})^{\frac{1}{1+\beta}}}V.\end{aligned}$$ A stable thermodynamic system requires a positive and finite entropy, which requests that the temperature satisfy $$0<T<T_0.$$ By the definition of $c_V$ and the formulae of temperature and entropy, one can rewrite $c_V$ as, $$c_V=\frac{1}{\beta T_0}\frac{c^{\frac{1}{\beta}}V}{[1-(\frac{T}{T_0})^{\frac{1+\beta}{\beta}}]^{\frac{2+\beta}{1+\beta}}}(\frac{T}{T_0})^{\frac{1}{\beta}},$$ Thus, When $\beta>0$ and $0<T<T_0$, one can get a positive $c_V$. Correspondingly, we can obtain the expression of pressure, $$p=-c^{\frac{1}{1+\beta}}[1-(\frac{T}{T_0})^{\frac{1+\beta}{\beta}}]^{\frac{\beta}{1+\beta}},$$ It can be seen that the pressure is only the function of temperature, so $(\frac{\partial p}{\partial V})_T>0$ is satisfied. In a word, in the case of $\beta>0$ and $0<T<T_0$, the system we consider is thermodynamically stable. Thermodynamic parameters and its relation with quantum stabilities ================================================================== In the first two sections, we have studied the stability of a system filled with Spinor Quintom DE fluid from the classical thermodynamic point of view. For this part, we will derive a class of thermal quantities as functions either of entropy or volume, then we may discuss the relation with quantum perturbation and which constraint is much stronger. From the expressions of energy density (EQ.\[density\]) and pressure (EQ.\[pressure\]), we can get, $$\begin{aligned} &&\rho+P=\sqrt[1+\beta]{c(1+(\frac{\sigma}{V})^{1+\beta})}-\sqrt[1+\beta]{\frac{c}{1+(\frac{\sigma}{V})^{1+\beta}}}\nonumber\\ &&=\sqrt[1+\beta]{c}(\sqrt[1+\beta]{1+(\frac{\sigma}{V})^{1+\beta}}-\frac{1}{\sqrt[1+\beta]{1+(\frac{\sigma}{V})^{1+\beta}}}).\end{aligned}$$ Besides, From the definition of entropy $$S\equiv\frac{\rho+P}{T}V,$$ we can derive a defining equation of tempertature for an adiabatic process, $$\label{T}T\equiv\frac{\rho+P}{S}V.$$ Then we have the temperature $$\begin{aligned} T_{(V)}&=&\frac{\sqrt[1+\beta]{c}}{S}(\sqrt[1+\beta]{V^{1+\beta}+\sigma^{1+\beta}}\nonumber\\&&-\frac{V^2}{\sqrt[1+\beta]{V^{1+\beta}+\sigma^{1+\beta}}}).\end{aligned}$$ In addition, the EoS $W_{V}$, squared speed of sound $C^2_{s(V)}$ and entropy $S_{V}$ read respectively, $$\begin{aligned} W_{(V)}&=&\frac{P}{\rho}=-\frac{V^{1+\beta}}{V^{1+\beta}+\sigma^{1+\beta}},\\ C^2_{s(V)}&=&\frac{\partial P}{\partial\rho}=\frac{V^{1+\beta}}{\sigma^{1+\beta}},\\ S_{(V)}&=&\frac{C^{\frac{1}{1+\beta}}}{S}(\sqrt[1+\beta]{V^{1+\beta}+\sigma^{1+\beta}}\nonumber\\&&-\frac{V^2}{\sqrt[1+\beta]{V^{1+\beta}+\sigma^{1+\beta}}}).\end{aligned}$$ The combination among the integrability condition $$\frac{\partial^2S}{\partial T\partial V}=\frac{\partial^2S}{\partial V\partial T},$$ the Maxwell Relation $$\frac{\partial T}{\partial V}=-\frac{\partial P}{\partial S},$$ and EQ.(\[T\]), can lead to the relation, $$dP=-\frac{\rho+P}{S}dS.$$ And setting $\beta=1$ in EQ. (\[density\]) and EQ. (\[pressure\]), one has $$\begin{aligned} \rho+P&=&-\sqrt{c}\frac{\frac{\sigma^2}{V^2}}{\sqrt{1+(\frac{\sigma}{V})^2}} \nonumber\\ &=&\frac{c}{P}-P,\end{aligned}$$ so $$\frac{PdP}{P^2-c}=\frac{dS}{S}.$$ Finally we can get the thermal quantities as functions of entropy. $$\begin{aligned} P_{(S)}&=&-\sqrt{c}\sqrt{1-(\frac{S}{S_*})^2},\\ \rho_(S)&=&\frac{\sqrt{c}}{\sqrt{1-(\frac{S}{S_*})^2}},\\ W_{(S)}&=&(\frac{S}{S_*})^2-1,\\ C^2_{S(S)}&=&1-(\frac{S}{S_*})^2.\end{aligned}$$ Based on the above expressions of these quantities, we may analyze the quantum stability in connection with perturbations which is one important issue of a DE model. Usually systems with negative kinetic modes from ghost fields suffer from the quantum instabilities which may induce some supersonic phenomenon. However, in our Spinor Quintom DE model, we do not introduce any ghost field, and is it to say that this model will not perform any quantum instability? To study this issue, we would like to redefine the spinor as $\psi_N\equiv a^{\frac{3}{2}}\psi$. Then perturbing the spinor field, one gives the perturbation equation as follows [@Cai:2008gk], $$\begin{aligned} \label{perteq} &&\frac{d^2}{d\tau^2}\delta\psi_N-\nabla^2\delta\psi_N+\nonumber\\ &&a^2\left[ V'^2+i\gamma^0 (HV'-3HV''\bar\psi\psi) \right]\delta\psi_N\nonumber\\ &&=-2a^2V'V''\delta(\bar\psi\psi)\psi_N\nonumber\\ &&-i\gamma^\mu\partial_\mu[a V''\delta(\bar\psi\psi)]\psi_N~,\end{aligned}$$ where $\tau$ is the conformal time defined by $d\tau\equiv dt/a$. From the perturbation equation above, we can read that the sound speed is equal to $1$ which eliminates the instability of the system in short wavelength. Thus to what degree the system is stable in both quantum and classical level, and which constraint is much stronger. Furthermore, whether there are some instability from the unrenormalizable quantum effect. Such issues we may discuss in detail in our future work. Conclusion and Discussions ========================== To summarize, we have investigated the thermodynamics of Quintom DE dominant thermodynamical system in spinor field. Firstly, we show the conditions in which the total entropy may not decrease with time not only in Phantom and Quintessence phase but also at the transition time and the final approximative de Sitter phase. We set up the similar conditions to a Quintom universe with two scalar fields without coupling potential term. In the second place, we, using general thermodynamics, explore the thermodynamic stability of a system full of the DE fluid combined Spinor Quintom with GCG, and we conclude that in a certain range of temperature, i.t. $0<T<T_0$, this system remains thermodynamically stable without any limitation on pressure. We also derive a class of thermal quantities as functions either of entropy or volume, then we may discuss the relation with quantum perturbation. 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--- abstract: 'A brief review of the background for experiments at future $e^{+}e^{-}$ linear colliders is done. Two sources of background are discussed: the beam delivery system and the interaction point. The abundance of background muons, neutrons, photons and $e^{+}e^{-}$ pairs is quoted for different sub-detectors and both background sources. The background caused by the [beamstrahlung]{} is described in more detail. Space distributions are sketched and the impact on calorimeters is discussed for the background neutrons.' author: - | Oleg Markin[^1]\ [*Institute for Theoretical and Experimental Physics, Moscow*]{} title: Backgrounds at future linear colliders --- Introduction ============ The background expected in experiments at future $e^{+}e^{-}$ linear colliders substantially differs from that was at the LEP because of higher energy and much more narrow beam. Hits caused by background particles seriously complicate the event reconstruction. The negative influence of the backgrounds on energy reconstruction comprises both worse pattern recognition and pile-up. Therefore the presence of the background should be carefully taken into account throughout the computer simulation of benchmark processes. After all, the load of detectors by background signals determines the detector design as well as the choice of technologies for the detecting systems.[^2] Backgrounds from the [beam delivery system]{} ============================================= For the purpose of collimating both $e^+$ and $e^-$ beams of the International Linear Collider (ILC), three metallic primary collimators will be situated at each side from the Interaction Point (IP). There are also protective masks in the beam tunnel. These elements serve to purify the beam, but as they are placed rather close to IP, the interaction of beam particles with these collimators and masks gives rise to background. The first of the collimators has an aperture $8\sigma_x\times65\sigma_y$, where $\sigma_x$ and $\sigma_y$ are the standard deviations of the transverse density of beam. That collimator is placed about 1,500 m from IP, while two other collimators are about 1,300 m and 1,000 m from IP, respectively. The presence of the collimators leads to the loss of beam particles, this is shown in fig. \[loss\] for positrons as predicted by two different MC codes. The magnitude of the loss is about 0.1%; the lost positrons/electrons are being converted to $\gamma$s, electron-positron pairs, muon-antimuon pairs and neutrons. These (background) particles contribute to so-called the Beam Delivery System (BDS) background. Besides, the BDS background includes photons originated from bremsstrahlung at gas molecules in the beam tunnel and from the synchrotron radiation. In addition to the BDS background, there is an IP-related background. ![ []{data-label="loss"}](loss.png){width=".8\textwidth"} The BDS neutrons can be created in photo-nuclear reactions. Muons can be produced in electromagnetic showers initiated by beam particles. The probability of the muon production is proportional to the squared charge of nuclei involved in the process, and amounts about $4\cdot10^{-4}$ [@Burkhardt_99]. Thus, the process results in about 4,000 muons for the 0.1% loss of $10^{10}$ positrons/electrons per bunch-crossing (BX). A good way to minimize the number of those muons is to sweep them away by the magnetic field created by iron spoilers built purposely, cf. fig. \[spoiler\]. There are two possible locations and different constructions of the spoilers: wall- and donut-shape; fig. \[sp\_effect\] demonstrates the effect of the two constructions. ![ []{data-label="spoiler"}](spoiler.png){width=".9\textwidth"} ![ []{data-label="sp_effect"}](sp_effect.png){width=".8\textwidth"} The spoilers drastically reduce the muon density, making it several thousand times lower at the entrance to the experimental hall. A side effect of the spoilers is increasing of the number of neutrons, see table in the fig. \[num\_of\_part\]. The neutrons from the BDS background are not a problem since a concrete wall will be placed at the entrance to the experimental hall. The vertex system (VTX) is the only sub-detector not affected by the muon spoilers. The energy distribution of the BDS muons and neutrons is shown in fig. \[distr\]. The BDS muons and neutrons have rather flat radial distribution within three meters from the beam axis, see fig. \[distr\]. The BDS photons and positrons/electrons are concentrated near or inside the beam pipe. The BDS background dominates in the muon detector while in the tracker and the VTX it gives the number of hits that does not exceed the IP background[@Denisov_06]. ![ []{data-label="num_of_part"}](num_of_part.png){width=".8\textwidth"} ![ []{data-label="distr"}](energy_distr.png "fig:"){width=".49\textwidth"} ![ []{data-label="distr"}](radial_distr.png "fig:"){width=".49\textwidth"} Backgrounds related to the interaction point ============================================ Hard photons ------------ At high-energy colliders the electromagnetic field of each bunch causes focusing of bunches of the opposite beam. That leads to bending of [electron/positron]{} trajectories near the IP, which results in emission of hard photons. At the ILC, this process (called [beamstrahlung]{}) will give about two photons per a beam particle, with the average energy of [beamstrahlung]{} photons being several percents of the beam energy [@Schulte_96]. The spectrum of the [beamstrahlung]{} photons is shown in fig. \[besh\_spectrum\]. The [beamstrahlung]{} photons are strongly focused in forward direction and, therefore, do not significantly contribute to the detector background. However, the [beamstrahlung]{} photons produce [electron-positron]{} pairs that do contribute to the detector background either directly or through their backscatters. The photons can also produce neutrons when they hit components of the [beam delivery system]{}, such as collimators and the [beamstrahlung]{} dump. ![ []{data-label="besh_spectrum"}](besh_spectrum.png){width=".9\textwidth"} Besides, hard photons can be created in the ordinary initial state radiation. Both the [initial state radiation]{} and the [beamstrahlung]{} reduce the luminosity of colliders in the region close to nominal energy, cf. fig. \[luminosity\]. A prominent source of hard photons is the collision of two beam particles resulting in [bremsstrahlung]{}, called the radiative Bhabha scattering, see fig. \[rbs\]. The photons created in the [radiative Bhabha scattering]{} are also focused in the forward direction, and escape through the beam pipe, but the remnant [electrons/positrons]{} contribute to the detector background. ![ []{data-label="luminosity"}](luminosity.png){width=".6\textwidth"} ![ []{data-label="rbs"}](rbs.png){width=".6\textwidth"} Soft [electrons/positrons]{} ---------------------------- The basic process that gives rise to the detector background is the creation of [electron-positron]{} pairs additional to beam particles. An [electron-positron]{} pair can be created by a [beamstrahlung]{} photon in the strong electromagnetic field of beams. This is referred to as the coherent pair production. Under conditions of the Compact Linear Collider (CLIC), the newly born pairs constitute several percents of beam particles. At the ILC, another way of pair creation dominates, which is referred to as the incoherent pair production, when new [electron-positron]{} pairs are created as a result of interaction of just two particles. Those can be either two real [beamstrahlung]{} photons or a single real [beamstrahlung]{} photon plus an [electron/positron]{}, or a couple of [electrons/positrons]{}, cf. fig. \[pairs\]. At the ILC with $\sqrt{s}=$0.5 TeV, there should be about 76,000 pairs per BX, with the average energy 2.5 GeV per an [electron/positron]{} [@Maruyama_05]. The energy spectrum of the pairs is shown in fig. \[pair\_spectrum\]. The interaction of photons can also give a couple of quarks followed by hadronization into minijets. ![ []{data-label="pairs"}](pairs.png){width=".8\textwidth"} ![ []{data-label="pair_spectrum"}](pair_spectrum.png){width=".5\textwidth"} An additional to [beamstrahlung]{} source of off-energy [electrons/positrons]{} is the radiative Bhabha scattering. For that process, the spectrum of remnant beam particles is rather flat at the low energy end [@Schulte_99], see fig. \[brsh\_spectrum\]. Under the ILC-like conditions, after emission of a photon, the average energy of the scattered [electron/positron]{} is equal to 50 GeV on average, and the number of corresponding electron-positron pairs is about $4\cdot 10^{4}$ per BX [@Wagner_01]. ![ []{data-label="brsh_spectrum"}](brsh_spectrum.png){width=".6\textwidth"} The [electrons/positrons]{} can be either focused or defocused by the beam field, which depends on the absolute value of particle momentum and its direction. The remnant [electrons/positrons]{} from [radiative Bhabha scattering]{} follow the appropriate beam and are thus focused. They will be dumped inside quadrupole magnets. The [electrons/positrons]{} from the [beamstrahlung]{} pair production can follow the positive beam direction as well as the negative one, which predetermines focusing [@Schulte_99]. Most of them initially have a small $\theta$-angle with respect to the beam axis. The final angle depends on the deflection by the beam field. The bigger transverse momentum $p_t$ they have, the smaller $\theta$-angle they finally acquire. Therefore, only low-$p_t$ [electrons/positrons]{} reach the detector where their range is limited by the main magnetic field, though. For the main part, those [electrons/positrons]{} curl up and move longitudinally towards the quadrupoles. They may hit the forward detectors (LumiCal and BeamCal), the mask, the quadrupoles and the beam tube, producing secondaries, photons in that number, which may convert to pairs inside sub-detectors. In the Time Projection Chamber (TPC) for example, the [electrons/positrons]{} from the pairs produced by the secondary photons are actually seen as lines (helices with tiny radius) parallel to the field direction [@Vogel_08]. The number of such tracks is about 1,400 per BX, which also includes backscatters from ECal and charged particles from the hadronic reactions in minijets. Neutrons -------- A large amount of [electrons/positrons]{} from pairs are deflected by the beam field and can hit the beam pipe and the quadrupole from inside. Thus there are two space origin of the IP-induced background: (i) directly the IP and (ii) backscatters from the beam pipe itself and the very forward region, particularly the BeamCal. The latter serves as an active absorber: it provides shielding of the BDS elements from the incoherent pairs. Neutrons are produced by photons and [electrons/positrons]{} that create electromagnetic showers where a photo-nuclear reaction may take place with a resonance at the photon energy about 10 GeV. For an ILC-like collider, the pairs created by [beamstrahlung]{} photons produce about 70,000 neutrons per BX when they hit the mask and quadrupoles [@Wagner_01]. The remnant [electrons/positrons]{} from [radiative Bhabha scattering]{} produce even more, about $3\cdot10^5$ neutrons per BX. Fortunately, they do not reach most of sub-detectors since they are produced at larger distance from IP, cf. the table in fig. \[neut\_det\]. ![ []{data-label="neut_det"}](neut_det.png){width=".7\textwidth"} The neutrons created by the incoherent pairs at the BeamCal dominate the background in the endcaps of the Hadronic Calorimeter (HCal). The level of this background is illustrated in fig. \[neut\_hcal\]. For the CLIC beam parameters (presumably at 3 GeV), the line corresponding to the background from incoherent pairs in the right plot in fig. \[CLIC\_endcap\] actually represents the neutrons. The cell occupancy in the HCal endcaps at 40 cm from the beam axis is very high, $\cal O$ $(10^{-2})$ hits per BX, steeply dropping outward. In the ECal endcaps, the dominating contribution to cell occupancy is concerned with the minijet background. It decreases with a radial distance from the beam axis (i.e. with the polar angle of a final-state particle) not as fast as the neutron background in the HCal endcaps, cf. the left plot in fig. \[CLIC\_endcap\]. ![ []{data-label="neut_hcal"}](neut_hcal.png){width=".5\textwidth"} ![ []{data-label="CLIC_endcap"}](ecal_endcap.png "fig:"){width=".47\textwidth"} ![ []{data-label="CLIC_endcap"}](hcal_endcap.png "fig:"){width=".49\textwidth"} The signal of clusters created by particles from minijets can be rejected making use of precise time stamping. That has been studied throughout the preparation of the CLIC CDR by including the background from $\gamma\gamma \to $ hadrons to simulation of several benchmark processes. For neutrons such a rejection procedure based on the time stamp might be less effective. Moreover, the substantial difference exists between the neutron background and the backgrounds from pairs or $\gamma\gamma \to $ hadrons since neutrons can deposit their energy through low-energy recoil protons in scintillator, which can produce ionization signals much exceeding those of MIP, amplifying thus the calorimeter response to the neutrons. Still, the background from neutrons created by the incoherent pairs could have a significant impact on the reconstruction of events only at very small polar angles because of the steep radial dependence, cf. fig. \[CLIC\_endcap\]. Neutrons are also abundantly produced in water and concrete of both beam and [beamstrahlung]{} dumps. Some amount of these neutrons return to the beam tunnel, but most of them are absorbed in the tunnel wall. The rest, about 33,000 neutrons per BX, hit the detector yoke, but few of them reach sub-detectors [@Wagner_01]. [99]{} A.I. Drozhdin, N.V. Mokhov, N. Nakao, S.I. Striganov, L. Keller, *Supression of Muon Backgrounds Generated in the ILC Beam Delivery System*, SLAC-PUB-12741 (2007). H. Burkhardt, *Background in future linear $e^+e^-$ colliders*, CERN-SL-99-057 AP, CLIC Note 416 (1999). D.S. Denisov, N.V. Mokhov, S.I. Striganov, M.A. Kostin, I.S. Tropin, *Machine-Related Backgrounds in the SiD Detector at ILC*, FERMILAB-FN-0790-AD (2006). D. Schulte, *Study of Electromagnetic and Hadronic Background in the Interaction Region of the TESLA Collider*, Doctoral thesis, TESLA 1997-08 (1996). D. Schulte, *Beam-Beam interaction*, a lecture\ www.linearcollider.org/files/ilc$\_$school/Lecture\_15\_Daniel\_Schulte.pdf T. Maruyama, *Energy flow comparison between 20 mrad and 2 mrad crossings*, a talk at LCWS2005. D. Schulte, *Beam-Beam Simulations with GUINEA-PIG*, CERN-PS-99-014-LP, CLIC-Note-387 (1999). G. Wagner, *Neutron Background Studies at TESLA Collider*, LC-DET-2001-148(2001). A. Vogel, *Beam-Induced Backgrounds in Detectors at the ILC*, Doctoral thesis, DESY-THESIS-08-036 (2008). D. Dannheim and A. Sailer, *Beam-induced Backgrounds in the CLIC Detectors*, LCD-Note-2011-021. [^1]: e-mail: [email protected] [^2]: This review represents studies of background completed to the autumn of 2012.

--- abstract: 'We consider the problem of coding over the multi-user Interference Channel (IC). It is well-known that aligning the interfering signals results in improved achievable rates in certain setups involving more than two users. We argue that in the general interference problem, senders face a tradeoff between communicating their message to their corresponding decoder or cooperating with other users by aligning their signals. Traditionally, interference alignment is carried out using structured codes such as linear codes and group codes. We show through an example that the usual structured coding schemes used for interference neutralization lack the necessary flexibility to optimize this tradeoff. Based on this intuition, we propose a new class of codes for this problem. We use the example to show that the application of these codes gives strict improvements in terms of achievable rates. Finally, we derive a new achievable region for the three user IC which strictly improves upon the previously known inner bounds for this problem.' author: - Farhad Shirani - title: 'Trade-off between Communication and Cooperation in the Interference Channel' --- Introduction ============ interference channel problem describes a setup where multiple pairs of transmitters and receivers share a communication medium. Each receiver is only interested in decoding the message from its corresponding transmitter. However, since the channel is shared, signals from other senders interfere with the desired signal at each decoder. The presence of interfering signals adds new dimensions to this problem in terms of strategies that can be used as compared to point-to-point (PtP) communication. For example, the encoders can *cooperate* with each other by choosing their channel inputs in a way that would facilitate their joint communication. It turns out that, often, this cooperation requires an encoder to employ a strategy which may be sub-optimal from its own PtP communications perspective. In this paper, we investigate this tradeoff and develop a new class of codes which allow for more efficient cooperation between the transmitters. Characterizing the capacity region for the general IC has been a challenge for decades. Even in the simplest case of the two user IC, the capacity region is only known in special cases [@Sato][@Costa]. The best known achievable region for the IC was due to Han and Kobayashi [@HK]. However, recently it was shown that the Han-Kobayashi (HK) rate region is suboptimal [@chandra][@IC]. Particularly, when there are more than two transmitter-receiver pairs, the natural generalization of the HK strategy can be improved upon by inducing structure in the codebooks used in the scheme [@IC]. Structured codes such as linear codes and group codes enable the encoders to align their signals more efficiently. This in turn reduces interference at the decoders. Such codebook structures have also proven to give gains in other multi-terminal communication problems [@KM]-[@MD]. The idea of interference alignment was proposed for managing interference when there are three or more users. Initially, the technique was proposed by Maddah-Ali et. al. [@MA] for the MIMO X channel, and for the multi-user IC by Jafar and Cadambe [@CJ]. The interference alignment strategy was developed for cases of additive interference and uniform channel inputs over finite fields. This was extended to arbitrary interference settings and input distributions in [@IC]. However, it turns out that alignment is not always beneficial to the users in terms of achievable rates. Consider the example in Figure \[fig:1\]. Intuitively, it would be beneficial to align the input from users 1 and 2 to reduce interference at decoder 3. However, if users 1 and 2 align their signals, it becomes harder for decoder 2 to distinguish between the two inputs. One might suggest that the problem could be alleviated if users 1 and 2 designed their codebooks in a way that they would “look” aligned at decoder 3 based on $P_{Y_3|X_1,X_2,X_3}$, but at the same time they would seem different at decoder 2 based on $P_{Y_2|X_1,X_2}$. In this paper we show that linear codes lack the necessary flexibility for such a strategy. Based on this intuition, we propose a new class of structured codes. Using these codes we derive an achievable rate region which improves upon the best known achievable region for the three user IC given in [@IC]. ![A setup where interference alignment is beneficial to user 3 but harmful for user 2.[]{data-label="fig:1"}](fig1){height="1.3in"} The rest of the paper is organized as follows: Section \[sec:not\] gives the notation used in the paper as well as the problem statement. In section \[sec:trade\], we consider two different examples of three user IC. In the first example - where interference alignment is strictly beneficial - we prove that not only structured codes are useful for alignment but that any arbitrary coding scheme which achieves optimality must possess certain linearity properties. In the second example, we show the existence of the tradeoff discussed above and prove that linear codes are suboptimal for that example. Section \[sec:code\], gives the new codebook constructions and proves that these new codes outperform the linear coding scheme in [@IC]. In Section \[sec:RR\], we provide a new general achievable rate region for this problem. Section \[sec:conclude\], concludes the paper. Problem Statement and Notation {#sec:not} ============================== In this section, we give the notation used in the paper and provide the problem statement. Throughout the paper, we denote random variables by capital letters such as $X, U$, their realizations by small letters $x,u$, and their corresponding alphabets (finite) by sans-serif typeface $\mathsf{X}$, $\mathsf{U}$, respectively. Small letters such as $l, k$ are used to represent numbers. The field of size $q$ is denoted by $\mathbb{F}_q$. We represent the field addition by $\oplus$ and the addition on real numbers by $+$. For $m\in \mathbb{N}$, We define the set of numbers $[1,m]\triangleq \{1,2,\dotsb, m\}$. Vectors are represented by the bold type-face such as $\mathbf{u}, \mathbf{b}$. For a random variable $X$, $A_{\epsilon}^n(X)$ denotes the set of $\epsilon$-typical sequence of length $n$ with respect to the probability distribution $P_X$, where we use the definition of frequency typicality. Let $q$ be a prime number. For $l\in\mathbb{N}$, consider $U_i,i\in[1,m]$ i.i.d random variables with distribution $P_U$ defined on the field $\mathbb{F}_q$. $U^{\otimes{l}}$ denotes a random variable which has the same distribution as $\sum_{i\in [1,l]}U_i$ where the summation is over $\mathbb{F}_q$. We proceed with formally defining the three user IC problem. A three user IC consists of three input alphabets $\mathsf{X}_i, i\in \{1,2,3\}$, three output alphabets $\mathsf{Y}_i, i\in \{1,2,3\}$, and a transition probability matrix $P_{\mathbf{Y}|\mathbf{X}}$. A code for this setup is defined as follows. A three user IC code $(n,\mathsf{M}_1,\mathsf{M}_2,\mathsf{M}_3, \mathbf{e},\mathbf{d})$ consists of (1) Three sets of message indices $\mathsf{M}_i$ (2) Three encoder mappings $e_i:\mathsf{M}_i\to \mathsf{X}_i^n,i\in[1,3]$, without loss of generality, these maps are assumed to be injective (3) and three decoding functions $d_i:\mathsf{Y}_i^n\to \mathsf{M}_i, i\in [1,3]$. We define the codebook corresponding to the encoding map $e_i$ as $\mathbb{C}_i=\{e_i(m_i)|m_i\in \mathsf{M}_i\},i\in[1,3]$. The rate of user $i$ is defined as $r_i=\frac{1}{n}\log{|\mathbb{C}_i|}$. A rate-triple $(R_1,R_2,R_3)$ is said to be achievable if for every $\epsilon>0$, there exists a code $(n,\mathsf{M}_1,\mathsf{M}_2,\mathsf{M}_3, \mathbf{e},\mathbf{d})$ such that $(1)$ $r_i\geq R_i-\epsilon, i \in [1,3]$, and (2) $P(d(\mathbf{Y}^n)=\mathbf{M}|\mathbf{e}(\mathbf{M})=\mathbf{X}^n)\geq 1-\epsilon$. We make frequent use of coset codes and Nested Linear Codes (NLC) which are defined next. A $(k,n)$ coset code $\mathcal{C}$ is characterized by a generator matrix $G_{k\times n}$ and a dither $\mathbf{b}^n$ defined on the field $\mathbb{F}_q$. The code is defined as $\mathcal{C}\triangleq\{\mathbf{u}G\oplus \mathbf{b}|\mathbf{u}\in \mathbb{F}_q^k\}$. The rate of the code is given by $R=\frac{k}{n}\log{q}$. \[def:PNLC\] For natural numbers $k_i<k_o,k'_o<n$, let $G_{k_i\times n}, \Delta{G}_{(k_o-k_i)\times n}$ and $\Delta{G'}_{(k'_o-k_i)\times n}$ be matrices on $\mathbb{F}_q$. Define $\mathcal{C}_i, \mathcal{C}_o$ and $\mathcal{C}'_o$ as the linear codes generated by $G$, $[G|\Delta{G}]$ and $[G|\Delta{G}']$, respectively. $\mathcal{C}_o$ and $\mathcal{C}'_o$ are called a pair of NLC’s with inner code $\mathcal{C}_i$. We denote the outer rates as $r_o=\frac{k_o}{n}$ and $r'_o=\frac{k'_o}{n}$ and the inner rate $r_i=\frac{k_i}{n}$. The Interference Alignment tradeoff {#sec:trade} =================================== In this section, we investigate the interference alignment tradeoff mentioned in the introduction in more detail. We show that in certain three user interference setups, on the one hand, alignment is beneficial to one of the users, while on the other hand, the rates achieved by the aligning users is reduced due to the alignment. We investigate the phenomenon in two examples. The first example involves a three user interference setup. In this example, the first two encoders use linear codes to manage the interference for the third user. This gives a strictly improved achievable rate region. It is well-known that interference alignment can be induced efficiently by the application of structured codes. Additional to this, we show the stronger statement that the only ensemble of codes which achieve the desired rate-triples in this example, are the ones with specific linearity properties. Next, we build upon the first example to create a setup where alignment is beneficial to one of the users and harmful for the other one. This second example provides the motivation for our new codebook constructions in the next section. Example 1 --------- Consider the example shown in Figure \[fig:goodalignment\]. All of the inputs are $q$-ary and the additions are defined on the field $\mathbb{F}_q$. The three outputs of the channel are $Y_i=X_i\oplus N_1\oplus N_3, i\in\{1,2\}$, $Y_3=X_1\oplus X_2\oplus X_3\oplus N_3$. We are interested in achieving the following rates for the first and second users: ![A Three User IC Where Alignment Is Strictly Beneficial[]{data-label="fig:goodalignment"}](Eg1){height="1.3in"} $$R_1=R_2=\log{q}-H(N_1\oplus N_3).$$ Given these rates, we want to maximize $R_3$. The following lemma shows that linear codes achieve the optimum $R_3$ for this setup. Furthermore, we show that if an ensemble of codes achieves the optimum $R_3$, then the codes corresponding to the first two users are “almost” the same coset code. \[Lem:eg1\] For a given family of codes $(n,\mathsf{M}_1,\mathsf{M}_2,\mathsf{M}_3, \mathbf{e},\mathbf{d}), n \in \mathbb{N}$ satisfying the rate and error constraints at decoders 1 and 2, user $3$ can achieve the rate $R_3=H(N_1\oplus N_3)-H(N_3)$ iff there exists a dither $\mathbf{b}$ such that for every random variable $N$ defined on $\mathbb{F}_q$ with positive entropy, the following holds: $$P(e_1(M_1)\oplus e_2(M_2))\in \mathbb{C}_1\bigcup \mathbb{C}_2\oplus A_\epsilon^n(N)\oplus \mathbf{b})\to 1, \text{ as } n\to \infty.$$ Equivalently, the optimal rate is achieved iff there exists another family of codes $(n,\mathsf{M}'_1,\mathsf{M}'_2,\mathsf{M}'_3, \mathbf{e}',\mathbf{d}'), n \in \mathbb{N}$ for which 1) $P(\mathbf{e'}_i(M'_i)\in \mathbb{C}_i\oplus A_\epsilon^n(N))\to 1$ as $n\to \infty$, 2) $\mathbb{C}'_1=\mathbb{C}'_2$ is a coset code, and 3) they also achieve the rate triple $(R_1,R_2,R_3)=(\log{q}-H(N_1\oplus N_3),\log{q}-H(N_1\oplus N_3),H(N_1\oplus N_3)-H(N_3))$. We provide an outline of the proof in Appendix \[App:eg1\]. The lemma proves that even if we expand our search to arbitrary $n$-length codebook constructions (as opposed to the usual random codebook generation based on single-letter distributions), coset codes are the only efficient ensemble of codes for the classes of interference channels under consideration up to small perturbations. This is a stronger assertion than the well-known result that linear codes are useful for aligning the interfering signals. The lemma can be used to provide a converse result proving that schemes involving random unstructured codes (e.g. the generalized version of the single-letter HK scheme), can’t achieve the desired rate-triple without directly analyzing the bounds corresponding to their achievable rate region as done in [@IC]. Example 2 --------- Next, we consider an example where interference alignment results in a tradeoff between two of the users. Consider the setup in Figure \[fig:eg2\]. Similar to the previous example, all input alphabets, output alphabets, and additions are defined on the field $\mathbb{F}_q$. The outputs of the channel are $Y_1=X_1\oplus_{q}N_1\oplus_{q}N_2\oplus_{q}N_3$, $Y_2=X_1\oplus_{q}X_2\oplus_{q}N_2\oplus N_3$, and $Y_3=2X_1\oplus_{q}X_2\oplus_{q}X_3\oplus_{q}N_3$. Following our arguments in the previous example, for user 3 to be able to transmit its messages at rate $R_3=H(X_3\oplus N_3)-H(N_3)$, the inputs for users $1$ and $2$ must align. However, if these two users align their inputs, user 2 would not be able to decode its message which is being corrupted by its aligned interfering signal coming from user 1. Hence, we have a tradeoff. We proceed with evaluating the rate-triples achievable in this example. The following lemma proves that we must have $R_1+R_2\leq \log{q}-H(N_2\oplus N_3)$, otherwise the rate-triple $(R_1,R_2,R_3)$ is not achievable. ![A Three User IC Where Alignment Results in a Tradeoff[]{data-label="fig:eg2"}](Eg2){height="1.3in"} \[Lem:lem2\] Given that $(R_1,R_2,R_3)$ is achievable, we must have $R_1+R_2\leq log{q}-H(N_2\oplus N_3)$. Since the rate triple is achievable, there exists a family of codes $(n,\mathsf{M}_1,\mathsf{M}_2,\mathsf{M}_3, \mathbf{e},\mathbf{d})$ for which the codewords sent by the second user, $X^n_2$, is decoded at decoder 2 with error probability approaching 0. Assuming errorless decoding at decoder 2, the decoder has access to $X^n_2, X_1^n\oplus X_2^n\oplus N_2^n$. The decoder can subtract $X_2^n$ from $X_1^n\oplus X_2^n\oplus N_2^n$ to get $X_1^n\oplus N_2^n$. Now, since by assumption decoder 1 can decode $X_1^n$ from $X_1^n\oplus N_1^n\oplus N_2^n$ with error going to 0, decoder 2 can use $X_1^n\oplus N_2^n$ to recover $X_1^n$ with small error. So decoder 2 has access to $M_1$ and $M_2$. By the converse of the point-to-point channel coding theorem, we must have $\frac{1}{n}\log{|\mathsf{M}_1\times\mathsf{M}_2|}\leq \log{q}-H(N_2)$, which completes the proof. We want to achieve the rate $R_1=\log{q}-H(N_1\oplus N_2\oplus N_3)$. In other words, encoder 1 is to operate at PtP optimality. The goal is to optimize the linear combination $R_2+R_3$. We argue that the linear coding scheme presented in [@IC] can’t achieve the triple $(R_1,R_2,R_3)$ for $R_2+R_3>H(N_1\oplus N_2\oplus N_3)-H(N_3)$. \[Lem:lem3\] Given $R_1=\log{q}-H(N_1)$, the scheme in [@IC] can’t achieve $R_2+R_3>H(N_1\oplus N_2\oplus N_3)-H(N_3)$. We provide the intuition behind the proof here. Let us use two NCL’s $\mathcal{C}_1$ and $\mathcal{C}_2$ as defined in Definition \[def:PNLC\] to transmit the messages at encoders 1 and 2. Let the rate of $\mathcal{C}_j, j\in \{1,2\}$ be $r_j$ and let the inner code have rate $r_i$. If we assume that the coding scheme exists which achieves the rate-triple, then by the proof of Lemma \[Lem:lem2\], one should be able to recover $X_1, X_2$ from $X_1\oplus X_2\oplus N_2\oplus N_3$ with small error probability. Also, at decoder $3$, the decoder can reconstruct $X_3$ with low error probability and by subtraction it can have $2X_1\oplus X_2\oplus N_3$. Note that in the linear coding scheme, both $2X_1\oplus X_2$ and $X_1\oplus X_2$ come from randomly and uniformly generated linear codes of rate $r_1+r_2-r_i$. So, given that $X_1$ and $X_2$ can be recovered from $X_1\oplus X_2\oplus N_2\oplus N_3$, decoder 3 must be able to recover $X_1$ and $X_2$ from $2X_1\oplus X_2\oplus N_3$. Then similar to the proof of Lemma \[Lem:eg1\], by the point-to-point channel coding converse, we must have $R_1+R_2+R_3<\log{q}-H(N_3)$. The arguments in the proof of the previous lemma suggest that NLC’s lack the necessary flexibility when it comes to determining the size of different linear combinations of such codes. We explain this in more detail. Consider two NLC’s, $\mathcal{C}$ and $\mathcal{C}'$, with rates $r_o$ and $r'_o$, respectively, and with inner code rate $r_i$. The rate of any linear combination of the two, $\alpha \mathcal{C}\oplus \beta\mathcal{C}', \alpha,\beta\in\mathbb{F}_q\backslash\{0\}$, is equal to $r_o+r'_o-r_i$. Whereas in settings such as the one at hand, it is desirable to have different rates for different values of $\alpha$ and $\beta$. In this setup, decoder 2 requires $\mathcal{C}_1\oplus \mathcal{C}_2$ to be large (since by Lemma \[Lem:lem2\] in order to increase $R_2$ it needs to increase the rate of this linear combination) and decoder 3 wants the size of the interfering codebook $2\mathcal{C}_1\oplus \mathcal{C}_2$ to be small, so that it can decode the interference. In the next section, we provide a new class of codes. The new construction allows for different rates for different linear combinations of such codes. This in turn results in higher achievable sum-rates. A New Class of Code Constructions {#sec:code} ================================= In this section, we present our new coding constructions. These new codes are called Quasi Linear Codes (QLC). They are not linearly closed but maintain a degree of linearity. In order to construct a QLC, we first construct a linear code. Then, we take a subset of that codebook to transmit the messages. More precisely, QLC’s are defined as follows: \[def:Qlingen\] A $(k,n)$ QLC on the field $\mathbb{F}_q$, is characterized by a generator matrix $G_{k\times n}$, a dither $\mathbf{b}^n$ and a set $\mathsf{U}$. The codebook is defined as $$\mathcal{C}\triangleq\{\mathbf{u}G\oplus \mathbf{b}|\mathbf{u}\in \mathsf{U}\}.$$ For injective $G$ on $\mathsf{U}$, the rate of the code is given by $R=\frac{1}{n}\log{|\mathcal{C}|}=\frac{1}{n}\log{|\mathsf{U}|}$. In this paper we only consider the cases when $\mathsf{U}$ is a cartesian product of typical sets: $$\mathcal{C}\triangleq \{\sum_{i\in [1,m]}\mathbf{u}_iG_i\oplus \mathbf{b}|\mathbf{u}_i\in A_{\epsilon}^{k_i}(U_i)\}.$$ A pair of Nested Quasi Linear Codes (NQLC) is defined below: \[Def:PQLC\] For natural numbers $k_1,k_2,\dotsb, k_m$, let $G_{k_i\times n}, i\in [1,m]$ be matrices, and $\mathbf{b}, \mathbf{b}'$ dithers all defined on $\mathbb{F}_q$ . Also, let $(U_1,U_2,\dotsb, U_m)$ and $(U'_1,U'_2,\dotsb, U'_m)$ be a pair of random vectors on $\mathbb{F}_q$. The pair of QLC’s characterized by the matrices $G_{k_i\times n}, i\in [1,m]$ and each of the two dithers and vectors of random variables are called a pair of NQLC’s. As explained in the previous section, our motivation for defining NQLC’s, is to construct codes such that different linear combinations of those codes have different rates. The next lemma shows that NQLC’s have this property. Let $\mathcal{C}$ and $\mathcal{C}'$ be two QLC’s as defined in Definition \[Def:PQLC\], whose generator matrices and dithers are taken randomly and uniformly from $\mathbb{F}_q$. Then, $\alpha\mathcal{C}_1\oplus \beta\mathcal{C}_2$ has rate close to $\sum_{i\in [1,m]}\frac{k_i}{n}H(\alpha U_i\oplus \beta U'_i)$ for large $n$ with high probability. The proof follows from the injectiveness of the $G_i$’s and the usual typicality arguments and is omitted. Having defined QLC’s, we return to our interference channel setup in Example 2. We claim that NQLC’s can achieve a sum-rate $R_2+R_3$ which is higher than $H(N_1\oplus N_2\oplus N_3)-H(N_3)$. \[Lem:newrate\] There exists achievable rate-triples $(R_1,R_2,R_3)=(\log{q}-H(N_1\oplus N_2\oplus N_3), r_2, r_3)$ such that $r_2+r_3>H(N_1\oplus N_2\oplus N_3)-H(N_3)$. Refer to Appendix \[App:newrate\]. So far we have proved that NQLC’s outperform NLC’s in this specific example. It is straightforward to show that NQLC’s are a generalization of NLC’s. To see this, consider an arbitrary pair of NLC’s with the parameters as in definition \[def:PNLC\]. These two codes are a pair of NQLC’s with parameters $m=3$, $U_1, U_2$ and $U'_1,U'_3$ uniform, $U_3$ and $U'_2$ constants and $k_1=k'_1=k_i$ and $k_2=k_o-k_i, k'_3=k'_o-k_i$. So, any rate region achievable by NLC’s is also achievable using NQLC’s. New Achievable Rate Region {#sec:RR} ========================== In this section, we provide a general achievable rate region for the three user IC. The scheme is similar to the one presented in [@IC] (Theorem 2). The main difference is that here instead of NLC’s we use NQLC’s. The random variables involved in the coding scheme are depicted in Figure \[fig:rvs\]. Note that in contrast with the scheme in [@IC], decoder 2 reconstructs a linear combination of $U_1$ and $U_2$. By setting $\alpha_2=0, \beta_2=1$, we recover the random variables in [@IC]. The next theorem provides the achievable rate region. ![The LHS random variables are the ones sent by each encoder, the RHS random variables are the ones decoded at each decoder.[]{data-label="fig:rvs"}](rvs){height="1in"} For a given three user IC problem with $q$-ary inputs and outputs, define the set $\mathcal{R}_{3\text{-IC}}$ as the set of rate triples $(R_1,R_2,R_3)$ such that there exist 1) a joint probability distribution $P_{U_1,X_1}P_{U_2,X_3}P_{X_3}$, 2) A vector of positive reals $(K_1,K_2,L_1,L_2,T_1,T_2)$, and 3) a vector of parameters $(m,n,k_1,k_2,\dotsb,k_m)$ and pair of vectors of random variables $(V_{ij})_ {j\in [1,m]}, i\in\{1,2\}$, such that the following inequalities are satisfied: $$\begin{aligned} &R_1=L_1+T_1, R_2=L_2+\frac{I(U_2;\alpha_2U_1\oplus \alpha_2U_2)}{H(U_2)}T_2\label{eq:rates}\\ &r_{1,0}-T_1\geq \log{q}-H(U_1), r_{0,1}-T_2\geq \log{q}-H(U_2), \label{eq:packS}\\ &K_1+r_{1,0}-T_1\geq \log{q}+H(X_1)-H(X_1,U_1)\\ &K_2+r_{0,1}-T_2\geq \log{q}+H(X_2)-H(X_2,U_2)\label{eq:packF}\\ &r_{0,1}\leq \log{q}-H(U_1|X_1,Y_1)\label{eq:cov1S}\\ &r_{0,1}+L_1+K_1\leq \log{q}+H(X_1)-H(U_1,X_1|Y_1)\\ &L_1+K_1\leq I(X_1;U_1Y_1)\label{eq:cov1F}\\ &r_{\alpha_2,\beta_2}\leq \log{q}-H(\alpha_2U_1\oplus \beta_2|X_2,Y_2)\label{eq:cov2S}\\ &r_{\alpha_2,\beta_2}\!\!+L_2+K_2\leq \log{q}\!+\!H(X_2)-H(\alpha_2U_1\!\!\oplus \!\!\beta_2U_2,X_2|Y_2)\\ &L_2+K_2\leq I(X_2;U_2Y_2)\label{eq:cov2F}\\ &r_{\alpha_3,\beta_3}\!\!+L_3+K_3\leq \log{q}\!+\!H(X_3)-H(\alpha_3U_1\!\!\oplus \!\!\beta_3U_2,X_3|Y_3)\label{eq:cov3S}\\ &R_3\leq I(X_3;Y_3,\alpha_3U_1\oplus \beta_3U_2)\label{eq:cov3F}\end{aligned}$$ ,where $r_{\alpha,\beta}\triangleq \sum_{i\in[1,m]}\frac{k_i}{n}H(\alpha V_1\oplus \beta V_2),\forall \alpha,\beta\in \mathbb{F}_q$. A rate triple $(R_1,R_2,R_3)$ is achievable if it belongs to $cl(\mathcal{R}_{3\text{-IC}})$. We provide an outline of the proof. The coding scheme is similar to the one in [@IC]. Except that 1) decoder 2 also decodes a linear combination $\alpha_2 U_{1}+\beta_2 U_2$, 2) The underlying codes for $U_1$ and $U_2$ are QNLC’s instead of nested coset codes, and 2) There is an outer code on $U_2$ which allows decoder $2$ to decode $U_2$ from $\alpha_2 U_{1}+\beta_2 U_2$. As a result the rate region is similar to the one in [@IC] except for a few changes. Bounds (\[eq:packS\])-(\[eq:packF\]) ensure the existence of jointly typical codewords at each encoder. These bounds are the same with the ones in [@IC]. Bounds (\[eq:cov1S\])-(\[eq:cov1F\]) ensure errorless decoding at decoder $1$, they also remain the same. Inequalities (\[eq:cov2S\])-(\[eq:cov2F\]) correspond to the error events at decoder 2, these bounds are altered to ensure reconstruction of $\alpha_2U_1+\beta_2U_2$, also the rate $R_2$ is changed and the linear coding rate $T_2$ is multiplied by $\frac{I(U_2;\alpha_2U_1\oplus \alpha_2U_2)}{H(U_2)}$, which is due to the outer code. Lastly, (\[eq:cov3S\])-(\[eq:cov3F\]) are for the error events at decoder 3, which is also similar to the ones in [@IC]. For ease of notation, we have dropped the time-sharing random variable $Q$. The scheme can be enhanced by adding the variable in the standard way. By taking $\alpha_2=0$ and $\beta_2=1$ and choosing the NQLC parameters so that the codes become a pair of NLC’s we recover the bound in [@IC] as expected. Following the generalizations in [@IC], this coding scheme can be enhanced by adding additional layers containing the public message codebooks corresponding to the HK strategy. Conclusion {#sec:conclude} ========== The problem of three user IC was considered. We showed that there is an inherent tradeoff in the general IC. The users can choose to communicate their messages by using optimal PtP strategies or cooperate with other users to facilitate their communication. It was shown that the previously used coding structures are unable to optimize this tradeoff. New coding structures were proposed. It was shown through an example that these new structures give strict improvements. Using these new codebooks, an achievable region for the three user IC was derived which improves upon the previous known inner bounds for this problem. Proof of Lemma \[Lem:eg1\] {#App:eg1} ========================== Assume the family $(n,\mathsf{M}_1,\mathsf{M}_2,\mathsf{M}_3, \mathbf{e},\mathbf{d}), n \in \mathbb{N}$ achieves the rate-triple. Let $M_i,i\in \{1,2,3\}$ be uniform random variables defined on sets $\mathsf{M}_i$. In the first step, we argue that the size of the set ${\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbb{C}_3\oplus A_{\epsilon}^n(N_3)}$ is close to ${|\mathbb{C}_1\oplus \mathbb{C}_2\oplus A_{\epsilon}^n(N_3)||\mathbb{C}_3|}$. More precisely, we prove the following claim: For every $\epsilon\in \mathbb{R}^+$, there exists a sequence of numbers $\alpha_{n,\epsilon}\in \mathbb{R}^+, n\in\mathbb{N}$ such that the following inequality holds: $$\frac{1}{n}\log{|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbb{C}_3\oplus A_{\epsilon}^n(N_3)|}\geq \frac{1}{n}\log{(|\mathbb{C}_1\oplus \mathbb{C}_2\oplus A_{\epsilon}^n(N_3)||\mathbb{C}_3|)}-\alpha_{n,\epsilon},$$ and $\alpha_{n,\epsilon}$ goes to 0 as $n\to \infty$ and $\epsilon \to 0$. Intuitively, if the size of $\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbb{C}_3\oplus A_{\epsilon}^n(N_3)$ is much smaller than $|\mathbb{C}_1\oplus \mathbb{C}_2\oplus A_{\epsilon}^n(N_3)||\mathbb{C}_3|$, that means there exists a large number of sets of vectors $\mathbf{c_1},\mathbf{c}_2,\mathbf{c}_3,\mathbf{n}_3$, with different $\mathbf{c}_3$’s for which the sum is equal. This causes a large error probability in decoder 3 since the decoder is unable to distinguish between these sets of vectors. More precisely, let $\mathbf{n_t}$ be a type on vectors in $A_{\epsilon}^n(N_3)$, and let $\mathbf{c}_3\in \mathbb{C}_3$, define $\mathsf{B}_{\mathbf{c}_3, \mathbf{n}_t}$ as follows, $$\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}=\{\mathbf{c}_1\oplus\mathbf{c}_2\oplus \mathbf{n}_3| \exists \mathbf{c}'_1,\mathbf{c}'_2,\mathbf{c}'_3, \mathbf{n}'_3\in \mathbb{C}_1\times\mathbb{C}_2\times\mathbb{C}_3\times \mathcal{P}_t, \text{ such that } \mathbf{c}'_1\oplus\mathbf{c}'_2\oplus\mathbf{c}'_3\oplus\mathbf{n}'_3=\mathbf{c}_1\oplus\mathbf{c}_2\oplus\mathbf{c}_3\oplus\mathbf{n}_3, \mathbf{c}'_3\neq\mathbf{c}_3\},$$ where $\mathcal{P}_t$ is the set of all vectors $\mathbf{n}_3\in A_{\epsilon}^n(N_3)$ with type $\mathbf{n}_t$. That is $\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}$ is the set of $(\mathbf{c}_1,\mathbf{c}_2,\mathbf{n}_3)$’s for which the decoder has non-zero error probability for decoding $\mathbf{c}_3$ or another codeword $\mathbf{c}'_3$. From set theory, we have the following: $$\begin{aligned} |\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbb{C}_3\oplus A_{\epsilon}^n(N_3)|&\geq |\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbb{C}_3\oplus \mathcal{P}_t|\\ &= |\bigcup_{\mathbf{c}_3}\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbf{c}_3\oplus \mathcal{P}_t|\\ &\geq |\bigcup_{\mathbf{c}_3}\left(\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbf{c}_3\oplus \mathcal{P}_t - \bigcup_{\mathbf{c}'_3\neq \mathbf{c}_3} \mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbf{c}'_3\oplus \mathcal{P}_t \right)|\\ &=\sum_{\mathbf{c}_3} |\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbf{c}_3\oplus \mathcal{P}_t - \bigcup_{\mathbf{c}'_3\neq \mathbf{c}_3} \mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbf{c}'_3\oplus \mathcal{P}_t|\\ &=\sum_{\mathbf{c}_3\in\mathbb{C}_3}(|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t|-|\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}|)\\ &=|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t||\mathbb{C}_3|-\sum_{\mathbf{c}_3\in\mathbb{C}_3}|\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}|\\ &=|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t||\mathbb{C}_3|-{|\mathbb{C}_3|}\sum_{\mathbf{c}_3\in\mathbb{C}_3}\frac{|\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}|}{|\mathbb{C}_3|}\\ &=\left(|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t|-E(|\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}|)\right){|\mathbb{C}_3|} \\ &=\left(|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t|(1-\frac{E(|\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}|)}{|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t|})\right){|\mathbb{C}_3|}\label{lastneq} .\end{aligned}$$ On the other hand, as $n\to \infty$, the error probability at decoder $3$ goes to 0. This means that $P\left(\mathbf{d}_3(\mathbf{c}_1\oplus \mathbf{c}_2\oplus \mathbf{e}(M_3) \oplus \mathbf{n}_3)\neq M_3 \right)$ goes to 0. Consequently, there exists a family of types type $\mathbf{n}_t$ such that $P\left(\mathbf{d}_3(\mathbf{c}_1\oplus \mathbf{c}_2\oplus \mathbf{e}(M_3) \oplus \mathbf{n}_3)\neq M_3 |\mathbf{n}_t \right)$ goes to 0. There exists a sequence $\delta_n$ which approaches 0 at the limit such that: $$\begin{aligned} \delta_n &\geq P\left(\mathbf{d}_3(\mathbf{c}_1\oplus \mathbf{c}_2\oplus \mathbf{e}(M_3) \oplus \mathbf{n}_3)\neq M_3 |\mathbf{n}_t \right)\\ &{\geq} \frac{1}{2}P\left(\mathbf{c}_1\oplus\mathbf{c}_2\oplus \mathbf{n}_3\in \mathsf{B}_{\mathbf{e}(M_3),\mathbf{n}_t}|\mathbf{n}_t \right) \\ &{\geq} \frac{1}{2}\sum_{\mathbf{c}_3\in \mathbb{C}_3}\frac{|\mathsf{B}_{\mathbf{c}_3,\mathbf{n}_t}|}{|\mathbb{C}_3||\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t|}\\ &=\frac{1}{2}\frac{E(|\mathsf{B}_{\mathbf{e}_3(M_3),\mathbf{n}_t}|)}{|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t|} \end{aligned}$$ Inserting this last inequality in (\[lastneq\]) we have, $$\begin{aligned} &|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t||\mathbb{C}_3|-\sum_{\mathbf{c}_3\in\mathbb{C}_3}|\mathsf{B}_{\mathbf{c}_3}|\\ &\geq \left(|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathcal{P}_t|(1-2\delta_n) \right){|\mathbb{C}_3|}\end{aligned}$$ Observe that $\mathcal{P}_t$ and $A_{\epsilon}^n(N_3)$ have the same exponential rate. Note that $|\mathbb{C}_1\oplus \mathbb{C}_2\oplus A_{\epsilon}^n(N_3)|\geq|\mathbb{C}_1\oplus A_{\epsilon}^n(N_3)|$ and since decoder 1 can decode $X_1$ with probability of error approaching 0, we can use the same argument to show the following: $$\begin{aligned} &\frac{1}{n}\log{|\mathbb{C}_1\!\!\oplus A_{\epsilon}^n(N_3)|}\!\!\to\!\! \frac{1}{n}\log{|\mathbb{C}_1||A_{\epsilon}^n(N_3)|}\to \!\log{q}\!-\!\!H(N_1)\oplus H(N_3)\nonumber \\ &\Rightarrow\frac{1}{n}\log{|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbb{C}_3\oplus A_{\epsilon}^n(N_3)|}\geq \\&\qquad\qquad\left(\log{q}-H(N_1)+H(N_3)\right)+\left(H(N_1)-H(N_3)\right)=\log{q}.\nonumber\end{aligned}$$ But we know that $\frac{1}{n}\log{|\mathbb{C}_1\oplus \mathbb{C}_2\oplus \mathbb{C}_3\oplus A_{\epsilon}^n(N_3)|}\leq\log{q}$. So, we should have equality at all of the inequalities. Hence, $\frac{1}{n}\log_q{|\mathbb{C}_1\oplus \mathbb{C}_2\oplus A_{\epsilon}^n(N_3)|}\to \frac{1}{n}\log_q{|\mathbb{C}_1\oplus A_{\epsilon}^n(N_3)|}$. In order to have $|\mathbb{C}_1\oplus \mathbb{C}_2|$ close to $|\mathbb{C}_1|$, we must have the properties stated in the lemma. Proof of Lemma \[Lem:newrate\] {#App:newrate} ============================== We provide a coding scheme based on NQLC’s which achieves the rate vector. Consider two ternary random variables $V_1$ and $V_2$ such that $H(V_1\oplus V_2)>H(2V_1\oplus V_2)$. We will show the achievability of the following rate-triple: $$\begin{aligned} & R_1=\log{q}-H(N_1\oplus N_2\oplus N_3)\\ & R_2=(\frac{H(V_1\oplus V_2)}{H(V_1)}-1)(\log{q}-H(N_1\oplus N_2\oplus N_3))\\ & R_3=\log{q}-H(N_3)-\frac{H(2V_1\oplus V_2)}{H(V_1)}\left(\log{q}-H(N_1\oplus N_2\oplus N_3)\right).\end{aligned}$$ Note that in this case $R_2+R_3=H(N_1\oplus N_2\oplus N_3)-H(N_3)+\frac{(H(V_1\oplus V_2)-H(2V_1\oplus V_2))}{H(V_1)}\left(\log{q}-H(N_1\oplus N_2\oplus N_3)\right)$. Choose random variable $V_3$ such that $R_3=H(V_3\oplus N_3)-H(N_3)$.\ **Codebook Generation:** Construct a family of pairs NQLC’s with length $n$ and parameters $m=1$, $k_1=\frac{(\log{q}-H(N_1\oplus N_2\oplus N_3))}{H(V_1)}n$, $U_1=V_1$, and $U'_1=V_2$ by choosing the dither $\mathbf{b}$ and generator matrix $G_1$ randomly and uniformly on $\mathbb{F}_q$. For a fixed $n\in \mathbb{N}$, Let $\mathcal{C}^n_1$ and $\mathcal{C}^n_2$ be the corresponding pair of NQLC’s. Let $\Phi_i=2^{nR_i}$ for $i\in \{1,2\}$. Choose $\Phi_i$ of the codewords in $\mathcal{C}_i^n$ randomly and uniformly, and index these sequences using the indices $[1,\Phi_i]$. Also, generate an unstructured codebook $\mathcal{C}_3$ randomly and uniformly with rate $R_3$ based on the single-letter distribution $P_{V_3}$. Index $\mathcal{C}_3$ by $[1,2^{nR_3}]$.\ **Encoding:** Upon receiving message index $M_i$ encoder $i$ sends the sequence in $\mathcal{C}_1$ which is indexed $M_i$ for $i\in\{1,2\}$. Let the codewords sent by encoder $i, i\in \{1,2\}$ be denoted by $\mathbf{v}_iG_1\oplus \mathbf{b}_i$. Encoder 3 sends the codeword in $\mathcal{C}_3$ indexed by $M_3$. Let the codeword sent by the third decoder be denoted by $\mathbf{c}_3$.\ **Decoding:** Decoder 1 receives $X^n_1\oplus N^n_1\oplus N^n_2\oplus N^n_3$. Using typicality decoding, the decoder can decode the message as long as $\frac{k_1}{n}H(V_1)\leq \log{q}-H(N_1\oplus N_2\oplus N_3)$. Decoder 2 receives $X^n_1\oplus X^n_2\oplus N^n_2\oplus N^n_3=(\mathbf{v}_1\oplus \mathbf{v}_2)G_1\oplus \mathbf{b}_1\oplus \mathbf{b}_2\oplus N^n_2\oplus N^n_3$. It can decode $\mathbf{v}_1,\mathbf{v}_2$ jointly as long as 1) $ \frac{k_1}{n}H(V_1\oplus V_2)<\log{q}-H(N_1\oplus N_3)$, and 2) $R_1+R_2\leq \frac{k_1}{n}H(V_1\oplus V_2)$. The first condition ensures that $\mathbf{v}_1\oplus \mathbf{v}_2$ can be recovered with probability of error going to 0 as $n\to \infty$. After recovering $\mathbf{v}_1\oplus \mathbf{v}_2$, the decoder needs to jointly decode $\mathbf{v}_1,\mathbf{v}_2$ (for reasons explained in Lemma \[Lem:lem2\]). This is a noiseless additive MAC problem and condition 2 ensures errorless decoding. Note that in condition 2, the coefficient $\frac{k_1}{n}$ is present since $\mathbf{v}_1$ is of length $k_1$. Also, The term $H(V_1\oplus V_2)$ is the capacity of the MAC channel. Decoder 3 receives $X^n_1\oplus X^n_2\oplus X^n_3\oplus N^n_3=(2\mathbf{v}_1\oplus \mathbf{v}_2)G_1\oplus \mathbf{b}_1\oplus \mathbf{b}_2\oplus \mathbf{c}_3^n\oplus N^n_3$. The decoder can recover $2\mathbf{v}_1\oplus \mathbf{v}_2$ as long as $\frac{k_1}{n}H(2V_1\oplus V_2)<\log{q}-H(X_3\oplus N_3)$. Then, the decoder subtracts $2X_1^n\oplus X_2^n$ to get $X^n_3\oplus N^n_3$. It can decode $X_3$ as long as $R_3\leq H(V_3\oplus N_3)-H(N_3)$. It is straightforward to check the rate given at the beginning satisfy all of these bounds. [1]{} H. Sato, “The capacity of the Gaussian interference channel under strong interference,” IEEE Trans. Inf. Theory, vol. 27, no. 6, pp. 786-788, Nov. 1981. M. H. M. Costa, and A. El Gamal, “The capacity region of the discrete memoryless interference channel with strong interference,” IEEE Trans. Inf. Theory, vol. 33, no. 5, pp. 710-711, 1987. T. S. Han, and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Trans. Inf. Theory, vol. 27, no. 1, pp. 49-60, 1981. C. Nair, K. Xia, and M. Yazdanpanah, “Sub-optimality of the Han–Kobayashi Achievable Region for Interference Channels”, arXiv preprint arXiv:1502.02589, 2015 M. Maddah-Ali, A. Motahari, and A. Khandani, “Communication over MIMO X channels: Interference alignment, decomposition, and performance analysis,” Information Theory, IEEE Transactions on, vol. 54, no. 8, pp. 3457-3470, Aug 2008. V. Cadambe, and S. Jafar, “Interference alignment and degrees of freedom of the k -user interference channel,” IEEE Trans. on Info. Th., vol. 54, no. 8, pp. 3425-3441, 2008. J. Korner, K. Marton, How to encode the modulo-two sum of binary sources, IEEE Transactions on Information Theory , vol. 25, no. 2, pp. 219-221, 1979. A. Padakandla, S.S. Pradhan, Achievable rate region for three user discrete broadcast channel based on coset codes, in IEEE International Symp. on Information Theory. IEEE, pp.1277,1281, 7-12 July 2013 A. Padakandla, A.G. Sahebi, S.S. Pradhan, A new achievable rate region for the 3-user discrete memoryless interference channel, Proceedings of IEEE International Symposium on Information Theory, July, 2012 F. Shirani, S.S. Pradhan, “An Achievable Rate-Distortion Region for Multiple Descriptions Source Coding Based on Coset Codes,” arXiv:1602.01911, 2016

--- abstract: 'We develop an approach to liquid thermodynamics based on collective modes. We perform extensive molecular dynamics simulations of noble, molecular and metallic liquids and provide the direct evidence that liquid energy and specific heat are well-described by the temperature dependence of the Frenkel (hopping) frequency. The agreement between predicted and calculated thermodynamic properties is seen in the notably wide range of temperature spanning tens of thousands of Kelvin. The range includes both subcritical liquids and supercritical fluids. We discuss the structural crossover and inter-relationships between structure, dynamics and thermodynamics of liquids and supercritical fluids.' address: - '$^1$ School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK' - '$^2$ Institute for High Pressure Physics, RAS, 142190, Moscow, Russia' author: - 'L. Wang$^{1}$' - 'C. Yang$^{1}$' - 'M. T. Dove$^{1}$' - 'Yu. D. Fomin$^{2}$' - 'V. V. Brazhkin$^{2}$' - 'K. Trachenko$^{1}$' title: 'Direct links between dynamical, thermodynamic and structural properties of liquids: modelling results' --- Introduction ============ It is an interesting fact that the liquid state has proven to be difficult to describe by theory throughout the history of condensed matter research [@frenkel; @boonyip; @march; @march1; @enskog; @baluca; @ziman; @zwanzig; @hansen1; @faber; @hansen2; @landau]. The problem extends beyond condensed matter and exists in other areas where strong interactions are combined with dynamical disorder such as field theory. In a weakly-interacting system such as a dense gas, the potential energy is much smaller than the kinetic energy. These systems are amenable to perturbation treatment giving corrections to the non-interacting case [@enskog]. Perturbation approaches have been widely explored to calculate liquid thermodynamic properties but have not been able to agree with experiments. For example, the analysis of tractable models such as van der Waals or hard-spheres systems returns the gas-like result for the liquid constant-volume specific heat $c_v=\frac{3}{2}k_{\rm B}$ [@landau; @pre; @prl]. This is in contrast to experimental results showing that $c_v$ of monatomic liquids close to the melting point is nearly identical to the solid-like result, $c_v=3k_{\rm B}$ and decreases to about $2k_{\rm B}$ at high temperature [@grimvall; @wallace]. As expected on general grounds, the perturbation approach does not work for strongly-interacting systems. Strong interactions are successfully treated in solids, crystals or glasses, where the harmonic model is a good starting point and gives the most of the vibrational energy. However, this approach requires fixed reference points around which the energy expansion can be made. With small vibrations around mean atomic positions, solids meet this requirement but liquids seemingly do not: liquid ability to flow implies that the reference lattice is non-existent. Therefore, liquids seemingly have no simplifying features such as small interactions of gases or small displacements of solids [@landau]. In other words, liquids have no small parameter. One might adopt a general approach not relying on approximations and seek to directly calculate the liquid energy for a model system where interactions and structure are known. This meets another challenge: because the interactions are both strong and system-dependent, the resulting energy and other thermodynamic functions will also be strongly system-dependent, precluding their calculation in general form and understanding using basic principles, in contrast to solids and gases [@landau]. Consistent with this somewhat pessimistic view, the discussion of liquid thermodynamic properties has remained scarce. Indeed, physics textbooks have very little, if anything, to say about liquid specific heat, including textbooks dedicated to liquids [@frenkel; @boonyip; @march; @march1; @enskog; @baluca; @ziman; @zwanzig; @hansen1; @faber; @hansen2; @landau]. As recently reviewed [@ropp], emerging evidence advances our understanding of the thermodynamics of the liquid state. The start point is the early theoretical idea of J Frenkel [@frenkel] who proposed that liquids can be considered as solids at times smaller than liquid relaxation time, $\tau$, the average time between two particle rearrangements at one point in space. This implies that phonons in liquids will be similar to those in solids for frequencies above the Frenkel frequency $\omega_{\rm F}$: $$\omega>\omega_{\rm F}=\frac{1}{\tau} \label{omega}$$ The above argument predicts that liquids are capable of supporting shear modes, the property hitherto attributable to solids only, but only for frequencies above $\omega_{\rm F}$. We note that low-frequency modes in liquids, sound waves, are well-understood in the hydrodynamic regime $\omega\tau<1$ [@hydro], however Eq. (1) denotes a distinct, solid-like elastic regime of wave propagation where $\omega\tau>1$. In essence, this suggests the existence of a cutoff frequency $\omega_{\rm F}$ above which particles in the liquid can be described by the same equations of motion as in, for example, solid glass. Therefore, liquid collective modes include both longitudinal and transverse modes with frequency above $\omega_{\rm F}$ in the solid-like elastic regime and one longitudinal hydrodynamic mode with frequency below $\omega_{\rm F}$ (shear mode is non-propagating below frequency $\omega_{\rm F}$ as discussed below). Recall the earlier textbook assertion [@landau] that a general thermodynamic theory of liquids can not be developed because liquids have no small parameter. How is this fundamental problem addressed here? According to Frenkel’s idea, liquids behave like solids with small oscillating particle displacements serving as a small parameter. Large-amplitude diffusive particle jumps continue to play an important role, but do not destroy the existence of the small parameter. Instead, the jumps serve to modify the phonon spectrum: their frequency, $\omega_{\rm F}$, sets the minimal frequency above which the small-parameter description applies and solid-like modes propagate. It has taken a long time to verify this picture experimentally. The experimental evidence supporting the propagation of high-frequency modes in liquids currently includes inelastic X-ray, neutron and Brillouin scattering experiments but most important evidence is recent and follows the deployment of powerful synchrotron sources of X-rays [@copley; @pilgrim; @burkel; @pilgrim2; @ruocco; @water; @rec-review; @hoso; @hoso3; @mon-na; @mon-ga; @sn; @disu1; @disu2; @grim; @scarponi; @water-fast; @water-tran]. Early experiments detected the presence of high-frequency longitudinal acoustic propagating modes and mapped dispersion curves which were in striking resemblance to those in solids [@copley]. These and similar results were generated at temperature just above the melting. The measurements were later extended to high temperatures considerably above the melting point, confirming the same result. It is now well established that liquids sustain propagating modes with wavelengths extending down towards interatomic separations, comparable to the wave vectors of phonons in crystals at the Brillouin zone boundaries [@pilgrim; @burkel; @pilgrim2; @ruocco; @water; @rec-review; @hoso; @hoso3; @mon-na; @mon-ga; @sn; @disu1; @disu2]. More recently, the same result has been asserted for supercritical fluids [@water; @disu1; @disu2]. Importantly, the propagating modes in liquids include acoustic transverse modes. These were first seen in highly viscous fluids (see, e.g., Refs. [@grim; @scarponi]), but were then studied in low-viscosity liquids on the basis of positive dispersion [@pilgrim; @burkel; @pilgrim2; @rec-review] (the presence of high-frequency transverse modes increases sound velocity from the hydrodynamic to the solid-like value). These studies included water [@water-fast], where it was found that the onset of transverse excitations coincides with the inverse of liquid relaxation time [@water-tran], as predicted by Frenkel [@frenkel]. More recently, high-frequency transverse modes in liquids were directly measured in the form of distinct dispersion branches and verified on the basis of computer modeling [@hoso; @mon-na; @mon-ga; @sn; @hoso3], and the striking similarity between dispersion curves in liquids and their crystalline (poly-crystalline) counterparts was noted. We note that the contribution of high-frequency modes is particularly important for liquid thermodynamics because these modes make the largest contribution to the energy due to quadratic density of states. The above discussion calls for an important question about liquid thermodynamics. In solids, collective modes, phonons, play a central role in the theory, including the theory of thermodynamic properties. Can collective modes in liquids play the same role, in view of the earlier Frenkel proposal and recent experimental evidence? We have started exploring this question [@prb] just before the high-frequency transverse modes were directly measured and subsequently developed it in a number of ways [@ropp]. This involves calculating the liquid energy as the phonon energy where transverse modes propagate above $\omega_{\rm F}$ in Eq. (\[omega\]). The main aim of this paper is to provide direct computational evidence to the phonon theory of liquid thermodynamics and its predictions. We achieve this by calculating the liquid energy and $\omega_{\rm F}$ in extensive molecular dynamics simulations. In the next chapter, we briefly discuss the main steps involved in calculating the liquid energy. We then proceed to calculating the liquid energy and Frenkel frequency independently from molecular dynamics simulations using several methods which agree with each other. We do this for three systems chosen from different classes of liquids: noble, metallic and molecular, and find good agreement between predicted and calculated results in the wide range of temperature and pressure. The range includes both subcritical liquids and supercritical state below the Frenkel line where transverse waves propagate. We calculate and analyze liquid energy and $c_v$ using several different methods. Finally, we discuss how our results offer insights into inter-relationships between structure, dynamics and thermodynamics in liquids and supercritical fluids. Phonon approach to liquid thermodynamics ======================================== Calculating liquid energy ------------------------- We summarize the main result of calculation of the liquid energy on the basis of propagating modes. A detailed discussion can be found in a recent review [@ropp]. According to the previous discussion, the propagating modes in liquids include two transverse modes propagating in the solid-like elastic regime with frequency $\omega>\omega_{\rm F}$. The energy of these modes, together with the energy of the longitudinal mode gives the liquid vibrational energy. In addition to vibrations, particles in the liquids undergo diffusive jumps between quasi-equilibrium positions as discussed above. Adding the energy of these jumps to the phonon energy in the Debye model gives the total energy of thermal motion in the liquid [@ropp; @prb]: $$E_{\rm T}=NT\left(3-\left(\frac{\omega_{\rm F}}{\omega_{\rm D}}\right)^3\right) \label{harmo}$$ where $N$ is the number of particles and $\omega_{\rm D}$ is transverse Debye frequency and the subscript refers to thermal motion. Here and below, $k_{\rm B}=1$. At low temperature, $\tau\gg\tau_{\rm D}$, where $\tau_{\rm D}$ is the Debye vibration period, or $\omega_{\rm F}\ll\omega_{\rm D}$. In this case, Eq. (\[harmo\]) gives the specific heat $c_v=\frac{1}{N}\frac{dE}{dT}$ close to 3, the solid-like result. At high temperature when $\tau\rightarrow\tau_{\rm D}$ and $\omega_{\rm F}\rightarrow\omega_{\rm D}$, Eq. (\[harmo\]) gives $c_v$ close to 2. The decrease of $c_v$ from 3 to 2 with temperature is consistent with experimental results in monatomic liquids [@grimvall; @wallace]. The decrease of $c_v$ is also seen in complex liquids [@dexter]. Eq. (\[harmo\]) attributes the experimental decrease of $c_v$ with temperature to the reduction of the number of transverse modes above the frequency $\omega_{\rm F}=\frac{1}{\tau}$. The comparison of this effect with experiments can be more detailed if $c_v$ is compared in the entire temperature range where it decreases from $3$ to $2$. This meets the challenge that $\omega_{\rm F}$ in Eq. (\[harmo\]) is not directly available in the cases of interest. $\omega_{\rm F}$ ($\tau$) is measured is dielectric relaxation or NMR experiments in systems responding to electric or magnetic fields only. These liquids are often complex and do not include simple model systems that are widely studied theoretically such as liquid Ar. Importantly, the range of measured $\omega_{\rm F}$ does not extend to high frequency comparable to $\omega_{\rm D}$, and it is in this range where liquid $c_v$ undergoes an important change from 3 to 2 as discussed above. $\omega_{\rm F}$ can be calculated from the Maxwell relationship $\omega_{\rm F}=\frac{G_\infty}{\eta}$, where $G_\infty$ is the instantaneous shear modulus and $\eta$ is viscosity taken from a different experiment [@ropp]. More recently, it has been suggested [@puosi] that taking the shear modulus at a finite high frequency (rather than infinite frequency) agrees better with the modelling data. Apart from rare estimations [@puosi; @wallace-G], $G_\infty$ is not available. In practice, the comparison of experimental $c_v$ and $c_v$ predicted as $\frac{dE}{dT}$ with $E$ given by Eq. (\[harmo\]) is done by keeping $G_\infty$ as a free parameter, obtaining a good agreement between experimental and predicted $c_v$ and observing that $G_\infty$ lies in the range of several GPa typical for liquids [@prb; @ropp]. In the last few years, Eq. (\[harmo\]) and its extensions to include the phonon anharmonicity and quantum effects of phonon excitations was shown to account for the experimental $c_v$ of over 20 different systems, including metallic, noble, molecular and network liquids [@ropp]. In view of the persisting problem of liquid thermodynamics, it is important to test Eq. (\[harmo\]) directly by linking the liquid energy ($c_v$) on one hand and $\omega_{\rm F}$ on the other and testing the theory in a precise way. This, together with achieving consistency with other approaches to calculate the liquid energy, is one of the objectives of this study. Importantly, this programme includes supercritical fluids as well as subcritical liquids, as discussed below. Thermodynamics of supercritical fluids -------------------------------------- If the system is below the critical point (see Figure 1), the temperature increase eventually results in boiling and the first-order transition, with $c_v$ discontinuously decreasing to about $\frac{3}{2}$ in the gas phase. The intervening phase transition excludes the state of the liquid where $c_v$ can gradually reduce to $\frac{3}{2}$ and where interesting physics operates. However, this becomes possible above the critical point. This brings us to the interesting discussion of the supercritical state of matter. Theoretically, little is known about the supercritical state, apart from the general assertion that supercritical fluids can be thought of as high-density gases or high-temperature fluids whose properties change smoothly with temperature or pressure and without qualitative changes of properties. This assertion followed from the known absence of a phase transition above the critical point. We have recently proposed that this picture should be modified, and that a new line, the Frenkel line (FL), exists above the critical point and separates two states with distinct properties (see Figure \[frenline\]) [@pre; @prl; @phystoday; @ufn]. Physically, the FL is not related to the critical point and exists in systems where the critical point is absent. The main idea of the FL lies in considering how the particle dynamics change in response to pressure and temperature. Recall that particle dynamics in the liquid can be separated into solid-like oscillatory and gas-like diffusive components. This separation applies equally to supercritical fluids as it does to subcritical liquids. Indeed, increasing temperature reduces $\tau$, and each particle spends less time oscillating and more time jumping; increasing pressure reverses this and results in the increase of time spent oscillating relative to jumping. Increasing temperature at constant pressure or density (or decreasing pressure at constant temperature) eventually results in the disappearance of the solid-like oscillatory motion of particles; all that remains is the diffusive gas-like motion. This disappearance represents the qualitative change in particle dynamics and gives the point on the FL in Figure \[frenline\]. Most important system properties qualitatively change either on the line or in its vicinity [@pre; @prl; @phystoday; @ufn]. In a given system, the FL exists at arbitrarily high pressure and temperature, as does the melting line. Quantitatively, the FL can be rigorously defined by pressure and temperature at which the minimum of the velocity autocorrelation function (VAF) disappears [@prl]. Above the line defined in such a way, velocities of a large number of particles stop changing their sign and particles lose the oscillatory component of motion. Above the line, VAF is monotonically decaying as in a gas [@prl]. For the purposes of this discussion, the significance of the FL is that the phonon approach to liquids and Eq. (\[harmo\]) apply to supercritical fluids below the FL to the same extent as they apply to subcritical liquids. Indeed, the presence of an oscillatory component of particle motion below the FL implies that $\tau$ is a well-defined parameter and that transverse modes propagate according to Eq. (\[omega\]). The ability of the supercritical system to sustain solid-like rigidity at frequency above $\omega_{\rm F}$ suggested the term “rigid” liquid to differentiate it from the “non-rigid” gas-like fluid above the FL [@pre; @prl]. Therefore, the FL separates the supercritical state into two states where transverse modes can and cannot propagate. This is supported by direct calculation of the current correlation functions [@condmat] showing that propagating and non-propagating transverse modes are separated by the Frenkel line. Interestingly, Eq. (\[harmo\]) can serve as a thermodynamic definition of the FL: the loss of the oscillatory component of particle motion at the FL approximately corresponds to $\tau\rightarrow\tau_{\rm D}$ (here, $\tau_{\rm D}$ refers to Debye period of transverse modes) or $\omega_{\rm F}\rightarrow\omega_{\rm D}$. According to Eq. (\[harmo\]), this gives $c_v$ of about 2. Using the criterion $c_v=2$ gives the line that is in remarkably good coincidence with the line obtained from the VAF criterion above [@prl]. Simulation details ================== We have considered liquids from three important system types: noble Ar, molecular CO$_2$ and metallic Fe. We have used the molecular dynamics (MD) simulation package DL\_POLY [@dlpoly] and simulated systems with $4576 - 8000$ particles with periodic boundary conditions. The interatomic potential for Ar is the pair Lennard-Jones potential [@ar], known to perform well at elevated pressure and temperature. For CO$_2$ and Fe, we have used interatomic potentials optimized tested in the liquid state at high pressure and temperature. The potential for CO$_2$ is the rigid-body nonpolarizable potential based on a quantum chemistry calculation, with the partial charges derived using the Distributed Multipole Analysis method [@min]. Fe was simulated using the many-body embedded-atom potential [@iron]. In the case of CO$_2$, the electrostatic interactions were evaluated using the smooth particle mesh Ewald method. The MD systems were first equilibrated in the constant pressure and temperature ensemble at respective state points for 20 ps. System properties were subsequently simulated at different temperatures and averaged in the constant energy and volume ensemble for 30 ps. We are interested in properties of real dense strongly-interacting liquids with potential energy comparable to kinetic energy and hence have chosen fairly high densities: $\rho=1.5$ g/cm$^3$ and $\rho=1.9$ g/cm$^3$ for Ar, $\rho=8$ g/cm$^3$ and $\rho=11$ g/cm$^3$ for Fe and $\rho=1.34$ g/cm$^3$ for CO$_2$. The lowest temperature in each simulation was the melting temperature at the corresponding density, $T_m$. The highest temperature significantly exceeded the temperature at the Frenkel line at the corresponding density, $T_{\rm F}$, taken from the earlier calculation of the Frenkel line in Ar [@prl], Fe [@scirep] and CO$_2$ [@yang]. As discussed above, the temperature range between $T_m$ and $T_{\rm F}$ corresponds to the regime where transverse modes progressively disappear and where Eq. (\[harmo\]) applies. We have simulated $100-700$ temperature points at each pressure depending on the system. The number of temperature points was chosen to keep the temperature step close to 10 K. As discussed above, Eq. (\[harmo\]) applies to subcritical liquids as well as to supercritical fluids below the Frenkel line. Accordingly, our simulations include the temperature range both below and above the critical temperature. This will be discussed in more detail below. Results and discussion ====================== Liquid energy and heat capacity ------------------------------- We have calculated $\omega_{\rm F}$ in (\[harmo\]) from its definition in (\[omega\]), as $\omega_{\rm F}=\frac{1}{\tau}$. $\tau$ can be calculated in a number of ways. Most common methods calculate $\tau$ as decay time of the self-intermediate scattering or other functions by the factor of $e$ or as the time at which the mean-squared displacement crosses over from ballistic to diffusive regime [@overlap]. These methods give $\tau$ in agreement with a method employing the overlap function depending on the cutoff parameter $a_c$ provided $a_c=0.3a$, where $a$ is the inter-molecuar distance [@overlap]. We use the latter method and calculate $\tau$ at 13-20 temperature points at each density depending on the system. At each density, we fit $\tau$ to the commonly used Vogel-Fulcher-Tammann dependence and use $\omega_{\rm F}=\frac{1}{\tau}$ to calculate the liquid energy predicted from the theory. The predicted $c_v$ is calculated as $c_v=\frac{1}{N}\frac{dE}{dT}$ where $E$ is given by Eq. (\[harmo\]): $$c_v=3-\left(\frac{\omega_{\rm F}}{\omega_{\rm D}}\right)^3-\frac{3T\omega_{\rm F}^2}{\omega_{\rm D}^3}\frac{d\omega_{\rm F}}{dT} \label{cv}$$ where $N$ is the number of atoms for Ar and Fe and the number of molecules for CO$_2$. The first two terms in (\[cv\]) give $c_v=2$ when $\omega_{\rm F}$ tends to its high-temperature limit of $\omega_{\rm F}$. The last term reduces $c_v$ below 2 by a small amount because $\frac{d\omega_{\rm F}}{dT}$ is close to zero at high temperature [@ropp]. We now compare the calculated energy and $c_v$ with those directly computed in the MD simulations. We note that the energy in Eq. (\[harmo\]) is the energy of thermal phonon motion, $E_T$, which contributes to the total liquid energy as $$E=E_0+E_{\rm T} \label{toten}$$ where $E_0$ is liquid energy at zero temperature and represents temperature-independent background contribution due to the interaction energy. In comparing the calculated $E_{\rm T}$ in Eq. (\[harmo\]) with the energy from MD simulations, we therefore subtract the constant term from the MD energy. The comparison of $c_v=\frac{1}{N}\frac{dE}{dT}$ is performed directly because the constant term does not contribute to $c_v$. We have also calculated $c_v$ using the fluctuations formula for the kinetic energy $K$ in the constant energy ensemble: $\langle K^2\rangle-\langle K\rangle^2=1.5(k_{\rm B} T)^2N(1-1.5k_{\rm B}/c_v)$ [@frensim]. Both methods agree well, as follows from Figures \[ar\]a and \[ar\]b. There is only one adjustable parameter in Eq. (\[harmo\]), $\omega_{\rm D}$, which is expected to be close to transverse Debye frequency. $\omega_{\rm F}$ is independently calculated from the MD simulation as discussed above. In Figures \[ar\] and \[fe\] we compare the energy and $c_v$ calculated on the basis of Eqs. (\[harmo\]) and (\[cv\]) and compare them with those computed in MD simulations. Blue circle in each figure shows the critical temperature. We observe good agreement between predicted and calculated properties in a temperature range including both subcritical and supercritical temperature. This involved using $\tau_{\rm D}\approx 0.6$ ps ($\rho=8$ g/cm$^3$) and $\tau_{\rm D}\approx 0.2$ ps ($\rho=11$ g/cm$^3$) for Fe, $\tau_{\rm D}\approx 0.9$ ps ($\rho=1.5$ g/cm$^3$) and $\tau_{\rm D}\approx 0.3$ ps ($\rho=1.9$ g/cm$^3$) for Ar and $\tau_{\rm D}\approx 0.5$ ps for CO$_2$, in reasonable order-of-magnitude agreement with experimental $\tau_{\rm D}$ of respective crystalline systems as well as maximal frequencies seen in experimental liquid dispersion curves (see, e.g., [@hoso3]). We note the expected trend of $\tau_{\rm D}$ reducing with density. At high temperature where $\omega_{\rm F}\approx\omega_{\rm D}$, Eq. (\[cv\]) predicts $c_v$ close to 2, noting that the last term gives only a small contribution to $c_v$ because $\omega_{\rm F}$ becomes slowly varying at high temperature. Consistent with this prediction, we observe the decrease of $c_v$ from 3 to 2 in Figures \[ar\] and \[fe\]. The agreement between the predicted and calculated results supports the interpretation of the decrease of $c_v$ with temperature discussed in the Introduction: $\omega_{\rm F}$ decreases with temperature, and this causes the reduction of the number of transverse modes propagating above $\omega_{\rm F}$ and hence the reduction of $c_v$. For CO$_2$, the same mechanism operates except we need to account for degrees of freedom in a molecular system. We first consider the case of solid CO$_2$. The MD interatomic potential treats CO$_2$ molecules as rigid linear units, contributing the kinetic term of 2.5 to the specific heat per molecule including 1 from the rotational degrees of freedom of the linear molecular and 1.5 from translations (here, we have noted that CO$_2$ molecules librate and rotate in the solid at low and high temperature, respectively [@rotat]). Noting the potential energy contributes the same term due to equipartition, the specific heat becomes 5 per molecule. This implies that for molecular CO$_2$, Eqs. (\[harmo\]) modifies as $E_{\rm T}=NT\left(5-\left(\frac{\omega_{\rm F}}{\omega_{\rm D}}\right)^3\right)$, where $N$ is the number of molecules and $\omega_{\rm F}$ is related to the jump frequency of molecules and which gives $c_v=5$ in the solid state where $\omega_{\rm F}$ is infinite. We use the modified equation to calculate the energy and $c_v$ and compare them to those computed from the MD simulation in Figure \[co2\]. Consistent with the above discussion, we observe that $c_v$ for CO$_2$ calculated directly from the MD simulations is close to 5 at low temperature just above melting. At this temperature, $\omega_{\rm F}\ll\omega_{\rm D}$, giving the solid-like value of $c_v$ as in the case of monatomic Ar and Fe. As temperature increases, two transverse modes of inter-molecular motion progressively disappear, resulting in the decrease of $c_v$ to the value of about $c_v=4$, in agreement with $c_v$ calculated from the theoretical equation for $E_{\rm T}$. We note that the temperature range in which we compare the predicted and calculated properties is notably large (e.g., $200-8000$ K for Ar, and $2000-55000$ K for Fe). This range is 10-100 times larger than those typically considered earlier [@ropp]. The higher temperatures for Fe might appear as unusual, however we note that liquid iron as well as supercritical iron fluid remains an unmodified system up to very high temperature: the first ionization potential of Fe is 7.9 eV, or over 90,000 K. Hence the considered temperature range is below the temperature at which the system changes its structure and type of interactions. The very wide temperature range reported here is mostly related to the large part of the temperature interval in Figures \[ar\]-\[co2\] being above the critical point where no phase transition intervenes and where the liquid phase exists at high temperature, in contrast to subcritical liquids where the upper temperature is limited by the boiling line. The agreement between predicted and calculated properties in such a wide temperature range adds support to the phonon approach to liquid thermodynamics we propose. We make three points regarding the observed agreement between the calculated and predicted results. First, the collective modes contributing to the thermal energy in (\[harmo\]) are considered to be harmonic. The anharmonicity can be accounted for in the Grüneisen approximation, however this involves an additional parameter [@ropp]. We attempted to avoid introducing additional parameters and sought to test Eq. (\[harmo\]) which contains only one parameter, $\omega_{\rm D}$. Second, Eq. (\[harmo\]) involves the Debye model and quadratic density of states (DOS). This approximation is justified since the Debye model is particularly relevant for disordered isotropic systems such as glasses [@landau], which are known to be nearly identical to liquids from the structural point of view. Furthermore, the experimental dispersion curves in liquids are very similar to those in solids such as poly-crystals [@mon-na; @mon-ga; @sn]. Therefore, the Debye model can be used in liquids to the same extent as in solids. One important consequence of this is that the high-frequency range of the phonon spectrum makes the largest contribution to the energy, as it does in solids including disordered solids. We also note that liquid DOS can be represented as the sum of solid-like and gas-like components in the two-phase thermodynamic model [@goddard], and the solid-like component can be extracted from the liquid DOS calculated in MD simulations. This can provide more information about the DOS beyond Debye approximation. Third, Eq. (\[harmo\]) assumes a lower frequency cutoff for transverse waves, $\omega_{\rm F}=\frac{1}{\tau}$, as envisaged by Frenkel in (\[omega\]). Our recent detailed analysis of the Frenkel equations shows that the dispersion relationship for liquid transverse modes is $\omega=\sqrt{c_s^2k^2-\frac{1}{4\tau^2}}$, where $c_s$ is the shear speed of sound and $k$ is wavenumber [@ropp]. Here, $\omega$ gradually crosses over from $0$ to its solid-like branch $\omega=c_sk$ when $\omega\gg\omega_{\rm F}=\frac{1}{\tau}$. In this sense, using a lower frequency cutoff in (\[harmo\]) might be thought of as an approximation. However, we have recently shown [@chenxing] that the square-root dependence of $\omega$ gives the liquid energy that is identical to (\[harmo\]). Structural crossover and its relationship to dynamical and thermodynamic properties ----------------------------------------------------------------------------------- The results in the previous sections support the picture in which the decrease of liquid $c_v$ from 3 to 2 is related to reduction of the energy of transverse modes propagating above $\omega_{\rm F}$ as described by Eq. (\[cv\]). According to Eq. (\[cv\]), $c_v=2$ corresponds to complete disappearance of transverse modes at the FL when $\omega_{\rm F}\approx\omega_{\rm D}$ (the disappearance is supported by the direct calculation of transverse modes on the basis of current correlation functions [@condmat]). Importantly, $c_v=2$ marks the crossover of $c_v$ because the evolution of collective modes is qualitatively different below and above the FL [@ropp]. Below the line, transverse modes disappear starting from the lowest frequency $\omega_{\rm F}$. Above the line, the remaining longitudinal mode starts disappearing starting from the highest frequency $\frac{2\pi c}{L}$, where $L$ is the particle mean free path (no oscillations can take place at distance smaller than $L$). This gives qualitatively different behavior of the energy and $c_v$ below and above the FL, resulting in their crossover at the FL [@ropp]. Interestingly, the thermodynamic crossover at $c_v=2$ implies a structural crossover. Indeed, the energy per particle in a system with pair-wise interactions is $$E=\frac{3}{2}k_{\rm B}T+4\pi \rho \int\limits_0^{\infty} r^2 U(r)g(r)dr \label{ene}$$ where $\rho=N/V$ is number density and $g(r)$ is radial distribution function. According to Eq. (\[toten\]), the liquid energy is $E=E_0+E_{\rm T}$, where $E_{\rm T}$ is given by Eq. (\[harmo\]). If the system energy undergoes the crossover at the FL where $c_v=2$, Eq. (\[ene\]) implies that $g(r)$ should also undergo a crossover. Therefore, the structural crossover in liquids can be predicted on the basis of the thermodynamic properties. We also expect the structural crossover at the FL to be related to the dynamical crossover on general grounds. As discussed above, below the FL particles oscillate around quasi-equilibrium positions and occasionally jump between them. The average time between jumps is given by liquid relaxation time, $\tau$ (Figure \[atoms\] schematically shows a local jump event from its surrounding “cage”.) This means that a static structure exists during $\tau$ for a large number of particles below the FL, giving rise to the well-defined medium-range order comparable to that existing in disordered solids [@salmon]. On the other hand, the particles lose the oscillatory component of motion above the FL and start to move in a purely diffusive manner as in gases. This implies that the features of $g(r)$ are expected to be gas-like. As a result, $g(r)$ medium-range peaks are expected to have different temperature dependence below and above the FL. This behavior was observed in Ar in MD simulations in the short-range structure [@jcp]. More recently, the crossover in supercritical Ne in the medium range at the FL was ascertained on the basis of X-ray scattering experiments [@clemens]. In Figure \[rdf\]a we plot pair distribution functions (PDFs) of Ar at density $\rho=1.9$ g/cm$^3$ in a wide temperature range. Using the FL criterion $c_v=2$ gives the temperature at the FL, $T_{\rm F}$, of about 4000 K at that density, which we find to be consistent with the criterion of the disappearance of the minimum of the velocity autocorrelation function [@prl]. The PDF was calculated with the distance step of $0.05$ Å, giving 600 PDF points. We observe PDF peaks in the medium range order up to about 20 Å at low temperature. The peaks reduce and broaden with temperature. To study this in more detail, we plot the peak heights vs temperature in Figure \[rdf\]b. We observe that the medium-range third and fourth peaks persist well above the critical temperature ($T_c=151$ K for Ar): the highest temperature simulated corresponds to $53T_c$. This interestingly differs from the traditional expectation that the structure of the matter so deep in the supercritical state has gas-like features only. At temperature above $T_{\rm F}$, the height of the fourth peak becomes comparable to its temperature fluctuations (calculated as the standard deviation of the peak height over many structures separated in time by 1 ps at each temperature) by order of magnitude. The fifth and higher-order peaks disappear before the highest temperature in the simulated range is reached. We plot the peak heights in Figure \[rdf\]b in the double-logarithmic plot because we expect to see an approximate power-law decay of the peak heights at low temperature. Indeed, PDF in solids can be represented as a set of Gaussian functions with peaks heights $h$ depending on temperature as $h\propto\frac{1}{\sqrt{T}}\exp\left(-\frac{\alpha}{T}\right)$ where $\alpha$ is a temperature-independent factor [@frenkel; @marad]. This temperature dependence of $h$ was also quantified in MD simulations [@stan]. $h$ decrease mostly due to the factor $\frac{1}{\sqrt{T}}$ whereas the effect of the exponential factor on $h$ is small and serves to reduce the rate at which $h$ decrease [@stan]. This implies that in solids, $\log h\propto-\log T$ approximately holds. In liquids, we expect the same relationship to hold below the FL where $\tau\gg\tau_{\rm D}$, corresponding to a particle oscillating many times before diffusively moving to the next quasi-equilibrium position. Indeed, the ratio of the number of diffusing particles $N_{\rm dif}$ to the total number of particles $N$ in the equilibrium state is $\frac{N_{\rm dif}}{N}=\frac{\tau_{\rm D}}{\tau}$ [@ropp] at any given moment of time. $\frac{N_{\rm dif}}{N}$ is small when $\tau\gg\tau_{\rm D}$ below the FL and can be neglected. Hence, $\log h\propto-\log T$ applies to liquids at any given moment of time below the FL where $\tau\gg\tau_{\rm D}$. This also applies to longer observation times if $h$ is averaged over $\tau$ [@ropp]. We note that the above result, $h\propto\frac{1}{\sqrt{T}}$, involves the assumption that the energy of particle displacements is harmonic (see, e.g., Ref. [@frenkel]). Anharmonicity becomes appreciable at high temperature, however the anharmonic energy terms are generally small compared to the harmonic energy. This is witnessed by the closeness of high-temperature $c_v$ to its harmonic result for both solids and high-temperature liquids [@andr; @bolm]. We therefore expect that $\log(h-1)\propto-\log T$ approximately holds in the low-temperature range below the FL as in solids but deviates from the linearity around the crossover at the FL where $\tau\rightarrow\tau_{\rm D}$ and where the dynamics becomes gas-like (the calculated PDF in Fig. \[rdf\]a is normalized to 1 where no correlations are present at large distances; hence we plot $h-1$ in order to compare it with the theoretical result $h\propto\frac{1}{\sqrt{T}}$ which tends to zero when no correlations are present at high temperature). We note that the crossover is expected to be broad because $\tau\gg\tau_{\rm D}$ applies well below the FL only. A substantial diffusive motion takes place in the vicinity of the line where $\frac{N_{\rm dif}}{N}$ can not be neglected, affecting the linear relationship. Consistent with the above prediction, we observe the linear regime at low temperature in Figure \[rdf\]b, followed by the deviation from the straight lines taking place around 3000 K for the 2nd peak, 5000 K for the 3rd peak and 4000 K for the 4th peak, respectively. The smooth crossover in the 3000-5000 K range is centered around 4000 K, consistent with the temperature at the Frenkel line discussed above. We also note that 4000 K corresponds to the specific heat $c_v=2$ in Figure \[ar\]b, in agreement with the earlier discussion. Summary ======= As discussed in the Introduction, liquids have been viewed as inherently complicated systems lacking useful theoretical concepts such as a small parameter [@landau]. Together with recent experimental evidence and theory [@ropp], the modelling data presented here and its quantitative agreement with predictions are beginning to change this traditional perspective. Our extensive molecular dynamics simulations of liquid energy and specific heat provide direct evidence for the link between dynamical and thermodynamic properties of liquids. We have found this to be the case for several important types of liquids at both subcritical and supercritical conditions spanning thousands of Kelvin. This supports an emerging picture that liquid thermodynamics can be understood on the basis of high-frequency collective modes. A more general implication is that, contrary to the prevailing view, liquids are emerging as systems amenable to theoretical understanding in a consistent picture as is the case in solid state theory. In addition to the link between dynamical and thermodynamic properties, we have discussed how these properties are related to liquid structure. This research utilised MidPlus computational facilities supported by QMUL Research-IT and funded by the EPSRC Grant EP/K000128/1. 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